each density $n_{s}$ in the range $n_{s}\in[0,(1-m_{\eta}-m_{s})/2]$. The
reference SzSLNSN subspaces 1A and 1B considered here have a fixed density
$m_{s}$ in the interval $m_{s}\in[0,(1-m_{\eta})]$ and a maximum and a minimum
density $n_{s}\rightarrow(1-m_{\eta}-m_{s})/2$ and $n_{s}\rightarrow 0$,
respectively.

The $\beta=c,\eta n$ bands for which $N_{\beta}>0$ of the $\eta$-Bethe states
that span the four reference SzSLNSN subspaces 1A, 1B, 2, and 3 under
consideration have occupancies such that $N_{\beta}^{h}=2S_{\eta}=m_{\eta}L$.
Hence the contributions to the charge current of such $\beta=c,\eta n$ bands
are free of virtual elementary charge currents cancelling, which much
simplifies the calculation of that current. The main effect of the virtual
elementary current cancelling processes occurring for $u>0$ in the remaining
set of SzSLNSN subspaces of a SzS subspace corresponding to intermediate
values of the densities $l_{\eta}\in[m_{\eta},1]$,
$n_{\eta}\in[0,(l_{\eta}-m_{\eta})/2]$, $m_{s}\in[0,(1-l_{\eta})]$, and
$n_{s}\in[0,(1-l_{\eta}-m_{s})/2]$ that label these subspaces is that the
corresponding set of largest charge current absolute values, Eq. (60), are in
the TL continuous functions of such densities. They smoothly vary between the
largest charge current absolute values of the reference SzSLNSN subspaces 1A,
1B, 2, and 3 considered here.

For such limiting reference subspaces the use of the TBA equations, Eqs. (111)
and (112) of Appendix A, allows the derivation of the general simplified
current expression, Eq. (53), for $\eta$-Bethe states with $\beta=c,\eta n,sn$
bands compact distributions of the general form, Eq. (51). Often such current
expressions have though only simple analytical form in the $u\rightarrow 0$
and $u\gg 1$ limits. Within the set of limiting reference SzSLNSN subspaces
1A, 1B, 2, and 3, the problem is most complex for the reference SzSLNSN
subspace 1A, its analysis being addressed in more detail below in Section
IV.3. In addition to the direct use of the equivalent charge current
$\langle\hat{J}_{LWS}(l_{\rm r},L_{\eta},S_{\eta},u)\rangle$ expressions, Eq.
(53) and Eq. (120) of Appendix A, to derive the coefficient
$C_{l_{\eta},n_{\eta},m_{s},n_{s}}$ in Eq. (60), one can use its limiting
expressions, Eq. (58), which are valid in the $m_{\eta}\ll 1$ limit and in the
$(1-m_{\eta})\ll 1$ limit provided that $l_{\eta}\rightarrow 1$.

For a reference SzSLNSN subspace 1A of a SzS subspace as defined above one has
that $n_{c}\rightarrow(1-m_{\eta})$, $n_{\eta n}\rightarrow 0$ for
$n=1,...,\infty$, $n_{s1}\rightarrow(1-m_{\eta}-m_{s})/2$, and
$n_{sn}\rightarrow 0$ for $n>1$. Hence the coefficient
$C_{l_{\eta},n_{\eta},m_{s},n_{s}}$, Eq. (60), only depends on $u$ and on the
subspace fixed densities $m_{\eta}$ and $m_{s}$ and is thus called here
$C_{m_{\eta},m_{s}}$. As further discussed in Section IV.3, one finds that in
this subspace that coefficient has the limiting behaviors,

$\displaystyle C_{m_{\eta},m_{s}}$ $\displaystyle=$ $\displaystyle 4\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow 0\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 0$ (61) $\displaystyle=$ $\displaystyle
2\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in[0,1]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 1-m_{\eta}$ $\displaystyle=$
$\displaystyle 2\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow
1\hskip 5.69046pt{\rm and}\hskip 5.69046ptm_{s}\rightarrow 0\,,$

for $u\rightarrow 0$ and,

$\displaystyle C_{m_{\eta},m_{s}}$ $\displaystyle=$ $\displaystyle
2\left(1+{1\over 2}\left({\ln 2\over
u}\right)^{2}+{\cal{O}}(u^{-4})\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\rightarrow 0\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ (62) $\displaystyle=$ $\displaystyle
2\left(1+{1\over 2}\left({1-m_{s}\over
u}\right)^{2}+{\cal{O}}(u^{-4})\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\rightarrow 0\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 1$ $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow 1\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 0\,,$

up to $u^{-3}$ order, which is a good approximation for approximately $u>3/2$.
(Corresponding expansions of the coefficient $C_{m_{\eta},m_{s}}$ up to
$u^{-3}$ order and valid for the whole $m_{\eta}\in[0,1]$ interval are given
in Section IV.3.) For $u>0$ and $m_{s}\in[0,(1-m_{\eta})]$, the coefficient
$C_{m_{\eta},m_{s}}$ smoothly increases upon increasing $m_{\eta}$ from
$m_{\eta}=0$ until reaching a maximum value at an $u$-dependent intermediate
density $m_{\eta}$. Upon further increasing $m_{\eta}$, it is a continuous
decreasing function of $m_{\eta}$. The largest $C_{m_{\eta},m_{s}}$ value
refers for any density $m_{\eta}\in[0,1]$ to the reference SzSLNSN subspace 1
for which $m_{s}\rightarrow 0$.

The reference SzSLNSN subspace 1B of a SzS subspace is such that
$n_{c}\rightarrow(1-m_{\eta})$, $n_{\eta n}\rightarrow 0$ for
$n=1,...,\infty$, $N_{sn}=1$ for $n=(L-2S_{\eta}-2S_{s})/2$, and
$n_{sn^{\prime}}\rightarrow 0$ for $n^{\prime}=1,...,\infty$ (including for
$n^{\prime}=n$ within the TL.) In the case of this subspace, the coefficient
$C_{l_{\eta},n_{\eta},m_{s},n_{s}}$, Eq. (60), is for $u>0$,
$m_{\eta}\in[0,1]$, and $m_{s}\in[0,(1-m_{\eta})]$ found to be independent of
$u$ and $m_{s}$, so that it is here denoted by $C_{m_{\eta}}$. It is found to
read,

$C_{m_{\eta}}={2\over\pi}{\sin(\pi m_{\eta})\over m_{\eta}(1-m_{\eta})}\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in[0,1]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\in[0,(1-m_{\eta})]\,.$ (63)

It increases and decreases upon increasing $m_{\eta}$ within the ranges
$m_{\eta}\in[0,1/2]$ and $m_{\eta}\in[1/2,1]$, respectively, reaching a
maximum value $8/\pi$ at $m_{\eta}=1/2$. Its limiting behaviors are,

$\displaystyle C_{m_{\eta}}$ $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow 0\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\in[0,1]$ (64) $\displaystyle=$ $\displaystyle
8/\pi\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}=1/2\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\in[0,1/2]$ $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow 1\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 0\,.$

For the reference SzSLNSN subspace 2 of a SzS subspace one has that
$n_{c}\rightarrow 0$, $n_{\eta 1}\rightarrow(1-m_{\eta})/2$, $n_{\eta
n}\rightarrow 0$ for $n>1$, and $n_{sn}\rightarrow 0$ for $n=1,...,\infty$.
The corresponding coefficient $C_{l_{\eta},n_{\eta},m_{s},n_{s}}$, Eq. (60),
only depends on $u$ and $m_{\eta}$, so that we call it $C_{m_{\eta}}$. Its
limiting behaviors are found to be given by,

$\displaystyle C_{m_{\eta}}$ $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,\hskip
5.69046ptm_{\eta}\rightarrow 0\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ (65) $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,\hskip
5.69046ptm_{\eta}\rightarrow 1\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ $\displaystyle=$ $\displaystyle{\pi\over
2u}\hskip 5.69046pt{\rm for}\hskip 5.69046ptu\gg 1\,,\hskip
5.69046ptm_{\eta}\rightarrow 0\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ $\displaystyle=$ $\displaystyle{1\over u}\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\gg 1\,,\hskip 5.69046ptm_{\eta}\rightarrow
1\,,\hskip 5.69046pt{\rm and}\hskip 5.69046ptm_{s}\rightarrow 0\,.$

For $u>0$ the coefficient $C_{m_{\eta}}$ is a continuous function of
$m_{\eta}$ with a maximum value at a $u$-dependent intermediate density
$m_{\eta}$.

The reference SzSLNSN subspace 3 of a SzS subspace is such that
$n_{c}\rightarrow 0$, $N_{\eta n}=1$ for $n=(L-2S_{\eta})/2$, $n_{\eta
n^{\prime}}\rightarrow 0$ for $n^{\prime}=1,...,\infty$ (including for
$n^{\prime}=n$ within the TL), and $n_{sn}\rightarrow 0$ for $n=1,...,\infty$.
The coefficient $C_{l_{\eta},n_{\eta},m_{s},n_{s}}$, Eq. (60), again only
depends on $u$ and $m_{\eta}$. It is here denoted by $C_{m_{\eta}}$. Its
values in the $u\rightarrow 0$ and $u\gg 1$ limits are for the whole
$m_{\eta}\in[0,1]$ range given by,

$\displaystyle C_{m_{\eta}}$ $\displaystyle=$
$\displaystyle{2\sin\left({\pi\over 2}m_{\eta}\right)\over m_{\eta}}\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,\hskip
5.69046ptm_{\eta}\in[0,1]\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ (66) $\displaystyle=$
$\displaystyle{\cal{O}}(1/L)=0\hskip 5.69046pt{\rm in}\hskip 5.69046pt\hskip
5.69046pt{\rm the}\hskip 5.69046pt{\rm TL}\hskip 5.69046pt{\rm for}\hskip
5.69046ptu\gg 1\,,\hskip 5.69046ptm_{\eta}\in[0,1[\,,\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 0\,,$

respectively. In the $u\rightarrow 0$ limit it is thus a decreasing function
of $m_{\eta}$ with limiting values,

$\displaystyle C_{m_{\eta}}$ $\displaystyle=$ $\displaystyle\pi\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,\hskip
5.69046ptm_{\eta}\rightarrow 0\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0$ (67) $\displaystyle=$ $\displaystyle 2\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,\hskip
5.69046ptm_{\eta}\rightarrow 1\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 0\,.$

As mentioned above, the largest charge current absolute value, Eq. (60), and
thus the corresponding coefficient $C_{l_{\eta},n_{\eta},m_{s},n_{s}}$ of all
remaining reference SzSLNSN subspaces contained in a SzS subspace is a
continuous and smooth function of the densities $l_{\eta}\in[m_{\eta},1]$,
$n_{\eta}\in[0,(l_{\eta}-m_{\eta})/2]$, $m_{s}\in[0,(1-l_{\eta})]$, and
$n_{s}\in[0,(1-l_{\eta}-m_{s})/2]$. Such a coefficient
$C_{l_{\eta},n_{\eta},m_{s},n_{s}}$ varies between the limiting values of
those of the limiting reference SzSLNSN subspaces 1A, 1B, 2, and 3. From
analysis and comparison of the whole corresponding set of largest charge
current absolute value $|\langle\hat{J}_{LWS}^{\rm max}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle|$, Eq. (60), one finds that the larger
coefficients $C_{l_{\eta},n_{\eta},m_{s},n_{s}}$ are, on the one hand
concerning the density $l_{\eta}\in[m_{\eta},1]$, reached for some of the
reference SzSLNSN subspaces contained in the reference SzSLN 1 for which
$l_{\eta}\rightarrow m_{\eta}$. Concerning the density
$n_{s}\in[0,(1-m_{\eta}-m_{s})/2]$ of the subset of reference SzSLNSN
subspaces contained in the reference SzSLN 1 for which $l_{\eta}\rightarrow
m_{\eta}$ and thus $n_{\eta}\rightarrow 0$ and $m_{s}\in[0,(1-m_{\eta})]$, one
finds in turn that the larger coefficients $C_{l_{\eta},n_{\eta},m_{s},n_{s}}$
in Eq. (60) are reached for the SzSLNSN subspaces 1A for which
$n_{s}\rightarrow(1-m_{\eta}-m_{s})/2$ where $m_{s}\in[0,(1-m_{\eta})]$. This
largest charge current absolute value of each SzS subspace is thus that used
in the upper bound procedures considered in the following and within the
canonical ensemble in Section V. Actually, the absolute largest coefficient
$C_{l_{\eta},n_{\eta},m_{s},n_{s}}$ is found to be that of the SzSLNSN
subspace 1A of all SzS subspaces for which $m_{s}\rightarrow 0$.

In Appendix C the effect of varying $u$ on the physical microscopic processes
behind the largest charge current absolute value of a SzS subspace reported in
this section is addressed. (That such an effect is discussed in an Appendix
follows from the remaining studies of this paper accounting for it but not
needing its detailed analysis, which requires a relative long account that
would affect the information flow on the main issues addressed in the
following. The information provided in Appendix C is though physically
important, its presentation in that Appendix contributing to the further
understanding of the microscopic mechanisms behind the 1D Hubbard model charge
transport properties.)

As discussed in Appendix B for $m_{\eta}^{z}=0$ and in Appendix C for
$m_{\eta}^{z}\in[0,1]$, the physics is very different (i) at $u=0$ and in the
$u\rightarrow 0$ limit and (ii) for finite $u$. In the latter Appendix it is
shown that due to the $u\rightarrow 0$ unbinding of the $\eta$-spin singlet
pairs that at finite $u$ are bound within the composite $\eta n$
pseudoparticles, the charge carriers that interchange position with the
$M_{\eta}$ unpaired physical $\eta$-spins $1/2$ that couple to charge probes
are different in the $u\rightarrow 0$ limit and for finite $u$. Their numbers
read $N_{c}+2\Pi_{\eta}=L-2S_{\eta}$ and $N_{\rho}=N_{c}+N_{\eta}$,
respectively.

Another issue discussed in Appendix C refers to the similarities and
differences relative to the case of the spin stiffness and currents of the
spin-$1/2$ $XXX$ chain, which are studied in Refs. CPC ; CP  by the upper-
bound method used in this paper. There is an advantage of that upper-bound
method relative to more common numerical approximations used to address the
stiffness problem and the charge currents contributing to it. It is that such
a method directly refers to a representation in terms of which the microscopic
mechanisms under consideration in the discussions of Appendix C refer to
elementary processes of the fractionalized particles whose configurations
generate the exact energy and momentum eigenstates. Indeed, in terms of
processes of the physical particles, the electrons, the present quantum
problem is non-perturbative, so that the microscopic mechanisms that control
the charge currents corresponds to a much more complex many-particle problem.
This is one of the reasons why such mechanisms remain hidden under the use of
standard numerical techniques, which usually rely on the electron
representation of the problem.

Other techniques that rely on the direct use of the BA quantum numbers without
accounting for their relation to the integrable models physical particles
Kawa-98 ; ANI-05  also pause technical problems. This occurs for instance
within the use of phenomelogical spinon and antispinon representations of the
BA quantum numbers whose relation to the physical particles remains undefined
ANI-05 . Moreover, in the case of some integrable models such as the present
1D Hubbard model, divergences emerge at the densities at which the Mazur’s
inequality is inconclusive in the stiffness expressions obtained from the
second derivative of the energy eigenvalues relative to the uniform vector
potential obtained from the BA. These divergences can though be avoided. This
is accomplished if one rather expresses the stiffness in terms of charge
current operator expectation values, as within the method used in this paper.

###  IV.3 The SzS subspaces largest charge current absolute value

In this section we provide further useful information on the largest charge
current absolute value $|\langle\hat{J}_{LWS}^{\rm max}(l_{\rm
r},S_{\eta},u)\rangle|$ of a reference SzSLNSN subspace 1A, which for any
fixed density $m_{s}$ is the largest charge current absolute value of the
corresponding SzS subspace. (Here $L_{\eta}$ was removed from
$|\langle\hat{J}_{LWS}(l_{\rm r},L_{\eta},S_{\eta},u)\rangle|$ because
$L_{\eta}=2S_{\eta}$ for a reference SzSLN subspace 1 where the SzSLNSN
subspaces 1A are contained.)

For the 1D Hubbard model in a reference SzSLN subspace 1 the general
$\eta$-Bethe-state charge current expression, Eq. (40), simplifies to,

$\langle\hat{J}_{LWS}(l_{\rm
r},S_{\eta},u)\rangle=\sum_{j=1}^{L}\,N_{c}^{h}(q_{j})\,\,J_{c}^{h}(q_{j})\hskip
5.69046pt{\rm where}\hskip 5.69046ptJ_{c}^{h}(q_{j})={2t\sin k^{c}(q_{j})\over
2\pi\rho_{c}(k^{c}(q_{j}))}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi,\pi]\,.$ (68)

Moreover, for $\eta$-Bethe states with $c$ and $sn$ bands compact
distributions of general form, Eq. (51), belonging to the reference SzSLN
subspace 1, the charge current expression, Eq. (53), further simplifies for
densities $|m_{\eta}^{z}|\in[0,1]$ and $m_{\eta}\in[|m_{\eta}^{z}|,1]$,

$\langle\hat{J}_{LWS}(l_{\rm
r},S_{\eta},u)\rangle={\tau\,L\,t\over\pi}\sum_{\iota=\pm}(\iota)\cos
k^{c}(q_{Fc,\tau}^{\iota})\,.$ (69)

Thus this applies to any SzSLNSN subspace contained in a reference SzSLN
subspace 1 whose densities $m_{s}\in[0,(1-m_{\eta})]$ and
$n_{s}\in[0,(1-m_{\eta}-m_{s})]/2$ have fixed values.

In the following we consider the reference SzSLNSN subspaces 1A of interest
for our upper-bound procedures for which
$n_{s}\rightarrow(1-m_{\eta}-m_{s})/2$ where the density $m_{s}$ has a fixed
value in the interval $m_{s}\in[0,(1-m_{\eta})]$. The derivation of the
$c$-band current spectrum $J_{c}^{h}(q_{j})$ in Eq. (68) involves the solution
of the TBA equations, Eqs. (111) and (112), to derive the momentum rapidity
function $k^{c}(q_{j})$ and related distribution $2\pi\rho_{c}(k^{c}(q_{j}))$
of the $\eta$-Bethe states that span the reference SzSLN subspaces 1A of a SzS
subspace. This is simplest to be accomplished in terms of $u^{-1}$ expansions
of such a function and distribution, which for densities $m_{\eta}\in[0,1]$
and $m_{s}\in[0,(1-m_{\eta})]$ leads to the following universal expansion for
$J_{c}^{h}(q_{j})$ up to $u^{-2}$ order,

$J_{c}^{h}(q_{j})=2t\sin q_{j}-2t\,{n_{\eta s}\over u}\sin
2q_{j}+6t\,\left({n_{\eta s}\over u}\right)^{2}\left(1-{3\over
2}\sin^{2}q_{j}\right)\sin q_{j}\,.$ (70)

Here,

$\displaystyle n_{\eta s}$ $\displaystyle=$
$\displaystyle(1-m_{\eta}-m_{s})\,g_{s}\hskip 5.69046pt{\rm where}$
$\displaystyle g_{s}$ $\displaystyle=$ $\displaystyle\ln 2\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm_{s}\rightarrow 0\hskip 5.69046pt{\rm and}\hskip
5.69046ptg_{s}=1\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{s}\rightarrow
1-m_{\eta}\,.$ (71)

$g_{s}=g_{s}(m_{s})\in[\ln 2,1]$ is in these equations a continuous increasing
function of the spin density $m_{s}$.

For $j>2$ orders $u^{-j}$ the calculations become a more complex technical
problem. Analysis of the interplay of the TBA equations with those that define
the charge operator expectation values reveals that the $c$-band current
spectrum $J_{c}^{h}(q_{j})$ expansion terms of such $j>2$ orders are state
dependent. As an example, two current spectra expansions up to $u^{-3}$ order
are derived in Appendix G for $\eta$-Bethe states that span two reference
SzSLNSN subspaces 1A with limiting spin densities $m_{s}=0$ and
$m_{s}\rightarrow 1-m_{\eta}$, respectively. For the former spin density, the
obtained expansion refers to $\eta$-Bethe states with compact distributions of
general form, Eq. (51), belonging to the reference SzSLNSN subspaces 1A under
consideration and $\eta$-Bethe states generated from those by a finite number
of $c$-band particle-hole processes. It reads,

$\displaystyle J_{c}^{h}(q_{j})$ $\displaystyle=$ $\displaystyle 2t\sin
q_{j}-2t\,{(1-m_{\eta})\ln 2\over u}\sin 2q_{j}+6t\,{((1-m_{\eta})\ln
2)^{2}\over u^{2}}\left(1-{3\over 2}\sin^{2}q_{j}\right)\sin q_{j}$ (72)
$\displaystyle-$ $\displaystyle 4t\,{((1-m_{\eta})\ln 2)^{3}\over
u^{3}}\left(1-{8\over 3}\sin^{2}q_{j}\right)\sin 2q_{j}$ $\displaystyle+$
$\displaystyle{3t\zeta(3)\over 16\,u^{3}}\left((1-m_{\eta})\left(1+{4\over
3}\sin^{2}q_{j}\right)+{3\tau\over
2\pi}\sum_{\iota=\pm}(\iota)\cos(q_{Fc,\tau}^{\iota})\left(\sin q_{j}-{1\over
3}\sin(q_{Fc,\tau}^{\iota})\right)\right)\sin 2q_{j}\,.$

That its terms up to $u^{-2}$ order and of $u^{-3}$ order do not depend and
depend on the limiting momenta $q_{Fc,\tau}^{\iota}$ associated with the
states with compact and asymmetrical $c$-band distributions considered here is
consistent with their state independence and state dependence, respectively.

The expansion up to $u^{-3}$ order of $J_{c}^{h}(q_{j})$ obtained in Appendix
G for the reference SzSLNSN subspaces 1A with spin $m_{s}\rightarrow
1-m_{\eta}$ is of second order in $(1-m_{\eta}-m_{s})\ll 1$ and is given by,

$\displaystyle J_{c}^{h}(q_{j})$ $\displaystyle=$ $\displaystyle 2t\sin
q_{j}-2t\,{(1-m_{\eta}-m_{s})\over u}\sin
2q_{j}+6t\,{(1-m_{\eta}-m_{s})^{2}\over u^{2}}\left(1-{3\over
2}\sin^{2}q_{j}\right)\sin q_{j}$ (73) $\displaystyle+$ $\displaystyle{4t\over
3}\,{(1-m_{\eta}-m_{s})\over u^{3}}\sin^{2}q_{j}\,\sin
2q_{j}+{\cal{O}}((1-m_{\eta}-m_{s})^{3})\,.$

The reference SzSLNSN subspaces 1A associated with the $(1-m_{\eta}-m_{s})\ll
1$ limit is the only one for which the terms up to second order in
$(1-m_{\eta}-m_{s})$ of $J_{c}^{h}(q_{j})$ are state independent for any $u>0$
value. Note that the terms of $u^{-3}$ order in Eqs. (72) and (73),
respectively, have a completely different form for the two reference SzSLNSN
subspaces 1A for which $m_{s}=0$ and $m_{s}\rightarrow 1-m_{\eta}$,
respectively.

The use in the general expansion, Eq. (70), of $J_{c}^{h}(q_{j})$ up to
$u^{-2}$ order valid for the densities intervals $m_{\eta}\in[0,1]$ and
$m_{s}\in[0,(1-m_{\eta})]$ of the reference SzSLN subspaces 1A of the $n_{\eta
s}=(1-m_{\eta})\,\ln 2$ and $n_{\eta s}=(1-m_{\eta}-m_{s})$ values of the
function $g_{s}$ given in Eq. (71) specific to its $m_{s}=0$ and
$m_{s}\rightarrow 1-m_{\eta}$ reference SzSLNSN subspaces 1A recovers the
terms up to $u^{-2}$ order in Eqs. (72) and Eq. (73), respectively. The
calculations reported in Appendix G to derive the expansions given in these
equations are more complex for the $m_{s}=0$ reference SzSLNSN subspaces 1A
than for that for which $m_{s}\rightarrow 1-m_{\eta}$. For the former
$m_{s}=0$ subspace, the distribution $2\pi\rho_{c}(k)$ in the
$J_{c}^{h}(q_{j})$ expression in Eq. (68) and the function $q_{c}(k)$ that is
the inverse of the momentum rapidity functional $k^{c}(q)$ also appearing in
that expression are in Appendix G expanded in powers of $u^{-j}$ for all
$j=1,...,\infty$ orders.

From the use of Eq. (70) in the general expression for
$\langle\hat{J}_{LWS}(l_{\rm r},S_{\eta},u)\rangle$ in Eq. (68) with compact
distributions of general form, Eq. (51), one finds the following expansion up
to $u^{-2}$ order of the charge current expression, Eq. (69), valid for the
reference SzSLN subspace 1A,

$\displaystyle\langle\hat{J}_{LWS}(l_{\rm r},S_{\eta},u)\rangle$
$\displaystyle=$
$\displaystyle{\tau\,L\,t\over\pi}\sum_{\iota=\pm}(\iota)\cos(q_{Fc,\tau}^{\iota})+{\tau\,L\,t\over\pi}{n_{\eta
s}\over u}\sum_{\iota=\pm}(\iota)\sin^{2}(q_{Fc,\tau}^{\iota})$ (74)
$\displaystyle-$ $\displaystyle{\tau\,3\,L\,t\over 2\pi}\left({n_{\eta s}\over
u}\right)^{2}\sum_{\iota=\pm}(\iota)\sin^{2}(q_{Fc,\tau}^{\iota})\cos(q_{Fc,\tau}^{\iota})\,.$

Here $q_{Fc,\tau}^{\iota}$ with $\tau=\pm$ and $\iota=\pm$ are the $c$ band
limiting occupancy momenta in Eq. (52) for $\beta=c$. Terms of $u^{-3}$ order
of the charge current expression, Eq. (69), are derived in Appendix G for the
$m_{s}=0$ and $m_{s}\rightarrow 1-m_{\eta}$ reference SzSLNSN subspaces 1A and
the whole $m_{\eta}\in[0,1]$ range, with the results,

$\displaystyle\langle\hat{J}_{LWS}^{(3)}(l_{\rm r},S_{\eta},u)\rangle$
$\displaystyle=$ $\displaystyle{\tau\,2\,L\,t\over\pi}\left({(1-m_{\eta})\ln
2\over u}\right)^{3}\sum_{\iota=\pm}(\iota)\left(1-{4\over
3}\sin^{2}(q_{Fc,\tau}^{\iota})\right)\sin^{2}(q_{Fc,\tau}^{\iota})$ (75)
$\displaystyle-$ $\displaystyle{\tau\,3\zeta(3)\,L\,t\over
32\pi\,u^{3}}\sum_{\iota=\pm}(\iota)\\{(1-m_{\eta})\left(1+{2\over
3}\sin^{2}(q_{Fc,\tau}^{\iota})\right)$ $\displaystyle-$
$\displaystyle{\tau\over
2\pi}\sum_{\iota^{\prime}=\pm}(\iota^{\prime})\cos(q_{Fc,\tau}^{\iota^{\prime}})\left(\sin(q_{Fc,\tau}^{\iota^{\prime}})-2\sin(q_{Fc,\tau}^{\iota})\right)\\}\sin^{2}(q_{Fc,\tau}^{\iota})\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{s}=0\,,$

and

$\displaystyle\langle\hat{J}_{LWS}^{(3)}(l_{\rm r},S_{\eta},u)\rangle$
$\displaystyle=$ $\displaystyle-{\tau\,(1-m_{\eta}-m_{s})\,L\,t\over
3\pi\,u^{3}}\sum_{\iota=\pm}(\iota)\sin^{4}(q_{Fc,\tau}^{\iota})$ (76)
$\displaystyle+$ $\displaystyle{\cal{O}}((1-m_{\eta}-m_{s})^{3})\hskip
5.69046pt{\rm for}\hskip 5.69046pt(1-m_{\eta}-m_{s})\ll 1\,,$

respectively. (The expansion term of $u^{-3}$ order, Eq. (76), only includes
contributions up to second order in $(1-m_{\eta}-m_{s})\ll 1$.)

The following expansion of the limiting occupancy momenta
$q_{Fc,\tau}^{\iota}$ maximizes the $u^{-3}$ order expansion of the charge
current absolute value, Eq. (69), for all densities ranges
$|m_{\eta}^{z}|\in[0,1]$, $m_{\eta}\in[|m_{\eta}^{z}|,1]$, and
$m_{s}\in[0,(1-m_{\eta})]$ of the reference SzSLN subspaces 1A,

$\displaystyle q_{Fc,-}^{\iota}$ $\displaystyle=$ $\displaystyle{\pi\over
2}+\iota\,\pi m_{\eta}+{2n_{\eta s}\over u}+{\cal{O}}(u^{-3})\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in\left[0,{1\over 2}-\delta_{\eta
s}^{u}\right]$ $\displaystyle=$ $\displaystyle\pi-(1-\iota)\pi m_{\eta}\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in\left[{1\over 2}-\delta_{\eta
s}^{u},{1\over 2}\right]\,,$ $\displaystyle q_{Fc,+}^{\iota}$ $\displaystyle=$
$\displaystyle\pi-(1-\iota)\pi(1-m_{\eta})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[{1\over 2},{1\over 2}+\delta_{\eta s}^{u}\right]$
(77) $\displaystyle=$ $\displaystyle{\pi\over
2}+\iota\,\pi(1-m_{\eta})+{2n_{\eta s}\over u}+{\cal{O}}(u^{-3})\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in\left[{1\over 2}+\delta_{\eta
s}^{u},1\right]\,,$

where $\iota=\pm$ and,

$\delta_{\eta s}^{u}={(1-m_{s})\,g_{s}\over\pi u}+{\cal{O}}(u^{-3})\,.$ (78)

The corresponding largest charge current absolute value of general form, Eq.
(60), of such a reference SzSLN subspace can be written as,

$|\langle\hat{J}_{LWS}^{\rm max}(l_{\rm
r},S_{\eta},u)\rangle|=C_{m_{\eta},m_{s}}\,t\,L\,m_{\eta}\,(1-m_{\eta})\,,$
(79)

where $C_{m_{\eta},m_{s}}$ stands for the coefficient whose limiting values
are provided in Eqs. (61) and (62). It reaches as a function of $m_{\eta}$ and
for $u>0$ and $m_{s}\in[0,(1-m_{\eta})]$ a maximum value at an $u$-dependent
intermediate density $m_{\eta}$, being a continuous increasing and decreasing
function of $m_{\eta}$ below and above that density, respectively. At fixed
$m_{\eta}$, its largest value is reached for the $m_{s}=0$ reference SzSLNSN
subspace 1A.

The coefficient $C_{m_{\eta},m_{s}}$ limiting values valid for $u\rightarrow
0$, which from Eq. (61) read,

$\displaystyle C_{m_{\eta},m_{s}}$ $\displaystyle=$ $\displaystyle 4\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow 0\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 0$ (80) $\displaystyle=$ $\displaystyle
2\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in[0,1]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\rightarrow 1-m_{\eta}$ $\displaystyle=$
$\displaystyle 2\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow
1\hskip 5.69046pt{\rm and}\hskip 5.69046ptm_{s}\rightarrow 0\,,$

follow from those in Eq. (197) of Appendix G for the largest charge current
absolute value derived in that Appendix for $m_{\eta}\ll 1$ and
$(1-m_{\eta})\ll 1$.

Moreover, from the use of the expansion up to $u^{-2}$ order of
$|\langle\hat{J}_{LWS}^{\rm max}(l_{\rm r},S_{\eta},u)\rangle|$, Eq. (198) of
Appendix G, one finds that up to that order the coefficient
$C_{m_{\eta},m_{s}}$ in Eq. (79) is for densities $m_{\eta}\in[0,1]$ and
$m_{s}\in[0,(1-m_{\eta})]$ of reference SzSLN subspaces 1A given by,

$\displaystyle C_{m_{\eta},m_{s}}$ $\displaystyle=$
$\displaystyle{2\over\pi}{\sin(\pi m_{\eta})\over
m_{\eta}(1-m_{\eta})}\left(1-{7\over 2}\left({n_{\eta s}\over
u}\right)^{2}\left(1-{8\over 7}\cos(\pi m_{\eta})-{3\over 7}\sin^{2}(\pi
m_{\eta})\right)\right)$ (81) $\displaystyle\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[0,{1\over 2}-\delta_{\eta s}^{u}\right]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\hskip 5.69046ptm_{s}\in[0,(1-m_{\eta})]$
$\displaystyle=$ $\displaystyle{2\over\pi}{\sin^{2}(\pi m_{\eta})\over
m_{\eta}(1-m_{\eta})}\left(1+{2n_{\eta s}\over u}\left(1+{3\over 2}{n_{\eta
s}\over u}\cos(2\pi m_{\eta})\right)\cos^{2}(\pi m_{\eta})\right)$
$\displaystyle\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[{1\over 2}-\delta_{\eta s}^{u},{1\over
2}+\delta_{\eta s}^{u}\right]\hskip 5.69046pt{\rm and}\hskip 5.69046pt\hskip
5.69046ptm_{s}\in[0,(1-m_{\eta})]$ $\displaystyle=$
$\displaystyle{2\over\pi}{\sin(\pi m_{\eta})\over
m_{\eta}(1-m_{\eta})}\left(1-{7\over 2}\left({n_{\eta s}\over
u}\right)^{2}\left(1+{8\over 7}\cos(\pi m_{\eta})-{3\over 7}\sin^{2}(\pi
m_{\eta})\right)\right)$ $\displaystyle\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[{1\over 2}+\delta_{\eta s}^{u},1\right]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\hskip 5.69046ptm_{s}\in[0,(1-m_{\eta})]\,.$

Both for $m_{\eta}\in[0,1/2-\delta_{\eta s}^{u}]$ and
$m_{\eta}\in[1/2+\delta_{\eta s}^{u},1]$ the terms of orders $u^{-1}$,
$u^{-3}$, and remaining odd orders $u^{-j}$ where $j=5,7,...$ of this
coefficient expansion exactly vanish.

Finally, from the use of Eqs. (199) and (200) of Appendix G one finds that the
terms of $u^{-3}$ order of the coefficient $C_{m_{\eta},m_{s}}$ in Eq. (79)
are for the $m_{s}=0$ and $m_{s}\rightarrow 1-m_{\eta}$ reference SzSLNSN
subspaces 1A and the whole density interval $m_{\eta}\in[0,1]$ given by,

$\displaystyle C_{m_{\eta},m_{s}}^{(3)}$ $\displaystyle=$ $\displaystyle
0+{\cal{O}}(u^{-4})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[0,{1\over 2}-\delta_{\eta s}^{u}\right]\hskip
5.69046pt{\rm and}\hskip 5.69046ptm_{s}=0$ (82) $\displaystyle=$
$\displaystyle{\sin^{2}(2\pi m_{\eta})\over\pi
m_{\eta}\,(1-m_{\eta})}\\{2\left({(1-m_{\eta})\ln 2\over
u}\right)^{3}\left(1-{4\over 3}\sin^{2}(2\pi m_{\eta})\right)$
$\displaystyle+$ $\displaystyle{3\zeta(3)\over
16\,u^{3}}\left((1-m_{\eta})\left(1+{2\over 3}\sin^{2}(2\pi
m_{\eta})\right)+{1\over\pi}\left(1-{1\over 2}\cos(2\pi
m_{\eta})\right)\sin(2\pi m_{\eta})\right)\\}$ $\displaystyle+$
$\displaystyle{\cal{O}}(u^{-4})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[{1\over 2}-\delta_{\eta s}^{u},{1\over
2}+\delta_{\eta s}^{u}\right]\hskip 5.69046pt{\rm and}\hskip 5.69046ptm_{s}=0$
$\displaystyle=$ $\displaystyle 0+{\cal{O}}(u^{-4})\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm_{\eta}\in\left[{1\over 2}+\delta_{\eta
s}^{u},1\right]\hskip 5.69046pt{\rm and}\hskip 5.69046ptm_{s}=0\,,$

and

$\displaystyle C_{m_{\eta},m_{s}}^{(3)}$ $\displaystyle=$ $\displaystyle
0+{\cal{O}}(u^{-4})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[0,{1\over 2}-\delta_{\eta s}^{u}\right]\hskip
5.69046pt{\rm and}\hskip 5.69046ptm_{s}\rightarrow 1-m_{\eta}$ (83)
$\displaystyle=$ $\displaystyle{\sin^{4}(2\pi m_{\eta})\over 3\pi
u^{-3}}{(1-m_{\eta}-m_{s})\over
m_{\eta}\,(1-m_{\eta})}+{\cal{O}}((1-m_{\eta}-m_{s})^{3})$ $\displaystyle{\rm
for}\hskip 5.69046ptm_{\eta}\in\left[{1\over 2}-\delta_{\eta s}^{u},{1\over
2}+\delta_{\eta s}^{u}\right]\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}\rightarrow 1-m_{\eta}$ $\displaystyle=$ $\displaystyle
0+{\cal{O}}(u^{-4})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in\left[{1\over 2}+\delta_{\eta s}^{u},1\right]\hskip
5.69046pt{\rm and}\hskip 5.69046ptm_{s}\rightarrow 1-m_{\eta}\,,$

respectively. (The terms in Eq. (83) only include the contributions up to
second order in $(1-m_{\eta}-m_{s})\ll 1$.)

##  V Charge stiffness upper bounds within the canonical ensemble

The function,

$F^{\rm UB}(m_{\eta},u)\equiv{|\langle\hat{J}_{LWS}^{\rm max}(l_{\rm
r},S_{\eta},u)\rangle|\over 2S_{\eta}}=t\,(1-m_{\eta})\,C_{m_{\eta},m_{s}}\,,$
(84)

where $C_{m_{\eta},m_{s}}$ is the coefficient in Eqs. (79)-(83), is a
continuous and decreasing function of $m_{\eta}$ for $u>0$,
$m_{\eta}\in[0,1]$, and $m_{s}\in[0,(1-m_{\eta})]$. It has limiting behaviors,

$\displaystyle F^{\rm UB}(m_{\eta},u)$ $\displaystyle=$ $\displaystyle
4t\,(1-m_{\eta})\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow
0\,,\hskip 5.69046ptm_{s}\rightarrow 0\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptu\rightarrow 0$ (85) $\displaystyle=$ $\displaystyle
2t\,(1-m_{\eta})\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in[0,1]\,,\hskip 5.69046ptm_{s}\rightarrow
1-m_{\eta}\,,\hskip 5.69046pt{\rm and}\hskip 5.69046ptu\rightarrow 0$
$\displaystyle=$ $\displaystyle 2t\,(1-m_{\eta})\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm_{\eta}\rightarrow 1\,,\hskip 5.69046ptm_{s}\rightarrow
0\,,\hskip 5.69046pt{\rm and}\hskip 5.69046ptu\rightarrow 0$ $\displaystyle=$
$\displaystyle{2t\sin(\pi m_{\eta})\over\pi m_{\eta}}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm_{\eta}\in[0,1]\,,\hskip
5.69046ptm_{s}\in[0,(1-m_{\eta})]\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptu\rightarrow\infty\,.$

Its derivative with respect to $m_{\eta}$ is such that,

$\displaystyle{\partial F^{\rm UB}(m_{\eta},u)\over\partial m_{\eta}}$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\rightarrow 0\,,\hskip 5.69046ptm_{s}\in[0,1]\,,\hskip
5.69046pt{\rm and}\hskip 5.69046ptu>0\,,$ $\displaystyle{\partial F^{\rm
UB}(m_{\eta},u)\over\partial m_{\eta}}$ $\displaystyle<$ $\displaystyle
0\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\in]0,1]\,,\hskip
5.69046ptm_{s}\in[0,(1-m_{\eta})]\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptu>0\,.$ (86)

It has the limiting behaviors,

$\displaystyle{\partial F^{\rm UB}(m_{\eta},u)\over\partial m_{\eta}}$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\rightarrow 0\,,\hskip 5.69046ptm_{s}\in[0,1]\,,\hskip
5.69046pt{\rm and}\hskip 5.69046ptu\rightarrow 0\,,$ (87) $\displaystyle=$
$\displaystyle-4t\hskip 5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}\rightarrow
1\,,\hskip 5.69046ptm_{s}\rightarrow 0\,,\hskip 5.69046pt{\rm and}\hskip
5.69046ptu\rightarrow 0\,,$ $\displaystyle=$ $\displaystyle-{2t\over
m_{\eta}}\left({\sin(\pi m_{\eta})\over\pi m_{\eta}}-\cos(\pi
m_{\eta})\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}\in[0,1]\,,\hskip 5.69046ptm_{s}\in[0,(1-m_{\eta})]\,,\hskip
5.69046pt{\rm and}\hskip 5.69046ptu\rightarrow\infty\,.$

A first stiffness upper bound,

$\displaystyle D^{*}(T)$ $\displaystyle=$
$\displaystyle{(2S_{\eta}^{z})^{2}\over
2LT}\,\,\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm r}}p_{l_{\rm
r},S_{\eta},S_{\eta}^{z}}(F^{\rm UB}(m_{\eta},u))^{2}$ (88) $\displaystyle=$
$\displaystyle{t^{2}(m_{\eta}^{z})^{2}L\over
2T}\,\,\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm r}}p_{l_{\rm
r},S_{\eta},S_{\eta}^{z}}\,C_{m_{\eta},m_{s}}^{2}\,(1-m_{\eta})^{2}\,,$

is obtained within the canonical ensemble by replacing the moduli of the
expectation values $\langle\hat{J}_{LWS}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle$ of $\eta$-Bethe states with the same
$S_{\eta}$ value in the stiffness expression, Eq. (42), by the upper bound of
the largest absolute value of the charge current, Eq. (79).

For each fixed-$S_{\eta}^{z}$ and $S_{s}^{z}=0$ canonical ensemble, the
largest value of $F^{\rm UB}(m_{\eta},u)$ in the $S_{\eta}$ summation of Eq.
(88) is that referring to the minimum $S_{\eta}$ and $S_{s}$ values,
$S_{\eta}=|S_{\eta}^{z}|=m_{\eta}^{z}\,L/2$ and $S_{s}=|S_{s}^{z}|=0$,
respectively, such that $m_{\eta}=m_{\eta}^{z}$ and $m_{s}=m_{s}^{z}=0$. This
follows from the function $F^{\rm UB}(m_{\eta},u)$ smoothly decreasing upon
increasing $m_{\eta}$. The same applies upon increasing $m_{s}$ at finite $u$.
A second stiffness upper bound is then reached by replacing in Eq. (88) the
function $F^{\rm UB}(m_{\eta},u)$ by its largest value,

$F^{\rm UB}(m_{\eta}^{z},u)=t\,C_{m_{\eta}^{z},0}\,(1-m_{\eta}^{z})\hskip
5.69046pt{\rm for}\hskip 5.69046ptm_{\eta}^{z}\in[0,1]\,.$ (89)

Here $C_{m_{\eta}^{z},0}$ is obtained by replacing $m_{\eta}$ and $m_{s}$ by
$m_{\eta}^{z}$ and $m_{s}^{z}=0$, respectively, in the expression of the
coefficient $C_{m_{\eta},m_{s}}$. The state summations in Eq. (88) can then be
performed exactly for all finite temperatures $T>0$. Indeed, the probability
distribution $p_{l_{\rm r},S_{\eta},S_{\eta}^{z}}$ in each
fixed-$S_{\eta}^{z}$ canonical ensemble is normalized as,

$\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm r}}p_{l_{\rm
r},S_{\eta},S_{\eta}^{z}}=1\,.$ (90)

Such state summations account for the subspace dimensions and thus as well for
the full $S_{s}^{z}=0$ subspace dimension. For $T>0$ the resulting (larger)
upper bound $D^{**}(T)\geq D^{*}(T)\geq D(T)$, then becomes,

$D^{**}(T)={t^{2}\,C_{m_{\eta}^{z},0}^{2}\,L\over
2T}\,(1-m_{\eta}^{z})^{2}\,(m_{\eta}^{z})^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptm_{\eta}^{z}\in[0,1]\hskip 5.69046pt{\rm and}\hskip
5.69046ptm_{s}^{z}=0\,.$ (91)

For $m_{\eta}^{z}\ll 1$ and $m_{s}^{z}=0$ its values continuously vary from,

$D^{**}(T)={16\,t^{2}\,L\over 2T}\,(m_{\eta}^{z})^{2}\,,$ (92)

for $u\rightarrow 0$ to,

$D^{**}(T)={4\,t^{2}\,L\over 2T}\,(m_{\eta}^{z})^{2}\,,$ (93)

for $u\gg 1$ whereas for $(1-m_{\eta}^{z})\ll 1$ and $m_{s}^{z}=0$ it is given
by,

$D^{**}(T)={4\,t^{2}\,L\over 2T}\,(1-m_{\eta}^{z})^{2}\,,$ (94)

for all $u>0$ values.

In the $u\rightarrow\infty$ limit, it has the following simple expression for
$m_{s}^{z}=0$ and the whole $m_{\eta}^{z}\in[0,1]$ interval,

$D^{**}(T)={(2t/\pi)^{2}\,\sin^{2}(\pi m_{\eta}^{z})\,L\over
2T}={(2t/\pi)^{2}\,\sin^{2}(\pi(1-m_{\eta}^{z}))\,L\over 2T}\hskip
5.69046pt{\rm for}\hskip 5.69046pt\hskip 5.69046ptm_{\eta}^{z}\in[0,1]\hskip
5.69046pt{\rm and}\hskip 5.69046ptm_{s}^{z}=0\,.$ (95)

The charge stiffness of the 1D Hubbard model was studied in Ref. PDSC-00  for
large $u$, where it was shown to exactly vanish in the
$m_{\eta}^{z}\rightarrow 0$ limit. As found in that reference, for
$m_{\eta}^{z}$ finite the charge stiffness of the 1D Hubbard and that for
spinless fermions alone are different even in the $u\rightarrow\infty$ limit,
as illustrated in Fig. 1 of that reference for a finite system. Such a
different behavior persists in the TL and is due to the spinless-fermion phase
shifts imposed by the spins $1/2$ Karlo-97 .

The coefficient $c_{\rm c}$ in the upper bound, Eq. (6), then smoothly varies
from $c_{\rm c}=16$ for $u\rightarrow 0$ to $c_{\rm c}=4$ for $u\gg 1$ whereas
the coefficient $c_{\rm c}^{\prime}$ in the upper bound, Eq. (7), reads
$c_{\rm c}^{\prime}=4$ for the whole $u>0$ range. This completes our finding
of a vanishing charge stiffness in the TL, $L\to\infty$, within the canonical
ensemble for any fixed range or even distribution of $S_{\eta}^{z}$, or any
distribution of $m_{\eta}^{z}$ shrinking sufficiently fast that
$\langle(m_{\eta}^{z})^{2}\rangle L\to 0$.

##  VI Stiffness upper bounds within the grand-canonical ensemble for
$T\rightarrow\infty$

The average value of the square of the charge current $|\langle\hat{J}(l_{\rm
r},L_{\eta},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}$, Eq. (39), in a
fixed-$S_{\eta}^{z}$ and $S_{s}^{z}=0$ subspace that contains the set of SzS
subspaces with $\eta$-spin values
$S_{\eta}=|S_{\eta}^{z}|,|S_{\eta}^{z}|+1,|S_{\eta}^{z}|+2,...$, reads,

$\displaystyle\left\langle|\langle\hat{J}(l_{\rm
r},L_{\eta},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}}={(2S_{\eta}^{z})^{2}\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{l_{\rm
r}}{|\langle\hat{J}_{LWS}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}d_{\rm
subspace}^{LWS}(L_{\eta},S_{\eta})}$ (96) $\displaystyle=$
$\displaystyle{(2S_{\eta}^{z})^{2}\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{l_{\rm
r}}{|\langle\hat{J}_{LWS}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{S_{s}=0}^{(L-L_{\eta})/2}{L\choose
L_{\eta}}\times\left({L_{\eta}\choose L_{\eta}/2-S_{\eta}}-{L_{\eta}\choose
L_{\eta}/2-S_{\eta}-1}\right)\times\left({L-L_{\eta}\choose(L-L_{\eta})/2-S_{s}}-{L-L_{\eta}\choose(L-L_{\eta})/2-S_{s}-1}\right)}$
$\displaystyle=$
$\displaystyle{(2S_{\eta}^{z})^{2}\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{l_{\rm
r}}{|\langle\hat{J}_{LWS}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{S_{s}=0}^{(L-L_{\eta})/2}{L\choose
L_{\eta}}\times\left(\sum_{\\{N_{\eta n}\\}}\,\prod_{n=1}^{\infty}\,{L_{\eta
n}\choose N_{\eta
n}}\right)\times\left(\sum_{\\{N_{sn^{\prime}}\\}}\,\prod_{n^{\prime}=1}^{\infty}\,{L_{sn^{\prime}}\choose
N_{sn^{\prime}}}\right)}\,.$

The subspaces dimensions appearing here are defined in Appendix E and useful
information on the corresponding states summations was given in Section III.2.

We denote by $|\langle\hat{J}_{LWS}\rangle|_{A(l_{\eta},m_{\eta},n_{\rho})}$ a
SzSLN subspace current absolute value average. It is given by,

$|\langle\hat{J}_{LWS}\rangle|_{A(l_{\eta},m_{\eta},n_{\rho})}={\sum_{l_{n_{\rho}}^{\star}}|\langle\hat{J}_{LWS}(l_{n_{\rho}}^{\star},L_{\eta},S_{\eta},u)\rangle|\over\sum_{S_{s}=0}^{(L-L_{\eta})/2}{L\choose
L_{\eta}}\times\left(\sum_{\\{(N_{\eta
n})_{l_{\eta},m_{\eta},n_{\rho}}\\}}\,\prod_{n=1}^{\infty}\,{L_{\eta n}\choose
N_{\eta
n}}\right)\times\left({L-L_{\eta}\choose(L-L_{\eta})/2-S_{s}}-{L-L_{\eta}\choose(L-L_{\eta})/2-S_{s}-1}\right)}\,.$
(97)

Here $l_{n_{\rho}}^{\star}$ denotes $l_{\rm r}^{\star}$ for the set of
$\eta$-Bethe states with fixed values for the densities $l_{\eta}$,
$m_{\eta}$, and $n_{\rho}$ that span a SzSLN subspace and the sum
$\sum_{l_{n_{\rho}}^{\star}}$ runs over all $c$-band, $\eta n$-bands of
$n=1,...,\infty$ branches, and $sn$-band occupancy configurations of spin
$S_{s}=0,1,...,(L-L_{\eta})/2$ that generate such $\eta$-Bethe states. They
have the same numbers $M_{\eta}=2S_{\eta}$ of unpaired $\eta$-spins $1/2$ and
$N_{\rho}=N_{c}+N_{\eta}$, Eq. (43), of charge pseudoparticles where
$N_{c}=L-L_{\eta}$. The SzSLN subspace dimension
$\sum_{l_{n_{\rho}}^{\star}}1$ is given in the denominator on the right-hand
side of Eq. (97). Hence the summation $\sum_{\\{(N_{\eta
n})_{l_{\eta},m_{\eta},n_{\rho}}\\}}$ runs over all sets of $\eta n$
pseudoparticle numbers $\\{N_{\eta n}\\}$ that obey both the sum rules
$\sum_{n=1}^{\infty}n\,N_{\eta n}=(L_{\eta}-2S_{\eta})=\Pi_{\eta}$, Eq. (18)
for $\alpha=\eta$, and $N_{\eta}=\sum_{n=1}^{\infty}N_{\eta
n}=(L_{\eta}-N_{\eta 1}^{h})/2$, Eq. (30) for $\alpha=\eta$.

For finite $u$ a reference SzSLN subspace largest charge current absolute
value is proportional to $M_{\eta}\,N_{\rho}$. That in Eq. (60) it is written
as proportional to $m_{\eta}\,(1-m_{\eta})$ and thus to
$2S_{\eta}\,(L-2S_{\eta})$ follows from the expression given in that equation
applying both to $u\rightarrow 0$ and to finite $u$. Indeed and as justified
in Appendix C, the carriers of charge are different for $u\rightarrow 0$ and
finite $u$, respectively. As a result, such a largest charge current absolute
value can be written as proportional to $2S_{\eta}\,(L-2S_{\eta})$ and
$M_{\eta}\,N_{\rho}$ for $u\rightarrow 0$ and finite $u$, respectively. (If
one requires it to apply both to the $u\rightarrow 0$ limit and to finite $u$,
then it should be written as given in Eq. (60).)

That each SzSLN subspace of a SzS subspace is spanned by $\eta$-Bethe states
with exactly the same number $N_{\rho}=N_{c}+N_{\eta}$ of charge
pseudoparticles simplifies the form of the current absolute values average,
Eq. (97). Its expression can for $u>0$ be written in the general form,

$|\langle\hat{J}_{LWS}\rangle|_{A(l_{\eta},m_{\eta},n_{\rho})}=J_{\rho}\,4t\,m_{\eta}\,\sqrt{2N_{\rho}}={1\over
L}\,J_{\rho}\,4t\,M_{\eta}\,\sqrt{2N_{\rho}}\,.$ (98)

The coefficient $J_{\rho}$ obeys the inequality $J_{\rho}\leq 1$, being of the
order of unity. While for $u>0$ the largest charge current absolute value of a
reference SzSLN subspace is proportional to $M_{\eta}\,N_{\rho}$, such a
subspace average current absolute value, Eq. (97), is proportional to
$M_{\eta}\,\sqrt{N_{\rho}}$. That
$|\langle\hat{J}_{LWS}\rangle|_{A(l_{\eta},m_{\eta},n_{\rho})}\propto
M_{\eta}\,\sqrt{N_{\rho}}$ stems from the energy and momentum eigenstates that
span the SzSLN subspace being generated by all possible occupancy
configurations of the $N_{\rho}=N_{c}+N_{\eta}$ charge pseudoparticles.

For all SzSLN subspaces contained in a SzS subspace the density
$m_{\eta}=m_{\eta}^{z}$ in Eq. (98) has a fixed value. This combined with
$J_{\rho}\leq 1$ being of the order of the unity reveals that the SzSLN
subspace in a SzS subspace whose current absolute value average, Eq. (97), is
largest is that for which $N_{\rho}$ reaches its maximum value. One finds from
$N_{\rho}=L-(L_{\eta}+N_{\eta 1}^{h})/2$ where $N^{h}_{\eta
1}=2S_{\eta}+\sum_{n=2}^{\infty}2(n-1)N_{\eta n}$, Eq. (24) for $\alpha n=\eta
1$, that the latter maximum value refers to the SzS subspace minimum
$N^{h}_{\eta 1}$ value, which for the corresponding fixed $\eta$-spin
$S_{\eta}$ reads $N^{h}_{\eta 1}=2S_{\eta}$. This gives
$N_{\rho}=L-(L_{\eta}+2S_{\eta})/2$. For general SzSLN subspaces of a SzS
subspace for which $N^{h}_{\eta 1}=2S_{\eta}$ and thus $N_{\rho}=L-\Pi_{\eta}$
one has that $N_{\eta n}=0$ for $n>1$, so that
$N_{\rho}=L-(L_{\eta}+2S_{\eta})/2=N_{c}+N_{\eta 1}$ where
$N_{c}=L-2S_{\eta}-2N_{\eta 1}$ and thus $N_{\rho}=L-2S_{\eta}-N_{\eta 1}$.
Further maximizing $N_{\rho}$ at the SzS subspace fixed $S_{\eta}$ value
corresponds to minimizing $L_{\eta}$, which gives $L_{\eta}=2S_{\eta}$ and
thus $N_{\eta 1}=0$. This corresponds to reference SzSLN subspace 1 of the SzS
subspace under consideration for which $l_{\eta}\rightarrow m_{\eta}$ and thus
$n_{\eta}\rightarrow 0$, so that it is indeed spanned by a subset of
$\eta$-Bethe states for which $N_{\rho}=N_{c}$. For such states $N_{\rho}$
reaches its maximum value, $N_{\rho}=L-2S_{\eta}$. A reference SzSLN subspace
1 current absolute value average, Eq. (97), can be written as,

$|\langle\hat{J}_{LWS}\rangle|_{A(m_{\eta})}=J_{\rho
1}\,4t\,m_{\eta}\,\sqrt{2N_{\rho}}={1\over L}\,J_{\rho
1}\,4t\,M_{\eta}\,\sqrt{2N_{\rho}}\,.$ (99)

Since $l_{\eta}=m_{\eta}$ and $n_{\rho}=1-l_{\eta}=1-m_{\eta}$, the index
$A(l_{\eta},m_{\eta},n_{\rho})$ in the general SzSLN subspace current absolute
value average, Eq. (98), was for the particular case of the SzSLN subspace 1
denoted by $A(m_{\eta})$ in Eq. (99).

The spin degrees of freedom do not couple directly to charge probes and the
charge currents do not depend on the spin-singlet $sn$ pseudoparticle
occupancy configurations associated with the $sn$-bands momentum distribution
functions $N_{sn}(q_{j})$. However, for finite $u$ the charge current spectra
of the $\eta$-Bethe states that span a reference SzSLN subspace 1 depend on
the spin density $m_{s}$ and overall spin $sn$ pseudoparticle density $n_{s}$.
As a consequence, the corresponding coefficient $J_{\rho}$ in Eq. (98) also
depends on the densities $m_{s}$ and $n_{s}$.

One finds that finite-$u$ $\eta$-Bethe states contained in a reference SzSLN
subspace 1 with exactly the same $c$ pseudoparticle occupancy configurations
have for any fixed density $m_{s}$ in the interval $m_{s}\in[0,(1-m_{\eta})]$
the largest charge current absolute values for the SzSLNSN subspaces 1A for
which the density $n_{s}\in[0,(1-m_{\eta}-m_{s})/2]$ in Eq. (48) has its
largest value, $n_{s}=n_{s}^{\rm max}=(1-m_{\eta}-m_{s})/2$. Only in the
$u\rightarrow\infty$ limit in which all spin configurations are degenerate
have these states the same charge currents absolute values. Limiting examples
are (i) the SzSLNSN subspace 1A and (ii) the SzSLNSN subspace 1B. Both such
SzSLNSN subspaces of a reference SzSLN subspace 1 have fixed densities
$l_{\eta}\rightarrow m_{\eta}$, $n_{\eta}\rightarrow 0$, $m_{\eta}\in[0,1]$,
and $m_{s}\in[0,(1-m_{\eta})]$. Their density $n_{s}$ is given by (i) its
maximum value $n_{s}\rightarrow(1-m_{\eta}-m_{s})/2$ and (ii) minimum value
$n_{s}\rightarrow 0$, respectively.

We thus consider here a subspace contained in a reference SzSLN subspace 1
that we call SzSLN${}_{N_{1}}$ subspace. It is spanned by $\eta$-Bethe states
with spin values $S_{s}=0,1,...,L-L_{\eta}$ whose overall number of $sn$
pseudoparticles reads $N_{s}=N_{s1}=(L-L_{\eta}-2S_{s})/2$ for each such a
spin value. Hence the SzSLN${}_{N_{1}}$ subspace corresponds to the set of
reference SzSLNSN subspaces 1A, each with a fixed density
$m_{s}\in[0,(1-m_{\eta})]$. Its current absolute value average thus reads,

$|\langle\hat{J}_{LWS}\rangle|_{A_{1N}(m_{\eta})}={\sum_{l_{n_{\rho}}^{*}}|\langle\hat{J}_{LWS}(l_{n_{\rho}}^{*},L_{\eta},S_{\eta},u)\rangle|\over\sum_{S_{s}=0}^{(L-L_{\eta})/2}{L\choose
L_{\eta}}\times\left(\sum_{\\{(N_{\eta
n})_{l_{\eta},m_{\eta},n_{\rho}}\\}}\,\prod_{n=1}^{\infty}\,{L_{\eta n}\choose
N_{\eta n}}\right)\times{(L-2S_{\eta}+2S_{s})/2\choose 2S_{s}}}\,,$ (100)

where ${(L-2S_{\eta}+2S_{s})/2\choose 2S_{s}}$ is the number of independent
$s1$-band occupancy configurations for each of the spin values
$S_{s}=0,1,...,L-L_{\eta}$. The SzSLN${}_{N_{1}}$ dimension
$\sum_{l_{n_{\rho}}^{*}}1$ is given by the denominator on the right-hand side
of Eq. (100). The current absolute value average, Eq. (100), can be written
as,

$|\langle\hat{J}_{LWS}\rangle|_{A(m_{\eta},n_{s}^{\rm max})}=J_{\rho
1N}\,4t\,m_{\eta}\,\sqrt{2N_{\rho}}={1\over L}\,J_{\rho
1N}\,4t\,M_{\eta}\,\sqrt{2N_{\rho}}\,.$ (101)

The difference relative to the current absolute value average, Eq. (98), of
the reference SzSLN subspace 1 where the SzSLN${}_{N_{1}}$ subspace is
contained is that $J_{\rho 1N}\geq J_{\rho 1}$.

Each SzS subspace only contains one reference SzSLN subspace 1. Since a
reference SzSLN subspace 1 only contains one SzSLN${}_{N_{1}}$ subspace, a SzS
subspace also only contains one SzSLN${}_{N_{1}}$ subspace. Let
$\langle\hat{J}(l_{\rm r}^{\diamond},S_{\eta},S_{\eta}^{z},u)\rangle$ denote
the currents of the energy and momentum eigenstates that span a reduced
subspace of the fixed-$S_{\eta}^{z}$ and $S_{s}^{z}=0$ subspace obtained by
replacing each of its SzS subspaces by the corresponding SzSLN${}_{N_{1}}$
subspace. Here $l_{\rm r}^{\diamond}$ stands for all quantum numbers other
than $S_{\eta}$, $S_{\eta}^{z}$, and $u>0$ needed to uniquely define each such
an energy and momentum eigenstate. An important quantity for our upper-bound
procedures is the average value of the current square $|\langle\hat{J}(l_{\rm
r}^{\diamond},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}$ in the reduced subspace
under consideration, which reads,

$\left\langle|\langle\hat{J}(l_{\rm
r}^{\diamond},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}\,1N}={(2S_{\eta}^{z})^{2}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm
r}^{\diamond}}{|\langle\hat{J}_{LWS}(l_{\rm
r}^{\diamond},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{S_{s}=0}^{(L-2S_{\eta})/2}{L\choose
2S_{\eta}}\times{(L-2S_{\eta}+2S_{s})/2\choose 2S_{s}}}\,.$ (102)

Here $\sum_{l_{\rm r}^{\diamond}}1=\sum_{S_{s}=0}^{(L-2S_{\eta})/2}{L\choose
2S_{\eta}}\times{(L-2S_{\eta}+2S_{s})/2\choose 2S_{s}}$ is the dimension of
that reduced subspace.

$J_{\rho 1}=$max$\\{J_{\rho}\\}$ is in Eq. (99) for the reference SzSLN
subspace 1 the largest coefficient $J_{\rho}$ in Eq. (98) of all SzSLN
subspaces contained in a SzS subspace with density $m_{\eta}=m_{\eta}^{z}$.
Moreover, the inequality $J_{\rho 1N}\geq J_{\rho 1}$ involving the
coefficients of the current absolute value averages in Eqs. (99) and (101) is
valid for all fixed densities $m_{\eta}\in[0,1]$ of the corresponding
reference SzSLN subspace 1 and SzSLN${}_{N_{1}}$ subspace belonging to the
same SzS subspace. A consequence of such properties is that the following
inequality involving the average values of the square of the charge current in
Eqs. (96) and (102) holds,

$\left\langle|\langle\hat{J}(l_{\rm
r}^{\diamond},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}\,1N}\geq\left\langle|\langle\hat{J}(l_{\rm
r}^{\star},L_{\eta},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}}\,.$
(103)

For high temperature $T\rightarrow\infty$, the $T>0$ expression of the charge
stiffness, Eq. (42), simplifies to,

$\displaystyle D(T)$ $\displaystyle=$ $\displaystyle{(2S_{\eta}^{z})^{2}\over
2LT}\,\,{\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}\sum_{l_{\rm
r}}{|\langle\hat{J}_{LWS}(l_{\rm
r},L_{\eta},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{L_{\eta}=2|S_{\eta}^{z}|}^{L}\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L_{\eta}/2}d_{\rm
subspace}^{LWS}(L_{\eta},S_{\eta})}$ (104) $\displaystyle=$
$\displaystyle{\left\langle|\langle\hat{J}(l_{\rm
r},L_{\eta},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}}\over
2LT}\,.$

A high temperature $T\rightarrow\infty$ charge stiffness upper bound,

$\displaystyle D^{\diamond}(T)$ $\displaystyle=$
$\displaystyle{(2S_{\eta}^{z})^{2}\over
2LT}\,\,{\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm
r}^{\diamond}}{|\langle\hat{J}_{LWS}(l_{\rm
r}^{\diamond},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{S_{s}=0}^{(L-2S_{\eta})/2}{L\choose
2S_{\eta}}\times{(L-2S_{\eta}+2S_{s})/2\choose 2S_{s}}}$ (105)
$\displaystyle=$ $\displaystyle{\left\langle|\langle\hat{J}(l_{\rm
r}^{\diamond},L_{\eta},S_{\eta},S_{\eta}^{z},u)\rangle|^{2}\right\rangle_{S_{\eta}^{z}\,1N}\over
2LT}\ \,,$

such that $D(T)\leq D^{\diamond}(T)$ then follows from the inequality, Eq.
(103).

As mentioned above, a SzSLN${}_{N_{1}}$ subspace can be divided into a set of
reference SzSLNSN subspaces 1A, each with a fixed density
$m_{s}\in[0,(1-m_{\eta})]$. In Appendix H it is shown that a corresponding
charge stiffness upper bound only involving the $S_{s}=0$ contributions from
the $m_{s}=0$ reference SzSLNSN subspace 1A is larger than that given in Eq.
(105). This gives our ultimate charge stiffness upper bound within the grand
canonical ensemble for the TL and high temperature $T\rightarrow\infty$,

$\displaystyle D^{\diamond\diamond}(T)$ $\displaystyle=$
$\displaystyle{(2S_{\eta}^{z})^{2}\over
2LT}\,\,{\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm
r}^{\diamond}}{|\langle\hat{J}_{LWS}(l_{\rm
r}^{\diamond},S_{\eta},u)\rangle|^{2}\over(2S_{\eta})^{2}}\over\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}{L\choose
2S_{\eta}}}\,,$ (106)

where $J_{c}^{h}(q_{j})$ is the general current spectrum in Eq. (68) for
$S_{s}=0$. Up to $u^{-2}$ order it is given in Eq. (70) for $S_{s}=0$. It thus
reads,

$J_{c}^{h}(q_{j})=2t\sin q_{j}-2t\,{(1-m_{\eta})\ln 2\over u}\sin
2q_{j}+6t\,\left({(1-m_{\eta})\ln 2\over u}\right)^{2}\left(1-{3\over
2}\sin^{2}q_{j}\right)\sin q_{j}\,.$ (107)

The general current spectrum in Eq. (68) has up to $u^{-2}$ order the same
universal form for all $\eta$-Bethe states that span the reference SzSLNSN
subspaces 1A, Eq. (70) for $m_{s}\in[0,(1-m_{\eta})]$ and Eq. (107) at
$m_{s}=0$.

In Appendix H the current spectrum, Eq. (107), is used in the charge stiffness
upper bound, Eq. (106), to derive the following exact expansion up to $u^{-2}$
order of that upper bound valid in the TL for $m^{z}_{\eta}\in[0,1/2]$,

$D(T)\leq D^{\diamond\diamond}(T)=\frac{c_{\rm
gc}\,t^{2}}{2T}(m^{z}_{\eta})^{2}\hskip 5.69046pt{\rm where}\hskip
5.69046ptc_{\rm gc}=2\pi^{2}\left(1+\left(\frac{\ln
2}{2u}\right)^{2}\right)\,.$ (108)

On the one hand, this expression applies to the $m^{z}_{\eta}\ll 1$ limit. The
charge stiffness Mazur’s lower bound has been derived for $T\rightarrow\infty$
in Ref. ZNP-97 . In the $S_{s}^{z}=0$ case considered in the upper-bound
studies of this paper, one finds that the charge stiffness Mazur’s lower bound
$D^{\rm Mz}(T)$ can be written as given in Eq. (217) of Appendix H. From the
combined use of that equation and Eq. (108) one finds that in the
$m^{z}_{\eta}\ll 1$ limit of more interest for our study and up to
${\cal{O}}(u^{-2})$ order the charge stiffness is of the form
$D(t)=\frac{c_{u}\,t^{2}}{2T}(m^{z}_{\eta})^{2}$ where the coefficient $c_{u}$
obeys the double inequality,

$\displaystyle 2\left(1-\left({1/\sqrt{2}\over 2u}\right)^{2}\right)$
$\displaystyle\leq$ $\displaystyle c_{u}\leq 2\pi^{2}\left(1+\left(\frac{\ln
2}{2u}\right)^{2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptm^{z}_{\eta}\ll 1\,.$ (109)

Here $1/\sqrt{2}\approx 0.707$ and $\ln 2\approx 0.693$ have near values.

On the other hand, the use of the current spectrum, Eq. (107), in the upper
bound, Eq. (106), trivially leads in the $n_{e}=(1-m^{z}_{\eta})\ll 1$ limit
to,

$D(T)=D^{\diamond\diamond}(T)=\frac{c_{\rm
gc}^{\prime}\,t^{2}}{2T}(1-m^{z}_{\eta})\hskip 5.69046pt{\rm where}\hskip
5.69046ptc_{\rm gc}^{\prime}=2\,.$ (110)

In the $n_{e}=(1-m^{z}_{\eta})\ll 1$ limit the upper bound, Eq. (106), equals
up to ${\cal{O}}(u^{-2})$ order the charge stiffness, so that the expression,
Eq. (110), gives the exact asymptotic behavior in that limit of the charge
stiffness for $T\rightarrow\infty$ in the TL.

On the one hand, the coefficient $c_{\rm gc}$ in the upper bound, Eq. (8),
smoothly slightly increases from $c_{\rm gc}=2\pi^{2}\approx 19.74$ for
$u\rightarrow\infty$ upon decreasing $u$ at least down to $u\approx 3/2$
within the $u>3/2$ range for which its ${\cal{O}}(u^{-2})$ order expansion
remains a good approximation. At $u\approx 3/2$ it reads $c_{\rm
gc}=2\pi^{2}(1+(\ln 2/3)^{2})\approx 20.79$. On the other hand, the
coefficient $c_{\rm gc}^{\prime}$ in the upper bound, Eq. (9), reads $c_{\rm
gc}^{\prime}=2$ up to ${\cal{O}}(u^{-2})$.

This completes our finding of a vanishing charge stiffness in the TL,
$L\to\infty$, within the grand-canonical ensemble for $T\rightarrow\infty$ and
any fixed range or even distribution of $S_{\eta}^{z}$, or any distribution of
$m_{\eta}^{z}$ shrinking sufficiently fast that
$\langle(m_{\eta}^{z})^{2}\rangle\to 0$.

##  VII Concluding remarks

At $U=0$ the charge stiffness $D(T)$ of the 1D Hubbard model is a simple
problem in terms of the non-interacting electron representation. It is found
to be finite at $m_{\eta}^{z}=0$, both at zero and finite temperature.
$D(T)>0$ reaches a maximum value at $T=0$, max $D(T)=D(0)=2t/\pi$, behaving
for low and high temperature $T$ as $[D(0)-D(T)]\propto T^{2}>0$ and
$D(T)\propto 1/T$, respectively. (The qualitative difference of the $U=0$ and
$u>0$ physics and the related $T>0$ transition that occurs at $U=U_{c}=0$ is
an issue discussed in Appendix B for $m_{\eta}^{z}\rightarrow 0$ and
$m_{\eta}^{z}=0$ and in Appendix C for $m_{\eta}^{z}\in[0,1]$.)

In this paper strong evidence is provided that the charge stiffness of the 1D
Hubbard model vanishes at $m_{\eta}^{z}=0$ for $T>0$ and the whole $u>0$ range
in the TL within the canonical ensemble. For finite temperatures this leaves
out, marginally, the grand canonical ensemble in which
$\langle(m_{\eta}^{z})^{2}\rangle={\cal{O}}(1/L)$. However, the following
properties lead us to expect that our prediction remains valid at finite
temperatures in the grand-canonical ensemble case, in accord with the usual
expectation of the equivalence of ensembles in the TL.

First, we have specifically confirmed the validity of this expectation in the
limit of very high temperature $T\rightarrow\infty$. The corresponding high-
temperature charge stiffness upper bound, Eq. (108), confirms that for
$T\rightarrow\infty$ the charge stiffness of the 1D Hubbard model vanishes in
the TL in the chemical potential $\mu\rightarrow\mu_{u}$ limit where
$(\mu-\mu_{u})\geq 0$ and $2\mu_{u}$ is the Mott-Hubbard gap, Eq. (119) of
Appendix A. That upper bound was computed up to $u^{-2}$ order, which applies
for approximately $u>3/2$, yet it is expected that similar results apply for
$u>0$.

Second, at zero temperature and $m_{\eta}^{z}=0$ the charge Drude weight is
given in the TL by $D(0)=2t/\pi$ at $U=0$ and vanishes for $u>0$ Shastry-90 ;
Carmelo-92-C . That it vanishes at $T=0$ for $u>0$ reveals that a finite
charge stiffness $D(T)$ for $T>0$ at $m_{\eta}^{z}=0$ would result from
thermal fluctuations alone. That such fluctuations are largest at high
temperature thus provides strong evidence that our $T\rightarrow\infty$
results within the grand-canonical ensemble apply as well to all temperatures
$T\geq 0$.

Third, the large overestimate of the charge elementary currents we used in
deriving the charge stiffness upper bound, Eq. (91), is consistent with such
an expectation. Our canonical-ensemble charge stiffness upper bounds in Eqs.
(91)-(94) are also valid for $T\rightarrow\infty$. Their comparison with those
provided in Eqs. (108) and (110) within the grand-canonical ensemble confirms
an average charge stiffness upper bound overestimation factor $c_{\rm
oe}^{2}={\cal{O}}(L)$. For instance, $c_{\rm oe}^{2}$ changes for $u\gg 1$
from $c_{\rm oe}^{2}\approx(2/\pi^{2})L$ for $m_{\eta}^{z}\ll 1$ to $c_{\rm
oe}^{2}=n_{\rho}\,L=N_{\rho}$ for $m_{\eta}^{z}\rightarrow 1$. (For the SzSLN
subspace 1 that dominates the contributions to the charge stiffness, one has
that $1-m_{\eta}^{z}=n_{c}=n_{\rho}$.) In terms of the charge current absolute
values upper bounds derived for the canonical ensemble relative those
constructed for the grand canonical ensemble, this means for general SzSLN
subspaces an average overestimation factor $c_{\rm oe}\approx\sqrt{L}$ that
for $u\gg 1$ varies from $c_{\rm oe}\approx\sqrt{(2/\pi^{2})L}$ for
$m_{\eta}^{z}\ll 1$ to $c_{\rm oe}\approx\sqrt{n_{\rho}\,L}=\sqrt{N_{\rho}}$
for $m_{\eta}^{z}\rightarrow 1$. This huge overestimate of the charge
elementary BA currents used in the computation of the charge Drude weight
upper bound, Eq. (91), provides additional strong evidence that, as for high
temperature $T\rightarrow\infty$, the charge stiffness vanishes for finite
temperatures within the grand-canonical ensemble in the TL for chemical
potential $\mu\rightarrow\mu_{u}$ where $(\mu-\mu_{u})\geq 0$.

The use of the general formalism of hydrodynamics introduced in Refs.
Ilievski-17A ; Ilievski-17  provides further strong evidence that the charge
or spin stiffnesses vanish at finite temperatures within the grand canonical
ensemble when the corresponding chemical potentials vanish. (Within our
notation, in the case of the charge degrees of freedom the chemical potential
of such references refers to $(\mu-\mu_{u})$.) The analysis of Refs.
Ilievski-17A ; Ilievski-17  accounts for in the 1D Hubbard model the entire
space of macro-states being in a one-to-one correspondence with particle-hole
invariant commuting (fused) transfer matrices, pertaining to a discrete family
of unitary irreducible representations of the underlying quantum symmetry.
According to the authors of these references, this readily implies vanishing
finite-temperature charge or spin Drude weights when the corresponding
chemical potentials vanish, irrespective of the interaction strength.

The problem studied in this paper refers though to a controversial issue, as
different approaches yield contradictory results ZP-96 ; Kawa-98 ; PDSC-00 ;
PSZL-04 ; ANI-05 ; HPZ-11 ; Ilievski-17A ; Ilievski-17 ; Carmelo-13 ;
Karrasch-16 ; Karrasch-17 . This includes different methods based on the same
TBA. Indeed we believe that the problem is not the TBA but rather how to use
the TBA to access the stiffness of each specific solvable model. As mentioned
in Section I, our $u>0$ and $m_{\eta}^{z}=0$ predictions for $D(T)$ agree with
the conjectures of Ref. ZP-96  and the exact large-$u$ results of Ref. PDSC-00
. The latter disagree with the prediction of Ref. Kawa-98  that $D(T)$ should
be finite in the TL for $u>0$, $T>0$, and $m_{\eta}^{z}=0$. The exact
large-$u$ results of Ref. PDSC-00 , which find that $D(T)=0$ in the TL at
$m_{\eta}^{z}=0$, reveal that the results of Ref. Kawa-98  cannot be exact.
Indeed, it is shown that if $D(T)$ was finite in the TL for $T>0$ at
$m_{\eta}^{z}=0$, the pre-factor of the exponential of the Mott-Hubbard gap,
Eq. (119) of Appendix A, would be, at least, of the order of $t^{2}/U$, and
not of the order one, as found in Ref. Kawa-98 .

The method introduced in that reference for the 1D Hubbard model and used in
Ref. Zotos-99  for the spin-$1/2$ $XXZ$ chain relies on the TBA. In the case
of the spin-$1/2$ $XXZ$ chain, it leads to completely different results from
the phenomenological method of Ref. ANI-05 , which however relies on a spinon
and anti-spinon particle basis for the same TBA. The studies of Refs. SPA-11 ;
CPC ; CP  exclude the large spin stiffness found in Ref. ANI-05  for the
spin-$1/2$ $XXX$ chain spin stiffness for zero spin density and $T>0$ in the
TL.

On the one hand, the results of Refs. CPC ; CP  provide strong evidence that
those of Ref. Zotos-99  for the spin-$1/2$ $XXZ$ and $XXX$ chains spin
stiffness are correct. On the other hand, our present results reveal that the
results Ref. Kawa-98  for the charge stiffness of the half-filled 1D Hubbard,
which predict it to be finite at $m_{\eta}^{z}=0$ in the TL for $u>0$ and
$T>0$, are incorrect. This is despite such a prediction apparently relying on
the TBA-based method that has been used in Ref. Zotos-99  to derive the spin
stiffness of the spin-$1/2$ $XXZ$ and $XXX$ chains.

The possible error source of the predictions of Ref. Kawa-98  is revealed by
inspection of separate integrals of the individual summands occurring in the
integrands of Eq. (25) of that reference. One finds that such separate
integrals diverge at the hole concentration $m_{\eta}^{z}=0$ at which the
general Mazur’s inequality is inconclusive. This turns out to be a fatal
problem in that equation. Also in the case of the spin-$1/2$ $XXX$ chain the
separate integrals of the individual summands occurring in the integrands of
Eqs. (24) and (25) of Ref. ANI-05  diverge at zero spin density or which the
general Mazur’s inequality is again inconclusive. However, there is evidence
that such divergences can be removed in the case of models whose stiffness is
finite at zero temperature. This is the case of the spin-$1/2$ $XXX$ chain in
the zero spin density limit Zotos-99 . They are though a fatal problem for the
1D Hubbard model for $m_{\eta}^{z}\rightarrow 0$, $u>0$, and $T>0$, whose
charge stiffness vanishes at $T=0$ for $u>0$. This problem deserves though
further investigations.

Finally, the lack of charge ballistic transport in the 1D Hubbard model for
$u>0$ also found in this paper indicates that charge transport at high
temperatures is dominated by a diffusive contribution.

###### Acknowledgements.

We thank David K. Campbell, Pedro. D. Sacramento, and Xenophon Zotos for
discussions. J. M. P. C. and S. N. thank the support from C. S. R. C.
(Beijing) and J. M. P. C. and T. P. acknowledge the support of the ERC
Advanced Grant 694544 - OMNES. J. M. P. C. thanks the support by the
Portuguese FCT through the Grant UID/FIS/04650/2013. T. P. acknowledges the
support from the Grants of the Slovenian Research Agency (ARRS) P1-004 and
N1-0025.

##  Appendix A Functional representation for the TBA equations, $\eta$-Bethe
states energy eigenvalues, and charge current operator expectation values

Here some TBA results needed for the studies of this paper are provided. This
includes the 1D Hubbard model TBA equations within the functional
representation used in this paper. Furthermore, the model’s exact energy
eigenvalues and other energy scales related to them are also provided and the
validity of the $\beta=c,\eta$ band hole representation of the $\eta$-Bethe
states charge currents in Eq. (40) is confirmed.

Within the pseudoparticle momentum distribution functional notation used in
this paper, the TBA equations introduced in Ref. Takahashi  read,

$\displaystyle q_{j}$ $\displaystyle=$ $\displaystyle k^{c}(q_{j})+{2\over
L}\sum_{n=1}^{\infty}\sum_{j^{\prime}=1}^{L_{sn}}\,N_{sn}(q_{j^{\prime}})\arctan\left({\sin
k^{c}(q_{j})-\Lambda^{sn}(q_{j^{\prime}})\over nu}\right)$ (111)
$\displaystyle+$ $\displaystyle{2\over
L}\sum_{n=1}^{\infty}\sum_{j^{\prime}=1}^{L_{\eta n}}\,N_{\eta
n}(q_{j^{\prime}})\arctan\left({\sin k^{c}(q_{j})-\Lambda^{\eta
n}(q_{j^{\prime}})\over nu}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptj=1,...,L\,,$

and

$\displaystyle q_{j}$ $\displaystyle=$
$\displaystyle\delta_{\alpha,\eta}\sum_{\iota=\pm 1}\arcsin(\Lambda^{\alpha
n}(q_{j})-i\,\iota\,u)+{2\,(-1)^{\delta_{\alpha,\eta}}\over
L}\sum_{j^{\prime}=1}^{L}\,N_{c}(q_{j^{\prime}})\arctan\left({\Lambda^{\alpha
n}(q_{j})-\sin k^{c}(q_{j^{\prime}})\over nu}\right)$ (112) $\displaystyle-$
$\displaystyle{1\over
L}\sum_{n^{\prime}=1}^{\infty}\sum_{j^{\prime}=1}^{L_{\alpha
n^{\prime}}}\,N_{\alpha
n^{\prime}}(q_{j^{\prime}})\Theta_{n\,n^{\prime}}\left({\Lambda^{\alpha
n}(q_{j})-\Lambda^{\alpha n^{\prime}}(q_{j^{\prime}})\over u}\right)$
$\displaystyle{\rm for}\hskip 5.69046ptj=1,...,L_{\alpha n}\hskip
5.69046pt{\rm where}\hskip 5.69046pt\alpha=\eta,s\hskip 5.69046pt{\rm
and}\hskip 5.69046ptn=1,...,\infty\,.$

The sets of $j=1,...,L$ and $j=1,...,L_{\alpha n}$ quantum numbers $q_{j}$ in
Eqs. (111) and (112), respectively, which are defined in Eqs. (21) and (22),
play the role of microscopic momentum values of different TBA excitation
branches. The corresponding $\beta$-band momentum distribution functions
$N_{\beta}(q_{j})$ where $\beta=c,\eta n,sn$ read $N_{\beta}(q_{j})=1$ and
$N_{\beta}(q_{j})=0$ for occupied and unoccupied discrete momentum values,
respectively, the rapidity functional $\Lambda^{\alpha n}(q_{j})$ is the real
part of the complex rapidity, Eq. (28), and $\Theta_{n\,n^{\prime}}(x)$ is the
function,

$\displaystyle\Theta_{n\,n^{\prime}}(x)$ $\displaystyle=$
$\displaystyle\delta_{n,n^{\prime}}\Bigl{\\{}2\arctan\Bigl{(}{x\over
2n}\Bigl{)}+\sum_{l^{\prime}=1}^{n-1}4\arctan\Bigl{(}{x\over
2l^{\prime}}\Bigl{)}\Bigr{\\}}$ (113) $\displaystyle+$
$\displaystyle(1-\delta_{n,n^{\prime}})\Bigl{\\{}2\arctan\Bigl{(}{x\over|\,n-n^{\prime}|}\Bigl{)}+2\arctan\Bigl{(}{x\over
n+n^{\prime}}\Bigl{)}+\sum_{l^{\prime}=1}^{{n+n^{\prime}-|\,n-n^{\prime}|\over
2}-1}4\arctan\Bigl{(}{x\over|\,n-n^{\prime}|+2l^{\prime}}\Bigl{)}\Bigr{\\}}\,,$

where $n,n^{\prime}=1,...,\infty$. The indices $\alpha=\eta,s$ and numbers
$n=1,...,\infty$ refer to different TBA excitation branches that as discussed
in Sections II.4 and II.5 are associated with the $\alpha n$-pair
configurations within which $n=1,...,\infty$ $\eta$-spin singlet pairs are
bound. They refer to the internal degrees of freedom of the neutral composite
$\alpha n$ pseudoparticles considered in Section II.5.

Useful quantities directly related to the rapidity momentum functional
$k^{c}(q)$ and rapidity functionals $\Lambda^{\alpha n}(q)$ defined for each
$\eta$-Bethe state by Eqs. (111) and (112) are the general distributions
$2\pi\rho_{c}(k_{j})$ and $2\pi\sigma_{\alpha n}(\Lambda_{j})$ where
$\alpha=\eta,s$ and $n=1,...,\infty$. (They appear in the current spectra, Eq.
(41).) Such distributions are defined by the following derivatives,

$\displaystyle 2\pi\rho_{c}(k_{j})$ $\displaystyle=$ $\displaystyle{\partial
q^{c}(k)\over\partial k}|_{k=k_{j}}\hskip 5.69046pt{\rm and}\hskip
5.69046pt{\rm thus}\hskip 5.69046pt{1\over
2\pi\rho_{c}(k^{c}(q_{j}))}={\partial k^{c}(q)\over\partial q}|_{q=q_{j}}\,,$
$\displaystyle 2\pi\sigma_{\alpha n}(\Lambda_{j})$ $\displaystyle=$
$\displaystyle{\partial q^{\alpha
n}(\Lambda)\over\partial\Lambda}|_{\Lambda=\Lambda_{j}}\hskip 5.69046pt{\rm
and}\hskip 5.69046pt{\rm thus}\hskip 5.69046pt{1\over 2\pi\sigma_{\alpha
n}(\Lambda^{\alpha n}(q_{j}))}={\partial\Lambda^{\alpha n}(q)\over\partial
q}|_{q=q_{j}}\,.$ (114)

The functions $q^{c}(k)$ and $q^{\alpha n}(\Lambda)$ in these expressions
stand for the inverse functions of the rapidity momentum functional $k^{c}(q)$
and rapidity functionals $\Lambda^{\alpha n}(q)$, respectively.

The energy eigenvalues have for the hole concentration range
$m_{\eta}^{z}\in[0,1]$ and the spin density interval
$m_{s}^{z}\in[0,(1-m_{\eta}^{z})]$ the following form,

$E=\sum_{j=1}^{L}\left(N_{c}(q_{j})\,E_{c}(q_{j})+U/4-\mu_{\eta}\right)+\sum_{\alpha=\eta,s}\sum_{n=1}^{\infty}\sum_{j=1}^{L_{\alpha
n}}\,N_{\alpha n}(q_{j})\,E_{\alpha
n}(q_{j})+\sum_{\alpha=\eta,s}2\mu_{\alpha}\,(S_{\alpha}+S_{\alpha}^{z})\,.$
(115)

The $\alpha=\eta,s$ energy scales $2\mu_{\alpha}$ are here given by,

$2\mu_{\eta}=2\mu\hskip 5.69046pt{\rm and}\hskip
5.69046pt2\mu_{s}=2\mu_{B}\,h\,,$ (116)

where $\mu_{B}$ is the Bohr magneton and $h$ denotes a magnetic field. The
spectra $E_{c}(q_{j})$ and $E_{\alpha n}(q_{j})$ read,

$\displaystyle E_{c}(q_{j})$ $\displaystyle=$ $\displaystyle-2t\cos
k^{c}(q_{j})-U/2+\mu_{\eta}-\mu_{s}\,,$ $\displaystyle E_{\alpha n}(q_{j})$
$\displaystyle=$ $\displaystyle
n\,2\mu_{\alpha}+\delta_{\alpha,\eta}\left(2t\,\sum_{\iota=\pm
1}\Bigl{\\{}\sqrt{1-(\Lambda^{\eta n}(q_{j})-i\,\iota
nu)^{2}}\Bigr{\\}}-n\,U\right)$ (117) $\displaystyle{\rm for}\hskip
5.69046pt\alpha=\eta,s\hskip 5.69046pt{\rm where}\hskip
5.69046ptn=1,...,\infty\,,$

respectively. (The corresponding momentum eigenvalues of general $u>0$ energy
and momentum eigenstates are provided in Eq. (25).)

On the one hand, for the metallic phase densities ranges
$m_{\eta}^{z}\in]0,1]$ and $m_{s}^{z}\in[0,(1-m_{\eta}^{z})]$ the
$\alpha=\eta,s$ energy scales $2\mu_{\alpha}$ in Eq. (115) are related to the
unpaired physical $\eta$-spin $(\alpha=\eta)$ and unpaired physical spin
$(\alpha=s)$ energies relative to the ground state zero-energy level,

$\varepsilon_{\alpha,-1/2}=2\mu_{\alpha}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\varepsilon_{\alpha,+1/2}=0\hskip 5.69046pt{\rm for}\hskip
5.69046pt\alpha=\eta,s\,.$ (118)

On the other hand, for the $m_{\eta}^{z}=0$ Mott-Hubbard insulator phase,
$\varepsilon_{s,\pm 1/2}$ is given by Eq. (118) whereas it reads
$\varepsilon_{\eta,\pm 1/2}=(\mu_{u}\mp\mu)$ for the chemical potential
interval $\mu\in[-\mu_{u},\mu_{u}]$. The energy scale $\mu_{u}$ refers to the
Mott-Hubbard gap $2\mu_{u}$, which for the $S_{s}^{z}=0$ subspace considered
in some sections of this paper reads Lieb ; Lieb-03 ; Ovchi ,

$2\mu_{u}=U-4t+8t\int_{0}^{\infty}d\omega{J_{1}(\omega)\over\omega\,(1+e^{2\omega
u})}={16\,t^{2}\over
U}\int_{1}^{\infty}d\omega{\sqrt{\omega^{2}-1}\over\sinh\left({2\pi
t\omega\over U}\right)}\,.$ (119)

Here $J_{1}(\omega)$ is a Bessel function. Its limiting behaviors are
$2\mu_{u}\approx(8/\pi)\,\sqrt{t\,U}\,e^{-2\pi\left({t\over U}\right)}$ for
$u\ll 1$ Ovchi  and $2\mu_{u}\approx(U-4t)$ for $u\gg 1$.

The charge currents absolute values upper bounds and charge stiffness upper
bounds derived in this paper refer to the 1D Hubbard model in the
$S_{s}^{z}=0$ subspace for which $h=0$ and thus the energy scale
$2\mu_{s}=2\mu_{B}\,h$ in Eq. (116) vanishes, $2\mu_{B}\,h=0$. The other
energy scale $2\mu_{\eta}=2\mu$ in that equation involves the chemical
potential $\mu=\mu(m_{\eta}^{z})$. At $m_{\eta}^{z}=0$ it varies in the range
$\mu\in[-\mu_{u},\mu_{u}]$ in spite of the electronic density remaining
constant, which is a property of the corresponding $m_{\eta}^{z}=0$ and $u>0$
Mott-Hubbard insulator quantum phase. It is an odd function of the hole
concentration $m_{\eta}^{z}$. The interval $m_{\eta}^{z}\in]0,1[$ refers to
the metallic quantum phase for which $\mu=\mu(m_{\eta}^{z})$ is a continuous
function of $m_{\eta}^{z}$. It smoothly increases from $\mu=\mu_{u}$ for
$m_{\eta}^{z}\rightarrow 0$ to $\mu=(U+4t)/2$ for $m_{\eta}^{z}\rightarrow 1$
where the Mott-Hubbard gap, Eq. (119), obeys the inequality $2\mu_{u}<(U+4t)$.
It is finite for the whole $u>0$ range and vanishes at $U=0$.

Concerning the validity of the charge currents hole representation given in
Eq. (40), as mentioned in Section III.1, the 1D Hubbard model in a uniform
vector potential $\Phi/L$ whose Hamiltonian is given in Eq. (4) of Ref. Gu-07A
remains solvable by the BA. The coupling of the charge degrees of freedom to
the vector potential is described under the replacement on the left hand side
of Eq. (111) of $q_{j}$ by $q_{j}+\Phi/L$ and on the left hand side of Eq.
(112) for $\alpha n=\eta n$ of $q_{j}$ by $q_{j}-2n\,\Phi/L$. This implies
that $k^{c}(q_{j})$ and $\Lambda^{\eta n}(q_{j})$ remain having the same
expressions provided that their momentum variables $q_{j}$ are replaced by
$q_{j}+\Phi/L$ and $q_{j}-2n\,\Phi/L$, respectively. Hence the $\eta$-Bethe
states charge currents can be derived and expressed in the TL in terms of $c$
and $\eta n$-band pseudoparticle occupancies as follows,

$\displaystyle\langle\hat{J}_{LWS}(l_{\rm r},L_{\eta},S_{\eta},u)\rangle$
$\displaystyle=$ $\displaystyle-{dE\over
d(\Phi/L)}|_{\Phi=0}=\sum_{j=1}^{L}N_{c}(q_{j})\,\left(-{d\over
d(\Phi/L)}E_{c}(q_{j}+\Phi/L)|_{\Phi=0}\right)$ (120) $\displaystyle-$
$\displaystyle\sum_{n=1}^{\infty}\sum_{j=1}^{L_{\eta n}}\,N_{\eta
n}(q_{j})\,\left({d\over d(\Phi/L)}E_{\eta
n}(q_{j}-2n\,\Phi/L)|_{\Phi=0}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L}\,N_{c}(q_{j})\,\,J_{c}(q_{j})+\sum_{n=1}^{\infty}\sum_{j=1}^{L_{\eta
n}}\,N_{\eta n}(q_{j})\,\,J_{\eta n}(q_{j})\,.$

Here $E$ is in $dE/d(\Phi/L)$ the energy functional, Eq. (115) for
$S_{\alpha}+S_{\alpha}^{z}=0$ $\eta$-Bethe states, $E_{c}(q_{j})$ and $E_{\eta
n}(q_{j})$ are given in Eq. (117), and the current spectra $J_{c}(q_{j})$ and
$J_{\eta n}(q_{j})$ read,

$\displaystyle J_{c}(q_{j})$ $\displaystyle=$ $\displaystyle-{2t\sin
k^{c}(q_{j})\over 2\pi\rho_{c}(k^{c}(q_{j}))}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi,\pi]\,,$ $\displaystyle J_{\eta n}(q_{j})$
$\displaystyle=$ $\displaystyle-4nt\sum_{\iota=\pm 1}{\Lambda^{\eta
n}(q_{j})-i\,\iota nu\over 2\pi\sigma_{\eta n}(\Lambda^{\eta
n}(q_{j}))\sqrt{1-(\Lambda^{\eta n}(q_{j})-i\,\iota nu)^{2}}}\hskip
5.69046pt{\rm for}\hskip 5.69046ptq_{j}\in[-q_{\eta n},q_{\eta n}]\,,$ (121)

respectively.

That Eq. (120) can be rewritten as in Eq. (40) requires that,

$-\sum_{j=1}^{L}\,J_{c}(q_{j})-\sum_{n=1}^{\infty}\sum_{j=1}^{L_{\eta
n}}\,J_{\eta n}(q_{j})=0\,.$ (122)

In the present TL one replaces the discrete momentum values $q_{j}$ such that
$q_{j+1}-q_{j}=2\pi/L$ by a continuous momentum variable $q$, so that after
replacing sums by integrals and overall multiplication by $-1$ this equation
reads,

${L\over 2\pi}\int_{-\pi}^{\pi}dq\,J_{c}(q)+\sum_{n=1}^{\infty}{L\over
2\pi}\int_{-q_{\eta n}}^{q_{\eta n}}dq\,J_{\eta n}(q)=0\,.$ (123)

By combining Eqs. (114) and (121) one finds that the current spectra in Eq.
(121) can be written as,

$\displaystyle J_{c}(q)$ $\displaystyle=$ $\displaystyle 2t{d\over dq}\cos
k^{c}(q)\hskip 5.69046pt{\rm for}\hskip 5.69046ptq\in[-\pi,\pi]\,,$
$\displaystyle J_{\eta n}(q)$ $\displaystyle=$ $\displaystyle
4nt\sum_{\iota=\pm 1}{d\over dq}\sqrt{1-(\Lambda^{\eta n}(q)-i\,\iota
nu)^{2}}\hskip 5.69046pt{\rm for}\hskip 5.69046ptq\in[-q_{\eta n},q_{\eta
n}]\,.$ (124)

Another property used in the following is that for all $\eta$-Bethe states the
following relations,

$k^{c}(\pm\pi)=\pm\pi\hskip 5.69046pt{\rm and}\hskip 5.69046pt\Lambda^{\eta
n}(\pm q_{\eta n})=\pm\Lambda^{\eta n}_{\rm max}\,,$ (125)

apply. From the use of Eqs. (124) and (125) in Eq. (123) one confirms indeed
that,

$\displaystyle{L\over 2\pi}2t(\cos(\pi)-\cos(-\pi))+{L\over 2\pi}4nt$ (126)
$\displaystyle\times$ $\displaystyle\left(\sqrt{1-(\Lambda^{\eta n}_{\rm
max}-i\,nu)^{2}}+\sqrt{1-(\Lambda^{\eta n}_{\rm
max}+i\,nu)^{2}}-\sqrt{1-(\Lambda^{\eta n}_{\rm
max}+i\,nu)^{2}}-\sqrt{1-(\Lambda^{\eta n}_{\rm max}-i\,nu)^{2}}\right)=0\,,$

where the trivial equality $(-\Lambda^{\eta n}_{\rm max}\mp
i\,nu)^{2}=(\Lambda^{\eta n}_{\rm max}\pm i\,nu)^{2}$ was used. (The fact that
$\Lambda^{\eta n}_{\rm max}$ is given by $\infty$ is not needed for the
vanishing of the quantity in Eq. (126).)

##  Appendix B Qualitative difference of the $U=0$ and $u\rightarrow 0$
physics and the finite-$T$ transition at $U=U_{c}=0$ and $m_{\eta}^{z}=0$

Here the mechanisms behind the behavior of the $m_{\eta}^{z}=0$ charge
stiffness $D(T)$ in the TL under the $T>0$ transition that occurs at
$U=U_{c}=0$ are discussed. Such a transition is a generalization of the
$m_{\eta}^{z}=0$ zero-temperature quantum phase transition from a metal to a
Mott-Hubbard insulator that also occurs at $U=U_{c}=0$ and is driven by
correlation effects. In addition, it is confirmed that there is no
contradiction whatsoever between the charge stiffness of the 1D Hubbard model
vanishing within the canonical ensemble for $T>0$ in the TL both at
$m_{\eta}^{z}=0$ and as $m_{\eta}^{z}\rightarrow 0$ for $u>0$ and being finite
at $U=0$ in that limit and at the same hole concentration.

The general notation $|\nu,u\rangle$ for the energy eigenstates used in Eqs.
(2) and (3) refers to the whole $u\geq 0$ range. Here we use it for the
$m_{\eta}^{z}=0$ and $m_{\eta}^{z}\rightarrow 0$ energy and momentum
eigenstates. On the one hand, for $u>0$ the states $|\nu,u\rangle$ correspond
in this Appendix to the energy and momentum eigenstates $|l_{\rm
r},S_{\eta},0,u\rangle$, Eq. (13) for $m_{\eta}^{z}=0$ and
$m_{\eta}^{z}\rightarrow 0$. On the other hand, at $U=0$ the states
$|\nu,u\rangle$ are instead chosen to be the common eigenstates of the
momentum operator, current operator $\hat{J}$, and $U=0$ tight-binding
Hamiltonian,

$\hat{H}_{0}=-t\sum_{\sigma}\sum_{j=1}^{L}\left[c_{j,\sigma}^{{\dagger}}\,c_{j+1,\sigma}+{\rm
h.c.}\right]+2\mu\,{\hat{S}}_{\eta}^{z}\,.$ (127)

This Hamiltonian is that of the 1D Hubbard, Eq. (10), at $U=0$, which as given
in Eq. (29) commutes with the charge current operator, Eq. (11).

The unbinding in the $u\rightarrow 0$ limit of the $l=2,...,n$ $\eta$-spin
singlet pairs within each $u>0$ $\eta n$-pair configuration that follows from
the vanishing of the imaginary part $i\,(n+1-2l)\,u$ of the set of
corresponding $l=2,...,n$ complex rapidities with the same real part, Eq. (28)
for $\alpha=\eta$, has most severe consequences at hole concentration
$m_{\eta}^{z}=0$. As discussed in Section II.5 and Appendix C, that unbinding
along with the vanishing also as $u\rightarrow 0$ of the commutator
$[\hat{J},\hat{H}]$, Eq. (29), is associated with the rearrangement of
$\eta$-spin and spin degrees of freedom in terms of the noninteracting
electrons occupancy configurations that generate the $U=0$ common eigenstates
of the Hamiltonian, momentum operator, and charge current operator. Those are
the $U=0$ states $|\nu,u\rangle=|\nu,0\rangle$ considered here.

On the one hand and as discussed in Section II.5, the form $i\,(n+1-2l)\,u$ of
the complex $\eta n$ rapidities imaginary part and of the commutator
$[\hat{J},\hat{H}]$, Eq. (29), confirms that the $u>0$ physics survives for
any arbitrarily small value of $u$. On the other hand, in the $u\rightarrow 0$
limit the set of energy and momentum eigenstates,

$|l_{\rm r},S_{\eta},0,0^{+}\rangle\equiv\lim_{u\rightarrow 0}|l_{\rm
r},S_{\eta},0,u\rangle\,,$ (128)

remain being a complete basis in the $m_{\eta}^{z}=0$ and
$m_{\eta}^{z}\rightarrow 0$ subspaces of the noninteracting $U=0$ quantum
problem, yet are not eigenstates of the $U=0$ Hamiltonian $\hat{H}_{0}$, Eq.
(127). This is related to the states, Eq. (128), being eigenstates of the $c$
lattice $U(1)$ symmetry generator
${\tilde{N}}^{R}_{\eta}=\sum_{j=1}^{L}(1-{\tilde{n}}_{j,\uparrow}{\tilde{n}}_{j,\downarrow})$,
which counts the number of rotated-electron unoccupied plus doubly occupied
sites. (Here the $\sigma=\uparrow,\downarrow$ operators
${\tilde{n}}_{j,\sigma}$ are those in Eq. (12).) Indeed, that generator only
commutes with the 1D Hubbard model Hamiltonian for the $u>0$ range for which
it is well defined. At $U=0$ the quantum problem described by the Hamiltonian,
Eq. (127), lacks the corresponding global $c$ lattice $U(1)$ symmetry.

That the $U=0$ Hamiltonian, momentum operator, and charge current operator
eigenstates $|\nu,0\rangle$ have quantum numbers distinct from those of the
states $|l_{\rm r},S_{\eta},0,0^{+}\rangle$, Eq. (128), follows from the non-
perturbative character of the 1D Hubbard model and its corresponding different
global $SO(4)\otimes Z_{2}$ and $[SO(4)\otimes U(1)]/Z_{2}$ symmetries for
$U=0$ and $u>0$, respectively. Indeed, the $U=0$ and $u>0$ energy and momentum
eigenstates are in one-to-one correspondence to the representations of the
$SO(4)\otimes Z_{2}$ and $[SO(4)\otimes U(1)]/Z_{2}$ symmetry groups, which
have distinct structures.

The $m_{\eta}^{z}=0$ and $m_{\eta}^{z}\rightarrow 0$ common eigenstates
$|\nu,0\rangle$ of $\hat{H}_{0}$ and of the current operator can be written as
the following superposition of the states, Eq. (128),

$|\nu,0\rangle=\sum_{S_{\eta}=0}^{L/2}\sum_{l_{\rm r}}C_{l_{\rm
r},S_{\eta}}^{\nu,0}|l_{\rm r},S_{\eta},0,0^{+}\rangle\hskip 5.69046pt{\rm
where}\hskip 5.69046ptC_{l_{\rm r},S_{\eta}}^{\nu,0}=\langle l_{\rm
r},S_{\eta},0,0^{+}|\nu,0\rangle\,.$ (129)

The charge current operator $\hat{J}$ does not commute with both
$(\hat{\vec{S}}_{\eta})^{2}$ and the $c$ lattice $U(1)$ symmetry generator
${\tilde{N}}^{R}_{\eta}$, yet commutes with $\hat{S}_{\eta}^{z}$. Hence the
set of states $|l_{\rm r},S_{\eta},0,0^{+}\rangle$, Eq. (128), contributing to
each state $|\nu,0\rangle$ in the expansion of Eq. (129) have the same
eigenvalue $S_{\eta}^{z}$ but may have different $\eta$-spin values
$S_{\eta}=0,1,2,...$ and different ${\tilde{N}}^{R}_{\eta}$ eigenvalues
$L_{\eta}=2S_{\eta}+2\Pi_{\eta}=0,2,4,...$.

The use of the non-interacting basis associated with the representation of the
$U=0$ energy eigenstates $\\{|\nu,0\rangle\\}$ in terms of simple electron
occupancy configurations relative to the electronic vacuum, renders trivial
the problem of the calculation of the real part of the conductivity
$\sigma(\omega,T)$, Eq. (1), at $m_{\eta}^{z}=0$ and $U=0$. That the states
$|\nu,0\rangle$ are eigenstates of the current operator $\hat{J}$, implies
that $\langle\nu,0|\hat{J}|\nu^{\prime},0\rangle=0$ in Eq. (3) where
$\nu\neq\nu^{\prime}$, so that $\sigma_{reg}(\omega,T)=0$. As is well known,
Eq. (1) becomes then $\sigma(\omega,T)=2\pi\,D(T)\,\delta(\omega)$. Trivial
calculations relying onto the simple form in terms of electron creation
operators of the generators onto the electronic vacuum of the states
$|\nu,0\rangle$ lead to the known result that $D(T)>0$ reaches a maximum value
at $T=0$, max $D(T)=D(0)$, behaving for low and high temperature $T$ as
$[D(0)-D(T)]\propto T^{2}>0$ and $D(T)\propto 1/T$, respectively.

Turning on an arbitrarily small infinitesimal $u\ll 1$ value leads to the
emergence of the complex $\eta n$ rapidities imaginary part $i\,(n+1-2l)\,u$
associated with the rearrangement of the $\eta$-spin-singlet configurations
and brings about a nonzero commutator of the charge current operator with the
Hamiltonian, Eq. (29). This prevents the existence of common eigenstates of
these two operators. For any finite onsite repulsion value $u>0$, the energy
and momentum eigenstates are the states $|l_{\rm r},S_{\eta},0,u\rangle$ on
the right-hand side of Eq. (128), which are as well eigenstates of the
generator ${\tilde{N}}^{R}_{\eta}$. As found in this paper, for such
eigenstates of both the Hamiltonian, Eq. (10), and that generator one has
within the canonical ensemble that $D(T)=0$ for $u>0$ in the TL both at
$m_{\eta}^{z}=0$ and as $m_{\eta}^{z}\rightarrow 0$.

An interesting property is that the real-part of the conductivity sum rule
remains invariant under the finite-temperature $U=U_{c}=0$ transition, i.e. it
has no discontinuity at $U=0$. Indeed, it has the same value at $U=0$ and for
$u\rightarrow 0$, the following relations holding,

$\displaystyle\lim_{u\rightarrow 0}2\pi\,D(T)$ $\displaystyle=$
$\displaystyle\left[\int_{-\infty}^{\infty}d\omega\,\sigma_{reg}(\omega,T)\right]|_{U=0}=0\hskip
5.69046pt{\rm at}\hskip 5.69046ptm_{\eta}^{z}=0\,,$ $\displaystyle
2\pi\,D(T)|_{U=0}$ $\displaystyle=$ $\displaystyle\lim_{u\rightarrow
0}\int_{-\infty}^{\infty}d\omega\,\sigma_{reg}(\omega,T)>0\hskip 5.69046pt{\rm
at}\hskip 5.69046ptm_{\eta}^{z}=0\,.$ (130)

The singular behavior associated with the $U=U_{c}=0$ transition is rather
that $\\{D(T)=0$ and $\sigma_{reg}(\omega,T)>0\\}$ and $\\{D(T)>0$ and
$\sigma_{reg}(\omega,T)=0\\}$ for any arbitrarily small infinitesimal $u\ll 1$
and at $U=0$, respectively.

That the real part of the conductivity sum rule is finite and $D(T)=0$ for
$u\rightarrow 0$ implies that $\sigma_{reg}(\omega,T)$ does not vanish in the
$u\rightarrow 0$ limit and thus that some of the off-diagonal matrix elements
in the $\sigma_{reg}(\omega,T)$ expression between states $|l_{\rm
r},S_{\eta},0,0^{+}\rangle$ and $|l_{\rm r},S_{\eta}\pm 1,0,0^{+}\rangle$
allowed by the selection rule, Eq. (34), are non vanishing. This indeed occurs
provided that the $\eta$-spin value differs by $\delta S_{\eta}=\pm 1$ if
$S_{\eta}>0$ and by $\delta S_{\eta}=1$ if $S_{\eta}=0$. This is consistent
with the states $|l_{\rm r},S_{\eta},0,0^{+}\rangle$ not being eigenstates of
the current $\hat{J}$, in contrast to the $U=0$ energy eigenstates
$|\nu,0\rangle$. For the latter states all the off-diagonal current matrix
elements $\langle\nu,0|\hat{J}|\nu^{\prime},0\rangle=0$ in Eq. (3) vanish, so
that $\sigma_{reg}(\omega,T)=0$ and $D(T)>0$ at $U=0$, as given in Eq. (130).

Let $|\nu,0\rangle$ be a $U=0$ energy and momentum eigenstate that contributes
to the charge stiffness $D(T)$, Eq. (3), so that
$\langle\nu,0|\hat{J}|\nu,0\rangle\neq 0$. >From the use of Eq. (129) one may
express the current operator expectation value
$\langle\nu,0|\hat{J}|\nu,0\rangle$ of such a state as follows,

$\langle\nu,0|\hat{J}|\nu,0\rangle=\sum_{S_{\eta}=|S_{\eta}^{z}|}^{L/2}\sum_{l_{\rm
r}}\sum_{\iota=\pm 1}\left(C_{l_{\rm r},S_{\eta}}^{\nu,0}\right)^{*}C_{l_{\rm
r},S_{\eta}+\iota}^{\nu,0}\Theta(S_{\eta}+\iota)\,\langle l_{\rm
r},S_{\eta},0,0^{+}|\hat{J}|l_{\rm r},S_{\eta}+\iota,0,0^{+}\rangle\,.$ (131)

The equality given here confirms that the off-diagonal quantum overlap through
the current operator of the states $|l_{\rm r},S_{\eta},0,0^{+}\rangle$ and
$|l_{\rm r},S_{\eta}^{\prime},0,0^{+}\rangle$ whose $S_{\eta}$ and
$S_{\eta}^{\prime}$ values differ by $\delta S_{\eta}=\pm 1$ is fully
consistent with the $U=0$ Hamiltonian, momentum operator, and charge current
operator eigenstates $|\nu,0\rangle$ having finite current expectation value.

##  Appendix C Effect of varying $u=U/4t$ on the microscopic processes behind
the largest charge current absolute value of a SzS subspace

As reported in Section IV.2, for each SzS subspace as defined in Section III.2
the largest charge current absolute value is that of the corresponding
reference SzSLN subspace 1 introduced in Section IV.2. This is a result that
can be physically understood in terms of the microscopic mechanisms that
contribute to the charge currents of the $\eta$-Bethe states. Here we present
an analysis of the problem that relies on the exact properties considered in
Section III.1. In addition, we discuss the similarities and differences
relative to the case of the spin stiffness and currents of the spin-$1/2$
$XXX$ chain studied in Refs. CPC ; CP  by the general upper-bound method used
in this paper.

Following the analysis and results of Section III.1, the charge currents of
the $\eta$-Bethe states result from microscopic processes that are easiest to
be described in terms of original lattice occupancy configurations. The
corresponding charge carriers naturally describe the charge degrees of freedom
of the rotated electrons whose operators are given in Eq. (12). Those are
related to the electrons by a unitary transformation uniquely defined in Ref.
Carmelo-17 . Specifically, the $\eta$-Bethe-states charge currents result from
processes within which a number $M_{\eta}=2S_{\eta}$ of unpaired physical
$\eta$-spins $1/2$ that couple to charge probes interchange position with
$N_{\rho}$ charge pseudoparticles belonging to the $c$ and $\eta n$ branches.
This occurs upon the charge pseudoparticles moving along that lattice. This
relative motion is associated with the $c$ and $\eta n$ pseudoparticle
momentum $q_{j}$ and $\pi-q_{j}$, respectively, as given in Eq. (25).

The $M_{\eta}=2S_{\eta}$ unpaired physical $\eta$-spins $1/2$ couple to charge
probes. However, the charge current only flows upon them interchanging
positions with the charge $c$ and $\eta n$ pseudoparticles. Consistently, for
finite $u$ a SzSLN subspace largest charge current absolute value is
proportional to $M_{\eta}\,N_{\rho}$. (SzSLN subspaces are defined in Section
III.2.) This refers to $\eta$-Bethe states for which
$M_{\eta}=M_{\eta,+1/2}=2S_{\eta}$. Upon replacing the numbers of unpaired
physical $\eta$-spins $1/2$ and of charge pseudoparticles by those of unpaired
physical spins $1/2$ and of spin pseudoparticles, respectively, this factor
$M_{\eta}\,N_{\rho}$ is similar to that of the spin-$1/2$ $XXX$ chain largest
spin current absolute values CP .

The largest charge current absolute value of each SzSLN subspace is rather
written in Eq. (60) as proportional to $m_{\eta}\,(1-m_{\eta})$ and thus to
$2S_{\eta}\,(L-2S_{\eta})$. This follows from the expression given in that
equation applying both to $u=U/4t\rightarrow 0$ and to finite $u$. Indeed, as
justified in the following, such a largest charge current absolute value can
be written as proportional to $2S_{\eta}\,(L-2S_{\eta})$ and
$M_{\eta}\,N_{\rho}$ for $u\rightarrow 0$ and finite $u$, respectively. If one
requires it to be valid both for $u\rightarrow 0$ limit and to finite $u$,
then it should indeed be written as given in Eq. (60).

The charge current absolute value in that equation can be expressed as
$(C_{l_{\eta},n_{\eta},m_{s},n_{s}}/L)\,t\,M_{\eta}\,(N_{c}+2\Pi_{\eta})$
where $M_{\eta}=2S_{\eta}$ and $N_{c}+2\Pi_{\eta}=L-2S_{\eta}$. The latter
number involves that of a $\eta$-Bethe-state paired rotated $\eta$-spins
$1/2$, which reads $2\Pi_{\eta}=\sum_{n=1}^{\infty}2n\,N_{\eta n}$. As
discussed below for $m_{\eta}\in[0,1]$ and in Appendix B for $m_{\eta}=0$, the
physics is very different (i) both at $u=0$ and in the $u\rightarrow 0$ limit
and (ii) for finite $u$. The corresponding numbers of charge carriers will be
found to be different in these two cases and given by (i)
$N_{c}+2\Pi_{\eta}=L-2S_{\eta}$ and (ii) $N_{\rho}=N_{c}+N_{\eta}$,
respectively. This is why for finite $u$ the largest charge current absolute
value of each SzSLN subspace can be written as proportional to
$M_{\eta}\,N_{\rho}=M_{\eta}\,(N_{c}+N_{\eta})$ and thus expressed in terms of
the number of charge pseudoparticles $N_{\rho}=N_{c}+N_{\eta}$, Eq. (43),
which refers to the $N_{c}$ $c$ pseudoparticles and
$N_{\eta}=\sum_{n=1}^{\infty}N_{\eta n}\leq\Pi_{\eta}$ $\eta n$
pseudoparticles. The different charge carriers that within the present
representation emerge as $u\rightarrow 0$ is an issue that will be clarified
below.

On the one hand, each $c$ and $\eta n$ pseudoparticle occupies one and a
number $2n=2,4,6,...$, respectively, of original lattice sites. Hence the set
of charge pseudoparticles of a $\eta$-Bethe state occupy a number
$N_{c}+2\Pi_{\eta}=N_{c}+\sum_{n=1}^{\infty}2n\,N_{\eta n}$ of sites of that
lattice. On the other hand, each of the $M_{\eta}=2S_{\eta}$ unpaired physical
$\eta$-spins $1/2$ occupies a single site of the same lattice. The
corresponding exact sum rule, $M_{\eta}+N_{c}+2\Pi_{\eta}=L$, then ensures and
confirms that the charge degrees of freedom of all the $L$ original-lattice
sites are accounted for within the exact rotated-electron related
representation used in the studies of this paper.

The set of $2n$ original-lattice sites occupied by each $\eta n$
pseudoparticle refers to its internal degrees of freedom. It is its center of
mass that moves with momentum $\pi-q_{j}$. All $2n$ paired rotated
$\eta$-spins $1/2$ move coherently with it upon processes within which they
interchange position with the $M_{\eta}$ unpaired physical $\eta$-spins $1/2$
that singly occupy original-lattice sites.

For the $\eta$-Bethe states that span a SzSL subspace as defined in Section
III.2, both the number $M_{\eta}=2S_{\eta}$ of unpaired physical $\eta$-spins
and $L_{\eta}$ of $\eta$-spins are fixed. Hence the number
$L_{\eta}=N_{c}+2\Pi_{\eta}=L-2S_{\eta}$ of original-lattice sites occupied by
the charge pseudoparticles is fixed as well. In contrast, that of charge
pseudoparticles, $N_{\rho}=N_{c}+N_{\eta}$, varies from $N_{\rho}=1$ for the
present case of $S_{s}^{z}=0$ and thus $M_{\eta}=2S_{\eta}$ even, up to a
maximum value $N_{\rho}=N_{c}=L-2S_{\eta}$. The minimum value, $N_{\rho}=1$,
refers to a single composite $\eta n$ pseudoparticle with all the $\eta$-Bethe
state $n=\Pi_{\eta}=(L-2S_{\eta})/2$ $\eta$-spin singlet pairs bound within
it. Each of the possible different numbers
$N_{\rho}=N_{c}+N_{\eta}\in[1,(L-2S_{\eta})]$ of charge pseudoparticles refers
to a SzSLN subspace contained in the SzSL subspace under consideration.

In the case of the 1D Hubbard model in a SzSL subspace, a second aspect to
justify the largest charge current absolute value of that subspace must be
accounted for. This in addition to the charge current absolute values
increasing upon considering its SzSLN subspaces with an increasing number
$N_{\rho}$ of charge pseudoparticles that interchange position with the fixed
number $M_{\eta}=2S_{\eta}$ of unpaired physical $\eta$-spins $1/2$. It refers
to the current spectra and elementary currents absolute values
$|J_{c}(q_{j})|=|J_{c}^{h}(q_{j})|$ and $|j_{c}(q_{j})|=|j_{c}^{h}(q_{j})|$ of
the charge $c$ pseudoparticles relative to those of the $\eta n$
pseudoparticles, $|J_{\eta n}(q_{j})|=|J_{\eta n}^{h}(q_{j})|$ and $|j_{\eta
n}(q_{j})|=|j_{\eta n}^{h}(q_{j})|$, respectively. (The (i) $\beta=c,\eta n$
bands current spectra and (ii) $\beta=c,\eta n$ bands elementary currents
associated with the absolute values (i)
$|J_{\beta}(q_{j})|=|J_{\beta}^{h}(q_{j})|$ and (ii)
$|j_{\beta}(q_{j})|=|j_{\beta}^{h}(q_{j})|$ are given in (i) Eq. (41) and Eq.
(121) of Appendix A and (ii) in Eq. (56), respectively.)

At the fixed $m_{\eta}$ and $l_{\eta}$ densities of a SzSL subspace, such
spectra absolute values are in general functions of $u$ and of the densities
$n_{\rho}$, $m_{s}$, and $n_{s}$. It is a simple exercise to show that such
absolute values are larger for $\eta$-Bethe states for which $n_{s}=n_{s}^{\rm
max}=(1-l_{\eta}-m_{s})/2$. Hence in the following we consider the charge
currents of the latter states whose current spectra at fixed $m_{\eta}$ and
$l_{\eta}$ densities are functions of $u$ and of the densities $n_{\rho}$ and
$m_{s}$.

We recall that the $\eta$-Bethe states exact charge currents can be expressed
either in terms of the spectra $J_{\beta}(q_{j})$ associated with charge
$\beta=c,\eta n$ pseudoparticles, Eq. (120) of Appendix A, or with the related
spectra $J_{\beta}^{h}(q_{j})=-J_{\beta}(q_{j})$ associated with the
$\beta=c,\eta n$ bands holes, Eq. (40). Here and in the following analysis it
is more convenient to express such charge currents in terms of the
$\beta=c,\eta n$ pseudoparticles spectra $J_{\beta}(q_{j})$. The same applies
to the related alternative particle-like and hole-like elementary currents
$j_{\beta}(q_{j})$ and $j_{\beta}^{h}(q_{j})$, respectively.

It is useful to define the average values $|{\bar{J}}_{\beta}|$ and
$|{\bar{j}}_{\beta}|$ of the $\beta=c,\eta n$ quantities $|J_{\beta}(q_{j})|$
and $|j_{\beta}(q_{j})|$, respectively, which read,

$|{\bar{J}}_{\beta}|={1\over
2q_{\beta}}\int_{-q_{\beta}}^{q_{\beta}}dq\,|J_{\beta}(q)|\hskip 5.69046pt{\rm
and}\hskip 5.69046pt|{\bar{j}}_{\beta}|={1\over
2q_{\beta}}\int_{-q_{\beta}}^{q_{\beta}}dq\,|j_{\beta}(q)|\hskip 5.69046pt{\rm
where}\hskip 5.69046ptq_{c}=\pi\hskip 5.69046pt{\rm and}\hskip
5.69046ptq_{\eta n}=\pi n_{\eta n}\,,$ (132)

where within the TL the discrete momentum values $q_{j}$ such that
$q_{j+1}-q_{j}=2\pi/L$ were replaced by a corresponding continuous momentum
variable $q$. From manipulations of the TBA equations and related $\eta$-Bethe
states current spectra and elementary currents expressions, one finds that the
following exact inequalities hold for the whole $u\geq 0$ range,

$\displaystyle{\rm max}\\{|J_{c}(q_{j})|\\}$ $\displaystyle\geq$
$\displaystyle C_{u,n}\,{\rm max}\\{|J_{\eta n}(q_{j})|\\}\hskip 5.69046pt{\rm
and}\hskip 5.69046pt{\rm max}\\{|j_{c}(q_{j})|\\}\geq C_{u,n}\,{\rm
max}\\{|j_{\eta n}(q_{j})|\\}\,,$ $\displaystyle|{\bar{J}}_{c}|$
$\displaystyle\geq$ $\displaystyle C_{u,n}\,|{\bar{J}}_{\eta n}|\hskip
5.69046pt{\rm and}\hskip 5.69046pt|{\bar{j}}_{c}|\geq
C_{u,n}\,|{\bar{j}}_{\eta n}|\,,$ (133)

where $C_{u,n}\in[1/2n,1]$ is an increasing function of $u$ with limiting
behaviors,

$\displaystyle C_{u,n}$ $\displaystyle=$ $\displaystyle{1\over 2n}\hskip
5.69046pt{\rm for}\hskip 5.69046ptu\rightarrow 0\,,$ (134) $\displaystyle=$
$\displaystyle 1\hskip 5.69046pt{\rm for}\hskip
5.69046ptu\rightarrow\infty\,.$

According to such inequalities, for large $u$ values and all SzSLN subspaces
of a SzS subspace, the maximum and average values of $|J_{c}(q_{j})|$ and
$|j_{c}(q_{j})|$ are larger than those of $|J_{\eta n}(q_{j})|$ and $|j_{\eta
n}(q_{j})|$, respectively. Upon decreasing $u$, one finds that for
intermediate $u$ the maximum and average values of $|J_{c}(q_{j})|$ and
$|j_{c}(q_{j})|$ and those of $|J_{\eta n}(q_{j})|$ and $|j_{\eta n}(q_{j})|$,
respectively, remain of the same order.

Consistently, for large and intermediate $u$ values maximizing the number of
charge pseudoparticles $N_{\rho}=N_{c}+N_{\eta}$ that, upon moving in the
original lattice, interchange position with a fixed number
$M_{\eta}=2S_{\eta}$ of unpaired physical $\eta$-spins $1/2$ that couple to
charge probes, gives the largest charge current absolute value of a SzS
subspace. Indeed, for finite $u$ such an absolute value is proportional to
$M_{\eta}\,N_{\rho}=M_{\eta}\,(N_{c}+N_{\eta})$. This maximization procedure
then leads to the largest charge current absolute value being that of the
reference SzSLN subspace 1, for which $N_{\rho}=N_{c}+N_{\eta}$ reaches its
maximum value, $N_{\rho}=N_{c}=L-2S_{\eta}$.

Such a procedure is similar to that used for the spin-$1/2$ $XXX$ chain
concerning the largest spin current absolute value of a fixed-spin subspace CP
. There is though a different aspect relative to that chain. For it, all spin
pseudoparticles have internal degrees of freedom involving a number $2n$ of
paired spins $1/2$. The largest spin current absolute value is then reached
for a Bethe state populated by a number $(L-2S_{s})/2$ of $n=1$ spin
pseudoparticles whose internal degrees of freedom correspond to one spin-
singlet pair. Within a superficial comparison with the spin-$1/2$ $XXX$ chain,
one would then expect that the largest charge current absolute value of a 1D
Hubbard model SzS subspace would be that of a $\eta$-Bethe state populated by
a number $N_{\rho}=N_{\eta 1}=(L-2S_{\eta})/2$ of $\eta 1$ pseudoparticles
whose internal degrees of freedom correspond to one $\eta$-spin singlet pair.
This is the largest charge current absolute value of the reference SzSLN
subspace 2 as defined in Section IV.2.

However, for the 1D Hubbard model the charge pseudoparticles include both $c$
pseudoparticles without internal degrees of freedom and $\eta n$
pseudoparticles whose internal degrees of freedom involve a number $2n$ of
paired $\eta$-spins $1/2$. This is related to for $u>0$ the charge degrees of
freedom of the 1D Hubbard model being associated with a $U(2)=SU(2)\otimes
U(1)$ symmetry whereas the spin degrees of freedom of the spin-$1/2$ $XXX$
chain are associated only with a $SU(2)$ symmetry. As a result, the
$\eta$-Bethe state with largest charge current absolute value is rather found
to be populated by $N_{\rho}=N_{c}=L-2S_{\eta}$ $c$ pseudoparticles, which
thus is the largest charge current absolute value of the reference SzSLN
subspace 1.

As one decreases $u$ to reach small $u\ll 1$ values, the coefficient $C_{u,n}$
in Eqs. (133) and (134) tends to approach the value $1/2n$. The microscopic
mechanisms that justify why the largest charge current absolute value of the
reference SzSLN subspace 1 remains being the largest charge current absolute
value of the corresponding SzS subspace become now completely different from
those of the spin-$1/2$ $XXX$ chain. Before addressing that problem, we
provide useful information that confirms the validity of the inequalities, Eq.
(133), for small and intermediate values of $u$ for which the coefficient
$C_{u,n}$ remains being of the order of the unity. Specifically, we consider
densities given in Eqs. (45)-(50) in the vicinity of those of the reference
SzSLN subspaces 1 to 3 defined in Section IV.2.

The expressions of the $\beta=c,\eta n$ pseudoparticle spectra
$J_{\beta}(q_{j})$ and elementary currents $j_{\beta}(q_{j})$ given in the
following refer to $\eta$-Bethe states whose $c$ and $\eta n$ band
distributions are compact and asymmetrical, as given in Eq. (51). They refer
as well to $\eta$-Bethe states that in the TL are generated from those by a
finite number of $\beta=c,\eta n$ pseudoparticle processes. Furthermore, they
are also valid for $\eta$-Bethe states whose $c$ and $\eta n$ band
distributions are compact and symmetrical, Eq. (55), and $\eta$-Bethe states
generated from those by a finite number of $\beta=c,\eta n$ pseudoparticle
processes.

On the one hand, for the SzSLN subspaces for which the $J_{\beta}(q_{j})$ and
$j_{\beta}(q_{j})$ expressions are independent of $u$ for $u>0$, such
expressions are valid for all the classes of $\eta$-Bethe states under
consideration. On the other hand, for SzSLN subspaces for which they are $u$
dependent for $u>0$, their universality only applies to large $u$. Indeed,
upon decreasing $u$, the $J_{\beta}(q_{j})$ and $j_{\beta}(q_{j})$ expressions
become different for different $\eta$-Bethe states, as they become dependent
on the specific $q_{F\beta,\iota}^{\tau}$ values in Eq. (52) that define such
states. In addition to providing $J_{\beta}(q_{j})$ and $j_{\beta}(q_{j})$
expressions that are independent of $u$ for $u>0$ and are valid for the $u>0$
range, for simplicity in the case of $u$ dependent expressions for $u>0$ only
their universal expansions up to $u^{-1}$ order are given.

For $\eta$-Bethe states whose densities, Eqs. (45)-(50), are in the vicinity
of those of a reference SzSLNSN subspace 1B, both the $J_{\beta}(q_{j})$ and
$j_{\beta}(q_{j})$ expressions and the corresponding inequalities, Eq. (133),
are independent of $u$ for $u>0$ and remain valid for the whole $u>0$ range.
For these $\eta$-Bethe states the following expressions are found for $u>0$,
$m_{\eta}\in[0,1]$, and $m_{s}\in[0,(1-m_{\eta})]$,

$\displaystyle J_{c}(q_{j})$ $\displaystyle=$ $\displaystyle
j_{c}(q_{j})=-2t\sin(q_{j})\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi,\pi]\,,$ $\displaystyle J_{\eta n}(q_{j})$
$\displaystyle=$ $\displaystyle j_{\eta n}(q_{j})=-2t\,q_{j}\,{\sin(\pi
m_{\eta})\over\pi m_{\eta}}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi m_{\eta},\pi m_{\eta}]\,,$ $\displaystyle{\rm
max}\\{|J_{c}(q_{j})|\\}$ $\displaystyle=$ $\displaystyle{\rm
max}\\{|j_{c}(q_{j})|\\}=2t\hskip 5.69046pt{\rm and}\hskip
5.69046pt|{\bar{J}}_{c}|=|{\bar{j}}_{c}|={4t\over\pi}\,,$ $\displaystyle{\rm
max}\\{|J_{\eta n}(q_{j})|\\}$ $\displaystyle=$ $\displaystyle{\rm
max}\\{|j_{\eta n}(q_{j})|\\}=2t\,\sin(\pi m_{\eta})\hskip 5.69046pt{\rm
and}\hskip 5.69046pt|{\bar{J}}_{\eta n}|=|{\bar{j}}_{\eta n}|=t\sin(\pi
m_{\eta})\,.$ (135)

These exact expressions and values obey the inequalities, Eqs. (133) and
(134). (As discussed below, for the reference SzSLNSN subspace 1B one has that
$C_{u,n}=1$ for $u>0$ and $C_{u,n}=1/2n$ for $u\rightarrow 0$.)

For most SzSLN subspaces of a SzS subspace, the $\beta=c,\eta n$ spectra
$J_{\beta}(q_{j})$ and elementary currents $j_{\beta}(q_{j})$ depend on $u$.
This applies for densities, Eqs. (45)-(50), in the vicinity of those of a
reference SzSLNSN subspace 1A, for which the expansions of $J_{c}(q_{j})$ and
$J_{\eta n}(q_{j})$ are up to $u^{-1}$ order and for $m_{\eta}\in[0,1]$ and
$m_{s}\in[0,(1-m_{\eta})]$ given by,

$\displaystyle J_{c}(q_{j})$ $\displaystyle=$
$\displaystyle-2t\sin(q_{j})+2t\,{n_{\eta s}\over u}\sin(2q_{j})\hskip
5.69046pt{\rm for}\hskip 5.69046ptq_{j}\in[-\pi,\pi]$ $\displaystyle J_{\eta
n}(q_{j})$ $\displaystyle=$
$\displaystyle-2t\,q_{j}\left(1+{2\,(1-m_{s})\,g_{s}\over\pi\,u}\sin(\pi
m_{\eta})\right){\sin(\pi m_{\eta})\over\pi m_{\eta}}+{2t\over
u}\sin\left({q_{j}\over m_{\eta}}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi m_{\eta},\pi m_{\eta}]\,.$ (136)

As given in Eq. (71), $n_{\eta s}=(1-m_{\eta}-m_{s})\,g_{s}$ and the function
$g_{s}(m_{s})$ appearing here continuously increases from $g_{s}(0)=\ln 2$ to
$g_{s}(1-m_{\eta})=1$ as $m_{s}$ increases from $m_{s}=0$ to
$m_{s}=(1-m_{\eta})$, respectively.

Up to $u^{-1}$ order and for $m_{\eta}\in[0,1]$ and
$m_{s}\in[0,(1-m_{\eta})]$, the corresponding expansions for $j_{c}(q_{j})$
and $j_{\eta n}(q_{j})$ are found to read,

$\displaystyle j_{c}(q_{j})$ $\displaystyle=$ $\displaystyle
J_{c}(q_{j})-4t\,{n_{\eta s}\over
u}\,{\sin(\pi(1-m_{\eta}))\over\pi(1-m_{\eta})}\sin(q_{j})$ $\displaystyle=$
$\displaystyle-2t\sin(q_{j})+2t\,{n_{\eta s}\over u}\sin(2q_{j})-4t\,{n_{\eta
s}\over u}\,{\sin(\pi(1-m_{\eta}))\over\pi(1-m_{\eta})}\sin(q_{j})\hskip
5.69046pt{\rm for}\hskip 5.69046ptq_{j}\in[-\pi,\pi]$ $\displaystyle j_{\eta
n}(q_{j})$ $\displaystyle=$ $\displaystyle J_{\eta
n}(q_{j})-{2t\,(1-m_{\eta}-2m_{s})\over u\,(1-m_{\eta})}{\sin(2\pi
m_{\eta})\over 2\pi m_{\eta}}\sin\left({q_{j}\over m_{\eta}}\right)$ (137)
$\displaystyle=$
$\displaystyle-2t\,q_{j}\left(1+{2\,(1-m_{s})\,g_{s}\over\pi\,u}\sin(\pi
m_{\eta})\right){\sin(\pi m_{\eta})\over\pi m_{\eta}}$ $\displaystyle+$
$\displaystyle{2t\over
u}\left(1-{(1-m_{\eta}-2m_{s})\over(1-m_{\eta})}\,{\sin(2\pi m_{\eta})\over
2\pi m_{\eta}}\right)\sin\left({q_{j}\over m_{\eta}}\right)\hskip
5.69046pt{\rm for}\hskip 5.69046ptq_{j}\in[-\pi m_{\eta},\pi m_{\eta}]$
$\displaystyle{\rm where}\hskip 5.69046ptm_{\eta}\in[0,1]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm_{s}\in[0,(1-m_{\eta})]\,.$

To leading $(1/u)^{0}$ order, the $\beta=c,\eta n$ values of ${\rm
max}\\{|J_{\beta}(q_{j})|\\}$, ${\rm max}\\{|j_{\beta}(q_{j})|\\}$,
$|{\bar{J}}_{\beta}|$, and $|{\bar{j}}_{\beta}|$ remain being those given in
Eq. (135). For simplicity, their $u^{-1}$ corrections are not given here. The
exact expressions and values in Eqs. (136) and (137) again obey the
inequalities, Eqs. (133) and (134).

For densities in Eqs. (45)-(50) in the vicinity of those of the reference
SzSLN subspaces 2 and 3, one finds $J_{c}(q_{j})=j_{c}(q_{j})=2t\sin(q_{j})$
for $q_{j}\in[-\pi,\pi]$ and the whole $u\geq 0$ range. Moreover, $J_{\eta
n}(q_{j})=J_{\eta n}\sin q_{j}$ and $j_{\eta n}(q_{j})=j_{\eta n}\sin q_{j}$
where $J_{\eta n}\propto 1/u$ and $j_{\eta n}\propto 1/u$ for large $u$, so
that the inequalities, Eq. (133), are also obeyed in that limit.

As one decreases $u$ to reach $u\ll 1$ values, the coefficient $C_{u,n}$ in
Eqs. (133) and (134) approaches its limiting smallest value, $1/2n$.
Consistently, for some of the SzSLN subspaces contained in a SzS subspace the
maximum and average absolute values of the spectra $J_{\eta n}(q_{j})$ and
elementary currents $j_{\eta n}(q_{j})$ become larger than those of
$J_{c}(q_{j})$ and $j_{c}(q_{j})$, respectively. This holds particularly for
large $n$ values. The question is thus why does the largest charge current
absolute value of a SzS subspace remains that of its reference SzSLN subspace
1 for which $N_{\rho}=N_{c}$ and $N_{\eta}=\sum_{n=1}^{\infty}N_{\eta n}=0$?

To clarify this interesting issue, we account for again that for finite yet
arbitrarily small $u$ values the largest charge current absolute value of each
SzSLN subspace can be written as proportional to
$M_{\eta}\,N_{\rho}=M_{\eta}\,(N_{c}+N_{\eta})$. This number can be rewritten
as $M_{\eta}\,N_{\rho}=R_{\rho}\,M_{\eta}\,(N_{c}+2\Pi_{\eta})$ and thus
$M_{\eta}\,N_{\rho}=R_{\rho}\,2S_{\eta}\,(L-2S_{\eta})$. Here the exact
relation $N_{c}+2\Pi_{\eta}=L-2S_{\eta}$ was used and the coefficient
$R_{\rho}=N_{\rho}/(L-2S_{\eta})=(N_{c}+N_{\eta})/(N_{c}+2\Pi_{\eta})$ that
can be written as
$R_{\rho}=n_{\rho}/(1-m_{\eta})=(1-l_{\eta}+n_{\eta})/(1-l_{\eta}+2\pi_{\eta})$
varies in the range $R_{\rho}\in[1/(L-2S_{\eta}),1]$. Within the TL, that
interval reads $R_{\rho}\in[0,1]$ for $m_{\eta}<1$.

For the $\eta$-Bethe states that span a SzS subspace, the factor
$2S_{\eta}\,(L-2S_{\eta})$ in
$M_{\eta}\,N_{\rho}=R_{\rho}\,2S_{\eta}\,(L-2S_{\eta})$ has a fixed value.
Thus the changes in $M_{\eta}\,N_{\rho}$ all stem from $R_{\rho}$. On the one
hand, the maximum $R_{\rho}$ value, $R_{\rho}=1$, refers to $N_{\rho}=N_{c}$
and $N_{\eta}=\sum_{n=1}^{\infty}N_{\eta n}=0$. On the other hand, for
reference SzSLNSN subspaces for which $N_{\eta}>0$, increasing the population
of $\eta n$ pseudoparticles with a large number $n$ of $\eta$-spin singlet
pairs leads to a strong decreasing of the coefficient $R_{\rho}\in[0,1]$ from
its maximum value. It turns out that for small yet finite $u$ values such a
decreasing in the charge current factor
$M_{\eta}\,N_{\rho}=R_{\rho}\,2S_{\eta}\,(L-2S_{\eta})$ always compensates the
increasing of the spectra $J_{\eta n}(q_{j})$ and elementary currents $j_{\eta
n}(q_{j})$ absolute values upon increasing $n$. Indeed, for any arbitrarily
small yet finite value of $u$ the first of these two opposite processes
dominates, so that the largest charge current absolute value of a SzS subspace
remains that of its reference SzSLNSN subspace 1 for which $N_{\rho}=N_{c}$.

The small-$u$ increasing of the absolute values of $J_{\eta n}(q_{j})$ and
$j_{\eta n}(q_{j})$ upon increasing $n$ turns out to be part of a mechanism
that prepares and precedes a qualitative change of the physics that occurs
upon reaching the $u\rightarrow 0$ limit. In such a limit the competition
between the two above effects becomes critical. However, the largest charge
current absolute value of a SzS subspace remains that of its reference SzSLNSN
subspace 1, as confirmed in the following.

The $u=0$ problem is easiest to be described directly in terms of electron
configurations. However, the understanding of the corresponding finite-$u$
problem requires the description of the $u\rightarrow 0$ limit physics in
terms of the fractionalized particles emerging from the rotated electrons. In
the case of the 1D Hubbard model, the imaginary part $i\,(n+1-2l)\,u$ of each
set of $\eta$-spin $l=1,...,n$ rapidities $\Lambda^{\eta
n,l}(q_{j})=\Lambda^{\eta n}(q_{j})+i\,(n+1-2l)\,u$, Eq. (28) for
$\alpha=\eta$, with the same real part $\Lambda^{\eta n}(q_{j})$ depends on
the interaction $u$ and thus vanishes as $u\rightarrow 0$. This is in contrast
to the $n>1$ spin $l=1,...,n$ rapidities of the spin $1/2$ $XXX$ chain with
the same real part, which are associated with a $n$-band pseudoparticle CP .
For the Hubbard model, the imaginary part of such a set of $\eta$-spin
$l=1,...,n$ rapidities describes the bounding of a number $n$ of $\eta$-spin
singlet pairs within one $\eta n$ pseudoparticle.

In the case of $n>1$, the vanishing of the $l=1,...,n$ rapidities imaginary
part $i\,(n+1-2l)\,u$ that occurs in the $u\rightarrow 0$ limit has physical
consequences. It implies that the $\Pi_{\eta}$ $\eta$-spin singlet pairs
containing $2\Pi_{\eta}=\sum_{n=1}^{\infty}2n\,N_{\eta n}$ paired rotated
$\eta$-spins $1/2$ of each of the corresponding $\eta n$ pseudoparticles of a
$\eta$-Bethe state unbound. Furthermore, although the even number of
$2\Pi_{\eta}$ rotated $\eta$-spins $1/2$ remain contributing to $\eta$-spin-
singlet configurations, concerning their translational degrees of freedom they
behave as $2\Pi_{\eta}$ independent charge carriers. This means that the
$\eta$-spin singlet configurations associated with both the $n=1$ pair $\eta
1$ pseudoparticles and the $n>1$ pairs composite $\eta n$ pseudoparticles are
rearranged in the $u\rightarrow 0$ limit in such a away that such
pseudoparticles cease to exist in that limit. While the $\eta n$
pseudoparticles do not exist both for $u\rightarrow 0$ and at $u=0$, the $c$
pseudoparticles remain existing in the $u\rightarrow 0$ limit yet are ill
defined at $u=0$.

A physical consequence of such $c$ and $\eta 1$ pseudoparticle behaviors is
that for $u\rightarrow 0$ the charge currents of the $\eta$-Bethe states
rather result from microscopic processes within which a number
$M_{\eta}=2S_{\eta}$ of unpaired physical $\eta$-spins in the $\eta$-spin
multiplet configurations interchange position with $N_{c}$ $c$ pseudoparticles
and the $2\Pi_{\eta}=\sum_{n=1}^{\infty}2n\,N_{\eta n}$ rotated $\eta$-spins
$1/2$. Those behave as independent charge carriers in spite of remaining
participating in $\eta$-spin-singlet configurations. Hence in the
$u\rightarrow 0$ limit the largest charge current absolute value of each SzSLN
subspace can be written as
$(C_{l_{\eta},n_{\eta},m_{s}}/L)\,t\,M_{\eta}\,(N_{c}+2\Pi_{\eta})$. Now both
each of the $M_{\eta}=2S_{\eta}$ unpaired physical $\eta$-spins in the
$\eta$-spin multiplet configurations, $N_{c}$ $c$ pseudoparticles, and
$2\Pi_{\eta}$ rotated $\eta$-spins $1/2$ in $\eta$-spin-singlet configurations
occupy a single site of the original lattice. Indeed,
$M_{\eta}+N_{c}+2\Pi_{\eta}=L$.

The new physics emerging in the $u\rightarrow 0$ limit is accounted for the
exact BA solution. Indeed, the $\eta n$ band spectra $J_{\eta n}(q_{j})$ and
elementary currents $j_{\eta n}(q_{j})$ become in the $u\rightarrow 0$ limit
of the form,

$J_{\eta n}(q_{j})=2n\,J_{\eta}(q_{j})\hskip 5.69046pt{\rm and}\hskip
5.69046ptj_{\eta n}(q_{j})=2n\,j_{\eta}(q_{j})\,.$ (138)

Here $J_{\eta}(q_{j})$ and $j_{\eta}(q_{j})$ are the corresponding spectra and
elementary currents carried by each of the $2n$ rotated $\eta$-spins $1/2$ in
$\eta$-spin-singlet configurations resulting in the $u\rightarrow 0$ limit
from one $u>0$ $\eta n$ pseudoparticle. Since $N_{c}+2\Pi_{\eta}=L-2S_{\eta}$,
the number $N_{c}+2\Pi_{\eta}$ of $c$ pseudoparticles plus rotated
$\eta$-spins $1/2$ in $\eta$-spin-singlet configurations appearing in the
expression $(C_{l_{\eta},n_{\eta},m_{s}}/L)\,t\,M_{\eta}\,(N_{c}+2\Pi_{\eta})$
of the largest charge current absolute values of each SzSLN subspace is the
same for all $\eta$-Bethe states that span a SzS subspace.

The charge current operator expectation values general exact expression, Eq.
(120) of Appendix A, remains valid in the $u\rightarrow 0$ limit yet the
corresponding $\eta$-Bethe states are not energy eigenstates at $u=0$. This
issue and its consequences for the charge stiffness is addressed in Appendix B
in the case of the hole concentration $m_{\eta}^{z}\rightarrow 0$. The
expressions in Eq. (138) of the current spectra and elementary currents are
consistent with at $u=0$ the $\eta n$ pseudoparticles not existing. The $\eta
n$ pseudoparticle exists for any arbitrarily small yet finite $u$ value. Hence
within the $u\rightarrow 0$ limit the corresponding $2n=2,4,6,...$ rotated
$\eta$-spins $1/2$ that emerge from it continue all moving with momentum $\pi-
q_{j}$. This is why such $2n$ rotated $\eta$-spins $1/2$ have charge current
spectra $J_{\eta}(q_{j})$ and elementary currents $j_{\eta}(q_{j})$ in Eq.
(138) that depend on the same momentum value $q_{j}$.

The increase of the $\eta n$ pseudoparticle current spectrum (and elementary
current) absolute value upon decreasing $u$ is needed for its maximum value
$2n\,|J_{\eta}(q_{j})|$ (and $2n\,|j_{\eta}(q_{j})|$), which is reached in the
$u\rightarrow 0$ limit, be compatible with the current spectrum (and
elementary current) of the emerging $2n=2,4,6,...$ independent rotated
$\eta$-spins $1/2$ charge carriers reading $J_{\eta}(q_{j})$ (and
$j_{\eta}(q_{j})$.) That absolute value increase upon decreasing $u$ that
occurs in the womb of the $\eta n$ pseudoparticle is a needed preparation for
its dead upon it delivering in the $u\rightarrow 0$ limit such $2n=2,4,6,...$
independent charge carriers. Due to their emergence from the $\eta n$
pseudoparticle in the $u\rightarrow 0$ limit, the corresponding $u\rightarrow
0$ physics is qualitatively different from that of finite $u$.

From combination of Eqs. (133) and (138), one finds the following exact
inequality specific to the $u\rightarrow 0$ limit, which involves the emerging
rotated $\eta$-spins $1/2$ current spectra $J_{\eta}(q_{j})$ and elementary
currents $j_{\eta}(q_{j})$ in Eq. (138),

$\displaystyle{\rm max}\\{|J_{c}(q_{j})|\\}$ $\displaystyle\geq$
$\displaystyle{\rm max}\\{|J_{\eta}(q_{j})|\\}\hskip 5.69046pt{\rm and}\hskip
5.69046pt{\rm max}\\{|j_{c}(q_{j})|\\}\geq{\rm max}\\{|j_{\eta}(q_{j})|\\}\,,$
$\displaystyle|{\bar{J}}_{c}|$ $\displaystyle\geq$
$\displaystyle|{\bar{J}}_{\eta}|\hskip 5.69046pt{\rm and}\hskip
5.69046pt|{\bar{j}}_{c}|\geq|{\bar{j}}_{\eta}|\,.$ (139)

The number $N_{c}+2\Pi_{\eta}=L-2S_{\eta}$ of charge carriers that upon moving
in the original lattice interchange position with the $M_{\eta}=2S_{\eta}$ of
unpaired physical $\eta$-spins in the $\eta$-spin multiplet configurations is
the same for all the $\eta$-Bethe states that span a SzS subspace. This is a
necessary condition for the largest current absolute value remaining that of
the corresponding reference SzSLN subspace 1. As given in Eq. (139), for all
references SzSLNSN subspaces contained in such a subspace, the maximum and
average values of $|J_{c}(q_{j})|$ and $|j_{c}(q_{j})|$ are in the
$u\rightarrow 0$ limit larger than or equal to those of $|J_{\eta}(q_{j})|$
and $|j_{\eta}(q_{j})|$, respectively. This is why the largest charge current
absolute value of a SzS subspace remains that of its reference SzSLNSN
subspace 1 for which $N_{c}+2\Pi_{\eta}=N_{c}=L-2S_{\eta}$.

Hence, in contrast to finite $u$, in the $u\rightarrow 0$ limit the number of
charge carriers that upon moving in the original lattice interchange position
with $M_{\eta}=2S_{\eta}$ unpaired physical $\eta$-spins does not play any
role in the maximization of the charge current absolute value. The only
property that determines its maximization is the relative absolute values of
the charge carriers current spectra and elementary currents. According to the
exact inequalities, Eq. (139), the $c$ pseudoparticles indeed win such a
competition.

In the case of some SzSLN subspaces, for $u\rightarrow 0$ the expressions of
the spectra $J_{c}(q_{j})$ and $J_{\eta}(q_{j})$ and elementary currents
$j_{c}(q_{j})$ and $j_{\eta}(q_{j})$ in Eq. (139) depend on the specific
$q_{F\beta,\iota}^{\tau}$ values in Eq. (52). Those define the $\eta$-Bethe
states that have the largest charge current absolute value in that subspace.
The exception for many SzSLN subspaces is for those contained in SzS subspaces
for which $m_{\eta}\ll 1$ and $(1-m_{\eta})\ll 1$, respectively. For them,
such expressions are the same for all $\eta$-Bethe states belonging to the
same SzSLN subspace. And this applies both to $\eta$-Bethe states whose
distributions are asymmetrical, Eq. (51), and symmetrical, Eq. (55).

For SzS subspaces corresponding to the $m_{\eta}\ll 1$ and $(1-m_{\eta})\ll 1$
limits, the charge currents can be expressed in terms of the elementary
currents, as given in Eq. (58), with now $j_{\eta
n}(q_{j})=2n\,j_{\eta}(q_{j})$. For simplicity, in the present case of the
$u\rightarrow 0$ limit we only report in the following expressions for the
elementary currents $j_{c}(q_{j})$ and $j_{\eta}(q_{j})$ in $j_{\eta
n}(q_{j})=2n\,j_{\eta}(q_{j})$ that obey the inequalities, Eq. (139). Besides
controlling the exact expressions of the charge currents for $m_{\eta}\ll 1$
and $(1-m_{\eta})\ll 1$, Eq. (58), such elementary currents are useful for
some of the upper-bound procedures used in this paper. Related expressions
apply to the spectra $J_{c}(q_{j})$ and $J_{\eta}(q_{j})$ in $J_{\eta
n}(q_{j})=2n\,J_{\eta}(q_{j})$ also appearing in Eq. (139). Densities in Eqs.
(45)-(50) in the vicinity of those of the reference SzSLN subspaces 1 to 3
defined in Section IV.2 are again considered. Although the exact inequalities,
Eq. (139), apply to SzS subspaces corresponding to the $m_{\eta}\in[0,1]$ and
$m_{s}\in[0,(1-m_{\eta})]$ intervals, the limit $m_{\eta}\rightarrow 0$ for
which $m_{s}\in[0,1]$ plays a central role concerning the charge stiffness
issue clarified in this paper.

We start by considering the reference SzSLNSN subspace 1B for which
$n_{s}\rightarrow 0$, in contrast to that density maximum value
$n_{s}=n_{s}^{\rm max}=(1-l_{\eta}-m_{s})/2$ generally considered here.
Moreover, for this reference SzSLNSN subspace the coefficient $C_{u,n}$ in Eq.
(134) has as singular behavior, reading $C_{u,n}=1$ for finite $u$ and
$C_{u,n}=1/2n$ for $u\rightarrow 0$. Another property is that for it the
expression of the elementary currents is independent of the $\eta$-Bethe
states that span it, as it is the same for all such states. The elementary
current $j_{\eta n}(q_{j})$ is for $q_{j}\in[-\pi m_{\eta},\pi m_{\eta}]$
given by $j_{\eta n}(q_{j})=2t\,q_{j}\,\sin(\pi m_{\eta})/\pi m_{\eta}$ for
$u>0$, as provided in Eq. (135). However, following the singular behavior of
the coefficient $C_{u,n}$, its value is different in the $u\rightarrow 0$
limit. For densities in Eqs. (45)-(50) in the vicinity of those of a reference
SzSLNSN subspace 1B one finds in the $u\rightarrow 0$ limit that,

$\displaystyle j_{c}(q_{j})$ $\displaystyle=$
$\displaystyle-2t\sin(q_{j})\hskip 5.69046pt{\rm for}\hskip
5.69046ptq_{j}\in[-\pi,\pi]\,,$ $\displaystyle j_{\eta n}(q_{j})$
$\displaystyle=$ $\displaystyle 2n\,j_{\eta}(q_{j})\hskip 5.69046pt{\rm
where}\hskip 5.69046ptj_{\eta}(q_{j})=2t\sin\left({q_{j}\over 2}\right)\hskip
5.69046pt{\rm for}\hskip 5.69046ptq_{j}\in[-\pi m_{\eta},\pi m_{\eta}]\,,$
$\displaystyle{\rm max}\\{|j_{c}(q_{j})|\\}$ $\displaystyle=$ $\displaystyle
2t\hskip 5.69046pt{\rm and}\hskip 5.69046pt|{\bar{j}}_{c}|={4t\over\pi}\,,$
$\displaystyle{\rm max}\\{|j_{\eta}(q_{j})|\\}$ $\displaystyle=$
$\displaystyle 2t\,\sin\left({\pi\over 2}m_{\eta}\right)\hskip 5.69046pt{\rm
and}\hskip 5.69046pt|{\bar{j}}_{\eta}|={4t\over\pi
m_{\eta}}\left(1-\cos\left({\pi\over 2}m_{\eta}\right)\right)\,.$ (140)

These exact elementary current expressions and values obey the corresponding
inequality in Eq. (139).

Moreover, for densities in Eqs. (45)-(50) in the vicinity of those of a
reference SzSLN subspace 2 , the $c$ band elementary current is for the whole
$u\geq 0$ range independent of the $\eta$-Bethe states and reads
$j_{c}(q_{j})=2t\sin(q_{j})$, including in the present $u\rightarrow 0$ limit.
The elementary currents $j_{\eta}(q_{j})$ in $j_{\eta
n}(q_{j})=2n\,j_{\eta}(q_{j})$ have slightly different expressions for
different $\eta$-Bethe states belonging to the subspace under consideration.
For all such states, the corresponding elementary current inequality in Eqs.
(139) is obeyed. The exception is as mentioned above for reference SzSLN
subspaces 3 contained in SzS subspaces for which $m_{\eta}\ll 1$ and
$(1-m_{\eta})\ll 1$. In that case their expressions are for $u\rightarrow 0$
the same for all such states. Specifically, they read
$j_{\eta}(q_{j})=-t\sin(q_{j})$ for $m_{\eta}\rightarrow 0$ and
$j_{\eta}(q_{j})=-2t\sin(q_{j})$ for $m_{\eta}\rightarrow 1$, so that the
corresponding elementary current inequality in Eq. (139) is again obeyed. (For
a reference SzSLN subspace 2 one has that $l_{\eta}\rightarrow 1$, so that
$m_{s}\rightarrow 0$.)

For densities in Eqs. (45)-(50) in the vicinity of those of a reference SzSLN
subspace 3, the coefficient $C_{u,n}$ in Eq. (134) is a continuous function of
$u$. Furthermore, the expressions of the elementary currents are for the whole
$u\geq 0$ range independent of the $\eta$-Bethe states in that subspace. In
the $u\rightarrow 0$ limit they equal actually those given in Eq. (140), which
thus obey the corresponding inequality in Eqs. (139).

For densities in Eqs. (45)-(50) near those of a reference SzSLNSN subspace 1A,
which plays the major role in our studies, the elementary currents have in the
$u\rightarrow 0$ limit a universal form common to all $\eta$-Bethe states
again only for $m_{\eta}\ll 1$ and $(1-m_{\eta})\ll 1$. On the one hand, the
elementary currents $j_{c}(q_{j})$ and $j_{\eta n}(q_{j})$ associated with
$\eta$-Bethe states that span reference SzSLNSN subspaces 1A contained in SzS
subspaces with densities $m_{\eta}\rightarrow 0$ and $m_{s}\in[0,1]$ are in
the $u\rightarrow 0$ limit found to read,

$\displaystyle j_{c}(q_{j})$ $\displaystyle=$