instance, one may wonder: what is the analogue of an exceptional curve on a
del Pezzo surface? What corresponds to the famous statement about the 27 lines
on the cubic surface $\mathbb{B}_{6}$ on the topological side, the six-fold
cyclification $\mathcal{L}^{6}_{c}S^{4}$ of $S^{4}$? We investigate these
intriguing questions in section 4.6.

######  Remark 4.7 (Why $k\leq 8$ vs. $k\geq 9$).

The construction of the data
$\big{(}\mathfrak{h}_{k}^{*},\\{\epsilon_{0},\dots,\epsilon_{k}\\},(-,-),K_{k}^{*}\big{)}$
of Theorem 4.6 above extends beyond $k=8$ verbatim. However, the
identification of the root system for $k>8$ needs to be treated with care. For
$0\leq k\leq 8$, the degree
$\deg(\mathcal{L}_{c}^{k}S^{4})=(K_{k},K_{k})=(K_{k}^{*},K_{k}^{*})=9-k$ of
the cyclic loop space is positive, just like the degree of the del Pezzo
surface $\mathbb{B}_{k}$. From simple linear algebra of Lorentzian inner
products, we can see that $(K_{k},K_{k})>0$ implies that the inner product
induced on the subspace
$K_{k}^{\perp}=\\{x\in\mathfrak{h}_{k}\;|\;(x,K_{k})=0\\}$ of
$\mathfrak{h}_{k}$ by the Lorentzian inner product $(-,-)$ is negative-
definite. (If we switch the sign and use $-(-,-)$, it would be a more familiar
positive-definite inner product). This also implies that the root system
$R_{k}$ is finite; see [Ma74]. For $k\geq 9$, the inner product loses its
negative-definiteness, and the root system $R_{k}$ becomes infinite and can be
identified as the set of _real roots_ of a more general Kac-Moody algebra. In
fact, for $k=9$, the subspace $K_{k}^{\perp}$ is negative semi-definite of
nullity 1. For $k>9$, the orthogonal complement $K_{k}^{\perp}$ gets a
Lorentzian inner product. See the discussion of the $k\geq 9$ cases in Section
4.7.

##### Weyl group as symmetry of symmetries.

For each root $\alpha$, an element of the set $R_{k}$ (see expression (47))
define a reflection

$\sigma_{\alpha}:\beta\;\;\longmapsto\;\;\beta-2\frac{(\beta,\alpha)}{(\alpha,\alpha)}\alpha=\beta+(\beta,\alpha)\alpha,\qquad\beta\in\mathfrak{h}_{k}^{*},$

of the Lorentzian space $\mathfrak{h}_{k}^{*}$. The reflections
$\\{\sigma_{\alpha}\;|\;\alpha\in R_{k}\\}$ generate a group, known to be the
Weyl group $W(E_{k})$ of the root system $E_{k}$. It is also known that
$W(E_{k})$ is the group of all linear isometries of
$\mathbf{P}(\mathfrak{h}_{k})$ which preserve the element $K_{k}^{*}$ [Ma74].
Thus, the Weyl group, being an isometry group of (the dual of) the abelian Lie
algebra $\mathfrak{h}_{k}$ of infinitesimal symmetries of
$M(\mathcal{L}_{c}^{k}S^{4})$, is a “second derived” object with respect to
$\mathcal{L}_{c}^{k}S^{4}$: $W(E_{k})$ is the group of “symmetries of
symmetries” of $\mathcal{L}_{c}^{k}S^{4}$. This is typical for the role of
Weyl groups in Lie theory: a Lie group is usually a group of symmetries of a
certain mathematical object, the Cartan subalgebra is the maximal abelian Lie
algebra of infinitesimal symmetries of that object. The Weyl group is a group
of symmetries of the Cartan subalgebra.

##### The moduli space of $k$-fold cyclifications of $S^{4}$.

We can interpret an element $\omega$ of the linear dual space
$\mathfrak{h}_{k}^{*}\cong\mathbb{R}^{k+1}$ of the abelian Lie algebra
$\mathfrak{h}_{k}$ of infinitesimal symmetries of
$M(\mathcal{L}_{c}^{k}S^{4})$ as some sort of an extra, geometric ingredient
complementing the purely topological data carried by the real homotopy type of
the $k$-fold cyclification $\mathcal{L}_{c}^{k}S^{4}$. Indeed, an element
$\omega\in\mathfrak{h}_{k}^{*}$ is a weight, and as such, it tells us which
“spectral parameters” we might want to assign to the basic infinitesimal
symmetries $h_{0},h_{1},\dots,h_{k}$.

For instance, recall from Section 3.4.1 that $h_{0}$ comes from the action of
the real 1-torus $\mathbb{R}^{\times}$ by automorphisms of the real homotopy
type of $S^{4}$ and the resulting action of $\mathbb{R}^{\times}$ on
$\mathcal{L}_{c}^{k}S^{4}$. That action originates ultimately from the folding
self-maps of $S^{4}$. The value $\omega(h_{0})\in\mathbb{R}$ tells us how much
we shall value the effect of the folding self-maps of $S^{4}$. In this sense,
$\omega(h_{0})$ is akin to the size of $S^{4}$, such as its radius $R_{0}$, or
rather the logarithm $\log R_{0}$ thereof, since $\omega(h_{0})$ is not
necessarily positive. This value $\omega(h_{0})$ is analogous to the
logarithmic Planck scale $\log\ell_{p}$ in the 11-dimensional supergravity and
the generalized Kähler volume $\omega(H)=\int_{H}\omega$ of the line
$H=\mathbb{C}\mathbb{P}^{1}$ in $\mathbb{C}\mathbb{P}^{2}$ and its image in
$\mathbb{B}_{k}$ in the del Pezzo story; cf. [INV02, §3.1].

Similarly, as per Section 3.4.2, the element $h_{i}$ for each $i$, $1\leq
i\leq k$, comes from the action of $\mathbb{R}^{\times}$ by the folding self-
maps of the $i$th source circle of the cyclic loop space
$\mathcal{L}_{c}^{k}S^{4}$. Assigning $h_{i}$ a real value $\omega(h_{i})$
tells us how much we shall value the effect of the folding self-maps of the
$i$th source circle of $\mathcal{L}_{c}^{k}S^{4}$. In this way,
$\omega(h_{i})$ is analogous to the logarithm $\log R_{i}$ of the radius
$R_{i}$ of the $i$th source circle and the $i$th compactification circle in
M-theory wrapped on $T^{k}=(S^{1})^{k}$, as well as the generalized Kähler
volume $\omega(\mathcal{E}_{i})$ of the exceptional divisor $\mathcal{E}_{i}$
in $\mathbb{B}_{k}$ in the del Pezzo story; cf. [INV02, §3.1] again.

In this sense the choice of a weight $\omega\in\mathfrak{h}_{k}^{*}$ adds a
certain ingredient of _metric_ flavor, missing in the real homotopy model
$M(\mathcal{L}_{c}^{k}S^{4})$ of $\mathcal{L}_{c}^{k}S^{4}$. For example, an
arbitrary weight $\omega\in\mathfrak{h}_{k}^{*}$ will not be a real homotopy
invariant of $\mathcal{L}_{c}^{k}S^{4}$. The values $\omega(h_{i})$,
$i=0,1,\dots,k$, that is to say, the logarithmic radii of the target sphere
$S^{4}$ and the source circles $S^{1}$, may be thought of as the coordinates
of the weight $\omega$ in the space of all weights.

The Weyl group $W(E_{k})$, being the group of symmetries of the $E_{k}$ data
$(\mathbf{P}(\mathfrak{h}_{k}),(-,-),K_{k}^{*})$, acts on the vector space
$\mathfrak{h}_{k}^{*}$. Thus, it makes sense to identify the weights $\omega$
brought together by this action. We call the corresponding quotient orbifold
the _moduli space $\mathcal{M}_{k}$ of $k$-fold cyclifications of $S^{4}$_:

$\mathcal{M}_{k}=\mathcal{M}_{k}(\mathcal{L}^{k}S^{4}):=[\mathfrak{h}_{k}^{*}/W(E_{k})].$

This is a stacky quotient $[\mathfrak{h}_{k}^{*}/W(E_{k})]$, which is
different from the naive, topological quotient
$\mathfrak{h}_{k}^{*}/W(E_{k})$. Another reincarnation of the quotient
orbifold $[\mathfrak{h}_{k}^{*}/W(E_{k})]$ is the familiar homotopy quotient
$\mathfrak{h}_{k}^{*}/\\!\\!/W(E_{k})$, which may be realized via the Borel
construction, cf. (11), but in this context, the orbifold viewpoint would be
more common. The $(k+1)$-dimensional topological quotient
$\mathfrak{h}_{k}^{*}/W(E_{k})$ contains the $k$-dimensional quotient
$K_{k}^{\perp}/W(E_{k})$, which for $k\leq 8$ may be identified with the
closure of a _Weyl chamber_ for the Weyl group action in the Euclidean space
$K_{k}^{\perp}=\\{\omega\in\mathfrak{h}_{k}^{*}\;|\;\omega(K_{k})=0\\}\subset\mathfrak{h}_{k}^{*}$,
cf. [Hal15, Prop. 8.29].

######  Remark 4.8 (Further interpretation).

We highlight the following:

(i) Identifying the weights $\omega$ under a permutation of the radii of the
circles entering $\mathcal{L}_{c}^{k}S^{4}$ is similar to identifying
punctured Riemann surfaces under a permutation of the punctures, if we wish to
consider the moduli space of Riemann surfaces with unlabeled punctures. The
identification of weights under the Weyl group action is also analogous to
identifying the Planck scale and the sequence of radii of the circle
components in compactified M-theory under U-duality. In the del Pezzo story,
one also identifies generalized Kähler classes on a del Pezzo surface under
the action of the Weyl group, see [INV02, §3.1], and the fundamental Weyl
chamber in the Picard group of a del Pezzo surface plays a prominent role in
studying Cremona isometries [Do12, §8.2.8].

(iii) Ultimately, physical configurations have to be U-duality invariant, so
it makes sense to mod out by that symmetry. The Weyl group $W=W(E_{k})$ is
traditionally taken as a subgroup of the discrete $\mathbb{Z}$-form of the
U-duality group $E_{k}(\mathbb{R})$; see the discussion before (1). And since
$W$ already contains a substantial part of that symmetry, modding out by $W$
makes sense and is also ‘close’ to the ultimate moduli space, in the sense of
(1). We will consider this in more detail in [SV22].

###  4.5 The $E_{k}$ symmetry of type IIB

Given that the real homotopy type $IIB$ was worked out so as to match with
type $IIA$ via T-duality (23) and (24), we would like to choose a compatible
trivialization $T^{B}\cong\mathbb{G}_{m}\times\mathbb{G}_{m}$ of the maximal
$\mathbb{R}$-split torus of $\operatorname{Aut}M(IIB)$ and a compatible basis
of weights of the action of $T^{B}$ on $M(IIB)$.

We have been identifying the weights corresponding to the maximal
$\mathbb{R}$-split tori actions on $M(\mathcal{L}_{c}^{k}S^{4})$ in 3.6
starting from $k=0$ and moving up to higher $k$ with the use of Theorem 3.4,
which related the toroidal symmetries of $M(\mathcal{L}_{c}^{k+1}S^{4})$ to
those of $M(\mathcal{L}_{c}^{k}S^{4})$. The real homotopy type $IIB$ is
connected to this sequence by a single cyclification: $M(\mathcal{L}_{c}IIB)$
is closely related to $M(\mathcal{L}_{c}IIA)=M(\mathcal{L}_{c}^{2}S^{4})$,
expressing T-duality; see Equation 23. Thus, if $T^{B}_{c}$ is the maximal
$\mathbb{R}$-split torus of $\operatorname{Aut}M(\mathcal{L}_{c}IIB)$, then it
maps naturally to the maximal split torus $T^{B}$ of
$\operatorname{Aut}M(IIB)$, because $\mathbb{R}[w]$, where $w$ is as in the
extension (13), is $T^{B}_{c}$-invariant. Since the differential ideal
$(h_{7},\omega_{7},dh_{7},d\omega_{7})$ of $M(\mathcal{L}_{c}IIB)$ is also
$T^{B}_{c}$-invariant, $T^{B}_{c}$ acts on the quotient
$M(\mathcal{L}_{c}IIB)/(h_{7},\omega_{7},dh_{7},d\omega_{7})$ in (23). The
maximal $\mathbb{R}$-split torus $T^{A}_{c}$ of
$\operatorname{Aut}M(\mathcal{L}_{c}IIA)$ also keeps the differential ideal
$(h_{7},dh_{7})$ invariant and maps naturally to $T^{B}_{c}$. This way, we
have a map

$T^{A}_{c}\longrightarrow T^{B}_{c}\longrightarrow T^{B}.$

One can reverse these maps and get a natural map $T^{B}\to T^{A}_{c}$. This
also implies that the weights of $T^{A}_{c}$ pull back to the weights of
$T^{B}$.

Now, let us use these maps to create distinguished bases of the Lie algebra
$\mathfrak{h}_{B}$ of the maximal $\mathbb{R}$-split torus $T^{B}$ and the
dual space vector space $\mathfrak{h}_{B}^{*}$ of weights. We will use the
weights of the elements $h_{3}$ and $\omega_{3}$ as fundamental (corresponding
to spacetime fields $H_{3}$ and $F_{3}$ associated with the fundamental string
and its S-dual, the D1-brane). With that, we write

$t\cdot h_{3}=t^{\gamma_{\,1}}h_{3},\qquad
t\cdot\omega_{3}=t^{\gamma_{\,2}}\omega_{3},$

where

$\gamma_{\,1}:=\epsilon_{0}-\epsilon_{1},\quad\text{and}\quad\gamma_{\,2}:=\epsilon_{0}-\epsilon_{2},$

given that

$h_{3}\mapsto s_{1}g_{4},\quad\text{and}\quad\omega_{3}\mapsto s_{2}g_{4}$

in the correspondence between $M(\mathcal{L}_{c}IIB)$ and
$M(\mathcal{L}_{c}IIA)=M(\mathcal{L}_{c}^{2}S^{4})$. Here $\epsilon_{0}$,
$\epsilon_{1}$, and $\epsilon_{2}$ are the generating weights of
$\mathfrak{h}_{2}^{*}$, the Lie algebra of the maximal $\mathbb{R}$-split
torus $T^{A}_{c}$ of
$\operatorname{Aut}M(\mathcal{L}_{c}IIA)=\operatorname{Aut}M(\mathcal{L}_{c}^{2}S^{4})$.
We view these weights as being pulled back to $T^{B}$ (and $\mathfrak{h}_{B}$)
via the homomorphism $T^{B}\to T^{A}_{c}$. Since $\gamma_{\,1}=\beta_{0}$ and
$\gamma_{\,2}=\beta_{0}+\beta_{1}$ by Equations (38), $\gamma_{\,1}$ and
$\gamma_{\,2}$ form a basis of $\mathfrak{h}_{B}^{*}$ and, moreover, of the
weight lattice
$\mathbf{P}(\mathfrak{h}_{B}):=\mathbb{Z}\gamma_{1}\oplus\mathbb{Z}\gamma_{2}\subseteq\mathfrak{h}_{B}^{*}$.

We will use the inner product induced on $\mathfrak{h}_{B}^{*}$ from
$\mathfrak{h}_{2}^{*}$:

$(\gamma_{\,1},\gamma_{\,1}):=(\epsilon_{0}-\epsilon_{1},\epsilon_{0}-\epsilon_{1})=0,\quad(\gamma_{\,2},\gamma_{\,2}):=(\epsilon_{0}-\epsilon_{2},\epsilon_{0}-\epsilon_{2})=0,\quad(\gamma_{\,1},\gamma_{\,2}):=(\epsilon_{0}-\epsilon_{1},\epsilon_{0}-\epsilon_{2})=1\;.$

The dual basis of $\mathfrak{h}_{B}$ will be given as
$\\{\mathfrak{l}_{1}:=h_{0}-h_{1},\mathfrak{l}_{2}:=h_{0}-h_{2}\\}$, where we
used the images of the generators $h_{0},h_{1},h_{2}$ of $\mathfrak{h}_{2}$
under the linear map $\mathfrak{h}_{2}\to\mathfrak{h}_{B}$ linearizing the
above group homomorphism $T^{A}_{c}\to T^{B}$. Then one can check from
Equations (38) that there exists a unique element $-K_{B}\in\mathfrak{h}_{B}$
which acts as the degree operator (44) on the Quillen model $Q(IIB)$, namely,
the element

$-K_{B}=2\mathfrak{l}_{1}+2\mathfrak{l}_{2}.$

Indeed, $2(h_{0}-h_{1})+2(h_{0}-h_{2})$ acts on $s_{1}g_{4}$ in
$M(\mathcal{L}_{c}IIA)$ by scaling by $2(1-1)+2(1-0)=2$ and the same way on
$s_{2}g_{4}$. This implies that $-K_{B}$ acts on $h_{3}$ and $\omega_{3}$ in
$M(IIB)$ by a factor of 2, which is the degree of the dual elements in the
Quillen model $Q(IIB)$. The element $2(h_{0}-h_{1})+2(h_{0}-h_{2})$ acts on
$s_{2}w_{1}$ in $M(\mathcal{L}_{c}IIA)$ by zero, and this implies that
$-K_{B}$ acts on $\omega_{1}\in M(IIB)$ also trivially, just as it is supposed
to act on an degree-zero element of $Q(IIB)$. Equations (20) and the fact that
$\mathfrak{h}_{B}$ acts on $M(IIB)$ by derivations then imply that
$-K_{B}\in\mathfrak{h}_{B}$ acts on the remaining generators $\omega_{5}$,
$h_{7}$, and $\omega_{7}$ by 4, 6, and 6, respectively, again compatible with
acting as the degree operator on the Quillen model $Q(IIB)$.

As concerns uniqueness of an element realizing the degree operator, an
arbitrary element
$a_{1}\mathfrak{l}_{1}+a_{2}\mathfrak{l}_{2}\in\mathfrak{h}_{B}$ acts on
$\omega_{1}$ by a factor of $a_{1}-a_{2}$ and on $h_{3}$ by $a_{2}$. Thus, if
it acts by the Quillen-model degree, we must have $a_{1}-a_{2}=0$ and
$a_{2}=2$, whence $a_{1}=a_{2}=2$.

This way, as in Theorem 4.6, we create a root system $R_{B}$, which we might
denote by $E_{B}$, corresponding to the real homotopy type $IIB$. In contrast
with the type $IIA$ root system, which is empty (in the notation of Table 1):

$E_{1}=R_{1}=\big{\\{}\alpha\in\mathbf{P}(\mathfrak{h}_{1})\;|\;(\alpha,3\epsilon_{0}-\epsilon_{1})=0,\;(\alpha,\alpha)=-2\big{\\}}=\varnothing=A_{0}\,,$

we have:

######  Proposition 4.9 (Exceptional root data from the rational model for
type IIB).

The data, as in Theorem 4.6, associated to the Sullivan minimal model $M(IIB)$
of type IIB replicates the data determined by the del Pezzo surface
$\mathbb{B}^{\prime}_{1}=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$
and produces the root system

$E_{B}=R_{B}:=\big{\\{}\alpha\in\mathbf{P}(\mathfrak{h}_{B})\;|\;(\alpha,2\gamma_{1}+2\gamma_{2})=0,\;(x,x)=-2\big{\\}}=\big{\\{}\gamma_{1}-\gamma_{2},\;\gamma_{2}-\gamma_{1}\big{\\}}$

which is nonempty and may be identified as a root system of type $A_{1}$.

This justifies the type IIB row of Table 1.

###  4.6 27 “Lines” in the cyclic loop space $\mathcal{L}_{c}^{6}S^{4}$

In this section we show that our discovery of at least the toroidal part of
$E_{k}$ symmetry of the rational homotopy type of $\mathcal{L}_{c}^{k}S^{4}$
may lead to surprising consequences, such as the existence of 27 “lines” in
$\mathcal{L}_{c}^{6}S^{4}$: shortly,
$\dim\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=27$. Since 27 is also
remarkable as the dimension of a fundamental representation of the Lie algebra
$\mathfrak{e}_{6}$, this is suggestive of the possibility of extending the
rational homotopy symmetries of $\mathcal{L}_{c}^{k}S^{4}$ outside of the
toroidal part of $E_{k}$, if not to the whole Lie algebra $\mathfrak{e}_{k}$.

Given that the data
$\left(\mathfrak{h}_{6}^{*},\\{\epsilon_{0},\dots,\epsilon_{6}\\},(-,-),K_{6}^{*}\right)$
arising in Theorem 4.6 is exactly the same as that for the del Pezzo surface
$\mathbb{B}_{6}$, we can identify the 27 “lines” in
$\mathcal{L}_{c}^{6}S^{4}$. These lines are generated by the
$\mathbb{R}$-homotopy classes of 27 maps
$\mathbb{C}\mathbb{P}^{1}\to\mathcal{L}_{c}^{6}S^{4}$ (to be more precise,
linear combinations of such, i.e., 27 elements of
$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=\pi_{2}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{R}$)
supplied by the following result.

######  Theorem 4.10 (27 lines via rational homotopy of 6-fold cyclic loop
space).

The 27 exceptional vectors $\alpha\in\mathbf{P}(\mathfrak{h}_{6})$,
$(\alpha,\alpha)=(\alpha,K_{6}^{*})=-1$, give rise to 27 canonically defined
lines in the $\mathbb{R}$-vector space
$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$. Moreover, these lines freely
generate $\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$ and thus
$\dim\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=27$.

###### Proof.

The 27 exceptional vectors for the data
$\left(\mathfrak{h}_{6}^{*},\\{\epsilon_{0},\dots,\epsilon_{6}\\},(-,-),K_{6}^{*}\right)$
associated with the space $\mathcal{L}_{c}^{6}S^{4}$ by Theorem 4.6 are the
elements $\alpha$ of the weight lattice
$\mathbf{P}(\mathfrak{h}_{6})=\mathbb{Z}\epsilon_{0}\oplus\dots\oplus\mathbb{Z}\epsilon_{6}$
which pair to 1 with $-K_{6}^{*}$ or, equivalently, evaluate to 1 at
$-K_{6}\in\mathfrak{h}_{6}$: $\alpha(-K_{6})=1$. By the definition of the
“anticanonical class” $-K_{6}$, for any weight
$\alpha\in\mathbf{P}(\mathfrak{h}_{6})$, an element $x$ of the weight space
$Q_{\alpha}$ of the Quillen minimal model
$Q=\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})[1]$ has degree $\lvert
x\rvert=\alpha(-K_{6})$:

$\displaystyle hx$ $\displaystyle=\alpha(h)\cdot x$ $\displaystyle\text{for
any $h\in\mathfrak{h}_{6}$},$ $\displaystyle-K_{6}x$ $\displaystyle=\lvert
x\rvert\cdot x$ $\displaystyle\text{for $h=-K_{6}$}.$

Thus, the weights $\alpha\in\mathbf{P}(\mathfrak{h}_{6})$ which evaluate to 1
at $-K_{6}$ are precisely the weights of the degree-one component of
$\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})[1]$,

$\big{(}\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})[1]\big{)}_{1}=\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4}).$

The vector space $\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$ is linear
dual to the degree-two component $V^{2}$ of the generator vector space
$V=Q^{*}[1]=\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})^{*}$ of the
Sullivan minimal model $M(\mathcal{L}_{c}^{6}S^{4})=(S(V),d)$. Note that in
the weight decomposition

$V^{2}=\bigoplus_{\alpha\in\mathbf{P}(\mathfrak{h}_{6})}V^{2}_{\alpha}\;,$

the weights that actually occur, i.e., $V^{2}_{\alpha}\neq 0$, are exactly on
the following list:

$\displaystyle\epsilon_{1},\dots,\epsilon_{6},$
$\displaystyle\epsilon_{0}-\epsilon_{i}-\epsilon_{j},\hskip 83.08199pt1\leq
i<j\leq 6,$ (48) $\displaystyle
2\epsilon_{0}-\epsilon_{1}-\dots-\widehat{\epsilon_{i}}-\dots-\epsilon_{6},\quad
1\leq i\leq 6.$

Moreover, the corresponding weight spaces are all one-dimensional and
generated by the elements

$\displaystyle w_{1},\dots,w_{6},$ $\displaystyle s_{j}s_{i}g_{4},\hskip
48.36958pt1\leq i<j\leq 6,$ (49) $\displaystyle s_{6}\dots\widehat{s_{i}}\dots
s_{1}g_{7},\quad 1\leq i\leq 6,$

respectively, just as in the table which presented the 27 spacetime fields
earlier in this section. The corresponding one-dimensional linear-dual
subspaces
$(V^{2}_{\alpha})^{*}=(Q_{1})_{\alpha}\subset\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$
are canonically defined as weight spaces, and we have

$\displaystyle V^{2}$ $\displaystyle=\bigoplus_{\scalebox{0.55}{${\alpha\text{
on the list \eqref{27weights}}}$}}V^{2}_{\alpha}\;,$
$\displaystyle\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$
$\displaystyle=\bigoplus_{\scalebox{0.55}{${\alpha\text{ on the list
\eqref{27weights}}}$}}\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})_{\alpha}\;.$

These are the 27 lines in $\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$.
For each line, a generator, defined up to a nonzero real factor, is
represented by a real homotopy class in
$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$, a “line
$\mathbb{C}\mathbb{P}^{1}=S^{2}\to\mathcal{L}_{c}^{6}S^{4}$”. ∎

######  Remark 4.11 (Other cases).

A similar count of “lines in $\mathcal{L}_{c}^{k}S^{4}$” works for all $k$,
$0\leq k\leq 6$, with the same numbers as those for the exceptional vectors
$I_{k}\subset N_{k}$ and exceptional curves on del Pezzo surfaces
$\mathbb{B}_{k}$, cf. [Ma74, Theorem 4.3]. This count starts to break for
$k>6$, because some of the exceptional vectors will start having trivial
weight spaces. In particular, for $k=7$, instead of 28 pairs of lines in
$\mathbb{B}_{7}$, we will have 28 “lines,” 21 of which are paired to other 21
“lines” in $\mathcal{L}_{c}^{7}S^{4}$, with 7 “lines” missing a pair. We will
address this in an upcoming paper [SV22].

###  4.7 Cyclifying $\geq 9$ times and Kac-Moody algebras

Nothing prevents us from cyclifying the 4-sphere 9 times and more, just like
blowing up $\mathbb{C}\mathbb{P}^{2}$ at $k\geq 9$ points makes perfect sense
— the resulting surfaces are just no longer del Pezzo [Do08][Do20]. In our
case, it is interesting to see what “phase transition” is happening between
$k=8$ and 9. It is also reasonable to expect the emergence of Kac-Moody
algebras of type $E_{k}$ for $k\geq 9$ in relation to higher cyclifications
$\mathcal{L}_{c}^{k}S^{4}$. These Lie algebras play a role in further blowups
of $\mathbb{C}\mathbb{P}^{2}$, as well as in reductions of 11d supergravity to
dimensions 2, 1, and 0 [Ju86][N92]. We plan to address the relation to
algebraic geometry and physics in [SV22].

##### The Lie algebras $\mathfrak{e}_{k}$ of type $E_{k}$ for $k\geq 9$.

For $k\geq 9$, the Dynkin diagram

Dynkin diagram $E_{k}$

$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{k-2}$$\sigma_{k-1}$$\sigma_{0}$

corresponds to Kac-Moody algebras: the affine Lie algebra of type
$E_{9}=\widehat{E}_{8}=E_{8}^{(1)}$, the hyperbolic Kac-Moody algebra of type
$E_{10}$, and the Lorentzian Kac-Moody algebras of type $E_{k}$ for $k\geq
11$.

##### The $k$-fold cyclifications $\mathcal{L}_{c}^{k}S^{4}$ and root lattices
for $k\geq 9$.

The $E_{k}$ root system data arises from $\mathcal{L}_{c}^{k}S^{4}$ according
to Theorem 4.6: the maximal split real torus of
$\operatorname{Aut}\mathcal{L}_{c}^{k}S^{4}$ is $(k+1)$-dimensional; the dual
space $\mathfrak{h}_{k}^{*}$ of its Lie algebra has a natural basis
$\epsilon_{0},\epsilon_{1},\dots,\epsilon_{k}$, Lorentzian inner product, and
distinguished element $K_{k}^{*}$. The sublattice
$(K_{k}^{*})^{\perp}\subset\mathbf{P}(\mathfrak{h}_{k})$ is a root lattice of
type $E_{k}$, see table 2.

##### Exceptional vectors for $\mathcal{L}_{c}^{k}S^{4}$ for $k\geq 9$.

In contrast to the $k\leq 8$ case, for $k\geq 9$ the set

$I_{k}=\left\\{\alpha\in\mathbf{P}(\mathfrak{h}_{k})\;|\;(\alpha,K_{k}^{*})=(\alpha,\alpha)=-1\right\\}$

of exceptional vectors is infinite for $\mathcal{L}_{c}^{k}S^{4}$. Indeed, the
Weyl group $W_{k}$ acts on the dual of the Lie algebra $\mathfrak{h}_{k}$ of
the maximal $\mathbb{R}$-split torus $T^{k+1}$ of
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ by isometries preserving
$K_{k}^{*}=-3\epsilon_{0}+\epsilon_{1}+\dots+\epsilon_{k}$ and the lattice
$\mathbf{P}(\mathfrak{h}_{k})$. In particular, the Weyl group acts on the
exceptional vectors. Examples, such as in (48), show there are enough
exceptional vectors to generate the whole vector space $\mathfrak{h}_{k}^{*}$.
This implies that an element of $W_{k}$ is determined by its action on the
exceptional vectors. Then the fact that the group $W_{k}$ is infinite for
$k\geq 9$ implies that there are infinitely many exceptional vectors.

Recall from Theorem 4.10 that in the $k=6$ case, the 27 exceptional weight
spaces of the action of $\mathfrak{h}_{k}$ on the Quillen model
$Q(\mathcal{L}_{c}^{6}S^{4})=\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})[1]$,
i.e., on $\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$ are all one-
dimensional and present 27 lines in $\mathcal{L}_{c}^{6}S^{4}$.

On the contrary, for $k\geq 9$, only finitely many exceptional weights will be
populated, both in $Q(\mathcal{L}_{c}^{k}S^{4})$ and
$M(\mathcal{L}_{c}^{k}S^{4})$. This is because the underlying real vector
space $\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{k}S^{4})$ of the Quillen
model and the subspaces of bounded degree of the Sullivan model of
$M(\mathcal{L}_{c}^{k}S^{4})$ are finite-dimensional for all $k\geq 0$, as
follows from the identification of $M(\mathcal{L}_{c}^{k}S^{4})$ in Section
2.5.

The “phase transition” from $k\leq 8$ to $k\geq 9$ is summarized in the
following table.

Case | $\deg\mathcal{L}_{c}^{k}S^{4}$ | Inner product on $K_{k}^{\perp}$ |  $K_{k}$ and $K_{k}^{\perp}$ | Weyl group $W_{k}$ | Exceptnl vectors for $\mathcal{L}_{c}^{k}S^{4}$
---|---|---|---|---|---
$k\leq 8$ | $>0$ | negative-definite | $K_{k}\not\in K_{k}^{\perp}$ | finite | finitely many
$k=9$ | 0 | negative semi-definite | $K_{k}\in K_{k}^{\perp}$ | infinite | infinitely many
$k\geq 10$ | $<0$ | Lorentzian | $K_{k}\not\in K_{k}^{\perp}$ | infinite | infinitely many

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Hisham Sati, Mathematics, Division of Science, and
Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute,
New York University Abu Dhabi, UAE.
<EMAIL_ADDRESS>

Alexander A. Voronov, School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA, and
Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba 277-8583, Japan.
<EMAIL_ADDRESS>


# On Lieb-Robinson Bounds for the Double Bracket Flow

###### Abstract

We consider the possibility of developing a Lieb-Robinson bound for the double
bracket flow[3, 4]. This is a differential equation

$\partial_{B}H(B)=[[V,H(B)],H(B)]{}$

which may be used to diagonalize Hamiltonians. Here, $V{}$is fixed and
$H(0)=H{}$. We argue (but do not prove) that $H(B){}$need not converge to a
limit for nonzero real $B{}$in the infinite volume limit, even assuming
several conditions on $H(0){}$. However, we prove Lieb-Robinson bounds for all
$B{}$for the double-bracket flow for free fermion systems, but the range
increases _exponentially_ with the control parameter $B{}$.

##  Chapter 1 On Lieb-Robinson Bounds for the Double Bracket Flow

Matthew B. Hastings

00footnotetext: Keywords. Lieb-Robinson Bounds, Double bracket flow.

###  1.1 Some Cases Where Lieb-Robinson Bounds Can and Cannot be Proven

Lieb-Robinson bounds describe how an operator “spreads out" over space while
undergoing the Heisenberg equation of motion:

$\partial_{t}O(t)=i[H,O(t)].{}$

for some Hamiltonian $H{}$on some quantum lattice system. Crucially, one can
bound this spread in a way that is uniform in the size of the system if
$H{}$obeys suitable locality properties, so that the spread depends only on
the strength and range of terms in $H{}$. Indeed, while the first proof[10]
also depended on the local Hilbert space dimension, later it was possible to
derive bounds uniform in this dimension, as in [7].

For local Hamiltonians, the spread is at most linear in time, governed by a
Lieb-Robinson velocity. However, for more general Hamiltonians with
polynomially-decaying long-range interactions, one can derive in some cases a
weaker result[8, 6, 11, 5] where the spread is superlinear in time, but one
can still find bounds uniform in system size.

However, all these proofs use crucially the fact that the evolution considered
is unitary. If we instead consider an imaginary time version of the Heisenberg
equation:

$\partial_{\tau}O(\tau)=[H,O],{}$

then it is not possible to derive bounds uniform in system size valid for long
times. Indeed, a finite-time singularity may occur in the limit of infinite
system size[2]. The trouble is that for a system of volume $V{}$, an operator
$O{}$may have matrix elements between eigenstates whose energy differs by an
amount of order $V{}$, and the imaginary time evolution causes these matrix
elements to grow exponentially in $\tau{}$. It is however possible to prove
bounds for imaginary time evolution in the case of one-dimensional systems[1,
2]. Also, for local Hamiltonians in arbitrary dimensions, one may use a series
expansion[2] to prove locality bounds which are valid for $O(1){}$time, where
here $O(1){}$means a constant, independent of system size, but depending on
microscopic parameters of the Hamiltonian.

Another case where bounds for imaginary time evolution can be proven is for
free fermions. Suppose each site $i{}$has one Majorana operator
$\gamma_{i}{}$, with Majorana operators obeying canonical anticommutation
relations, with

$H=\sum_{i,j}\gamma_{i}h_{ij}\gamma_{j},{}$

where $h{}$is some anti-symmetric anti-Hermitian matrix. Suppose that there is
some pseudometric $\mathrm{dist}(i,j){}$between sites, and suppose
$h_{ij}=0{}$for $\mathrm{dist}(i,j)>R{}$for some range $R{}$and $\|h\|\leq
J{}$where $\|\ldots\|{}$denotes the operator norm. We have

$\gamma_{j}(\tau)=\gamma_{j}+2\tau\sum_{k}h_{kj}\gamma_{k}+\sum_{l,k}\frac{(2\tau)^{2}}{2!}h_{lk}h_{kj}\gamma_{l}+\ldots{}$

The $m{}$-th term in this series is a linear combination of Majorana
operators, with the coefficients having $\ell_{2}{}$norm bounded by $(\tau
J)^{m}/m!{}$and hence the $m{}$-th term has operator norm bounded also by
$(\tau J)^{m}/m!{}$. For $m\gg\tau J{}$, this is exponentially small in $m{}$,
and hence up to exponentially small error, $\gamma_{j}(\tau){}$is supported
within distance $O(R\tau J){}$of $j{}$.

###  1.2 The Double Bracket Flow

The double bracket flow[3, 4] is a differential equation that can be used to
diagonalize a Hamiltonian. Let $V{}$be some fixed Hermitian matrix and
consider the equation

$\partial_{B}H(B)=[[V,H(B)],H(B)],{}$

with initial conditions $H(0)=H{}$, where $B\geq 0{}$is a real parameter. As
$B\rightarrow\infty{}$, $H(B){}$converges to a fixed point, where it commutes
with $V{}$. Working in an eigenbasis of $V{}$with eigenvalues of $V{}$ordered
in non-decreasing order so that
$V=\mathrm{diag}(\lambda_{1},\lambda_{2},\ldots){}$with
$\lambda_{1}\geq\lambda_{2}\geq\ldots{}$, the only stable fixed points are
where the eigenvalues of $H(B){}$are also ordered in non-decreasing order.

Defining $\eta(B)=[V,H(B)]{}$, then $-i\eta{}$is Hermitian and the double
bracket flow is

$\partial_{B}H(B)=i[\eta(B),H(B)].{}$

So, it is equivalent to Heisenberg evolution of $H(B){}$under the
“Hamiltonian" $-i\eta(B){}$which itself depends on $H(B){}$.

A closely related equation has been studied in physics[12], where it is
sometimes called “Wegner’s flow equation". The difference is that the Wegner’s
flow equation is defined by $\partial_{B}H(B)=[[H_{0}(B),H(B)],H(B)],{}$where
$H_{0}(B){}$is the diagonal part of $H(B){}$in some basis. This flow has also
found applications in condensed matter physics[9].

Unfortunately, even if $V,H{}$are both local, the flow generates increasingly
complicated terms and in practice some truncation procedure is employed to
approximate the evolution using a simpler local Hamiltonian.

So, it is of interest to see whether some kind of locality result analogous to
a Lieb-Robinson bound might hold for this flow equation, to help control this
truncation. We discuss this in the next two sections.

Remark: the practical application of the double bracket flow is usually to
study the _ground state_. In this case, in practice truncation procedures are
employed which are (approximately) valid for the ground state. We do not
consider such issues at all here.

###  1.3 Lieb-Robinson Bounds for the Double Bracket Flow: Local Spin Systems

In this section, we consider arbitrary local spin systems. We assume that
there is some finite lattice of sites, $\Lambda{}$, with some metric
$\mathrm{dist}(\cdot,\cdot){}$, and the Hilbert space is a tensor product of
finite-dimensional Hilbert spaces associated with each site.

Assume $H(0)=H{}$and $V{}$both are _finite strength and range_ with _bounded
local geometry_. This means that $H=\sum_{X}h_{X}{}$where each $X{}$is
supported on a set of diamter at most $R{}$for some $R{}$(this is the finite
range), where $\|h_{X}\|\leq J{}$for some $J{}$(this is the finite strength)
and where for any site $i\in\Lambda{}$, the number of sites $j{}$with
$\mathrm{dist}(i,j)\leq R{}$is bounded by some constant (this is bounded local
geometry).

These conditions are sufficient to prove a Lieb-Robinson bound for Heisenberg
evolution under $H{}$. We may then ask: does $H(B){}$obey some locality bound?
For any finite $\Lambda{}$, the evolution equation has a unique solution, but
can one prove some bound on the locality of $H(B){}$, uniformly in
$|\Lambda|{}$?

We attempt to prove such a bound in the next two subsections, but we are
unable to do so, motivating the following conjecture (this conjecture clearly
has several additional assumptions and does not in any way follow from our
results but is motivated by them):

######  Conjecture 1.3.1.

There is some translation invariant Hamiltonian $H{}$on a $d{}$-dimensional
hypercubic lattice, and some other translationally invariant $V{}$, so that,
if one considers these $H{}$and $V{}$on a sequence of lattices of increasing
size, the solution of the double bracket flow does not converge to a
translationally invariant Hamiltonian in the infinite system size limit for
any real $B\neq 0{}$.

Indeed, the solution may fail to converge even if the Hilbert space on each
site is two-dimensional, $V{}$is the sum of Pauli $Z{}$on each site, and
$H=V+\epsilon\Delta{}$where $\Delta{}$is some translationally invariant
Hamiltonian and $\epsilon{}$is an arbitrarily small real scalar.

Remark: As is known, the double bracket flow is a gradient flow. Consider the
function $\operatorname*{{Tr}}((H(B)-V)^{2}){}$. Let
$H(B)=U(B)H(0)U(B)^{\dagger}{}$for unitary $U(B){}$. Consider infinitesimal
change in $U(B){}$: $U(B)\rightarrow U(B)+\eta U(B){}$for infinitesimal anti-
Hermitian $\eta{}$. The change in the potential is then
$-2\operatorname*{{Tr}}([\eta,H(B)]V)=-2\operatorname*{{Tr}}(\eta[H(B),V]){}$.
So, for the gradient flow on the manifold of $U(B){}$using a metric111Here we
mean a metric on $U(B){}$rather than on $H(B){}$. induced by the Hilbert-
Schmidt inner product is the double bracket flow. So, an interesting open
question is whether some modification of the double bracket flow can be
defined which will have some locality properties. It seems that one could
replace the Hilbert-Schmidt inner product by some other inner product to
improve the locality. For example, on a system of qudits, an orthonormal basis
for the Hilbert-Schmidt inner product may be obtained by products of
nontrivial (generalized) Pauli operators on sets of sites $X{}$. We might
change the metric so that it is still diagonal in this basis, but scales with
the diameter of $X{}$.

####  1.3.1 Power Series

One might first try to prove a locality bound for $B=O(1){}$by the same kind
of power series expansion technique that works for imaginary time Hamiltonian
evolution[2]. Unfortunately, this does not seem to work for the double bracket
flow as the expansion seems to have vanishing radius of convergence in the
infinite system size limit. Let us see what happens.

The calculation here is not intended to be a proof; rather, we show that some
subset of terms in a power series expansion may diverge for all $B\neq 0{}$,
at least under some worst case assumptions on certain commutators. It is
possible that a more refined power series analysis might be able to prove a
nontrivial result though; for example, perhaps these terms are canceled by
other terms or the worst case commutation bound may not hold. Thus we leave as
an interesting open question: does the power series have a nonzero radius of
convergence for some lattices in the infinite size limit, such as one-
dimensional systems?

Let $H{}$and $V{}$both be a sum of local terms on some quantum lattice system.
We can expand $H(B){}$as a power series in $B{}$for a finite lattice system.
Let $H_{k}(B){}$denote the $k{}$-th order term in $B{}$for $H(B){}$so that
$H_{0}(B)=H{}$. We then have

$\partial_{B}H_{k+1}(B)=\sum_{l=0}^{k}[[H_{k-l}(B),V],H_{l}(B)].{}$

Let us consider a particular term in the sum, where $l=0{}$. So we have

$H_{k+1}(B)=\int_{0}^{B}[[H_{k}(B),V],H_{0}(B)]\mathrm{d}B+\mathrm{other\,terms}.$
(1.1)

Suppose each term in $H_{0}{}$and each term in $V{}$is supported on some set
of diameter $O(1){}$and has norm at most $J{}$. Then, $H_{k}{}$is supported on
some set of diameter $O(k){}$and for some lattices, the double commutator may
be of order $J^{2}k^{2}{}$. Let $J_{k}(B){}$be a bound on the norm of these
terms of diameter $O(k){}$in $H_{k}{}$. Precisely, we imagine that $H_{k}{}$is
a sum over sites $j{}$of some term with norm at most $J_{k}(B){}$supported
within diameter $O(k){}$of $j{}$.

Then we have

$J_{k+1}(B)\leq\int_{0}^{B}J_{k}(B^{\prime})J^{2}k^{2}|\mathrm{d}B^{\prime}|,{}$

ignoring the contribution of these other terms. Replacing the inequality
$\leq{}$with an equality sign, and taking $J_{0}(0)=1{}$, $J_{k}(0)=0{}$for
$k>0{}$, the equation is easy to solve with

$J_{k}(B)=\frac{(J^{2}|B|)^{k}}{k!}((k-1)!)^{2},{}$

and so $\sum_{k}J_{k}(B){}$diverges at any $B\neq 0{}$.

####  1.3.2 Weak Perturbations

Having failed to prove convergence of the power series for any $B\neq 0{}$,
let us try something weaker. Throughout this subsection we consider real,
positive $B{}$.

Suppose $H{}$and $V{}$both have finite strength and range. Further, suppose
that $H(B)=V+\epsilon\Delta(B){}$for some small nonzero $\epsilon{}$and some
$\Delta(B){}$where $\Delta(0){}$which has finite strength and range. Then, the
double bracket flow becomes

$\partial_{B}\Delta(B)=[[V,\Delta(B)],V]+\epsilon[[V,\Delta(B)],\Delta(B)].$
(1.2)

The first term is a linear term, with all eigenvalues nonpositive, while the
second is nonlinear.

If we consider some simple case where, for example, we have a lattice of
qubits and $V{}$is a sum of Pauli $Z{}$operators on each site, then the linear
equation of motion can be solved readily. Let $a_{j}=(X_{j}+iY_{j})/2{}$where
$X_{j},Y_{j}{}$are Pauli $X,Y{}$operators on site $j{}$. We can expand
$\Delta(B){}$of $B{}$in a basis of products of single site operators
$Z,a,a^{\dagger}{}$. Given such a product in which $a{}$appears a total of
$n_{+}{}$times and $a^{\dagger}{}$appears a total of $n_{-}{}$times, say that
term has “charge $q{}$" where $q=n_{+}-n_{-}{}$. Then, any term with charge
$q{}$is an eigenoperator of Eq. 1.2 with eigenvalue $-4q^{2}{}$.

So, one might hope in this example that terms with with large $q{}$will decay
rapidly as $B{}$increases, while terms with small $q{}$operators will have a
small commutator with $V{}$. So, one might hope some $H(B){}$will obey some
locality properties, uniform in $|\Lambda|{}$.

Unfortunately, this does not seem to hold. Similar to the previous section,
the argument here involves focusing on certain terms in a power series
expansion. It is not intended to be a proof, but it is intended to give some
evidence to support 1.3.1.

We solve Eq. 1.2 as a power series in $\epsilon{}$. Let
$\Delta_{k}(B){}$denote the term of order $\epsilon^{k}{}$for $\Delta(B){}$.
We then have

$\partial_{B}\Delta_{k+1}(B)=[[V,\Delta_{k+1}(B)],V]+\epsilon\sum_{l=0}^{k}[[\Delta_{k-l}(B),V],\Delta_{l}(B)].{}$

Further, let $\Delta_{k,q}{}$denote the terms with given charge $q{}$in
$\Delta_{k}{}$. Then

$\partial_{B}\Delta_{k+1,q}(B)=-4q^{2}\Delta_{k+1,q}(B)+\epsilon\sum_{l=0}^{k}\sum_{r}[[\Delta_{k-l,q-r}(B),V],\Delta_{l,r}(B)].{}$

Let us focus on terms with $l=0{}$and $r=0{}$, writing

$\partial_{B}\Delta_{k+1,q}(B)=-4q^{2}\Delta_{k+1,q}(B)+\epsilon[[\Delta_{k,q}(B),V],\Delta_{0,0}(B)]+\mathrm{other\,terms}.$
(1.3)

The term $\Delta_{0,0}(B){}$is independent of $B{}$. Ignoring the other terms
in Eq. 1.3, this gives a linear equation of motion for $\Delta_{k+1,q}{}$in
terms of $\Delta_{k+1,q}{}$and $\Delta_{k,q}{}$. The term
$\Delta_{k,q}(B){}$is supported on some set of diameter $O(k){}$and for some
lattices, the double commutator may be of order $\epsilon kqJ^{2}{}$, where
$J{}$is proportional to the strength of $\Delta(0){}$. Let
$\delta_{k,q}(B){}$be a bound on the norm of these terms in $\Delta_{k,q}{}$.
Thus, in worst case we might expect a linear equation describing this norm:

$\partial_{B}\delta_{k+1,q}(B)=-4q^{2}\delta_{k+1,q}(B)+\epsilon
kqJ^{2}\delta_{k,q}(B).{}$

Equivalently, defining

$\tilde{\delta}_{k+1,q}(B)=\exp(4q^{2}B)\delta_{k+1,q}(B),{}$

we get

$\partial_{B}\tilde{\delta}_{k+1,q}(B)=\epsilon
kqJ^{2}\tilde{\delta}_{k,q}(B).{}$

This equation is readily solved with
$\tilde{\delta}_{k,q}(B)=\frac{(k-1)!}{k!}(\epsilon qJ^{2})^{k}B^{k},{}$and so
the sum over $k{}$has a finite-$B{}$singularity at $B=1/\epsilon qJ^{2}{}$.
Hence, $\delta_{k,q}{}$_also_ has a finite-$B{}$singularity at

$B_{q}\equiv(\epsilon qJ^{2})^{-1},{}$

at least in the approximations of this subsection.

We may take $q{}$large to get a finite-$B{}$singularity at arbitrarily small
$B{}$. Of course, the reader may object here: assuming $\Delta{}$has finite
range, then $\Delta_{0,q}{}$is nonvanishing only for charge $q=O(1){}$.
However, starting with terms with charge which is $O(1){}$, then Eq. 1.2 will
generate terms with any given $q{}$for any nonzero $B{}$, with a strength
proportional to $(\epsilon J^{2}B)^{q}/q!{}$. So, indeed we still expect a
finite-$B{}$singularity at the given $B_{q}{}$, and we may take $q{}$large to
get a singularity at arbitrarily small $B{}$.

###  1.4 Free Fermions

One case where we can prove a kind of Lieb-Robinson bound for all $B{}$is when
$H(B){}$and $V{}$both describe free fermions, with

$H(B)=\sum_{i,j}\gamma_{i}h(B)_{ij}\gamma_{j},{}$

where $h(0)=h{}$, and

$V=\sum_{i,j}\gamma_{i}v_{ij}\gamma_{j},{}$

for some anti-symmetric anti-Hermitian matrices $h(B),v{}$.

Then,

$\partial_{B}h(B)=4[[v,h(B)],h(B)].{}$

Suppose that $h_{ij}=0{}$for ${\rm dist}(i,j)>R{}$and $\|h\|=J{}$for some
$R,J{}$and similarly $v_{ij}=R{}$for ${\rm dist}(i,j)>0{}$and $\|v\|=J{}$.

Define for any matrix $m{}$the quantity $\|m\|_{r}{}$by

$\|m\|_{r}\equiv{\rm max}_{\psi,\phi,{\rm
dist}(\mathrm{supp}(\psi),\mathrm{supp}(\phi))>r}\langle\psi,m\phi\rangle,{}$

where $\psi,\phi{}$are vectors with $\ell_{2}{}$norm $1{}$and
$\mathrm{supp}(\cdot){}$denotes the support of a vector. Note that
$\|\cdot\|_{r}{}$is not a norm: while it obeys a triangle inequality,
$\|m\|_{r}{}$may be vanishing even if $m{}$is nonvanishing. Rather,
$\|m\|_{r}{}$may be thought of as measuring the strength of $m{}$at distance
$r{}$.

Our main result is _there is a light-cone, but the excitations may spread
exponentially quickly in $B{}$_. In particular

######  Lemma 1.4.1.

For any $\ell>0{}$, we have

$\|h(B)\|_{\ell}\leq
J\Bigl{(}\frac{8eJ^{2}B}{\log_{2}(\ell)-O(1)}\Bigr{)}^{\log_{2}(\ell)-O(1)}.{}$

So, for $\log_{2}(\ell){}$large compared to $8eJ^{2}B{}$, the quantity
$\|h(B)\|_{\ell}{}$is exponentially small in $\log_{2}(\ell){}$.

###### Proof.

For any $\psi,m,r{}$, we have $m\psi=\psi_{1}+\psi_{2}{}$where $\psi_{1}{}$is
supported within distance $r{}$of $\mathrm{supp}(\psi){}$with
$|\psi_{1}|\leq\|m\|\cdot|\psi|{}$and where
$|\psi_{2}|\leq\|m\|_{r}\cdot|\psi|{}$. So, we have in general

$\|m_{1}m_{2}\|_{r_{1}+r_{2}}\leq\|m_{1}\|_{r_{1}}\cdot\|m_{2}\|+\|m_{1}\|\cdot\|m_{2}\|_{r_{2}}.$
(1.4)

We have $\|h(B)\|=\|h\|=J{}$for all $B{}$since the flow equation describes a
unitary evolution. Hence, $\|h(B)\|_{r}\leq J{}$for all $B,r{}$.

We have, for small $\mathrm{d}B>0{}$:

$\langle\psi,h(B+\mathrm{d}B)\phi\rangle=\langle\psi,h(B)\phi\rangle+4\langle\psi,[[v,h(B)],h(B)]\phi\rangle\mathrm{d}B+\mathcal{O}(\mathrm{d}B^{2}).$
(1.5)

Let $|\psi|=|\phi|=1{}$. Suppose
$\mathrm{dist}(\mathrm{supp}(\psi),\mathrm{supp}(\phi))=2\ell+R{}$for some
$\ell{}$. The term $\langle\psi,[[v,h(B)],h(B)]\phi\rangle{}$is a sum of four
terms, given by the different ways of expanding the commutator. Consider a
typical term such as $\langle\psi,vh(B)h(B)\phi\rangle{}$. Applying Eq. 1.4
and using $\|v\|_{R}=0{}$, we have

$\|vh(B)\|_{\ell+R}\leq J\|h(B)\|_{\ell}{}$

and

$\displaystyle\|vh(B)h(B)\|_{2\ell+R}$
$\displaystyle\leq\|vh(B)\|_{\ell+R}\cdot\|h(B)\|+\|vh(B)\|\cdot\|h(B)\|_{\ell}$
(1.6) $\displaystyle\leq 2J^{2}\|h(B)\|_{\ell}.$

Summing over the four terms, we find

$|\langle\psi,[[v,h(B)],h(B)]\phi\rangle|\leq 8J^{2}\|h(B)\|_{\ell},$ (1.7)

so

$\|h(B+\mathrm{d}B)\|_{2\ell+R}\leq\|h(B)\|_{2\ell+R}+8J^{2}\|h(B)\|_{\ell}\mathrm{d}B+O(\mathrm{d}B^{2}).$
(1.8)

Define a sequence of scales $R_{k}{}$for $k=0,1,2,\ldots{}$by

$\displaystyle R_{0}$ $\displaystyle=R$ (1.9) $\displaystyle
R_{k+1}=2R_{k}+R,$

and taking a limit as $\mathrm{d}B\rightarrow 0^{+}{}$, we have

$\|h(B)\|_{R_{k+1}}\leq\int_{0}^{B}8J^{2}\|h(B)\|_{R(k)},$ (1.10)

with also

$\|h(B)\|_{R_{k}}\leq J{}$

for all $k{}$.

Replacing the inequality in Eq. 1.10 with an equality, the result is easily
solved since it is linear in the quantities $\|h(B)\|_{R(i)}{}$and we find

$\displaystyle\|h(B)\|_{R_{k}}$ $\displaystyle\leq J\frac{(8J^{2}B)^{k}}{k!}$
(1.11) $\displaystyle\leq J\Bigl{(}\frac{8eJ^{2}B}{k}\Bigr{)}^{k},$

where we used Stirling’s approximation. Hence, for $k\gg 8eJ^{2}B{}$for some
$c>0{}$, it is exponentially small.

We claim that

$R_{k}\leq(2^{k+1}-1)R.{}$

This follows inductively from $R_{0}\leq(2^{1}-1)\cdot R{}$and
$R_{k+1}=2R_{k}+R\leq(2^{k+2}-2)R+R=(2^{k+2}-1)R.{}$Hence, given any $\ell{}$,
we may choose $k=\log_{2}(\ell)-O(1){}$with $R_{k}\leq\ell{}$.

So, from Eq. 1.11,

$\|h(B)\|_{\ell}\leq
J\Bigl{(}\frac{8eJ^{2}B}{\log_{2}(\ell)-O(1)}\Bigr{)}^{\log_{2}(\ell)-O(1)}.{}$

∎

We expect that the bounds of this lemma can be tightened significantly for
$\ell{}$large compared to this “exponential light-cone". However, the
exponential spreading of Lemma 1.4.1 is the best possible in general. As a
model system, consider the family of Hamiltonians $h_{t}{}$in one-dimension,
depending on a parameter $t{}$, with sites indexed by an integer $k{}$, with
matrix elements

$\displaystyle(h_{t})_{2k,2k+1}$ $\displaystyle=(h_{t})_{2k+1,2k}=1+t,$ (1.12)
$\displaystyle(h_{t})_{2k+1,2k+2}$ $\displaystyle=(h_{t})_{2k+2,2k+1}=1-t,$

with all other matrix elements vanishing.

Introduce a basis of Fourier modes $\sum_{k}\exp(i\theta)|k\rangle{}$for
$\theta{}$periodic mod $2\pi{}$. Then, in this basis, $h_{t}{}$is a block two-
by-two matrix for each pair $\theta,\theta+\pi{}$:

$\begin{pmatrix}2\cos(\theta)&t\\\
t&-2\cos(\theta)\end{pmatrix}=t\sigma_{x}+2\cos(\theta)\sigma_{z},{}$

where $\sigma_{x},\sigma_{z}{}$are Pauli sigma matrices.

Let us pick some given $t{}$, and let $h=h_{t}{}$and $v=h_{-t}{}$. Each such
two-by-two matrix can be evolved independently under the double bracket flow.
Writing $h_{t}(B)=a(B)\sigma_{x}+b(B)\sigma_{z}{}$in a given block for given
$\theta{}$, the differential equation in a two-by-two block for
$a(B),b(B){}$may be computed explicitly. This evolution has two fixed points:
when $h_{t}(B)=v{}$or when $h_{t}=-v{}$. The first fixed point is stable while
the second is unstable. Any perturbation of the unstable fixed point grows
exponentially in $B{}$.

However, for $\theta{}$near $\pm\pi/2{}$, we have $\cos(\theta)\approx 0{}$,
and so we are near the unstable fixed point. Indeed, at large $B{}$, for
$|\log(\cos(\theta)|){}$large compared to $1/B{}$, then $h_{t}(B){}$evolves to
$v{}$in the given block, while for $\log(|\cos(\theta)|){}$small compared to
$1/B{}$, then $h_{t}(B){}$is close to $-v{}$.

So, $h_{t}(B){}$changes very rapidly as a function of $\theta{}$for
$\log(|\cos(\theta)|){}$of order $1/B{}$. This corresponds to an exponential
growth in $B{}$of the appropriate length scale in $h_{t}(B){}$after Fourier
transforming back to the lattice.

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