# Bootstrapping Bloch bands

Serguei Tchoumakov  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel,
38000 Grenoble, France  Serge Florens  Univ. Grenoble Alpes, CNRS, Grenoble
INP, Institut Néel, 38000 Grenoble, France

###### Abstract

Bootstrap methods, initially developed for solving statistical and quantum
field theories, have recently been shown to capture the discrete spectrum of
quantum mechanical problems, such as the single particle Schrödinger equation
with an anharmonic potential. The core of bootstrap methods builds on exact
recursion relations of arbitrary moments of some quantum operator and the use
of an adequate set of positivity criteria. We extend this methodology to
models with continuous Bloch band spectra, by considering a single quantum
particle in a periodic cosine potential. We find that the band structure can
be obtained accurately provided the bootstrap uses moments involving both
position and momentum variables. We also introduce several new techniques that
can apply generally to other bootstrap studies. First, we devise a trick to
reduce by one unit the dimensionality of the search space for the variables
parametrizing the bootstrap. Second, we employ statistical techniques to
reconstruct the distribution probability allowing to compute observables that
are analytic functions of the canonical variables. This method is used to
extract the Bloch momentum, a quantity that is not readily available from the
bootstrap recursion itself.

##  I Introduction

Despite the paradoxical name, bootstrapping methods constitute an interesting
toolkit to solve a large variety of theoretical physics models. The essence of
this idea lies in setting an infinite set of internal constraints that can be
derived from the studied problem, and that can be used to determine uniquely
its solution without ever computing it explicitly. While the initial bootstrap
program is aimed at constructing fundamental quantum field theories allowed by
Nature [1], later developments focused on practical solutions in statistical
and quantum mechanics [2]. Quite remarkably, enforcing conformal invariance
helps formulate a bootstrap principle to compute the critical exponents of the
three-dimensional (3D) Ising model [3] and solve some of its correlations
functions [4], while not feasible with other techniques. More recently, a
class of many-body quantum matrix models are shown to be amenable to a
bootstrap solution [5], from which the single particle Schrödinger equation
with quartic potential is a particular case [6]. Here, the bootstrap method
relies on the relationship between the statistical moments of quantum
operators, related to the positivity of a quadratic form, and that is used to
check whether the recursion between moments of an eigensolution of the
Hamiltonian is physical or not. This method has already been implemented to
various models in quantum mechanics [7, 8, 9, 10, 11].

Our main goal here is to extend the bootstrap methodology to solve periodic
quantum Hamiltonians with continuous spectra and to propose new tools to
improve the convergence and application-range of the bootstrap. We study in
detail the solution to a single quantum particle in a periodic cosine
potential. We show that the convergence of the bootstrap depends crucially on
the appropriate choice of constraints. Indeed, we find that the use of
statistical moments that mix position and momentum drastically improves the
convergence with respect to a bootstrap involving only the moments of a single
canonical variable. The band structure of the cosine potential problem is well
reproduced by the bootstrap, and convergence seems uniform. Due to the
amplification of numerical round-off errors in the bootstrap recursion,
arbitrary precision numerics need to be used in order to reach machine
precision for the band spectrum.

In previous bootstrap studies [3, 4, 5, 6], the energy spectra are obtained by
scanning a single moment for each energy eigenvalue to find values that
satisfy the bootstrap constraints. This exploration over a two-dimensional
(2D) space is numerically heavy and we devise a dimensional reduction method
that shrinks the search space from 2D to 1D. Finally, the bootstrap is often
used to compute the spectrum and the moments involved in the recursion
relation. The choice of moments is limited to those where the recursion
relation can be closed and thus prevents the evaluation of observables for
which no closed recursion relation exists. For example, we show that the
evaluation of the Bloch momentum of the periodic quantum Hamiltonian
necessitates to evaluate an infinite recursion. For this type of situation, we
propose a probabilistic method, based on the truncated Hamburger moment
problem, to compute arbitrary analytic functions of the canonical variables.
Our methodology is general enough that it can be extended to other bootstrap
studies [12, 13, 14].

The paper is organized as follows. In Sec. II, we derive all the useful exact
bootstrap identities for the model of a single quantum particle in a 1D cosine
potential. In Sec. III, we show a general dimensional reduction of the search
space for the bootstrap variables, which can apply to other bootstrap studies.
In Sec. IV, we construct a distribution that allows to compute the expectation
value of an arbitrary analytic function of the canonical variables,
illustrating how Bloch’s momentum can be reconstructed. General perspectives
close the manuscript.

##  II Bootstrapping identities

We focus here on the Hamiltonian describing a single quantum particle moving
in a cosine potential on the real line:

$\displaystyle\hat{H}=\hat{p}^{2}+2v\cos(\hat{x}),$ (1)

where $[\hat{x},\hat{p}]=i$ and $2v$ is the strength of the potential. We will
set throughout $\hbar=1$ and the particle mass $2m=1$. This simple Hamiltonian
is ubiquitous in physics, describing e.g. the motion of electrons in crystals
[15], as well as the dynamics of Josephson junctions [16].

The relation between averaged observables can be obtained using the following
identity [6]. In any eigenstate $|\Psi\rangle$ of $\hat{H}$ with energy $E$
and for any operator $\hat{O}$, the following average vanishes

$\displaystyle\langle[\hat{H},\hat{O}]\rangle=\langle\Psi|[\hat{H},\hat{O}]|\Psi\rangle=\langle\Psi|E\hat{O}-\hat{O}E|\Psi\rangle=0.$
(2)

Due to the periodicity of Hamiltonian (1), we expand (2) for the operators
$\hat{O}_{1}=\exp({in\hat{x}})$ and $\hat{O}_{2}=\exp(in\hat{x})\hat{p}$, with
$n$ an integer, leading to the respective identities:

$\displaystyle n^{2}\langle e^{in\hat{x}}\rangle+2n\langle
e^{in\hat{x}}\hat{p}\rangle=0,$ (3)

and,

$\displaystyle n^{2}\langle e^{in\hat{x}}\hat{p}\rangle+2n\langle
e^{in\hat{x}}\hat{p}^{2}\rangle+v(\langle e^{i(n-1)\hat{x}}\rangle-\langle
e^{i(n+1)\hat{x}}\rangle)=0.$ (4)

Since the averages are over an eigenstate of $\hat{H}$, we can also use the
identity $\langle\hat{O}\hat{H}\rangle=E\langle\hat{O}\rangle$ with
$\hat{O}=\exp({in\hat{x}})$:

$\displaystyle\langle e^{in\hat{x}}\hat{p}^{2}\rangle+v(\langle
e^{i(n-1)\hat{x}}\rangle+\langle e^{i(n+1)\hat{x}}\rangle)=E\langle
e^{in\hat{x}}\rangle.$ (5)

The identities in Eqs. (3-5) lead to the closed recursion on the expectation
values of the phase operator $\exp({in\hat{x}})$:

$\displaystyle n(4E-n^{2})\langle e^{in\hat{x}}\rangle$ (6)
$\displaystyle+2v\left((1-2n)\langle e^{i(n-1)\hat{x}}\rangle-(1+2n)\langle
e^{i(n+1)\hat{x}}\rangle\right)=0.$

One can readily show that
$\langle\exp({in\hat{x}})\rangle=\langle\cos(n\hat{x})\rangle$, so the
recursion relation can be computed for any $n$ once $E$ and
$\langle\cos(\hat{x})\rangle$ are set, since
$\langle\exp({i0\hat{x}})\rangle=1$ due to normalization of the probability
distribution. Note that formally the norm $\langle 1\rangle$ diverges in the
thermodynamic limit due to the non confining periodic potential, but
quantities such as $\langle e^{in\hat{x}}\rangle/\langle 1\rangle$ should be
well defined.

Also, it is useful to derive the recursion between averages in the form
$\langle\exp({in\hat{x}})\hat{p}^{s}\rangle$ involving both the phase operator
$\exp({in\hat{x}})$ and an arbitrary power of the momentum operator $\hat{p}$.
Following the same approach, we establish the following double recursion (see
Appendix A for details):

$\displaystyle(n^{2}-E)\langle e^{inx}p^{s}\rangle+2n\langle
e^{inx}p^{s+1}\rangle+\langle e^{inx}p^{s+2}\rangle$ $\displaystyle+v(\langle
e^{i(n+1)x}p^{s}\rangle+\langle e^{i(n-1)x}p^{s}\rangle)=0.$ (7)

Since $\langle\exp({in\hat{x}})\rangle$ is known for all $n$ from the single
recursion Eq. (6), and that $\langle\exp({in\hat{x}})\hat{p}\rangle$ can be
deduced from Eq. (3), we can use Eq. (7) to compute the moments in the form
$\langle\exp({in\hat{x}})\hat{p}^{s}\rangle$, for all $s>1$.

##  III Dimensional reduction of the search space

We now present how the bootstrap conditions are generally derived, before
addressing the details of the solution. We first define a general operator
$\hat{O}=\sum_{n=0}^{K}\sum_{s=0}^{L}a_{ns}\hat{p}^{s}\exp({in\hat{x}})$ from
a set of $(K+1)\times(L+1)$ arbitrary complex coefficients $a_{ns}$. The
operator $\hat{O}^{\dagger}\hat{O}$ is necessarily positive semi-definite, so
its expectation value in any state satisfies the positivity constraint
$\langle\hat{O}^{\dagger}\hat{O}\rangle\geq 0$ [17, 5, 6]. This implies that
the matrix $\mathcal{M}_{n\sigma,m\tau}=\langle
e^{-imx}\hat{p}^{\sigma+\tau}e^{in\hat{x}}\rangle$ should be positive semi-
definite. For averages over an exact eigenstate of $H$ the entries of
$\mathcal{M}_{n\sigma,m\tau}$ are obtained from Eq. (7), see Appendix A for
details. Thus, once the energy $E$ and the first moment
$\langle\cos(x)\rangle$ are set, the whole matrix $\mathcal{M}$ is determined.

The usual bootstrap consists in scanning all possible values of $E$ and
$\langle\cos(x)\rangle$ to check if $\mathcal{M}$ is positive semi-definite,
i.e. if its eigenvalues are all greater or equal to zero. Any pair
($E,\langle\cos(x)\rangle$) that leads to a non-positive $\mathcal{M}$ is non-
physical and the region of physical solutions gets more and more precise when
increasing $K$ or $L$, i.e. when increasing the number of constraint, although
the exact wavefunction is never computed explicitly. This procedure is
numerically expensive since it requires to scan all possible values of $E$ and
$\langle\cos(x)\rangle$ and we now show how the search space can be reduced to
the energy $E$ only.

The statistical moments in Eq. (7) are linearly related, so the matrix
$\mathcal{M}$ evaluated for $(E,\langle\cos(x)\rangle)$ satisfies:

$\displaystyle\mathcal{M}=\mathcal{M}^{(0)}+\langle\cos(x)\rangle\mathcal{M}^{(1)},$
(8)

where $\mathcal{M}^{(0)}$ and $\mathcal{M}^{(1)}$ are $\mathcal{M}$ matrices
computed for energy $E$ and respectively $\langle\cos(x)\rangle=0$ and
$\langle\cos(x)\rangle=1$. The eigenvalues $\lambda$ of $\mathcal{M}$ then
obey:

$\displaystyle\left(\mathcal{M}^{(0)}+\langle\cos(x)\rangle\mathcal{M}^{(1)}\right){\bf
v}=\lambda{\bf v},$ (9)

and the regions in the search space $(E,\langle\cos(x)\rangle)$ where
$\mathcal{M}$ is definite positive are delimited by boundaries where at least
one of the eigenvalues $\lambda$ vanishes. The value of
$\langle\cos(x)\rangle$ at these boundaries thus satisfies the eigenvalue
equation:

$\displaystyle[\mathcal{M}^{(1)}]^{-1}\mathcal{M}^{(0)}{\bf
v}=-\langle\cos(x)\rangle{\bf v}.$ (10)

The scan of all possible values of $\langle\cos(\hat{x})\rangle$ is thus
reduced to the $(K+1)\times(L+1)$ discrete eigenvalues of
$[\mathcal{M}^{(1)}]^{-1}\mathcal{M}^{(0)}$ for a given choice of $E$. Since
$|\langle\cos(\hat{x})\rangle|\leq 1$, we can further limit the search to the
few eigenvalues that are in the $[-1,1]$ interval. This technique accelerates
solving the bootstrap, which is especially useful to find the continuum of
eigenstates in the Bloch Hamiltonian (1). In general, a bootstrap with a
$d$-dimensional search space can be reduced to $d-1$ dimensions with this
technique.

We apply this technique to compute the spectrum of Eq. (1), as a function of
$\langle\cos(\hat{x})\rangle$, comparing to results from exact diagonalization
for a potential $v=1$, see Fig. 1. A similar spectrum was recently obtained in
the bootstrap study of gauge invariance of a charged particle on a circle [10,
9]. The computations are done by increasing the number of constrained
statistical moments, when increasing $K$, and for $L=1$, which implies that
the moments only include powers of $\hat{p}$ with $0\leq s\leq 2$. We find
that including statistical moments of the two conjugate variables is crucial
for the convergence of the bootstrap. When we perform the boostrap in position
or in momentum only, respectively with moments $\langle\exp(in\hat{x})\rangle$
($L=0$) or $\langle p^{s}\rangle$ ($K=0$), we find that some solutions
consistent with the bootstrap are absent in exact diagonalization, even for
large values of $K$ or $L$ respectively. For $L=1$ and large values of $K$, we
see in Fig. 1 the appearance of six Bloch bands that coincide with exact
diagonalization, in black and labeled ‘ED’. The four dots on the line of
constant energy $E=2.5$ illustrate four of the ten eigenvalues of Eq. (10) for
$K=4$. The two correct eigenvalues that correspond to the boundary are
uniquely determined by requiring that all but one eigenvalues $\lambda$ of
$\mathcal{M}$ are strictly positive. In this way, on a given domain of the
energy $E$, we bound the solutions consistent with the bootstrap by continuous
lines.

Figure 1:  Bootstrap solution of Eq. (1) for $v=1$ and an increasing number
$K$ of statistical moments, and compared to exact diagonalization (labelled
“ED”) [18]. The regions allowed by the bootstrap shrink to six Bloch bands for
large values of $K$. Four dots indicate the potential location of the boundary
at energy $E=2.5$ and for $K=4$, according to Eq. (10).

We observe in Fig. 1 that the bootstrap converges towards the spectrum in
exact diagonalization. For $K=8$, the average relative error is $\Delta
E/E\approx 10^{-3}$ on the considered range of energies, and goes down to
$\Delta E/E\approx 10^{-7}$ for $K=12$. These values are for the interaction
value $v=1$, and are typically less precise for larger values of $v$. There is
also loss in numerical precision when increasing $K$ and it can be necessary
to use symbolic calculus or arbitrary precision arithmetics.

##  IV Statistical approach to compute arbitrary observables

A limitation of the bootstrap is that it does not automatically output some
physical quantities, such as the Bloch wavevector – since
$\langle\hat{p}\rangle=0$. From Bloch’s theorem, exact eigenstates
$|\Psi\rangle$ of $\hat{H}$ in Eq. (1) satisfy $\langle
x|\Psi\rangle=\Psi(x)=e^{ikx}\Phi(x)$, with $\Phi(x)$ a $2\pi$ periodic
function, and $k$ the Bloch wavevector. We thus have
$e^{i2\pi\hat{p}}|\Psi\rangle=e^{i2\pi k}|\Psi\rangle$, so the Bloch
wavevector $k$ can be determined from the identity $\langle
e^{i2\pi\hat{p}}\rangle=e^{i2\pi k}$. The Bloch momentum $k$ is thus found
from the Taylor series

$\displaystyle e^{i2\pi k}=\langle
e^{i2\pi\hat{p}}\rangle=\sum_{s=0}^{\infty}\frac{(2i\pi)^{s}}{s!}\langle\hat{p}^{s}\rangle,$
(11)

that depends on the infinite set of moments $\langle\hat{p}^{s}\rangle$. The
evaluation of this expression from a recursion formula that involves
$\langle\hat{p}^{s}\rangle$ is unstable and is sensitive to the noise
generated by numerical errors on higher moment of $\hat{p}$. Instead, we
propose to evaluate $\langle e^{i2\pi\hat{p}}\rangle$ over the probability
distributions $\mathcal{P}(p)$ that reproduce exactly a finite number of known
moments,
$\\{\langle\hat{p}^{0}\rangle,\langle\hat{p}^{1}\rangle,...,\langle\hat{p}^{K}\rangle\\}$,
as $\langle\hat{p}^{s}\rangle=\int dp\mathcal{P}(p)p^{s}$.

In practice, there is an infinite set of probability distributions
$\mathcal{P}(p)$ that match the lowest moments. These probability
distributions can be expanded over the so-called canonical distributions, the
probability distributions that are non-zero only for a finite number of
discrete momenta $\\{p_{0},\ldots,p_{K}\\}$. These momenta are poles of the
Stieltjes transform of the underlying probability distribution, and are found
to be solutions of the following $(K+1)$-th order polynomial equation [19,
20]:

$\displaystyle h_{K}(p)+\nu h_{K-1}(p)=0,$ (12)

with $\nu$ a scalar labeling each canonical solution $\mathcal{P}_{\nu}$ and
$h_{K}(p)$ a unique set of orthogonal polynomials with respect to the
probability distribution $d\mathcal{P}$ [19] defined as:

$h_{K}(p)=\frac{1}{\Delta_{K}\Delta_{K-1}}\left|\begin{array}[]{cccc}\big{<}p^{0}\big{>}&\big{<}p^{1}\big{>}&\cdots&\big{<}p^{K}\big{>}\\\
\big{<}p^{1}\big{>}&\big{<}p^{2}\big{>}&\cdots&\big{<}p^{K+1}\big{>}\\\
\vdots&\vdots&\ddots&\vdots\\\
\big{<}p^{K-1}\big{>}&\big{<}p^{K}\big{>}&\cdots&\big{<}p^{2K-1}\big{>}\\\
1&p&\cdots&p^{K}\end{array}\right|,$ (13)

with $\Delta_{K}=\mathrm{det}[\big{<}p^{i+j}\big{>}]_{i,j=0\ldots K}$.

The probability distribution vanishes for momenta away from the solutions
$\\{p_{0},\ldots,p_{K}\\}$ of Eq. (12), so that the canonical probability
distributions obey:

$\mathcal{P}_{\nu}(p)=\sum_{i=0}^{K}\pi_{i}\delta(p-p_{i}).$ (14)

Both the support $\\{p_{0},\ldots,p_{K}\\}$ of $\mathcal{P}(p)$ and the
amplitudes $\pi_{i}\equiv\pi(p_{i})$ depend on the scalar $\nu$, which
parametrizes the set of canonical distributions. The amplitudes $\pi_{i}$ can
be deduced from a bootstrap condition, similar to the one used to analyze
magnetoresistance data in Refs. [21, 20]. Indeed, a physical probability
distribution $\mathcal{P}_{\nu}(p)$ must satisfy the consistency condition
that $\mathcal{M}_{\sigma\tau}=\langle\hat{p}^{\sigma+\tau}\rangle$ is
positive semi-definite for the $K$ lowest moments
$\\{\langle\hat{p}^{0}\rangle,\langle\hat{p}^{1}\rangle,\ldots,\langle\hat{p}^{K}\rangle\\}$.
We want to assess if this distribution has the amplitude $\pi(p^{\star})$ at
$p=p^{\star}$, as described by the Dirac distribution
$\pi(p^{\star})\delta(p-p^{\star})$. For this purpose, we subtract
$\pi(p^{\star})\delta(p-p^{\star})$ to the target probability distribution
$\mathcal{P}(p)$, defining the new matrix of moments
$\mathcal{M}^{\prime}_{\sigma\tau}=\mathcal{M}_{\sigma\tau}-\pi(p^{\star})D_{\sigma\tau}$,
where $D_{\sigma\tau}\equiv(p^{\star})^{\sigma+\tau}$. If
$\mathcal{M}^{\prime}$ is positive semi-definite, then we underestimate the
amplitude of $\pi(p^{\star})$, and if $\mathcal{M}^{\prime}$ is negative, then
we overestimate the amplitude of $\pi(p^{\star})$. Thus, the smooth envelope
function $\pi(p^{\star})$ describes the probability distribution $\mathcal{P}$
when one eigenvalue of $\mathcal{M}^{\prime}$ cancels, i.e.:

$\displaystyle\left(\mathcal{M}-\pi(p^{\star})D\right){\bf w}=0.$ (15)

Since $D$ is a matrix of rank $1$, it has an unique non-zero eigenvalue
$a(p^{\star})=\sum_{i=0}^{K}(p^{\star})^{2i}$ with associated eigenvector
$v(p^{\star})=(1,p^{\star},(p^{\star})^{2},\ldots,(p^{\star})^{K})$, and we
can easily project Eq. (15) on $v(p^{\star})$ and find the unique solution
[21]

$\displaystyle\pi(p^{\star})=\frac{|{\bf
v}(p^{\star})|^{2}}{a(p^{\star})}\left(\sum_{i=1}^{K}\frac{({\bf
e}_{i}\cdot{\bf v}(p^{\star}))^{2}}{\lambda_{i}}\right)^{-1},$ (16)

where $(\lambda_{i},{\bf e}_{i})$ are eigenvalues and eigenvectors of
$\mathcal{M}$. Since $\mathcal{M}$ is positive semi-definite, the eigenvalues
$(\lambda_{i})_{i=1..K}$ are also positive semi-definite. In practice, some
eigenvalue are very close to zero and appear negative due to floating point
errors. This can lead to numerical instability, that we avoid by evaluating
(16) with an absolute value on $\lambda_{i}$ [21].

Figure 2:  Reconstruction of the Bloch momentum for $v=1$ in the quantum
bootstrap with $K=5$ [18]. (a) Envelope function $\pi(p)$ of the canonical
probability distribution in the fundamental state at $k=0$. The dots
correspond to the canonical solution for $\nu=10$ in Eq. (12). (b) Bloch band
spectrum of the periodic quantum Hamiltonian obtained as the average (in
black) of the canonical solutions at $\nu=0$ and $\nu=\infty$ and the
corresponding variance (in thin dashed lines). We compare our result to the
exact diagonalization, in orange.

In Fig. 2(a) we show the smooth envelope function $\pi(p^{\star})$ (16) and
indicate with dots the canonical distribution $\mathcal{P}_{\nu}$ for $\nu=10$
(12), for the fundamental state at $k=0$ and a bootstrap with $K=5$. We did
not consider larger values of $K$ because of the considerable increase in the
calculation time. We observe that peaks in the probability distribution are
approximately equally spaced. This is reminiscent of the probability
distributions associated to eigenstates of Eq. (1). Indeed, according to
Bloch’s theorem, a eigensolution with Bloch wavevector $k$ can be expanded in
the form $\Psi_{k}(x)=e^{ikx}\sum_{n\in\mathbb{Z}}C_{n}e^{inx}$ which
corresponds to the probability distribution $\mathcal{P}(p)=\int
dx~{}|\Psi_{k}(x)|^{2}=\sum_{n\in\mathbb{Z}}|C_{n}|^{2}\delta(n+k-p)$, with
poles at momenta $p_{n}=n+k$ with $n$ an integer. We observe that in the
infinite set of canonical solutions derived from Eq. (12), only the solutions
at $\nu=0$ and $\nu=\infty$ have poles matching the exact solution.

The probability distribution is not uniquely determined from the lowest
moments
$\\{\langle\hat{p}^{0}\rangle,\langle\hat{p}^{1}\rangle,...,\langle\hat{p}^{K}\rangle\\}$
and is in general a linear combination over all $\nu\in\mathbb{R}$ of the
canonical probability distributions $\mathcal{P}_{\nu}$ obtained from Eq.
(12). Since only $\nu=0$ and $\nu=\infty$ reproduce the correct quantization
condition, we restrict the solution to the corresponding branches and estimate
the Bloch momentum as $k=\frac{1}{2}(k_{\nu=0}+k_{\nu=\infty})$ with a
precision $\Delta k=\frac{1}{2}|k_{\nu=0}-k_{\nu=\infty}|$. We then obtain the
Bloch spectrum in Fig. 2(b) that associates a Bloch momentum to each energy
eigenstate, comparing well with the result of exact diagonalization (in
orange). The thick black line show the average Bloch momentum we obtain, while
the gray dashed lines help represent the variance $\Delta k$. This
reconstruction method does not rely on a particular symmetry but only on
bootstrap and could extend to other studies for computing observables that are
not easy to access with a recursion [10, 9]. The bootstrap consistency
condition is equivalent to the truncated Hamburger moment problem, where one
search the probability distributions that are consistent with a finite
sequence of moments. We use this connection in our reconstruction technique
[19] and further investigation of this connection could prove useful for
future work on the bootstrap.

##  V Conclusion

In this work, we have explored the bootstrap method for the quantum mechanical
problem of a single particle in a cosine potential. While this problem is
sufficiently simple for a direct numerical diagonalization, implementation of
the bootstrap required the development of several tools that can be useful in
more ambitious contexts. For instance, we demonstrate that the dimension of
the search space can be reduced by one, thus helping accelerate the
calculations. In addition, we propose a probabilistic method to compute
observables that are not easily obtained from bootstrap equations. We conclude
this article by discussing some perspectives of bootstrap ideas for other
situations. In quantum mechanics, periodic potentials that involve a finite
number of resonators could be studied by the same method, although the
recursion is more complex to set up. It does not seem feasible to extend our
study to spatial dimensions above one however, because of the many degrees of
freedom involved. It might also be worthwhile to examine whether a matrix
model extension of the cosine potential can be solved by bootstrap. Another
interesting avenue concerns whether bootstrap could apply to random tight
binding lattices, where similar ideas have been developed with a focus on the
local environment [22, 23].

Acknowledgements.- We thank M. Debertolis for discussions. S. T. acknowledge
financial support from the European Union Horizon 2020 research and innovation
program under grant agreement No829044 (SCHINES).

##  Appendix A Recursion for statistical moments involving conjugate variables

We derive first the double recursion Eq. (7) used to find the mixed moments of
the problem. Starting with $\langle\hat{H}e^{inx}p^{s}\rangle=E\langle
e^{inx}p^{s}\rangle$, we have:

$\displaystyle\langle\hat{p}^{2}e^{in\hat{x}}\hat{p}^{s}+v(e^{i\hat{x}}+e^{-i\hat{x}})e^{in\hat{x}}\hat{p}^{s}\rangle=E\langle
e^{in\hat{x}}\hat{p}^{s}\rangle.$ (17)

Commuting $\hat{p}$ through $e^{in\hat{x}}$ gives:

$\displaystyle n^{2}\langle e^{in\hat{x}}\hat{p}^{s}\rangle+2n\langle
e^{in\hat{x}}\hat{p}^{s+1}\rangle+\langle e^{in\hat{x}}\hat{p}^{s+2}\rangle$
(18)
$\displaystyle+v\langle(e^{i(n+1)\hat{x}}+e^{i(n-1)\hat{x}})\hat{p}^{s}\rangle=E\langle
e^{in\hat{x}}\hat{p}^{s}\rangle,$ (19)

which is the wanted result.

Finally, since the bootstrap is based on the mixed operator
$\hat{O}=\sum_{n=0}^{K}\sum_{s=0}^{L}a_{ns}\hat{p}^{s}e^{in\hat{x}}$, this
requires the computation of the following averages:

$\displaystyle\mathcal{M}_{tm,sn}=\langle
e^{-im\hat{x}}\hat{p}^{t+s}e^{in\hat{x}}\rangle,$ (20)

that are not explicitly in the form $\langle e^{in\hat{x}}\hat{p}^{s}\rangle$
obtained from the recursion Eq. (7). Due to this complication, we have
computed the matrix $\mathcal{M}$ for $0\leq s+t\leq 1$ only. For $s+t=0$, one
has obviously $\langle e^{-im\hat{x}}e^{in\hat{x}}\rangle=\langle
e^{i(n-m)\hat{x}}\rangle$. For $s+t=1$, we get:

$\displaystyle\langle e^{-im\hat{x}}\hat{p}e^{in\hat{x}}\rangle$
$\displaystyle=n\langle e^{i(n-m)\hat{x}}\rangle+\langle
e^{i(n-m)\hat{x}}\hat{p}\rangle$ (21) $\displaystyle=\frac{n+m}{2}\langle
e^{i(n-m)\hat{x}}\rangle,$ (22)

using Eq. (3) and using the fact that $\langle\hat{p}\rangle=0$. Finally, for
$s+t=2$, we obtain by the same method:

$\displaystyle\langle e^{-im\hat{x}}\hat{p}^{2}e^{in\hat{x}}\rangle$
$\displaystyle=$ $\displaystyle\left(nm+E\right)\langle
e^{i(n-m)\hat{x}}\rangle$ $\displaystyle-v\left(\langle
e^{i(n-m+1)\hat{x}}\rangle+\langle e^{i(n-m-1)\hat{x}}\rangle\right).$

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