[challenge]: Certified bulk spectral-gap bounds for truncated Bose-Hubbard models on hyperbolic lattices

Released by

许湘灵（Xiangling Xu）, Inria Saclay; 王杰（Jie Wang）, AMSS-CAS

Contact email

xu.xiangling@inria.fr; wangjie212@amss.ac.cn

Method

Semidefinite programming / Noncommutative polynomial optimization

Background

Spectral gaps determine low-energy excitations and are closely related to stability and quantum phases, but rigorous thermodynamic-limit gap bounds are difficult to obtain. Exact diagonalization, DMRG, and tensor-network methods can give estimates, but they usually rely on finite geometries, extrapolations, or variational ansätze, and thus not certified values for the states in the thermodynamic-limit.

The recent work The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds introduced a hierarchy of semidefinite programs (SDPs) that produces certified upper bounds on the thermodynamic-limit bulk spectral gap $\Delta_{\mathrm{bulk}}$. The accompanying spin-$1/2$ code is available at wangjie212/SpectralGap. The hierarchy is based the state polynomial optimization (see Ref. 2), a novel technique that allows SDPs to deal with nonlinear constraints on the states.

At any relaxation level $(L,d)$, if the SDP is infeasible at $\gamma \geq 0$, then no ground state has bulk spectral gap at least $\gamma$. (The case $\gamma=0$ coincides with the set of ground states.) Consequently, at relaxation level $(L,d)$, maximizing the gap parameter $\gamma$ for which the SDP relaxation is feasible gives a monotonic convergent sequence

$$
\Gamma_{L,d} \searrow \Delta_{\mathrm{bulk}}.
$$

At a fixed assumed gap $\gamma$, the same SDP can give certified bounds

$$
O_{\min}(L,d;\gamma) \leq \omega(O) \leq O_{\max}(L,d;\gamma)
$$

for a chosen observable $O$ that are also monotonically convergent. That is, any ground state $\omega$ with bulk spectral gap of at least $\gamma$ must have its expectation value on $O$ bounded between $O_{\min}(L,d;\gamma)$ and $O_{\max}(L,d;\gamma)$. One can further incorporate symmetries for certified statements on symmetric ground states, which reduces the size of moment matrices. See also Ref. 3 for more techniques to improve the numerical values in practice.

The goal of this challenge is to extend to occupation-truncated bosonic algebras and apply it to Bose-Hubbard models on the three geometries shown in Figs. 2(a), 3(a), and 4(a) of A scalable superconducting circuit framework for emulating physics in hyperbolic space, which connects to Quantum phase transitions of interacting bosons on hyperbolic lattices.

Lattice geometries

A regular tiling with Schläfli symbol {p,q} is made of $p$-sided faces, with $q$ faces meeting at every vertex. The three graphs are:

1. the regular {8,3} lattice from Fig. 2(a): each face is an octagon and every site has three nearest neighbors;
2. the regular {12,4} lattice from Fig. 3(a): each face is a dodecagon and every site has four nearest neighbors;
3. the kagome-like graph from Fig. 4(a), constructed from the {8,3} lattice by the line-graph construction.

For the third graph, let $G_{8,3}$ denote the graph of the regular {8,3} tiling. Its line graph $L(G_{8,3})$ is defined as follows: every edge of $G_{8,3}$ becomes a site of $L(G_{8,3})$, and two new sites are connected when the corresponding parent edges share a parent vertex. Every site of $L(G_{8,3})$ has four nearest neighbors. This graph is called kagome-like because the three parent edges meeting at a parent vertex become a triangle in the line graph.

The number of nearest neighbors of a site will be denoted by $z$. Thus $z=3$ for {8,3}, and $z=4$ for {12,4} and the kagome-like line graph.

Bosonic algebras with truncated dimensions

A bosonic site has creation and annihilation operators $b_i^\dagger$ and $b_i$ acting on an infinite-dimensional local Fock space such that

$$
[b_i^\dagger, b_j^\dagger] = 0, \quad [b_i, b_j] = 0, \quad [b_i, b_j^\dagger] = \delta_{ij},
$$

and number operator $n_i=b_i^\dagger b_i$.
These operators are unbounded. The framework of Ref. 1 assumes finite-dimensional local matrix algebras, and therefore its certification and convergence theorem does not directly apply to the untruncated canonical commutation relation model. In this challenge, certified statements will only be claimed for finite occupation cutoffs.

Fix a cutoff $n_{\max}$ and restrict the allowed local occupations to

$$0,1,\ldots,n_{\max}.$$

The local Hilbert space is then

$$\mathcal H_i^{(n_{\max})}=\mathrm{span}\{|0\rangle,\ldots,|n_{\max}\rangle\},$$

with dimension $n_{\max}+1$. The truncated operators are defined by

$$b_i|r\rangle=\sqrt r|r-1\rangle,$$

$$b_i^\dagger|r\rangle=\sqrt{r+1}|r+1\rangle \text{ for } r<n_{\max}$$

and

$$b_i^\dagger|n_{\max}\rangle=0.$$

The number operator satisfies

$$n_i|r\rangle=r|r\rangle.$$

For every fixed cutoff $n_{\max}$, the local algebra is $M_{N+1}(\mathbb C)$, so the certification and convergence results of Ref. 1 apply to the corresponding cutoff Hamiltonian, the occupation-truncated Bose-Hubbard models.

Note that certificate obtained at cutoff $n_{\max}$ is a certificate for that cutoff Hamiltonian, but does not by itself give a certified result for the untruncated Bose-Hubbard model. In practice, one can impose the constraint

$$
\omega\left((b_i^\dagger)^{n_{\max}+1}b_i^{n_{\max}+1}\right)=0
$$

for every site $i$, which representing that the state $\omega$ is supported on the Foch space with particle number less than $n_{\max}$ on each $i$.

Bose-Hubbard model and occupation cutoffs

On each graph, study the Bose-Hubbard Hamiltonian

$$H=-t\sum_{\langle i,j\rangle}(b_i^\dagger b_j+b_j^\dagger b_i)+\frac{U}{2}\sum_i n_i(n_i-1)-\mu\sum_i n_i.$$

The first term allows bosons to move along graph edges. The parameter $U>0$ penalizes multiple occupation of one site, and the chemical potential $\mu$ controls the preferred particle density. Set $U=1$ when reporting numerical values.

As discussed before, we consider the truncated variant of the model at least at the cutoff

$$n_{\max}=1, 2,$$

and $n_{\max}=3$ whenever computationally feasible.

The Hamiltonian conserves total particle number and has a $U(1)$ symmetry. If this symmetry is imposed in the SDP, the moment matrix can be decomposed into particle-number charge sectors. Any result using this reduction should be identified as a statement about the $U(1)$-invariant ground-state sector.

When $t$ is small and the average occupation is close to an integer, the on-site repulsion can localize approximately one boson at each site. This is called a unit-filling Mott state. It has density near one, small local number fluctuations, weak hopping coherence, and a positive energy cost for locally adding or removing a particle. Increasing $t$ favors particle delocalization and superfluid behavior.

Target 1: exact atomic-limit check

As a sanity check, study

$$t=0,\ U=1,\ \mu=0.5.$$

When $t=0$, the Hamiltonian is a sum of independent one-site Hamiltonians. The local ground state is $|1\rangle$, so the full ground state is the tensor product with one boson at every site. Removing the boson from one site costs $0.5$, and adding a second boson also costs $0.5$. Therefore, for every cutoff $n_{\max}\geq2$,

$$\Delta_{\mathrm{bulk}}=0.5.$$

The exact local observables are

$$\omega(n_i)=1,$$

$$\omega((n_i-1)^2)=0,$$

and, for different sites $i,j$,

$$\omega(b_i^\dagger b_j+b_j^\dagger b_i)=0.$$

Calculate certified gap upper bounds $\Gamma_{L,d}$ for several accessible $L,d$ for the chosen cutoffs and check the expectation values of the aforementioned local observables. Note that as $t=0$, the lattice configuration should not affect the results.

Target 2: parameter and geometry comparison near the unit-filling Mott boundary

Choose a distinguished reference site $0$, consider the local density

$$\rho_0=\omega(n_0).$$

Consider also the local unit-filling deviation

$$F_0=\omega((n_0-1)^2).$$

For a general cutoff at $n_{\max}$, $F_0$ encodes the local particle-number variance and the deviation of the mean density from $1$ particle per site. A small $F_0$ suggests that the local occupation is concentrated near $1$.

Also consider the average nearest-neighbor hopping correlation

$$K_0=\frac{1}{z}\sum_{\langle j,0 \rangle}\omega(b_0^\dagger b_j+b_j^\dagger b_0),$$

where $\langle j,0 \rangle$ means that $j$ is a nearest neighbor of site $0$, and $z$ is the number of $0$ nearest-neighbor. The hopping contribution from the edges incident on site $0$ is $-tzK_0$, so a larger positive value suggests stronger local hopping coherence and delocalization.

The observables have the following qualitative interpretation:

• Mott-like behavior: $\rho_0$ close to one, small $F_0$, and small $K_0$;
• stronger local delocalization: larger $F_0$ and larger positive $K_0$.

Next, we choose our parameters motivated by the mean-field analysis and the phase diagrams in Figs. 3 and 4 of Ref. 5., extrapolated to our lattice configurations.

First fix

$$U=1,\qquad \mu/U=0.50,$$

and study

$$t/U=0.03,\ 0.05,\ 0.06.$$

Single-site mean-field theory predicts the following qualitative behavior:

• at $t/U=0.03$, all three graphs should be on the unit-filling Mott-like side;
• at $t/U=0.05$, the degree-$3$ graph {8,3} should remain more Mott-like, while the two degree-$4$ graphs should show stronger delocalization;
• at $t/U=0.06$, all three graphs should be outside the unit-filling Mott region.

Next fix

$$U=1,\qquad t/U=0.03,$$

and study

$$\mu/U=0.15,\ 0.50,\ 0.75.$$

Mean-field theory predicts that the configuration {8.3} has a wider unit-filling Mott interval than the other two configurations with graph degree $4$. The middle point $0.50$ is predicted to be Mott-like for all three configurations. At the other two points, {8.3} is predicted to be more Mott-like while the other two cases are predicted to show stronger delocalization.

For every graph, parameter point, and occupation cutoff, calculate:

• certified gap upper bounds $\Gamma_{L,d}$ for several accessible $L,d$;
• certified maximum and minimum values of $\rho_0$, $F_0$, and $K_0$ at

$$\gamma/U=0,0.05,0.10,$$

or other values suggested by the calculated $\Gamma_{L,d}$.

Then interpret the calculated results, finding the dependence of Mott-like phases with the parameter $t$ and $\mu$, the hyperbolic lattice configurations, and the occupation cutoff $n_{\max}$ if there is any.

Interpret how the certified gap and local Mott diagnostics depend on $t$, $\mu$, the lattice geometry, and the occupation cutoff.

Results to report

For every calculation, report the graph, model parameters, occupation cutoff, imposed symmetries (if any), $L$ and $d$, moment-matrix size, solver status, runtime, certified gap upper bound, and certified observable optima.

The discussion should state precisely:

• what gap values are excluded by the certificate;
• whether the result is unrestricted or symmetry-restricted;
• how results depend on the parameters, the lattice configurations, and the cutoff;
• how the gap and observable bounds change as $L$ and $d$ increase.

References

1. X. Xu et al., The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds.
2. I. Klep et al., State polynomials: positivity, optimization and nonlinear Bell inequalities
3. J. Wang et al., Scalable ground-state certification of quantum spin systems via structured noncommutative polynomial optimization
4. X. Xu et al., A scalable superconducting circuit framework for emulating physics in hyperbolic space.
5. X. Zhu et al., Quantum phase transitions of interacting bosons on hyperbolic lattices.
6. D. van Oosten, P. van der Straten, and H. T. C. Stoof, Quantum phases in an optical lattice.

No repository files, tests, or entry points are named. Start by reviewing the existing implementation alongside the cited SpectralGap work and the issue's references, then identify how truncated bosonic algebras, the three graph geometries, and SDP certification fit the project. Done means reproducible reports covering the requested parameters, cutoffs, symmetries, solver status, bounds, and convergence checks.