Bose–Hubbard model

In brief. The Bose–Hubbard model describes bosons hopping on a lattice with an on-site repulsion. At unit filling on the square lattice it has a quantum phase transition between a superfluid and a Mott insulator at $U_c/J=16.74(1)$. Ketboard compares tori of $L\times L$ sites at unit filling, energy per site in units of $J$, with no cutoff on the occupation of a site. The ground state has no sign problem, so Green-function Monte Carlo gives the exact energy up to a controlled bias where it has been computed. The tables on this page are generated from Ketboard and from those references.

Notation follows the conventions page. Reviewed entries live in Ketboard, not here.

Hamiltonian and instances

For bosons with annihilation operators $b_i$ and occupations $n_i=b_i^\dagger b_i$,

where $\langle i,j\rangle$ runs over nearest-neighbour bonds, each counted once.

• Energy. Energy per site $E/L^2$ in units of $J$. Without interactions the ground-state energy per site is $-4J$ at unit filling on the square lattice.
• Sector. The number of bosons $N$ is fixed: canonical, with no chemical-potential term. Unit filling means $N=L^2$. Grand-canonical energies at fixed $\mu$ contain $-\mu N$ and are another quantity.
• Boundary conditions. Periodic in both directions.
• Local Hilbert space. The occupation of a site is not truncated. A cutoff $n_{\max}$ changes the Hamiltonian, so a calculation with one belongs to an instance that states it: the submission form puts $n_{\max}$ in the key. Hard-core bosons ($n_{\max}=1$) are a different model.

Conventions to check. Capello, Becca, Fabrizio and Sorella write the hopping as $-\tfrac t2\sum_{\langle i,j\rangle}(b_i^\dagger b_j+\mathrm{h.c.})$ and place the two-dimensional transition at $U_c/t\sim8.5$ (arXiv:0801.0952, Eq. 1). Their $t$ is $2J$, so their $U/t$ is $U/2J$ and their energies in units of $t$ are half those in units of $J$. Several papers write the hopping as $-J\sum_{\langle i,j\rangle}b_ib_j^\dagger$. That is the convention above only if $\langle i,j\rangle$ runs over both orders of each pair, and the paper should say so.

Why the Bose–Hubbard model

A transition at the centre of the benchmark. At unit filling and large $U/J$ each site holds one boson, and the ground state is a Mott insulator with a gap. At small $U/J$ the bosons condense into a superfluid. The transition between the two belongs to the three-dimensional XY universality class. Its position, $J_c/U=0.05974(3)$, i.e. $U_c/J=16.74(1)$, comes from quantum Monte Carlo extrapolated to zero temperature (Capogrosso-Sansone et al., arXiv:0710.2703, as quoted by Denis and Carleo, arXiv:2404.07869). Near $U_c$ the correlations are long-ranged on any finite torus, and variational energies are hardest to converge there. In the V-score analysis of Denis and Carleo, the worst case is on the Mott side of the transition.

Unbounded occupations. Unlike spins or fermions, a site can hold any number of bosons. A neural state must represent configurations with multiply occupied sites without a cutoff, or the cutoff becomes part of the problem.

Exact references. For $J>0$ the off-diagonal matrix elements of $H$ in the occupation basis are non-positive. The ground state then has non-negative amplitudes, and projector and worldline quantum Monte Carlo methods have no sign problem. Green-function Monte Carlo (GFMC) energies are exact up to a controllable systematic bias (Denis and Carleo). Where they exist, a neural energy can be compared with the exact value, not only with other variational states.

Known energies

The tables are generated when the site is built, from the reviewed Ketboard entries and the reference energies stored with them; they change when an entry is added. Energies per site in units of $J$.

$8\times8$, against GFMC. The GFMC energies are by F. Becca, printed in Table 1 of Denis and Carleo.

2 instances are compared with an exact energy or a best estimate only, with no upper bound from another method.

At $U/J=17$ the neural energy is below the GFMC energy, by less than twice the GFMC error bar. A variational energy below an exact one measures the error bar or the residual bias of the GFMC estimate; it does not violate the variational principle.

Near the transition, $U/J=16.8$, from $8\times8$ to $20\times20$:

4 instances have neural entries but no upper bound from another method here.

Each depth of the residual backflow network that Denis and Carleo print is an entry. On the $20\times20$ torus the energy decreases with the depth $D$:

The lowest variational energy is a neural state: Neural backflow Jastrow (NBFJ), ResNet backflow, depth D = 8 (Denis and Carleo, arXiv:2404.07869v2).

$10\times10$, on both sides of the transition:

2 instances have neural entries but no upper bound from another method here.

Not recorded.

• Pei and Clark. Denis and Carleo compare their $10\times10$ energies with restricted-Boltzmann-machine variants of Pei and Clark: $-0.5721(3)$ (NQS-OH) at $U/J=16$ and $-0.4068(9)$ (NQS-B) at $U/J=20$, from data shared by S. R. Clark (arXiv:2404.07869, Table 1). Pei and Clark show their energies only in figures (arXiv:2402.15424), so these values are quoted, not entries.
• Jastrow baselines. The depth-zero rows of Table 2 of Denis and Carleo are a two-body Jastrow state without a neural network.
• Figure-only data. Energies, V-scores and condensate fractions shown only in figures: Denis and Carleo, Figs. 2–5; Saito, arXiv:1707.09723; Saito and Kato, arXiv:1709.05468; Choo et al., arXiv:1807.03325; McBrian, Carleo and Khatami, arXiv:1903.03076; Vargas-Calderón, Vinck-Posada and González, arXiv:2004.12556.
• Other energies. Grand-canonical energies, and energies computed by summing over the whole Hilbert space of a few sites without sampling.

Candidates not yet on Ketboard

• One-dimensional chains at unit filling. Exact diagonalization and DMRG references are available, but the neural energies so far are shown in figures.
• Bosons in a gauge field. The Hofstadter–Bose–Hubbard model with soft-core or hard-core bosons (Döschl et al., arXiv:2405.04472; Ledinauskas and Anisimovas, arXiv:2405.01981) has neural energies in figures, compared with exact diagonalization and DMRG.
• Z₂ lattice gauge theory with hard-core bosons (Kebrič et al., arXiv:2602.13192). The Supplementary Material tabulates DMRG and neural energies on a $5\times4$ cylinder, and two neural architectures on a $6\times6$ torus. These are hard-core bosons coupled to gauge fields, a model of its own.

Reporting a result for these instances

1. State the Hamiltonian with its hopping convention, the lattice $L\times L$ with periodic boundaries, and $N$.
2. State whether the occupation per site is truncated, and at which $n_{\max}$; a truncated calculation belongs to the instance with that cutoff.
3. Give the energy per site in units of $J$ with its error bar, from a dedicated evaluation of the final state, and the number of samples; the binning analysis applies.
4. Report the variance or the V-score if available (Wu et al., arXiv:2302.04919).
5. Give the printed digits exactly as in the paper, with the arXiv version, table and row, through the Ketboard form. "Add an instance like this one" on an instance page starts from its lattice, $U/J$ and $N$.

References

• B. Capogrosso-Sansone, Ş. G. Söyler, N. Prokof'ev, B. Svistunov, Monte Carlo study of the two-dimensional Bose-Hubbard model, arXiv:0710.2703 (2007). The critical point at unit filling.
• M. Capello, F. Becca, M. Fabrizio, S. Sorella, Mott transition in bosonic systems: insights from the variational approach, arXiv:0801.0952 (2008). Jastrow wave functions; the hopping written as $-\tfrac t2\sum$.
• H. Saito, Solving the Bose–Hubbard model with machine learning, arXiv:1707.09723 (2017).
• H. Saito, M. Kato, Machine learning technique to find quantum many-body ground states of bosons on a lattice, arXiv:1709.05468 (2017).
• K. Choo, G. Carleo, N. Regnault, T. Neupert, Symmetries and many-body excited states with neural-network quantum states, arXiv:1807.03325 (2018).
• K. McBrian, G. Carleo, E. Khatami, Ground state phase diagram of the one-dimensional Bose-Hubbard model from restricted Boltzmann machines, arXiv:1903.03076 (2019).
• V. Vargas-Calderón, H. Vinck-Posada, F. A. González, Phase diagram reconstruction of the Bose-Hubbard model with a restricted Boltzmann machine wavefunction, arXiv:2004.12556 (2020).
• D. Wu et al., Variational benchmarks for quantum many-body problems, arXiv:2302.04919 (2023). The V-score.
• M. Y. Pei, S. R. Clark, Specialising neural-network quantum states for the Bose Hubbard model, arXiv:2402.15424 (2024).
• Z. Denis, G. Carleo, Accurate neural quantum states for interacting lattice bosons, arXiv:2404.07869 (2024). Neural backflow Jastrow states up to $20\times20$, without truncation.
• E. Ledinauskas, E. Anisimovas, Universal performance gap of neural quantum states applied to the Hofstadter-Bose-Hubbard model, arXiv:2405.01981 (2024).
• F. Döschl, F. A. Palm, H. Lange, F. Grusdt, A. Bohrdt, Neural network quantum states for the interacting Hofstadter model with higher local occupations and long-range interactions, arXiv:2405.04472 (2024).
• M. Kebrič, F. Döschl, U. Borla, J. C. Halimeh, U. Schollwöck, A. Bohrdt, F. Grusdt, Matter-induced plaquette terms in a $\mathbb{Z}_2$ lattice gauge theory, arXiv:2602.13192 (2026).