Origin of the $T=2$ superfluid-boundary shift in the tensor-network annealing algorithm for two-dimensional thermal states

The tensor-network annealing algorithm of Kshetrimayum, Rizzi, Eisert and Orús (Phys. Rev. Lett. 122, 070502 (2019)) computes finite-temperature properties of two-dimensional quantum lattice systems in the thermodynamic limit by cooling a vectorized projected entangled-pair operator (PEPO) in small imaginary-time steps, truncating after each step with the simple update, and contracting observables with corner-transfer-matrix techniques. Its exactly solved 2D Ising benchmark shows the relative error growing near the critical point, and its Bose-Hubbard phase diagrams show truncation-sensitive boundary features — most explicitly a shift of the right-hand-side superfluid region at $T=2$ in the three-level (at most two bosons per site) Bose-Hubbard model at $U/J=100$, whose origin the authors explicitly deferred to future work. The open problem is to determine whether such phase-boundary shifts are genuine thermal physics or artifacts of the algorithmic truncation choices — most directly the local occupation cutoff to which the authors' footnote attaches, and potentially also the bond dimension $D$, the Suzuki-Trotter step $\Delta\beta$, or the CTM environment dimension $\chi$.

Background

Simulating quantum lattice systems at finite temperature in two spatial dimensions is substantially harder than ground-state work: the thermal (Gibbs) state $\rho=e^{-\beta H}$ at inverse temperature $\beta$ is a mixed state on an infinite lattice, and the correlations it carries grow precisely in the regimes of interest, near continuous phase transitions. Projected entangled-pair states (PEPS) approximate pure two-dimensional ground states by a network of site tensors joined by bonds of dimension $D$ (the bond dimension, which controls how much correlation the ansatz can hold); their operator analogue for mixed states is the projected entangled-pair operator (PEPO), the two-dimensional generalization of a matrix product operator (MPO). One standard route to thermal states is purification — evolving an enlarged pure state in imaginary time — as in the variational finite-temperature PEPS of Czarnik and Dziarmaga.

Kshetrimayum, Rizzi, Eisert and Orús introduced an alternative 'annealing' algorithm. Writing $\rho=e^{-\beta H/2}\,\mathbb{I}\,e^{-\beta H/2}$, they vectorize the mixed state into a pure state on a doubled Hilbert space (a Choi isomorphism previously used by the same school for two-dimensional steady states of open systems), start from the infinite-temperature state, and cool in small imaginary-time steps $\Delta\beta$ applied via a Suzuki-Trotter decomposition (a product-formula splitting of the exponential into nearest-neighbor gates) of $e^{-\Delta\beta H}$. After each step the network is truncated back to a PEPO of fixed bond dimension $D$ using the simple update — the truncation scheme of Jiang, Weng and Xiang, which approximates the environment of each bond by a mean-field-like weight and truncates by a singular-value decomposition — and observables are computed by corner-transfer-matrix (CTM) contraction with environment dimension $\chi$. The method is fast (update cost $O(d^4D^5+d^{12}D^3)$ for local dimension $d$) and works directly in the thermodynamic limit.

Two applications frame its accuracy. First, as a benchmark, the ferromagnetic Ising model on the square lattice — exactly solved by Onsager — has a thermal operator that factorizes over links into commuting two-site MPOs, i.e. an exact PEPO of bond dimension 2; the algorithm reproduces Onsager's magnetization curve well, but its relative error grows near the critical point, where CTM contraction errors are known to increase. Second, applied to the Bose-Hubbard model, $H=-J\sum_{\langle i,j\rangle}(a_i^\dagger a_j+\text{h.c.})-\mu\sum_i \hat n_i+\frac{U}{2}\sum_i \hat n_i(\hat n_i-1)$, which has Mott-insulating plateaus at integer density versus chemical potential $\mu$ and superfluid regions in between (order parameter $\rho_0=|\langle a_i\rangle|^2$), the algorithm computes phase diagrams for the hard-core model and for a three-level truncation (at most two bosons per site) at $U/J=100$, using $D=2$ and $D=3$ and $\Delta\beta=10^{-4}$. Quantum Monte Carlo can benchmark this bosonic model (e.g. Capogrosso-Sansone et al.) but suffers sign problems in frustrated and fermionic regimes, which is part of the motivation for tensor-network alternatives.

The paper leaves the accuracy of these phase diagrams uncontrolled and flags one feature explicitly: at $T=2$ the right-hand-side superfluid region (between the $n=1$ and $n=2$ Mott plateaus) of the three-level model appears shifted, and the authors state that 'the precise origin of this and whether this can be derived from the choice of truncation will be explored in detail in future work'. Relatedly, in the hard-core phase diagram the superfluid disappears at $T\approx1$ in $D=2$ simulations but at $T\approx2$ in $D=3$ ones: phase-boundary features of this algorithm are visibly sensitive to truncation choices, and no extrapolation or error quantification was provided. Later work has extended the algorithm (to frustrated magnets and quantum materials, to three dimensions, and to stochastic-reconfiguration updates) and alternative 2D thermal tensor-network families have matured, but none has established the controlled-error methodology or diagnosed the shift.

Problem Statement

Using the tensor-network annealing algorithm of Kshetrimayum, Rizzi, Eisert and Orús — imaginary-time cooling of the vectorized thermal operator in steps $\Delta\beta$, each Suzuki-Trotter gate application truncated to a PEPO of bond dimension $D$ by the simple update, with observables contracted by CTM techniques with environment dimension $\chi$ — the computed finite-temperature phase diagram of the three-level Bose-Hubbard model on the infinite square lattice (at most two bosons per site, $U/J=100$) shows a shift of the right-hand-side superfluid region at $T=2$, i.e. of the superfluid strip between the $n=1$ and $n=2$ Mott plateaus in the chemical potential $\mu$. The source paper explicitly defers 'the precise origin of this and whether this can be derived from the choice of truncation' to future work.

Determine, with controlled numerical evidence, whether this shift is a genuine finite-temperature effect or an artifact of the algorithmic truncation choices — most directly the local occupation cutoff to which the source footnote attaches, and, insofar as they bear on the diagnosis, the bond dimension $D$ of the simple update, the Suzuki-Trotter step $\Delta\beta$, and the CTM environment dimension $\chi$. Concretely, an answer should:

(a) locate the superfluid boundary $\mu^*(T)$ — through the order parameter $\rho_0=|\langle a_i\rangle|^2$ and the density $n(\mu)=\langle\hat n_i\rangle$ (particles per site), and preferably the superfluid stiffness (the phase rigidity, i.e. the free-energy cost of imposing a twist in the hopping phase), which the published algorithm does not compute — at fixed model parameters including $T=2$, and examine how it behaves under the truncation choices relevant to the diagnosis, most directly the local occupation cutoff and, where needed, the bond dimension $D$, the Suzuki-Trotter step $\Delta\beta$ and the CTM environment dimension $\chi$;

(b) decide from this convergence behavior, and/or from comparison with an independent controlled reference (sign-problem-free quantum Monte Carlo is available for this bosonic model; alternatively an update scheme with a proper environment, such as full, neighborhood or stochastic-reconfiguration updates), whether the shift is physical or algorithmic (the analogous truncation-sensitive features of the same algorithm — such as the hard-core superfluid disappearing at $T\approx1$ for $D=2$ but $T\approx2$ for $D=3$ — may be cited as corroboration where relevant, but the verdict required here concerns the $T=2$ shift).

References

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