[challenge]: 2D finite-temperature tensor networks

Released by

Wei Li, Institute of Theoretical Physics, Chinese Academy of Sciences

Contact email

w.li@itp.ac.cn

Method

PEPS Based Algorithm

Challenge issue

Background

Finite-temperature properties of strongly correlated systems — from the pseudogap of high-$T_c$ superconductors to the thermodynamic response of quantum magnets — hinge on an accurate treatment of thermal and quantum fluctuations together. Two-dimensional systems at finite temperature face a double curse: exponential Hilbert-space growth, plus thermal-state entanglement that grows rapidly at low temperature. Three technical routes exist today:

1. PEPO / purification — represent the Gibbs operator $e^{-\beta H}$ as a 2D tensor network via imaginary-time evolution or variational optimization (Czarnik et al.'s variational renormalization reached QMC-grade benchmarks on the Hubbard model).
2. PEPS — sample an ensemble of projected entangled-pair states, like METTS; extremely successful in 1D (MPS), but its 2D PEPS extension still struggles with sampling efficiency and entanglement control.
3. tanTRG — the current 2D finite-temperature benchmark: optimal imaginary-time evolution in the tangent space of the tensor-network manifold, with mild $O(D^3)$ complexity, demonstrated on the $10\times10$ Hubbard model.

The gap: 2D PEPO and METTS implementations remain immature — particularly in accuracy and efficiency at low temperature ($\beta J \gtrsim 1$) — and have not reached tanTRG's level.

Research objective

Extend the PEPO or METTS method to 2D and validate it on a finite-size benchmark where numerically exact reference data exists:

1. Implement a PEPO- or METTS-based algorithm for the transverse-field quantum Ising model on a $10\times10$ square lattice with open boundary conditions,
   $$H = -J \sum_{\langle i,j\rangle} \sigma_i^z \sigma_j^z - h \sum_{i} \sigma_i^x, \qquad J = 1,$$
   with the field chosen near the quantum critical point $h_c/J \approx 3.044$ — e.g. $h/J \in {2.5, 3.0, 3.5}$ — where the small gap and diverging correlation length make thermal tensor-network compression genuinely hard.
2. Compute the thermodynamics over $\beta J \in [0.1, 1.0]$ (the quantum critical fan): free energy density $f = -\ln Z/(\beta N)$, internal energy density $u = \langle H\rangle/N$, and specific heat $C = \beta^2(\langle H^2\rangle - \langle H\rangle^2)/N$. Bonus: uniform susceptibility $\chi = \frac{\beta}{N}\sum_{i,j}(\langle\sigma_i^z\sigma_j^z\rangle - \langle\sigma_i^z\rangle\langle\sigma_j^z\rangle)$.
3. Bonus — beat-or-match tanTRG: compare against tanTRG (or MPO-based LTRG) at the same $h$, on equal footing of achieved accuracy vs. computational cost (wall-clock time, memory, scaling in each method's own bond dimension) — MPO and PEPO bond dimensions are not directly comparable.

Verification plan

The transverse-field Ising model is sign-problem free, so QMC (SSE or worm) provides numerically exact finite-size ground truth at negligible cost; that makes every claim in this challenge checkable.

• QMC validation (mandatory): all reported quantities are validated against QMC reference data on the same $10\times10$ lattice; report relative errors of $u$ and $C$ over $\beta J \in [0.1, 1.0]$, plus the lowest temperature (largest $\beta J$) at which the method stays stable and the accuracy maintained there. Exact diagonalization on small lattices (e.g. $4\times4$) is recommended as a development-time sanity check.
• Convergence (mandatory). PEPO route: convergence of $u$ and $C$ in bond dimension $D$ for at least three values (e.g. $D \in {4, 6, 8}$ or higher), with convergence plots at representative temperatures ($\beta J = 0.1$ and $0.5$). METTS route: convergence in sample count with a statistical error analysis (binning or SEM); report the samples needed for relative statistical error below 1% on $u$ and 3% on $C$ at $\beta J = 0.8$.
• Reproducibility (mandatory): open source code (Python, Julia, or C++; ITensor / TensorKit / Quimb encouraged), technical documentation of the algorithm, contraction strategy, and imaginary-time scheme (Trotter or variational), and a test script that reproduces the benchmark results in one command.

Deliverables checklist

#	Deliverable	Mandatory
1	Thermodynamic curves $f(T)$, $u(T)$, $C(T)$ over $\beta J \in [0.1, 1.0]$	Yes
2	Convergence analysis (in $D$ or in sample count) with plots	Yes
3	Validation against QMC reference data (accuracy + low-$T$ reach)	Yes
4	Source code + technical document + one-command test script	Yes
5	tanTRG comparison: accuracy, timing, memory	Bonus
6	Uniform susceptibility $\chi(T)$	Bonus

Hints

• Exploit the $\mathbb{Z}_2$ spin-flip symmetry ($\sigma_i^z \to -\sigma_i^z$) in the PEPO/METTS construction to reduce the required bond dimension.
• For PEPO, prefer Trotter–Suzuki decomposition followed by variational optimization (rather than simple truncation) to compress the imaginary-time direction; for METTS, focus on controlling the 2D PEPS sampling variance.
• For the tanTRG baseline, start from the open-source ThermoTN codes (FiniteMPS.jl, FiniteLattices.jl), build on ITensor's tdvp/linsolve modules, or compare against the published data of Ref. [6].

Why this may lead to research output

2D finite temperature is a genuine bottleneck of tensor-network methods: tanTRG currently stands alone at the benchmark level, and a PEPO or METTS implementation that approaches (or beats) it in the quantum critical fan — with an honest, QMC-anchored accuracy-vs-cost comparison — would be a publishable methods paper and a reusable piece of community infrastructure for thermal simulations of frustrated magnets and correlated electrons, where QMC's sign problem removes the safety net this benchmark still provides.

References

1. P. Czarnik et al., Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature, Phys. Rev. B 94, 235142 (2016) — variational PEPO coarse-graining.
2. M. Zhang, H. Zhang, C. Wang, L. He, Scalable tensor network algorithm for thermal quantum many-body systems in two dimensions, Phys. Rev. B 111, 075146 (2025) — stochastic reconfiguration of vectorized PEPS thermal states.
3. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
4. E. M. Stoudenmire, S. R. White, Minimally entangled typical thermal state algorithms, New J. Phys. 12, 055026 (2010).
5. A. Wietek et al., Stripes, antiferromagnetism, and the pseudogap in the doped Hubbard model at finite temperature, Phys. Rev. X 11, 031007 (2021) — METTS and iPEPS purification on the 2D Hubbard model.
6. Q. Li et al., Tangent space approach for thermal tensor network simulations of the 2D Hubbard model, Phys. Rev. Lett. 130, 226502 (2023) — the tanTRG benchmark.
7. B.-B. Chen et al., Exponential thermal tensor network approach for quantum lattice models, Phys. Rev. X 8, 031082 (2018) — XTRG.
8. ThermoTN open-source thermal tensor network codes: https://github.com/ThermoTN (FiniteMPS.jl, FiniteLattices.jl, and utilities).

Start by reading the PEPO/METTS objective and the ThermoTN entry points FiniteMPS.jl and FiniteLattices.jl, along with the listed ITensor modules. Define the implementation and benchmark workflow around the 10×10 transverse-field Ising model, then verify that the thermodynamic curves, convergence analysis, QMC validation, source, documentation, and one-command test script are present.