Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo

TL;DR

Two-dimensional real-time quantum dynamics are challenging for classical methods. The authors develop PEPS–tVMC, a stable framework that removes PEPS gauge redundancies and employs single-layer contractions to precondition and solve the stochastic reconfiguration equations efficiently. They demonstrate four representative dynamical problems—chiral edge transport, fractional charge pumping, vison confinement, and superfluid flow—on sizable 2D lattices with long evolution times, achieving high fidelity against benchmarks where available. This work expands the practical reach of classical tensor-network simulations for low-energy 2D dynamics and provides a robust computational tool to benchmark and guide quantum simulators.

Abstract

Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable and efficient time-dependent variational Monte Carlo (tVMC) framework for PEPS. By analytically removing all gauge redundancies of the PEPS manifold and exploiting tensor locality, we obtain a numerically well-conditioned stochastic reconfiguration (SR) equation amenable to robust solution using the efficient Cholesky decomposition, enabling long-time evolution in previously inaccessible regimes. We demonstrate the power and generality of the method through four representative real-time problems in two dimensions: (I) chiral edge propagation in a free-fermion Chern insulator; (II) fractionalized charge transport in a fractional Chern insulator; (III) vison confinement dynamics in the Higgs phase of a Z2 lattice gauge theory; and (IV) superfluidity and critical velocity in interacting bosons. All simulations are performed on 12x12 or 13x13 lattices with evolution times T = 10 to 12 using modest computational resources (1 to 5 days on a single GPU card). Where exact benchmarks exist (case I), PEPS-tVMC matches free-fermion dynamics with high accuracy up to T = 12. These results establish PEPS-tVMC as a practical and versatile tool for real-time quantum dynamics in two dimensions. The method extends the reach of classical tensor-network simulations for studying elementary excitations in quantum many-body systems and provides a valuable computational counterpart to emerging quantum simulators.

Figures (7)

• Figure 1: PEPS structure and gauge freedom. (a) A $3\times3$ PEPS network with local tensors $A$. (b) A single tensor $A$ with physical index $p$ and virtual indices $(l,r,d,u)$. (c) Local gauge transformation on a virtual bond: inserting an invertible matrix $X$ and its inverse $X^{-1}$ between two adjacent tensors leaves the global PEPS wavefunction invariant. (d) Wavefunction amplitude $\braket{\bm{\mathbf{s}}|\Psi}$, computed via the contraction of the single layer network of sliced PEPS tensors: $a[\bm{\mathbf{x}}]_{lrdu} = A[\bm{\mathbf{x}}]^{p=\bm{\mathbf{s}}(\bm{\mathbf{x}})}_{lrdu}$.
• Figure 2: Chiral edge dynamics in a $12\times12$ Chern insulator. (a) Exact real-time evolution obtained from free-fermion numerics. The color scale shows the density deviation $\braket{n_i(t)} - \braket{n_i}_\text{gs}$. A density perturbation created near the boundary at $T=0$ propagates unidirectionally along the edge at later times $T=4, 8, 12$. (b) Same protocol computed using PEPS--tVMC. (c) Time evolution of the density at four selected sites red-marked in (a), chosen so that the chiral pulse passes them at $T=0, 4, 8, 12$. The error bars are of order $7\times10^{-4}$. All panels share the same vertical scale of 0.2.
• Figure 3: Charge pumping in the bosonic Hofstadter model at flux $\phi=\pi/2$. (a) Ground-state charge distribution on a $12\times12$ lattice with total particle number 15. A localized external flux is threaded through the central plaquette (marked by $\otimes$). The red horizontal bonds indicate the Peierls-phase path along which the time-dependent vector potential $A_{ij}(t)$ is ramped. The blue circle denotes the region over which the enclosed charge $Q_\text{in}$ is monitored. (b) Total charge within the blue circle during one flux-insertion cycle of duration $T_\text{flux} = 10$. (c) Evolution of real-space density, subtracted by $1/8$, during the pump.
• Figure 4: Real-time vison dynamics in the $\mathbb{Z}_2$ lattice gauge theory on a $12\times12$ lattice. Shown are snapshots of the plaquette operator $B_p(t)$, which highlights magnetic-flux (vison) excitations created at $T=0$. (a) Deconfined phase ($J=0.1, g=0.1$). (b) Higgs phase ($J=0.5, g=0.1$).
• Figure 5: (a) Real-time density response of a 2D bosonic superfluid to a moving obstacle (red dot) with velocity $\abs{\bm{\mathbf{v}}} = 0.5$. Snapshots show the deviation of the local density from 0.5 on a $13\times 13$ lattice for different times $T$ as the obstacle is dragged along the $x$ direction. (b) Same as (a), but with $|\bm{\mathbf{v}}|=5$. (c) Total energy $\braket{H_\text{sf} + H_\text{obs}(t)}$ as a function of time for both velocities. To assess the effect of the time step on the energy evolution, results obtained with $dt=0.01$ and $0.02$ are shown. For $|\mathbf v|=0.5$, the two time steps agree to within a relative energy difference of $2\times10^{-4}$ over the entire evolution.