Finite-Temperature Dynamical Phase Diagram of the $2+1$D Quantum Ising Model

TL;DR

The paperDevelops an efficient equilibrium quantum Monte Carlo framework to map finite-temperature dynamical phase diagrams of interacting quantum many-body systems by exploiting energy conservation after a quantum quench; long-time steady states are inferred from the conserved post-quench energy $E_q$ and the equilibrium phase diagram of the final Hamiltonian, without simulating unitary time evolution. Applied to the $2+1$D transverse-field Ising model, the method reveals dynamical phenomena such as cooling quenches and dynamical PM\rightarrow FM transitions, with finite-temperature dynamical phase boundaries mapped as functions of the initial temperature and quench field. The authors benchmark the QMC-based predictions against real-time exact diagonalization and tree tensor network simulations, finding good agreement on accessible system sizes, and discuss implications for quantum simulators and nonequilibrium universality in higher dimensions. This approach offers a scalable route to finite-temperature dynamical phase diagrams and can be extended to other lattice geometries and gauge theories, guiding experimental probes of dynamical scaling and relaxation in quantum many-body systems.

Abstract

Mapping finite-temperature dynamical phase diagrams of quantum many-body models is a necessary step towards establishing a framework of far-from-equilibrium quantum many-body universality. However, this is quite difficult due, in part, to the severe challenges in representing the volume-law entanglement that is generated under nonequilibrium dynamics at finite temperatures. Here, we address these challenges with an efficient equilibrium quantum Monte Carlo (QMC) framework for computing the finite-temperature dynamical phase diagram. Our method uses energy conservation and the self-thermalizing properties of ergodic quantum systems to determine observables at late times after a quantum quench. We use this technique to chart the dynamical phase diagram of the $2+1$D quantum Ising model generated by quenches of the transverse field in initial thermal states. Our approach allows us to track the evolution of dynamical phases as a function of both the initial temperature and transverse field. Surprisingly, we identify quenches in the ordered phase that cool the system as well as an interval of initial temperatures where it is possible to quench from the paramagnetic (PM) to ferromagnetic (FM) phases. Our method gives access to dynamical properties without explicitly simulating unitary time evolution, and is immediately applicable to other lattice geometries and interacting many-body systems. Finally, we propose a quantum simulation experiment on state-of-the-art digital quantum hardware to directly probe the predicted dynamical phases and their real-time formation.

Figures (7)

• Figure 1: Illustration of our method using the $2+1$D quantum Ising model on a square lattice. The system is prepared in a thermal ensemble $\hat{\rho}(h_\text{i},T_\text{i})$ at transverse-field strength $h_\text{i}$ and temperature $T_\text{i}$, which is then quenched by instantly changing the transverse-field value to $h_\text{f}$. Using QMC sampling, we determine the temperature $T_\text{f}$ of the thermal ensemble describing the long-time steady state. Knowledge of $h_\text{f}$, $T_\text{f}$ and the equilibrium phase diagram allows us to ultimately map the full finite-temperature dynamical phase diagram.
• Figure 2: The $2+1$D quantum Ising model on a square lattice is prepared in the thermal ensemble $\hat{\rho}_\text{i}(h_\text{i},T_\text{i})$ (red dot). The system is subsequently quenched by instantly changing the transverse-field value to $h_\text{f}$. The temperature $T_\text{f}$ (black circles) of the long-time steady state of the quenched system is obtained from the conservation of the quench energy. The data is obtained by solving the implicit Eq. \ref{['eq:cond']} for $24\times24$ systems using the bisection method and QMC simulations; see SM. The equilibrium critical line (orange line) is from Ref. Hesselmann_2016, where the cyan region corresponds to the FM phase.
• Figure 3: Finite-temperature dynamical phase diagrams (red line denotes $h_\text{c}^\text{d}(h_\text{i},T_\text{i}$); FM dynamical phase in cyan) for different values of $h_\text{i}$ (grey dashed line). The cyan region corresponds to systems with initial temperature $T_\mathrm{i}$, whose long-time steady states after a quench with a Hamiltonian with $h_\mathrm{f}$ will be in the FM phase. The orange line corresponds to the equilibrium critical line for systems with transverse field $h_\text{f}$ and temperature $T_\text{i}$. The indigo cross marks the equilibrium critical temperature at $h_\text{i}$. The lowest considered temperature is $T_\mathrm{i} = \frac{1}{L}$. The error bars on the dynamical critical line are obtained from the different intersection points of the upper and lower bound of $T_\mathrm{f}$ with the equilibrium critical line.
• Figure 4: Real-time evolution of observables after the quantum quench: (a) second moment of magnetization on a $4\times4$ lattice from ED, (b) nearest-neighbor correlator on a $8\times8$ lattice from TTNs. In TTNs, we restrict ourselves to $T_\text{i} = 0$, in ED, additionally, finite $T_\text{i}$ is considered. The real-time dynamics is compared with the predicted properties of the long-time steady state of the quenched systems (dashed lines). The temperature of the long-time steady state is obtained from solving Eq. \ref{['eq:cond']} implicitly using QMC on the same-sized lattices. For the $T_\mathrm{i} = 0$ predictions, the quench energy is obtained from TTNs, after which the final temperature is fixed using QMC.
• Figure S1: Finite-size scaling of the temperatures of the long-time steady states for four different quantum quenches. The color of each dataset corresponds to a different quench (see legend). The two upper quenches (green, orange) were selected because of their pronounced finite-size effects. The third quench from the top (pink), takes a PM initial state to an FM long-time steady state; we indicate the equilibrium critical temperature corresponding to $h_\text{f}$ (pink dashed line). Fitting a power law with constant offset to the data has small residuals. According to this fit, PM to FM transitions persist even in the thermodynamic limit. The dataset at the bottom (purple) corresponds to a cooling quench. We indicate the initial temperature of this quench (purple dashed line), visualizing the cooling effect becoming more pronounced with larger system size.