Imaginary-time evolution of interacting spin systems in the truncated Wigner approximation

Abstract

We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time, termed iTWA. The evolution of the canonical density matrix in imaginary time is mapped to a partial differential equation of its Wigner function. Truncation at the Fokker-Planck level leads to a set of stochastic differential equations, which can be efficiently simulated. We show that the iTWA can provide very good approximations to the ground state of a random and in general frustrated anti-ferromagnetic Ising Hamiltonian on a 3-regular graph, for which finding the exact ground state and approximations to it beyond a certain accuracy is NP hard. Furthermore in order to assess the ability of the method to properly account for leading-order quantum effects, we analyze the ground-state quantum phase transition of the nearest-neighbor, transverse-field Ising model in one and two spatial dimensions, finding very good agreement with the exact behaviour. The critical behavior obtained in iTWA follows the quantum-classical correspondence.

Figures (3)

• Figure 1: Average energy $\langle \hat{H}\rangle$ of the Gibbs state of the AF Ising Hamiltonian \ref{['eq:random-Ising']} on randomly chosen 3-regular graphs with $N=22$ nodes over $N_\mathrm{traj} = 84 \cdot 10^3$ trajectories as function of inverse temperature $\tau = 1/k_B T$. Shown are iTWA (solid, green) and exact results (ED, dashed, red) as well as the exact ground state energies $E_0=-23J, -25J, -27J$ (dot-dashed, black) together with illustrations of the 3-regular-graphs.
• Figure 2: Mean relative error $\Delta \epsilon = (\langle \hat{H} \rangle-E_0) / E_0$ of the average energy $\langle \hat{H} \rangle$ simulated via iTWA at an annealing time of $J\tau=10$ to the exact ground state energy $E_0$ estimated by Gurobi averaged over 200 randomly chosen 3-regular graphs with $N=100$ depending on the number of trajectories $N_\mathrm{traj}$. As error bar the standard deviation is given. Note that the underlying distribution function is not normal and is bounded from below (for all trajectories $E_0(\tau= 10 J^{-1}) >E_0$).
• Figure 3: Total squared magnetization $\langle m^2\rangle$ for the 1D (top) and 2D (bottom) TFIM for different number of spins with exact finite-size (green and blue dashed lines) in 1D and infinite-size solutions (red dashed line) in 1D and 2D. The stationary values were determined by averaging $\langle m^2\rangle$ over $\tau$. In 1D in the interval $\tau=3/J-5/J$ and in 2D in the interval $\tau = 1/J-3/J$. For the error bars the leave-one-out jackknife standard deviation was used to account for bias in single trajectories (1D: $N_\mathrm{traj} = 480\cdot10^3$, 2D: $N_\mathrm{traj} = 600\cdot10^3$) dca15e5b-b3f7-3417-8555-955fe36eb045.