Steady States of Infinite-Size Dissipative Quantum Chains via Imaginary Time Evolution

Abstract

Directly in the thermodynamic limit, we show how to combine local imaginary and real-time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESSs) of one-dimensional dissipative quantum lattices governed by a local Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly correlated subspace of the NESS, after which, real-time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the dissipative transverse field quantum Ising chain. We show that a crossover of an order parameter shown to be smooth in previous finite-size studies remains smooth in the thermodynamic limit.

Numerical simulation of critical dissipative non-equilibrium quantum systems with an absorbing state

“…Furthermore, since the universality class of the QCP is currently debated, [7,9,11], it is of considerable interest in its own right to make estimates of critical exponents, comparing these with previous estimates and known cases.To simulate the non-equilibrium dynamics of the QCP, we apply matrix product states (MPSs) and the timeevolving block-decimation (TEBD) algorithm [13][14][15]. This algorithm belongs to a more general class of tensor network (TN) methods, well established for the simulation of closed quantum systems in 1d, which have also been applied to dissipative quantum systems previously in a number of cases [16][17][18][19][20][21][22][23][24]. In the context of studying dissipative quantum dynamics, a key question for TN methods is whether different approaches, such as quantum trajectories (QTs) as opposed to the Lindblad master equation, can lead to substantially different accuracies.…”

Tensor Network Contractions

“…A TN operator is regarded as a mapping from the bra to the ket Hilbert space. Many algorithms explicitly employ the TN operator form, including the matrix product operator (MPO) for representing 1D many-body operators and mixed states, and for simulating 1D systems in and out of equilibrium [186,187,188,189,190,191,192,193,194,195,196], tensor product operator (also called projected entangled pair operators) in for higher-systems [140,141,143,197,198,199,200,201,202,203,204,205,206], and multiscale entangled renormalization ansatz [207,208,209].…”

“…The MPS or PEPS can be readily generalized from representations of states to those of operators called MPO [186,187,188,189,193,194,195,196] or projected entangled pair operator (PEPO) 7 [140,141,143,197,198,199,201,202,203,204]. Let us begin with MPO, which is also formed by the contraction of local tensors aŝ…”

Numerical linked-cluster expansion for the dissipative XYZ model on a triangular lattice

“…Because the steady-state solutions in the thermodynamic limit have been proven to be remarkably difficult, some approximations are imposed to the density matrix, such as the single-site and cluster Gutzwiller mean-field factorizations [12,18,16,[32][33][34], to unravel the many-body master equation. Besides, numerical methods based on tensor networks [35][36][37][38], corner space renormalization [39], variational principal [17,40,41], and the neural networks [42][43][44][45] have been proposed.…”