Time-evolving a matrix product state with long-ranged interactions

Abstract

We introduce a numerical algorithm to simulate the time evolution of a matrix product state under a long-ranged Hamiltonian. In the effectively one-dimensional representation of a system by matrix product states, long-ranged interactions are necessary to simulate not just many physical interactions but also higher-dimensional problems with short-ranged interactions. Since our method overcomes the restriction to short-ranged Hamiltonians of most existing methods, it proves particularly useful for studying the dynamics of both power-law interacting one-dimensional systems, such as Coulombic and dipolar systems, and quasi two-dimensional systems, such as strips or cylinders. First, we benchmark the method by verifying a long-standing theoretical prediction for the dynamical correlation functions of the Haldane-Shastry model. Second, we simulate the time evolution of an expanding cloud of particles in the two-dimensional Bose-Hubbard model, a subject of several recent experiments.

5 pages + 3 pages appendices, 4 figures

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• Time-step targeting time-dependent and dynamical density matrix renormalization group algorithms with ab initio Hamiltonians
• Full counting statistics in the Haldane-Shastry chain
• Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
• Simulating generic spin-boson models with matrix product states
• Phase transitions and adiabatic preparation of a fractional Chern insulator in a boson cold atom model
• Out-of-equilibrium dynamics in a quantum impurity model: numerics for particle transport and entanglement entropy
• Automated construction of $U(1)$-invariant matrix-product operators from graph representations
• Quantum phase transitions in effective spin-ladder models for graphene zigzag nanoribbons
• Cooling schemes for two-component fermions in layered optical lattices