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arXiv:1410.3342 · Physical Review B

Solving nonequilibrium dynamical mean-field theory using matrix product states

F. Alexander Wolf, Ian P. McCulloch, Ulrich Schollwöck

Published Dec 18, 2014• 121 citations
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Abstract

We solve nonequilibrium dynamical mean-field theory (DMFT) using matrix product states (MPS). This allows us to treat much larger bath sizes and by that reach substantially longer times (factor $\ensuremath{\sim}2--3$) than with exact diagonalization. We show that the star geometry of the underlying impurity problem can have substantially better entanglement properties than the previously favored chain geometry. This has immense consequences for the efficiency of an MPS-based description of general impurity problems: in the case of equilibrium DMFT, it leads to an orders-of-magnitude speedup. We introduce an approximation for the two-time hybridization function that uses time-translational invariance, which can be observed after a certain relaxation time after a quench to a time-independent Hamiltonian.

DOI
10.1103/physrevb.90.235131

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We solve nonequilibrium dynamical mean-field theory (DMFT) using matrix product states (MPS). This allows us to treat much larger bath sizes and by that reach substantially longer times (factor \ensuremath∼2−−3\ensuremath{\sim}2--3\ensuremath∼2−−3) than with exact diagonalization. We show that the star geometry of the underlying impurity problem can have substantially better entanglement properties than the previously favored chain geometry. This has immense consequences for the efficiency of an MPS-based description of general impurity problems: in the case of equilibrium DMFT, it leads to an orders-of-magnitude speedup. We introduce an approximation for the two-time hybridization function that uses time-translational invariance, which can be observed after a certain relaxation time after a quench to a time-independent Hamiltonian.
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Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.
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The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.