Hybrid infinite time-evolving block decimation algorithm for long-range multidimensional quantum many-body systems

Abstract

In recent years, the infinite time-evolving block decimation (iTEBD) method has been demonstrated to be one of the most efficient and powerful numerical schemes for time evolution in one-dimensional quantum many-body systems. However, a major shortcoming of the method, along with other state-of-the-art algorithms for manybody dynamics, has been their restriction to one spatial dimension. We present an algorithm based on a hybrid extension of iTEBD where finite blocks of a chain are first locally time evolved before an iTEBD-like method combines these processes globally. This in turn permits simulating the dynamics of many-body systems in spatial dimensions d 1 where the thermodynamic limit is achieved along one spatial dimension and where long-range interactions can also be included. Our work paves the way for simulating the dynamics of many-body phenomena that occur exclusively in higher dimensions and whose numerical treatments have hitherto been limited to exact diagonalization of small systems, which fundamentally limits a proper investigation of dynamical criticality. We expect the algorithm presented here to be of significant importance to validating and guiding investigations in state-of-the-art ion-trap and ultracold-atom experiments.

Out-of-equilibrium phase diagram of long-range superconductors

“…[34]. Indeed, the generally accepted explanation-based on results of long-range interacting quantum spin modelsfor why the dynamical critical point can be smaller than the equilibrium critical point has been that this is most likely due to interactions [18][19][20][21][22][23][24][25][26]. Nevertheless, as indicated in our previous work Ref.…”

“…The type of nonanalyticities that occur in the wake of a quench can either be regular or anomalous [18,20]. The former are the cusps that occur for quenches crossing the dynamical critical point, while the latter can occur for arbitrarily small quenches within the ordered phase of the system, and they have been shown to be connected to local spin excitations forming the lowest-lying excitations in the spectrum of the quench Hamiltonian [24,25,70].…”

“…In principle, this type of dynamical phase transition separates a ferromagnetic from a paramagnetic steady state in the wake of a quench, and this has been recently observed experimentally in trapped-ion setups [27]. However, in cases where the system has no finite-temperature phase transition and, therefore, always ends up in a paramagnetic steady state in the infinite-time limit, this dynamical phase transition can still be defined based on the decay behavior of the order parameter [24,25,[28][29][30]. For small enough quenches starting in an ordered state, the order parameter would decay asymptotically to zero without ever crossing zero.…”

“…Among the prominent thrusts of this research effort lies the phenomenon of dynamical phase transitions [8,9], which fall into two major categories when considering sudden quenches of a quantum many-body system. The first is characterized by a local order parameter, whose long-time behavior determines the dynamical phase of the steady state [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In principle, this type of dynamical phase transition separates a ferromagnetic from a paramagnetic steady state in the wake of a quench, and this has been recently observed experimentally in trapped-ion setups [27].…”

“…Indeed, it was then firmly established that the dynamical critical point, which separates different phases of DQPT, is in general distinct from the quantum equilibrium critical point and is strongly dependent on the initial condition [18,21,22]. Additionally, this dynamical critical point was shown to coincide with that of the type of dynamical phase transition based on a local order parameter [18,[20][21][22][23][24][25]. Also recently, the theory of DQPT has been extended to Floquet systems [38][39][40] and models with many-body localized phases [41,42].…”