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arXiv:1502.02980 · Physical Review B

Real-time dynamics of open quantum spin systems driven by dissipative processes

F. Hebenstreit, Debasish Banerjee, M. Hornung, Fu-Jiun Jiang, F. Schranz, U.‐J. Wiese

Published Jul 9, 2015• 11 citations
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Abstract

We study the real-time evolution of large open quantum spin systems in two spatial dimensions, whose dynamics is entirely driven by a dissipative coupling to the environment. We consider different dissipative processes and investigate the real-time evolution from an ordered phase of the Heisenberg or XY model towards a disordered phase at late times, disregarding unitary Hamiltonian dynamics. The corresponding Kossakowski-Lindblad equation is solved via an efficient cluster algorithm. We find that the symmetry of the dissipative process determines the time scales, which govern the approach towards a new equilibrium phase at late times. Most notably, we find a slow equilibration if the dissipative process conserves any of the magnetization Fourier modes. In these cases, the dynamics can be interpreted as a diffusion process of the conserved quantity.

DOI
10.1103/physrevb.92.035116

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We study the real-time evolution of large open quantum spin systems in two spatial dimensions, whose dynamics is entirely driven by a dissipative coupling to the environment. We consider different dissipative processes and investigate the real-time evolution from an ordered phase of the Heisenberg or XY model towards a disordered phase at late times, disregarding unitary Hamiltonian dynamics. The corresponding Kossakowski-Lindblad equation is solved via an efficient cluster algorithm. We find that the symmetry of the dissipative process determines the time scales, which govern the approach towards a new equilibrium phase at late times. Most notably, we find a slow equilibration if the dissipative process conserves any of the magnetization Fourier modes. In these cases, the dynamics can be interpreted as a diffusion process of the conserved quantity.
We consider a large quantum system with spins 12\frac{1}{2}21​ whose dynamics is driven entirely by measurements of the total spin of spin pairs. This gives rise to a dissipative coupling to the environment. When one averages over the measurement results, the corresponding real-time path integral does not suffer from a sign problem. Using an efficient cluster algorithm, we study the real-time evolution from an initial antiferromagnetic state of the two-dimensional Heisenberg model, which is driven to a disordered phase, not by a Hamiltonian, but by sporadic measurements or by continuous Lindblad evolution.
Using quantum Monte Carlo, we study the nonequilibrium transport of magnetization in large open strongly correlated quantum spin-12\frac{1}{2}21​ systems driven by purely dissipative processes that conserve the uniform or staggered magnetization, disregarding unitary Hamiltonian dynamics. We prepare both a low-temperature Heisenberg ferromagnet and an antiferromagnet in two parts of the system that are initially isolated from each other. We then bring the two subsystems in contact and study their real-time dissipative dynamics for different geometries. The flow of the uniform or staggered magnetization from one part of the system to the other is described by a diffusion equation that can be derived analytically.
A central goal in the study of modern condensed matter physics is the characterization of the dynamical properties of quantum systems. Many decades of effort towards this goal, studying a diverse range of (near-equilibrium) quantum matter, from Fermi liquids, to quantum two-level systems, to interacting spin models, and more, has revealed a remarkable pervasiveness of the simple dynamical description of these complex systems in terms of quasi-particles that carry spin, charge, and heat, and that are generally able to equilibrate systems. \n\nThis thesis is an examination of some exceptions to this rule. Specifically, we study a number of instances of quantum matter where equilibration phenomena happens at rather long time scales, or does not occur at all. Particular emphasis is laid on the role of dimensionality, disorder, and dissipation in engendering such novel dynamical behavior.\n\nFirst, we consider non-equilibrium dynamics in one-dimensional quasi-condensates. Low dimensionality inhibits scattering in these systems, and low-energy excitations are long-lived phase fluctuations that exhibit an enriched conformal symmetry. Utilizing this symmetry, we generalize sudden quenches typically used to study non-equilibrium dynamics to quenches along general relativistic and conformal trajectories. Gases never truly equilibrate after such a quench; instead, they evolve into a `prethermal' state with thermal-looking correlations and a chiral asymmetry. \n\nWe then study the problem of the dynamical transition driven by disorder, from an ergodic to a non-ergodic phase, in one-dimensional quantum spin chains. In particular, in XXZ chains with on-site disorder, we find a unique intermediate phase straddling the boundary of the dynamical phase transition, wherein rare-region effects lead to long-time tails in equilibration and vanishing DC conduction before the onset of non-ergodicity. We propose generalizations of such `Griffiths' behavior to arbitrary dimensions. We also study the dynamics of random-bond Heisenberg chains by developing a strong-disorder renormalization group protocol for these systems. We discuss how magnetic noise from such disordered systems contains signatures of their anomalous dynamical properties. \n\nNext, we re-examine the phenomenological theory of two-level systems in amorphous materials in the light of new experimental evidence that these states have large electric/magnetic dipole moments. We propose and justify an interpretation of the model as one of tunneling electrons slowed down by a large phonon drag and discuss the dynamical consequences of such polaronic effects. \n\nFinally, we discuss how magnetic noise measurements can be used to non-invasively access the anomalous properties of systems such as those discussed above. In particular, we examine how scattering properties of isolated magnetic impurities and non-local transport in a variety of two-dimensional materials can be probed experimentally using NV centers as noise magnetometers.
We investigate the real-time evolution of lattice bosons in two spatial dimensions whose dynamics is governed by a Markovian quantum master equation. We employ the Wigner-Weyl phase space quantization and derive the functional integral for open quantum many-body systems that determines the time evolution of the Wigner function. Using the truncated Wigner approximation, in which quantum fluctuations are only taken into account in the initial state whereas the dynamics is governed by classical evolution equations, we study the buildup of long-range correlations due to the action of non-Hermitean quantum jump operators that constitute a mechanism for dissipative cooling. Starting from an initially disordered state corresponding to a vortex condensate, the dissipative process results in the annihilation of vortex-antivortex pairs and the establishment of quasi-long-range order at late times. We observe that a finite vortex density survives the cooling process, which disagrees with the analytically constructed vortex-free Bose-Einstein condensate at asymptotic times. This indicates that quantum fluctuations beyond the truncated Wigner approximation need to be included to fully capture the physics of dissipative Bose-Einstein condensation.
We address the real-time dynamics of lattice quantum spin models coupled to single or multiple Markovian dissipative reservoirs using the method of closed hierarchies of correlation functions. This approach allows us to solve a number of quantum spin models exactly in arbitrary dimensions, which is illustrated explicitly with two examples of driven-dissipative systems. We investigate their respective nonequilibrium steady states as well as the full real-time evolution on unprecedented system sizes. Characteristic time scales are derived analytically, which allows us to understand the nontrivial finite-size scaling of the dissipative gap. The corresponding scaling exponents are confirmed by solving numerically for the full real-time evolution of two-point correlation functions.
We study the effects of dissipative boundaries in many-body systems at continuous quantum transitions, when the parameters of the Hamiltonian driving the unitary dynamics are close to their critical values. As paradigmatic models we consider fermionic wires subject to dissipative interactions at the boundaries, associated with pumping or loss of particles. They are induced by couplings with a Markovian baths, so that the evolution of the system density matrix can be described by a Lindblad master equation. We study the quantum evolution arising from variations of the Hamiltonian and dissipation parameters, starting at t=0t=0t=0 from the ground state of the Hamiltonian at, or close to, the critical point. Two different dynamic regimes emerge: (i) an early-time regime for times $t\ensuremath{\sim}L$, where the competition between coherent and incoherent drivings develops a dynamic finite-size scaling, obtained by extending the scaling framework describing the coherent critical dynamics of the closed system, to allow for the boundary dissipation; and (ii) a large-time regime for $t\ensuremath{\sim}{L}^{3}$ whose dynamic scaling describes the late quantum evolution leading to the $t\ensuremath{\rightarrow}\ensuremath{\infty}$ stationary states.