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arXiv:1812.06899 · SciPost Physics

Decoherence and relaxation of topological states in extended quantum Ising models

Hannes Weisbrich, Wolfgang Belzig, Gianluca Rastelli

Published Mar 26, 2019• 8 citations
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Abstract

We study the decoherence and the relaxation dynamics of topological states in an extended class of quantum Ising chains which can present a multidimensional ground state subspace. The leading interaction of the spins with the environment is assumed to be the local fluctuations of the transverse magnetic field. By deriving the Lindblad equation using the many-body states, we investigate the relation between decoherence, energy relaxation and topology. In particular, in the topological phase and at low temperature, we analyze the dephasing rates between the different degenerate ground states.

DOI
10.21468/scipostphys.6.3.037

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We study the decoherence and the relaxation dynamics of topological states in an extended class of quantum Ising chains which can present a multidimensional ground state subspace. The leading interaction of the spins with the environment is assumed to be the local fluctuations of the transverse magnetic field. By deriving the Lindblad equation using the many-body states, we investigate the relation between decoherence, energy relaxation and topology. In particular, in the topological phase and at low temperature, we analyze the dephasing rates between the different degenerate ground states.
We study the decoherence dynamics of a quantum Ising lattice of finite size with a transverse dissipative interaction, namely, the coupling with the bath is assumed perpendicular to the direction of the spins interaction and parallel to the external magnetic field. In the limit of small transverse field, the eigenstates and spectrum are obtained by a strong-coupling expansion, from which we derive the Lindblad equation in the Markovian limit. At temperature lower than the energy gap and for weak dissipation, the decoherence dynamics can be restricted to take only the two degenerate ground states and the first excited subspace into account. The latter is formed by pairs of topological excitations (domain walls or kinks), which are quantum delocalized along the chain due to the small magnetic field. We find that some of these excited states form a relaxation-free subspace, namely, they do not decay to the ground states.
We use two-site quantum nonlocality to identify the topological quantum phase transitions (TQPTs) of the extended Ising model driven by varying system parameters. We investigate how the system parameters, including the anisotropies of the nearest-neighbor and the next-nearest-neighbor spin pairs, the transverse magnetic field, and the three-spin interaction, affect the quantum nonlocality. We show that the nonlocality cannot mark any TQPTs while its first derivative can perfectly characterize the TQPTs. By making the influences of the thermal fluctuations and the site distance of spin pairs on the critical behavior of the TQPTs analysis, we show that the sufficiently low temperature has a slight impact on the features of nonlocality and its first derivative while the site distance of spin pairs can significantly alter the properties of nonlocality and its first derivative. We further present the energy spectra and the trajectories of the winding vectors of the model to demonstrate that the quantum nonlocality can be employed to successfully signalize the TQPTs.
We consider a linear quench from the paramagnetic to ferromagnetic phase in the quantum Ising chain interacting with a static spin environment. Both decoherence from the environment and nonadiabaticity of the evolution near a critical point excite the system from the final ferromagnetic ground state. For weak decoherence and relatively fast quenches the excitation energy, proportional to the number of kinks in the final state, decays like an inverse square root of a quench time, but slow transitions or strong decoherence makes it decay in a much slower logarithmic way. We also find that fidelity between the final ferromagnetic ground state and a final state after a quench decays exponentially with a size of a chain, with a decay rate proportional to average density of excited kinks and a proportionality factor evolving from 1.3 for weak decoherence and fast quenches to approximately 1 for slow transitions or strong decoherence. Simultaneously, correlations between kinks randomly distributed along the chain evolve from a near-crystalline antibunching to a Poissonian distribution of kinks in a number of isolated Anderson localization centers randomly scattered along the chain.
In this paper, we explore the decoherence dynamics of a probing spin coupled to a spin bath, where the spin bath is given by a controllable 1D transverse-field Ising chain. The 1D transverse-field Ising chain with free-end boundary condition is equivalent to a modified Kitaev model with nonlocal Majorana bound states in its topological phase. We find that the non-Markovian decoherence dynamics of the probing spin can manifest the topological structure of the spin chain. By controlling the external magnetic field on the Ising chain, we find a close relationship between the topological phase transition and the non-Markovian dynamics in the real-time domain. We also investigate the corresponding quantum entanglement dynamics in this topological system.
We discuss the quantum dynamics of an isolated composite system consisting of weakly interacting many-body subsystems. We focus on one of the subsystems, S, and study the dependence of its quantum correlations and decoherence rate on the state of the weakly-coupled complementary part E, which represents the environment. As a theoretical laboratory, we consider a composite system made of two stacked quantum Ising chains, locally and homogeneously weakly coupled. One of the chains is identified with the subsystem S under scrutiny, and the other one with the environment E. We investigate the behavior of S at equilibrium, when the global system is in its ground state, and under out-of-equilibrium conditions, when the global system evolves unitarily after a quench of the coupling between S and E. When S develops quantum critical correlations in the weak-coupling regime, the associated scaling behavior crucially depends on the quantum state of E, whether it is characterized by short-range correlations (analogous to those characterizing disordered phases in closed systems), algebraically decaying correlations (typical of critical systems), or long-range correlations (typical of magnetized ordered phases). In particular, different scaling behaviors, depending on the state of E, are observed for the decoherence of the subsystem S, as demonstrated by the different power-law divergences of the decoherence susceptibility that quantifies the sensitivity of the coherence to the interaction with E.
The quantum Ising chain in a transverse magnetic field is $t\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}e$ paradigm of a quantum phase transition, displaying universal quantum criticality at low temperatures. Insights from this model have informed studies of quantum criticality in numerous modern correlated-electron materials. There have been a number of experimental studies of the quantum Ising chain, most notably in the material CoNb2{}_{2}2​O6{}_{6}6​. It has recently become clear that there was a surprising discrepancy between the observations: the energy gap in the quantum disordered regime as measured by nuclear magnetic resonance was approximately twice that measured by other methods. Here, the authors show via careful analysis of the low-temperature excitations that this is not a discrepancy, rather, it is in fact to be expected from the subtle interactions between the excitations. Although the underlying spin excitations are bosonic, their interactions transmute them into fermions, and this is a key ingredient in understanding the experiments.