# Uniqueness regime for Markov dynamics on quantum lattice spin systems

@article{Crawford2015UniquenessRF, title={Uniqueness regime for Markov dynamics on quantum lattice spin systems}, author={Nick Crawford and Wojciech de Roeck and Marius Sch{\"u}tz}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2015}, volume={48} }

We consider a lattice of weakly interacting quantum Markov processes. Without interaction, the dynamics at each site is relaxing exponentially to a unique stationary state. With interaction, we show that there remains a unique stationary state in the thermodynamic limit, i.e. absence of phase coexistence, and the relaxation towards it is exponentially fast for local observables. We do not assume that the quantum Markov process is reversible (detailed balance) w.r.t. a local Hamiltonian.

## 11 Citations

### Dissipative Bose–Einstein condensation in contact with a thermal reservoir

- Physics
- 2016

We investigate the real-time dynamics of open quantum spin-1/2 or hardcore boson systems on a spatial lattice, which are governed by a Markovian quantum master equation. We derive general conditions…

### Stability of the uniqueness regime for ferromagnetic Glauber dynamics under non-equilibrium perturbations

- Mathematics
- 2016

In this paper, we prove a general result concerning finite-range, attractive interacting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particle system has a unique stationary measure and, in…

### An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory

- Physics
- 2015

Since its introduction by Hastings (Phys Rev B 69:104431, 2004), the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground-state phases.…

### An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory

- PhysicsLetters in Mathematical Physics
- 2016

Since its introduction by Hastings (Phys Rev B 69:104431, 2004), the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground-state phases.…

### Real-Time Dynamics of Open Quantum Systems and Anyonic Statistics in Doubled-Chern-Simons Theories

- Mathematics
- 2018

In the first part of this thesis we investigate real-time evolution of open quantum systems. We derive the Lindblad equation, as the Markovian master equation for their evolution. The Lindblad…

### Relaxation times of dissipative many-body quantum systems.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2015

We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system…

### Stability of the Uniqueness Regime for Ferromagnetic Glauber Dynamics Under Non-reversible Perturbations

- MathematicsAnnales Henri Poincaré
- 2018

We prove a general stability property concerning finite-range, attractive interacting particle systems on $$\{-\,1, 1\}^{{\mathbb {Z}}^d}$${-1,1}Zd. If the particle system has a unique stationary…

### Perturbation bounds of Markov semigroups on abstract states spaces

- Mathematics
- 2018

In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum…

### Stability Estimates of Markov Semigroups on Abstract States Spaces

- Mathematics
- 2020

In this paper, the stability and perturbation estimates for Markov $$C_0$$ C 0 -semigroups on abstract state spaces are explored using the Dobrushin ergodicity coefficient. Consequently, a linear…

### Stability of the Uniqueness Regime for Ferromagnetic Glauber Dynamics Under Non-reversible Perturbations

- Materials ScienceAnnales Henri Poincaré
- 2018

We prove a general stability property concerning finite-range, attractive interacting particle systems on {-1,1}Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}…

## References

SHOWING 1-10 OF 45 REFERENCES

### Locality in quantum and Markov dynamics on lattices and networks.

- PhysicsPhysical review letters
- 2004

For a broad class of dynamics, it is proved that ground or stationary state correlation functions can be written as a piece decaying exponentially in space plus a term set by matrix elements between the low-lying states.

### Construction and Ergodicity of Dissipative Dynamics for Quantum Spin Systems on a Lattice

- Mathematics, Physics
- 1998

We show that for a large class of interactions there exist translation-invariant dissipative dynamics which satisfy the detailed balance condition (in the associated noncommutative symmetric space),…

### Ergodicity of quantum cellular automata

- Mathematics
- 1995

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analoges of probabilistic cellular automata. We develop criteria for such a system to be…

### Uniqueness of the Ground State in Weak Perturbations of Non-Interacting Gapped Quantum Lattice Systems

- Mathematics, Physics
- 2005

We consider a general weak perturbation of a non-interacting quantum lattice system with a non-degenerate gapped ground state. We prove that in a finite volume the dependence of the ground state on…

### When is an interacting particle system ergodic?

- Mathematics
- 1993

We consider stochastic flip dynamics for an infinite number of Ising spins on the lattice ℤd. We find a sequence of constructive criteria for the system to be exponentially ergodic. The main idea is…

### Stochastic exclusion processes versus coherent transport

- Physics
- 2012

Stochastic exclusion processes play an integral role in the physics of non-equilibrium statistical mechanics. These models are Markovian processes, described by a classical master equation. In this…

### Lieb-Robinson bound and locality for general markovian quantum dynamics.

- PhysicsPhysical review letters
- 2010

The generalized bound is used to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.

### On Thermal Stability of Topological Qubit in Kitaev's 4D Model

- PhysicsOpen Syst. Inf. Dyn.
- 2010

We analyse stability of the four-dimensional Kitaev model — a candidate for scalable quantum memory — in finite temperature within the weak coupling Markovian limit. It is shown that, below a…

### Approach to equilibrium of Glauber dynamics in the one phase region

- Mathematics
- 1994

We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube Λ0, a Logarithmic Sobolev Inequality for…

### Stability of Local Quantum Dissipative Systems

- Mathematics
- 2015

Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In…