Marko Znidaric

Many-body localization in the Heisenberg XXZ magnet in a random field

We numerically investigate Heisenberg XXZ spin-1/2 chain in a spatially random static magnetic field. We find that tDMRG simulations of time evolution can be performed efficiently, namely the dimension of matrices needed to efficiently represent the time-evolution increases linearly with time and entanglement entropies for typical chain bipartitions increase logarithmically. As a result, we show that for large enough random fields infinite temperature spin-spin correlation function displays exponential localization in space indicating insulating behavior of the model.

Stability of quantum motion and correlation decay

We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster the decay of correlations. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a timescale 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a timescale 1/δ2, where δ is the strength of perturbation. A detailed discussion of a semiclassical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behaviour is recovered in the classical limit → 0, as the two limits δ → 0 and → 0 do not commute. In addition, we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on the celebrated example of a quantized kicked top.

Spin transport in a one-dimensional anisotropic Heisenberg model.

We analytically and numerically study spin transport in a one-dimensional Heisenberg model in linear-response regime at infinite temperature. It is shown that as the anisotropy parameter Δ is varied spin transport changes from ballistic for Δ<1 to anomalous at the isotropic point Δ=1, to diffusive for finite Δ>1, ending up as a perfect isolator in the Ising limit of infinite Δ. Using perturbation theory for large Δ a quantitative prediction is made for the dependence of diffusion constant on Δ.

Fidelity and purity decay in weakly coupled composite systems

We study the stability of unitary quantum dynamics of composite systems (for example: central system + environment) with respect to weak interaction between the two parts. Unified theoretical formalism is applied to study different physical situations: (i) coherence of a forward evolution as measured by the purity of the reduced density matrix, (ii) stability of time evolution with respect to small coupling between subsystems and (iii) Loschmidt echo measuring dynamical irreversibility. Stability has been measured either by fidelity of pure states of a composite system, or by the so-called reduced fidelity of reduced density matrices within a subsystem. Rigorous inequality among fidelity, reduced fidelity and purity is proved and a linear response theory is developed expressing these three quantities in terms of time correlation functions of the generator of interaction. The qualitatively different cases of regular (integrable) or mixing (chaotic in the classical limit) dynamics in each of the subsystems are discussed in detail. Theoretical results are demonstrated and confirmed in a numerical example of two coupled kicked tops.

Influence of dephasing on many-body localization

We study the effects of dephasing noise on a prototypical many-body localized system -- the XXZ spin 1/2 chain with a disordered magnetic field. At times longer than the inverse dephasing strength the dynamics of the system is described by a probabilistic Markov process on the space of diagonal density matrices, while all off-diagonal elements of the density matrix decay to zero. The generator of the Markovian process is a bond-disordered spin chain. The scaling variable is identified, and independence of relaxation on the interaction strength is demonstrated. We show that purity and von Neumann entropy are extensive, showing no signatures of localization, while the operator space entanglement entropy exhibits a logarithmic growth with time until the final saturation corresponding to localization breakdown, suggesting a many-body localized dynamics of the effective Markov process.

Is the efficiency of classical simulations of quantum dynamics related to integrability

Efficiency of time evolution of quantum observables, and thermal states of quenched Hamiltonians, is studied using time-dependent density-matrix renormalization group method in a family of generic quantum spin chains which undergo a transition from integrable to nonintegrable-quantum chaotic case as control parameters are varied. Quantum states (observables) are represented in terms of matrix-product operators with rank D(epsilon)(t), such that evolution of a long chain is accurate within fidelity error epsilon up to time t. We found that the rank generally increases exponentially D(epsilon)(t) proportional to exp(const t), unless the model system was integrable in which case we found polynomial increase.

Can quantum chaos enhance the stability of quantum computation

A combustor for a gas turbine includes a main fuel injector for receiving compressor discharge air and mixing the air with fuel for flow to a downstream catalytic section. The main fuel injector includes an array of venturis each having an inlet, a throat and a diffuser. A main fuel supply plenum between forward and aft plates supplies fuel to secondary annular plenums having openings for supplying fuel into the inlet of the venturis upstream of the throat. The diffusers transition from a circular cross-section at the throat to multiple discrete angularly related side walls at the diffuser exits without substantial gaps therebetween. With this arrangement, uniform flow distribution of the fuel/air, velocity and temperature is provided at the catalyst inlet.

Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation.

We study a one-dimensional XX chain under nonequilibrium driving and local dephasing described by the Lindblad master equation. The analytical solution for the nonequilibrium steady state found for particular parameters in a previous study [M. Žnidarič J. Stat. Mech. (2010) L05002] is extended to arbitrary coupling constants, driving, and a homogeneous magnetic field. All one-, two-, and three-point correlation functions are explicitly evaluated. It is shown that the nonequilibrium stationary state is not Gaussian. Nevertheless, in the thermodynamic and weak-driving limit it is only weakly correlated and can be described by a matrix product operator ansatz with matrices of fixed dimension 4. A nonequilibrium phase transition at zero dephasing is also discussed. It is suggested that the scaling of the relaxation time with the system size can serve as a signature of a nonequilibrium phase transition.

Thermalization and ergodicity in one-dimensional many-body open quantum systems

Using an approach based on the time-dependent density-matrix renormalization-group method, we study the thermalization in spin chains locally coupled to an external bath. Our results provide evidence that quantum chaotic systems do thermalize, that is, they exhibit relaxation to an invariant ergodic state which, in the bulk, is well approximated by the grand canonical state. Moreover, the resulting ergodic state in the bulk does not depend on the details of the baths. On the other hand, for integrable systems we found that the invariant state in general depends on the bath and is different from the grand canonical state.

Exact large-deviation statistics for a nonequilibrium quantum spin chain.

We consider a one-dimensional XX spin chain in a nonequilibrium setting with a Lindblad-type boundary driving. By calculating large-deviation rate function in the thermodynamic limit, a generalization of free energy to a nonequilibrium setting, we obtain a complete distribution of current, including closed expressions for lower-order cumulants. We also identify two phase-transition-like behaviors in either the thermodynamic limit, at which the current probability distribution becomes discontinuous, or at maximal driving, when the range of possible current values changes discontinuously. In the thermodynamic limit the current has a finite upper and lower bound. We also explicitly confirm nonequilibrium fluctuation relation and show that the current distribution is the same under mapping of the coupling strength Γ→1/Γ.