Random Matrix Spectral Form Factor in Kicked Interacting Fermionic Chains

We study quantum chaos and spectral correlations in periodically driven (Floquet) fermionic chains with long-range two-particle interactions, in the presence and absence of particle number conservation ($U(1)$) symmetry. We analytically show that the spectral form factor precisely follows the prediction of random matrix theory in the regime of long chains, and for timescales that exceed the so-called Thouless/Ehrenfest time which scales with the size $L$ as ${\cal O}(L^2)$, or ${\cal O}(L^0)$, in the presence, or absence of $U(1)$ symmetry, respectively. Using random phase assumption which essentially requires long-range nature of interaction, we demonstrate that the Thouless time scaling is equivalent to the behavior of the spectral gap of a classical Markov chain, which is in the continuous-time (Trotter) limit generated, respectively, by a gapless $XXX$, or gapped $XXZ$, spin-1/2 chain Hamiltonian.Comment: 6 pages, 1 figur

Fisher information approach to non-equilibrium phase transitions in quantum XXZ spin chain with boundary noise

We investigated quantum critical behaviours in the non-equilibrium steady state of a $XXZ$ spin chain with boundary Markovian noise using the Fisher information. The latter represents the distance between two infinitesimally close states, and its superextensive size scaling witnesses a critical behaviour due to a phase transition, since all the interaction terms are extensive. Perturbatively in the noise strength, we found superextensive Fisher information at anisotropy $|\Delta|\leqslant1$ and irrational $\frac{\arccos\Delta}{\pi}$ irrespective of the order of two non-commuting limits, i.e. the thermodynamic limit and the limit of sending $\frac{\arccos\Delta}{\pi}$ to an irrational number via a sequence of rational approximants. From this result we argue the existence of a non-equilibrium quantum phase transition with a critical phase $|\Delta|\leqslant1$. From the non-superextensivity of the Fisher information of reduced states, we infer that this non-equilibrium quantum phase transition does not have local order parameters but has non-local ones, at least at $|\Delta|=1$. In the non-perturbative regime for the noise strength, we numerically computed the reduced Fisher information which lower bounds the full state Fisher information, and is superextensive only at $|\Delta|=1$. Form the latter result, we derived local order parameters at $|\Delta|=1$ in the non-perturbative case. The existence of critical behaviour witnessed by the Fisher information in the phase $|\Delta|<1$ is still an open problem. The Fisher information also represents the best sensitivity for any estimation of the control parameter, in our case the anisotropy $\Delta$, and its superextensivity implies enhanced estimation precision which is also highly robust in the presence of a critical phase

Integrable quantum dynamics of open collective spin models

We consider a collective quantum spin-$s$ in contact with Markovian spin-polarized baths. Using a conserved super-operator charge, a differential representation of the Liouvillian is constructed to find its exact spectrum and eigen-modes. We study the spectral properties of the model in the large-$s$ limit using a semi-classical quantization condition and show that the spectral density may diverge along certain curves in the complex plane. We exploit our exact solution to characterize steady-state properties, in particular at the discontinuous phase transition that arises for unpolarized environments, and to determine the decay rates of coherences and populations. Our approach provides a systematic way of finding integrable Liouvillian operators with non-trivial steady-states as well as a way to study their spectral properties and eigen-modes.Comment: 5 pages, 4 figure

Heat transport in quantum harmonic chains with Redfield baths

We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the bath spectral function, the system-bath coupling strength, and the boundary conditions.Comment: 17 pages, 3 figure

PT-symmetric quantum Liouvillian dynamics

We discuss a combination of unitary and anti-unitary symmetry of quantum Liouvillian dynamics, in the context of open quantum systems, which implies a D2 symmetry of the complex Liovillean spectrum. For sufficiently weak system-bath coupling it implies a uniform decay rate for all coherences, i.e. off-diagonal elements of the system's density matrix taken in the eigenbasis of the Hamiltonian. As an example we discuss symmetrically boundary driven open XXZ spin 1/2 chains.Comment: Note [18] added with respect to a published version, explaining the symmetry of the matrix V [eq. (14)

Ballistic spin transport in a periodically driven integrable quantum system

We demonstrate ballistic spin transport of an integrable unitary quantum circuit, which can be understood either as a paradigm of an integrable periodically driven (Floquet) spin chain, or as a Trotterized anisotropic ($XXZ$) Heisenberg spin-1/2 model. We construct an analytic family of quasi-local conservation laws that break the spin-reversal symmetry and compute a lower bound on the spin Drude weight which is found to be a fractal function of the anisotropy parameter. Extensive numerical simulations of spin transport suggest that this fractal lower bound is in fact tight.Comment: 5 + 9 pages, 5 + 2 figure

Convergence radius of perturbative Lindblad driven non-equilibrium steady states

We address the problem of analyzing the radius of convergence of perturbative expansion of non-equilibrium steady states of Lindblad driven spin chains. A simple formal approach is developed for systematically computing the perturbative expansion of small driven systems. We consider the paradigmatic model of an open $XXZ$ spin 1/2 chain with boundary supported ultralocal Lindblad dissipators and treat two different perturbative cases: (i) expansion in system-bath coupling parameter and (ii) expansion in driving (bias) parameter. In the first case (i) we find that the radius of convergence quickly shrinks with increasing the system size, while in the second case (ii) we find that the convergence radius is always larger than $1$, and in particular it approaches $1$ from above as we change the anisotropy from easy plane ($XY$) to easy axis (Ising) regime