Quantum Ising Chain Research Articles

Excited-state entanglement and thermal mutual information in random spin chains

Entanglement properties of excited eigenstates (or of thermal mixed states) are difficult to study with conventional analytical methods. We approach this problem for random spin chains using a recently developed real-space renormalization group technique for excited states (``RSRG-X''). For the random XX and quantum Ising chains, which have logarithmic divergences in the entanglement entropy of their (infinite-randomness) critical ground states, we show that the entanglement entropy of excited eigenstates retains a logarithmic divergence while the mutual information of thermal mixed states does not. However, in the XX case the coefficient of the logarithmic divergence extends from the universal ground-state value to a universal interval due to the degeneracy of excited eigenstates. These models are noninteracting in the sense of having free-fermion representations, allowing strong numerical checks of our analytical predictions.

Edge modes in a frustrated quantum Ising chain

The ground state properties of an Ising chain with nearest ($J_{1}$) and next-nearest neighbor ($J_{2}$) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered phase), one of which has simple ferromagnetic or N\'eel order depending upon the sign of $J_{1}$, and the other is the double-staggered antiferromagnetic phase. The presence of the Majorana-like edge-modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near $J_{1}=0$ where it seems they have four edge modes like the exact case at $J_{1}=0$ itself.

Testing whether all eigenstates obey the eigenstate thermalization hypothesis.

We ask whether the eigenstate thermalization hypothesis (ETH) is valid in a strong sense: in the limit of an infinite system, every eigenstate is thermal. We examine expectation values of few-body operators in highly excited many-body eigenstates and search for "outliers," the eigenstates that deviate the most from ETH. We use exact diagonalization of two one-dimensional nonintegrable models: a quantum Ising chain with transverse and longitudinal fields, and hard-core bosons at half-filling with nearest- and next-nearest-neighbor hopping and interaction. We show that even the most extreme outliers appear to obey ETH as the system size increases and thus provide numerical evidences that support ETH in this strong sense. Finally, periodically driving the Ising Hamiltonian, we show that the eigenstates of the corresponding Floquet operator obey ETH even more closely. We attribute this better thermalization to removing the constraint of conservation of the total energy.

Many-body localization in a disordered quantum Ising chain.

Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition, a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this Letter, we explore two entanglement properties that are promising for the study of the many-body localization transition: the variance of the half-chain entanglement entropy of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a disordered quantum Ising chain and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a spin-glass transition at large disorder strength and provide evidence for it being a separate transition. We, thereby, give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [Huse etal., Phys. Rev. B 88, 014206 (2013).

Cumulants of time-integrated observables of closed quantum systems andPTsymmetry with an application to the quantum Ising chain

We study the connection between the cumulants of a time-integrated observable of a quantum system and the PT-symmetry properties of the non-Hermitian deformation of the Hamiltonian from which the generating function of these cumulants is obtained. This non-Hermitian Hamiltonian can display regimes of broken and of unbroken PT-symmetry, depending on the parameters of the problem and on the counting field that sets the strength of the non-Hermitian perturbation. This in turn determines the analytic structure of the long-time cumulant generating function (CGF) for the time-integrated observable. We consider in particular the case of the time-integrated (longitudinal) magnetisation in the one-dimensional Ising model in a transverse field. We show that its long-time CGF is singular on a curve in the magnetic field/counting field plane that delimits a regime where PT-symmetry is spontaneously broken (which includes the static ferromagnetic phase), from one where it is preserved (which includes the static paramagnetic phase). In the paramagnetic phase, conservation of PT -symmetry implies that all cumulants are sub-linear in time, a behaviour usually associated to the absence of decorrelation.

Finite-size scaling at first-order quantum transitions.

We study finite-size effects at first-order quantum transitions (FOQTs). We show that the low-energy properties show a finite-size scaling (FSS) behavior, the relevant scaling variable being the ratio of the energy associated with the perturbation driving the transition and the finite-size energy gap at the FOQT point. The size dependence of the scaling variable is therefore essentially determined by the size dependence of the gap at the transition, which in turn depends on the boundary conditions. Our results have broad validity and, in particular, apply to any FOQT characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. In this case, a phenomenological two-level theory provides exact expressions for the scaling functions. Numerical results for the quantum Ising chain in transverse and parallel magnetic fields support the FSS Ansätzes.

Stationary entropies after a quench from excited states in the Ising chain

We consider the asymptotic state after a sudden quench of the magnetic field in the transverse field quantum Ising chain starting from excited states of the pre-quench Hamiltonian. We compute the thermodynamic entropies of the generalised Gibbs and the diagonal ensembles and we find that the generalised Gibbs entropy is always twice the diagonal one. We show that particular care should be taken in extracting the thermodynamic limit since different averages of equivalent microstates give different results for the entropies.

Loschmidt echo and dynamical fidelity in periodically driven quantum systems

We study the dynamical fidelity and the Loschmidt echo , following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after n periods τ. We show that (the logarithm of the fidelity per site) reaches a steady value in the asymptotic limit , and we derive an exact analytical expression for this quantity. Remarkably, the steady-state value of shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional models.

Nonequilibrium quantum criticality in open systems: The dissipation rate as an additional indispensable scaling variable

We propose that nonequilibrium quantum criticality in open systems under the Born-Markov approximation can be described by a master equation of the Lindblad form. This master equation is derived from a system coupling weakly to a heat bath microscopically and is suggested to provide an approach to study dynamic quantum critical behavior of the system at finite temperatures. We find that the dissipation rate in the equation representing the coupling must be included in the scaling forms as an indispensable additional scaling variable in order to correctly describe the nonequilibrium quantum critical behavior, yet the equilibrium fixed point determines the nonequilibrium critical behavior in the weak coupling limit. Through numerically solving the Lindblad equation for the quantum Ising chain, we affirm these propositions by finite-time scaling forms with the dissipation rate. Nonequilibrium dynamic critical behavior of spontaneous emissions in dissipative open systems at zero temperature near their quantum critical points is discovered and is also described well by the scaling forms.

Free parafermions

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-Hermitian (but PT-symmetric) clock Hamiltonian is ‘free’, in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter’s and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a ‘free parafermion’. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.

Nonequilibrium dynamics of a noisy quantum Ising chain: Statistics of work and prethermalization after a sudden quench of the transverse field

We discuss the non-equilibrium dynamics of a Quantum Ising Chain (QIC) following a quantum quench of the transverse field and in the presence of a gaussian time dependent noise. We discuss the probability distribution of the work done on the system both for static and dynamic noise. While the effect of static noise is to smooth the low energy threshold of the statistic of the work, appearing for sudden quenches, a dynamical noise protocol affects also the spectral weight of such features. We also provide a detailed derivation of the kinetic equation for the Green's functions on the Keldysh contour and the time evolution of observables of physical interest, extending previously reported results (J. Marino, A. Silva, Phys. Rev. B 86, 060408 (2012)), and discussing the extension of the concept of prethermalization which can be used to study noisy quantum many body hamiltonians driven out-of-equilibrium.

Exact results for fidelity susceptibility of the quantum Ising model: the interplay between parity, system size, and magnetic field

We derive an exact closed-form expression for fidelity susceptibility of even- and odd-sized quantum Ising chains in a transverse field. To this aim, we diagonalize the Ising Hamiltonian and study the gap between its positive and negative parity subspaces. We derive an exact closed-form expression for the gap and use it to identify the parity of the ground state. We point out the misunderstanding in some of the former studies of fidelity susceptibility and discuss its consequences. Last but not least, we rigorously analyze the properties of the gap. For example, we derive analytical expressions showing its exponential dependence on the ratio between the system size and the correlation length.

Prethermalization in a Nonintegrable Quantum Spin Chain after a Quench

We study the dynamics of a quantum Ising chain after the sudden introduction of a nonintegrable long-range interaction. Via an exact mapping onto a fully connected lattice of hard-core bosons, we show that a prethermal state emerges and we investigate its features by focusing on a class of physically relevant observables. In order to gain insight into the eventual thermalization, we outline a diagrammatic approach which complements the study of the previous quasistationary state and provides the basis for a self-consistent solution of the kinetic equation. This analysis suggests that both the temporal decay towards the prethermal state and the crossover to the eventual thermal one may occur algebraically.

Nonequilibrium thermal transport in the quantum Ising chain

We consider two quantum Ising chains initially prepared at thermal equilibrium but with different temperatures and coupled at a given time through one of their end points. In the long-time limit the system reaches a non-equilibrium steady state. We discuss properties of this non-equilibrium steady state, and characterize the convergence to the steady regime. We compute the mean energy flux through the chain and the large deviation function for the quantum and thermal fluctuations of this energy transfer.

Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System

We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.

Linear response as a singular limit for a periodically driven closed quantum system

We address the issue of the validity of linear response theory for a closed quantum system subject to a periodic external driving. Linear response theory (LRT) predicts energy absorption at frequencies of the external driving where the imaginary part of the appropriate response function is different from zero. Here we show that, for a fairly general nonlinear many-body system on a lattice subject to an extensive perturbation, this approximation should be expected to be valid only up to a time t* depending on the strength of the driving, beyond which the true coherent Schrödinger evolution departs from the linear response prediction and the system stops absorbing energy from the driving. We exemplify this phenomenon in detail with the example of a quantum Ising chain subject to a time-periodic modulation of the transverse field, by comparing an exact Floquet analysis with the standard results of LRT. In this context, we also show that if the perturbation is just local, the system is expected in the thermodynamic limit to keep absorbing energy, and LRT works at all times. We finally argue more generally the validity of the scenario presented for closed quantum many-body lattice systems with a bound on the energy-per-site spectrum, discussing the experimental relevance of our findings in the context of cold atoms in optical lattices and ultra-fast spectroscopy experiments.

Understanding the entanglement entropy and spectra of 2D quantum systems through arrays of coupled 1D chains

We describe an algorithm for studying the entanglement entropy and spectrum of two-dimensional (2D) systems, as a coupled array of $N$ one-dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D (both in its disordered phase and near criticality), we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as ${c}_{\mathrm{eff}}\phantom{\rule{0.16em}{0ex}}\mathrm{log}(N)/6$ with ${c}_{\mathrm{eff}}\ensuremath{\approx}1$. Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of the 2D model. When short-range (area law) entanglement dominates we find (numerically and perturbatively) that this spectrum reflects the energy spectrum of a single quantum Ising chain.

Time-integrated observables as order parameters for full counting statistics transitions in closed quantum systems

The dynamical behavior of many-body systems is often richer than what can be anticipated from their static properties. Here we show that in closed quantum systems this becomes evident by considering the statistics of time-integrated observables. In particular, the analytic properties of their generating functions, as estimated by full counting statistics (FCS), allow one to identify FCS phases, i.e., phases with specific fluctuation properties of time-integrated observables, and to locate transitions between these phases. We discuss in detail the case of the quantum Ising chain in a transverse field. We show that this model displays a continuum of full counting statistics transitions, of which the static transition is just an end point. These singularities are not a consequence of particular choices of initial conditions or other external nonequilibrium protocols such as quenches in coupling constants. They can be probed generically through quantum jump statistics of an associated open problem, and for the case of the quantum Ising chain we outline a possible experimental realization of this scheme by digital quantum simulation with cold ions.

Dynamical phase transitions after quenches in nonintegrable models

We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a non-analytic function of time in the thermodynamic limit. This so-called `dynamical phase transition' was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for `interacting theories' indicate that non-analytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.

Majorana Fermion Realization of a Two-Channel Kondo Effect in a Junction of Three Quantum Ising Chains

It is shown that a junction of three critical quantum Ising chains (Delta junction) can be described as a two-channel Kondo model with a spin S=1/2 localized at the junction, which is composed of the respective Ising, zero energy boundary Majorana modes.