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arXiv:1310.4992 · Physical Review E

Three species of Schrödinger cat states in an infinite-range spin model

Bo Zhao, Cotty Kerridge, David A. Huse

Published Aug 5, 2014• 9 citations
arXivHTMLPDF

Abstract

We explore a transverse-field Ising model that exhibits both spontaneous symmetry breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrödinger cat states: superpositions of states with "up" and "down" spontaneous magnetization. This model exhibits two dynamical phase transitions within its ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels.

DOI
10.1103/physreve.90.022104

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We explore a transverse-field Ising model that exhibits both spontaneous symmetry breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrödinger cat states: superpositions of states with "up" and "down" spontaneous magnetization. This model exhibits two dynamical phase transitions within its ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels.
We study eigenstate thermalization and related signatures of quantum chaos in the one-dimensional ferromagnetic transverse-field Ising model with power-law interactions. The presence of long-range interactions allows for a finite-temperature phase transition despite the one-dimensional geometry of the model. Unlike previous studies of eigenstate thermalization in non-disordered systems with finite temperature phase transitions, our model possesses sufficiently many energy eigenstates below the critical energy density to allow us to make a definitive statement about the presence of eigenstate thermalization and chaotic level statistics in the broken-symmetry phase.
We study macroscopic thermalization in the ferromagnetic Ising model with a weak generic (highly nonlocal and many-body) random quantum perturbation. We prove that a single typical perturbation makes every initial state ∣Φ(0)⟩|Φ(0)\rangle∣Φ(0)⟩ in a specified energy shell thermalize: a measurement of a large class of macroscopic observables in the time-evolved state ∣Φ(t)⟩=e−iH~Lt∣Φ(0)⟩|Φ(t)\rangle=e^{-i\widetilde H_Lt}|Φ(0)\rangle∣Φ(t)⟩=e−iHL​t∣Φ(0)⟩ yields their thermal equilibrium values, within a prescribed precision and with overwhelming probability, at sufficiently large, typical times ttt. The result covers the full finite-temperature range, including the ordered phase, where we use plus boundary conditions. With periodic boundary conditions in the ordered phase, every energy eigenstate in the shell is shown to be an almost balanced macroscopic Schrödinger-cat state. Moreover, every pure initial state becomes such a cat at sufficiently large, typical times, with each branch in the corresponding macroscopic thermal equilibrium. The proof combines the theory of Roos, Sugimoto, Teufel, Tumulka, and Vogel with rigorous Ising large-deviation estimates.
We construct the finite-temperature dynamical phase diagram of the fully connected transverse-field Ising model from the vantage point of two disparate concepts of dynamical criticality. An analytical derivation of the classical dynamics and exact diagonalization simulations are used to study the dynamics after a quantum quench in the system prepared in a thermal equilibrium state. The different dynamical phases characterized by the type of nonanalyticities that emerge in an appropriately defined Loschmidt-echo return rate directly correspond to the dynamical phases determined by the spontaneous breaking of Z2{\mathbb{Z}}_{2}Z2​ symmetry in the long-time steady state. The dynamical phase diagram is qualitatively different depending on whether the initial thermal state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical phase diagram that can be directly related to its equilibrium counterpart, the latter gives rise to a divergent dynamical critical temperature at vanishing final transverse-field strength.
We report the existence of a large set of ferromagnetic scarred states in the one-dimensional transverse-field Ising model with long-range interactions, in a regime with no ferromagnetic phase at finite temperature. These scarred states are distributed over different spectral regions, surrounded by paramagnetic states. We show that simple initial conditions, consisting in a few small magnetic domains, selectively populate these scarred states. This leads to the appearance of a special dynamical phase, which we call scarred ferromagnetic phase. As a consequence, initial states with a small number of small magnetic domains evolve towards ferromagnetic equilibrium states, whereas initial states with larger domains or no magnetic structure relax to the expected thermal paramagnetic equilibrium state.
We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models which exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analogue of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in d=2d= 2d=2 dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body WKB" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our work represents a first step towards a rigorous classification of stable gapless quantum phases.
The eigenstate thermalization hypothesis (ETH) posits how isolated quantum many-body systems thermalize, assuming that individual eigenstates at the same energy density have identical expectation values of local observables in the limit of large systems. While the ETH apparently holds across a wide range of interacting quantum systems, in this work we show that it requires generalization in the presence of thermal first-order phase transitions. We introduce a class of all-to-all spin models, featuring first-order thermal phase transitions that stem from two distinct mean-field solutions (two ``branches'') that exchange dominance in the many-body density of states as the energy is varied. We argue that for energies in the vicinity of the thermal phase transition, eigenstate expectation values do not need to converge to the same thermal value. The system has a regime with coexistence of two classes of eigenstates corresponding to the two branches with distinct expectation values at the same energy density, and another regime with Schrodinger-cat-like eigenstates that are inter-branch superpositions; these two regimes are separated by an eigenstate phase transition. We support our results by semiclassical calculations and an exact diagonalization study of a microscopic spin model, and argue that the structure of eigenstates in the vicinity of thermal first-order phase transitions can be experimentally probed via non-equilibrium dynamics.