Aspects of Confinement in Low Dimensions

We briefly review some examples of confinement which arise in condensed matter physics. We focus on two instructive cases: the off-critical Ising model in a magnetic field, and an array of weakly coupled (extended) Hubbard chains in the Wigner crystal phase. In the appropriate regime, the elementary excitations in these 1+1 and quasi-one-dimensional systems are confined into `mesons'. Although the models are generically non-integrable, quantum mechanics and form factor techniques yield valuable information.Comment: Contribution to Ian Kogan memorial volume, World Scientifi

Feshbach Resonance in Optical Lattices and the Quantum Ising Model

Motivated by experiments on heteronuclear Feshbach resonances in Bose mixtures, we investigate s-wave pairing of two species of bosons in an optical lattice. The zero temperature phase diagram supports a rich array of superfluid and Mott phases and a network of quantum critical points. This topology reveals an underlying structure that is succinctly captured by a two-component Landau theory. Within the second Mott lobe we establish a quantum phase transition described by the paradigmatic longitudinal and transverse field Ising model. This is confirmed by exact diagonalization of the 1D bosonic Hamiltonian. We also find this transition in the homonuclear case.Comment: 5 pages, 4 figure

Collective Dynamics of Bose--Einstein Condensates in Optical Cavities

Recent experiments on Bose--Einstein condensates in optical cavities have reported a quantum phase transition to a coherent state of the matter-light system -- superradiance. The time dependent nature of these experiments demands consideration of collective dynamics. Here we establish a rich phase diagram, accessible by quench experiments, with distinct regimes of dynamics separated by non-equilibrium phase transitions. We include the key effects of cavity leakage and the back-reaction of the cavity field on the condensate. Proximity to some of these phase boundaries results in critical slowing down of the decay of many-body oscillations. Notably, this slow decay can be assisted by large cavity losses. Predictions include the frequency of collective oscillations, a variety of multi-phase co-existence regions, and persistent optomechanical oscillations described by a damped driven pendulum. These findings open new directions to study collective dynamics and non-equilibrium phase transitions in matter-light systems.Comment: 5 pages, 5 figure

Non-equilibrium steady states in the Klein-Gordon theory

We construct non-equilibrium steady states in the Klein-Gordon theory in arbitrary space dimension $d$ following a local quench. We consider the approach where two independently thermalized semi-infinite systems, with temperatures $T_{\rm L}$ and $T_{\rm R}$, are connected along a $d-1$-dimensional hypersurface. A current-carrying steady state, described by thermally distributed modes with temperatures $T_{\rm L}$ and $T_{\rm R}$ for left and right-moving modes, respectively, emerges at late times. The non-equilibrium density matrix is the exponential of a non-local conserved charge. We obtain exact results for the average energy current and the complete distribution of energy current fluctuations. The latter shows that the long-time energy transfer can be described by a continuum of independent Poisson processes, for which we provide the exact weights. We further describe the full time evolution of local observables following the quench. Averages of generic local observables, including the stress-energy tensor, approach the steady state with a power-law in time, where the exponent depends on the initial conditions at the connection hypersurface. We describe boundary conditions and special operators for which the steady state is reached instantaneously on the connection hypersurface. A semiclassical analysis of freely propagating modes yields the average energy current at large distances and late times. We conclude by comparing and contrasting our findings with results for interacting theories and provide an estimate for the timescale governing the crossover to hydrodynamics. As a modification of our Klein-Gordon analysis we also include exact results for free Dirac fermions.Comment: 42 pages, 7 figure

Quantum Phase Transitions in Bosonic Heteronuclear Pairing Hamiltonians

We explore the phase diagram of two-component bosons with Feshbach resonant pairing interactions in an optical lattice. It has been shown in previous work to exhibit a rich variety of phases and phase transitions, including a paradigmatic Ising quantum phase transition within the second Mott lobe. We discuss the evolution of the phase diagram with system parameters and relate this to the predictions of Landau theory. We extend our exact diagonalization studies of the one-dimensional bosonic Hamiltonian and confirm additional Ising critical exponents for the longitudinal and transverse magnetic susceptibilities within the second Mott lobe. The numerical results for the ground state energy and transverse magnetization are in good agreement with exact solutions of the Ising model in the thermodynamic limit. We also provide details of the low-energy spectrum, as well as density fluctuations and superfluid fractions in the grand canonical ensemble.Comment: 11 pages, 14 figures. To appear in Phys. Rev.

Magnetic Properties of the Second Mott Lobe in Pairing Hamiltonians

We explore the Mott insulating state of single-band bosonic pairing Hamiltonians using analytical approaches and large scale density matrix renormalization group calculations. We focus on the second Mott lobe which exhibits a magnetic quantum phase transition in the Ising universality class. We use this feature to discuss the behavior of a range of physical observables within the framework of the 1D quantum Ising model and the strongly anisotropic Heisenberg model. This includes the properties of local expectation values and correlation functions both at and away from criticality. Depending on the microscopic interactions it is possible to achieve either antiferromagnetic or ferromagnetic exchange interactions and we highlight the possibility of observing the E8 mass spectrum for the critical Ising model in a longitudinal magnetic field.Comment: 14 pages, 15 figure

Non-equilibrium quantum spin dynamics from classical stochastic processes

Following on from our recent work, we investigate a stochastic approach to non-equilibrium quantum spin systems. We show how the method can be applied to a variety of physical observables and for different initial conditions. We provide exact formulae of broad applicability for the time-dependence of expectation values and correlation functions following a quantum quench in terms of averages over classical stochastic processes. We further explore the behavior of the classical stochastic variables in the presence of dynamical quantum phase transitions, including results for their distributions and correlation functions. We provide details on the numerical solution of the associated stochastic differential equations, and examine the growth of fluctuations in the classical description. We discuss the strengths and limitations of the current implementation of the stochastic approach and the potential for further development

Stochastic Approach to Non-Equilibrium Quantum Spin Systems

We investigate a stochastic approach to non-equilibrium quantum spin systems based on recent insights linking quantum and classical dynamics. Exploiting a sequence of exact transformations, quantum expectation values can be recast as averages over classical stochastic processes. We illustrate this approach for the quantum Ising model by extracting the Loschmidt amplitude and the magnetization dynamics from the numerical solution of stochastic differential equations. We show that dynamical quantum phase transitions are accompanied by clear signatures in the associated classical distribution functions, including the presence of enhanced fluctuations. We demonstrate that the method is capable of handling integrable and non-integrable problems in a unified framework, including those in higher dimensions.Comment: 5 pages, 5 figure