Infinite-range Interactions Research Articles

Anomalous dynamical phase in quantum spin chains with long-range interactions

The existence or absence of non-analytic cusps in the Loschmidt-echo return rate is traditionally employed to distinguish between a regular dynamical phase (regular cusps) and a trivial phase (no cusps) in quantum spin chains after a global quench. However, numerical evidence in a recent study [J. C. Halimeh and V. Zauner-Stauber, arXiv:1610.02019] suggests that instead of the trivial phase a distinct anomalous dynamical phase characterized by a novel type of non-analytic cusps occurs in the one-dimensional transverse-field Ising model when interactions are sufficiently long-range. Using an analytic semiclassical approach and exact diagonalization, we show that this anomalous phase also arises in the fully-connected case of infinite-range interactions, and we discuss its defining signature. Our results show that the transition from the regular to the anomalous dynamical phase coincides with Z2-symmetry breaking in the infinite-time limit, thereby showing a connection between two different concepts of dynamical criticality. Our work further expands the dynamical phase diagram of long-range interacting quantum spin chains, and can be tested experimentally in ion-trap setups and ultracold atoms in optical cavities, where interactions are inherently long-range.

Black Hole on a Chip: Proposal for a Physical Realization of the Sachdev-Ye-Kitaev model in a Solid-State System

System of Majorana zero modes with random infinite range interactions -- the Sachdev-Ye-Kitaev (SYK) model -- is thought to exhibit an intriguing relation to the horizons of extremal black holes in two-dimensional anti-de Sitter (AdS$_2$) space. This connection provides a rare example of holographic duality between a solvable quantum-mechanical model and dilaton gravity. Here we propose a physical realization of the SYK model in a solid state system. The proposed setup employs the Fu-Kane superconductor realized at the interface between a three dimensional topological insulator (TI) and an ordinary superconductor. The requisite $N$ Majorana zero modes are bound to a nanoscale hole fabricated in the superconductor that is threaded by $N$ quanta of magnetic flux. We show that when the system is tuned to the surface neutrality point (i.e. chemical potential coincident with the Dirac point of the TI surface state) and the hole has sufficiently irregular shape, the Majorana zero modes are described by the SYK Hamiltonian. We perform extensive numerical simulations to demonstrate that the system indeed exhibits physical properties expected of the SYK model, including thermodynamic quantities and two-point as well as four-point correlators, and discuss ways in which these can be observed experimentally.

Phase diagram of bosons in a two-dimensional optical lattice with infinite-range cavity-mediated interactions

A high-finesse optical cavity allows the establishment of long-range interactions between bosons in an optical lattice when most cold-atom experiments are restricted to short-range interactions. Supersolid phases have recently been experimentally observed in such systems. Using both exact quantum Monte Carlo simulations and the Gutzwiller approximation, we study the ground-state phase diagrams of a two-dimensional Bose-Hubbard model with infinite-range interactions which describes such experiments. In addition to superfluid and insulating Mott phases, the infinite-range checkerboard interactions introduce charge-density waves and supersolid phases. We study here the system at various particle densities, elucidate the nature of the phases and quantum phase transitions, and discuss the stability of the phases with respect to phase separation. In particular we confirm the existence and stability of a supersolid phase detected experimentally.

Spectral properties and phase diagram of correlated lattice bosons in an optical cavity within bosonic dynamical mean-field theory

We use the Bose-Hubbard model with an effective infinite-range interaction to describe the correlated lattice bosons in an optical cavity. We study both static and spectral properties of such system within the bosonic dynamical mean-field theory (B-DMFT), which is the state of the art method for strongly correlated bosonic systems. Both similarities and differences are found and discussed between our results and these obtained within different theoretical methods and experiment.

Degenerated ground-states in a spin chain with pair interactions: a characterization by symbolic dynamics

In this paper we study a class of one-dimensional spin chain having a highly degenerated set of ground-state configurations. The model consists of spin chain having infinite-range pair interactions with a given structure. We show that the set of ground-state configurations of such a model can be fully characterized by means of symbolic dynamics. Particularly we found that the set ground-state configurations define what in symbolic dynamics is called sofic shift space. Finally we prove that this system has a non-vanishing residual entropy (the topological entropy of the shift space), which can be exactly calculated.

Bose–Einstein condensates in an optical cavity with sub-recoil bandwidth

This article provides a brief synopsis of our recent work on the interaction of Bose–Einstein condensates with the light field inside an optical cavity exhibiting a bandwidth on the order of the recoil frequency. Three different coupling scenarios are discussed giving rise to different physical phenomena at the borderline between the fields of quantum optics and many-body physics. This includes sub-recoil opto-mechanical cooling, cavity-controlled matter wave superradiance and the emergence of a superradiant superfluid or a superradiant Mott insulating many-body phase in a self-organized intra-cavity optical lattice with retarded infinite range interactions.

Critical relaxation with overdamped quasiparticles in open quantum systems

We study the late-time relaxation following a quench in a driven-dissipative quantum many-body system. We consider the open Dicke model, describing the infinite-range interactions between $N$ atoms and a single, lossy electromagnetic mode. We show that the dynamical phase transition at a critical atom-light coupling is characterized by the interplay between reservoir-driven and intrinsic relaxation processes. Above the critical coupling, small fluctuations in the occupation of the dominant quasiparticle-mode start to grow in time while the quasiparticle lifetime remains finite due to losses. Near the critical interaction strength we observe a crossover between exponential and power-law $1/\tau$ relaxation, the latter driven by collisions between quasiparticles. For a quench exactly to the critical coupling, the power-law relaxation extends to infinite times, but the finite lifetime of quasiparticles prevents ageing to appear. We predict our results to be accessible to quench experiments with ultracold bosons in optical resonators.

Excited-state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation

Excited state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions (QPTs) to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-1/2 system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.

Numerical study of fermion and boson models with infinite-range random interactions

We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the $N$-site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of $N=\infty$ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature. The exact diagonalization results for the fermion Green's function also appear to converge well to the $N=\infty$ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

Equivalence of a mixing condition and the LSI in spin systems with infinite range interaction

We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincaré inequality based on a directional Poincaré inequality, which we derive through an averaging procedure. We show that this decay of correlations is equivalent to the Dobrushin–Shlosman mixing condition. With this, we also state and provide a partial answer to a conjecture regarding the relationship between the relaxation rates of non-ferromagnetic and ferromagnetic systems. Finally, we show that for a symmetric, ferromagnetic system with zero boundary conditions, a weaker decay of correlations can be bootstrapped.

Quantum Monte Carlo study of long-range transverse-field Ising models on the triangular lattice

Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\alpha$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\alpha$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,-\frac{M}{2}, -\frac{M}{2})$ with unsaturated $M < 1$ in a process known as "order by disorder" similar to the nearest neighbour antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest neighbour and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.

Efficient Localization Bounds in a Continuous N-Particle Anderson Model with Long-Range Interaction

We establish strong dynamical and exponential spectral localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. For the first time in the mathematical literature, the uniform decay bounds on the eigenfunction correlators (EFCs) at low energies are proved, in the multi-particle continuous configuration space, in the (symmetrized) norm-distance, which is a natural distance in the multi-particle configuration space, and not in the Hausdorff distance. This results in uniform bounds on the EFCs in arbitrarily large but bounded domains in the physical configuration space, and not only in the actually infinite space, as in prior works on multi-particle localization in Euclidean spaces.

Anisotropic four-state clock model in the presence of random fields.

A four-state clock ferromagnetic model is studied in the presence of different configurations of anisotropies and random fields. The model is considered in the limit of infinite-range interactions, for which the mean-field approach becomes exact. Both representations of Cartesian spin components and two Ising variables are used, in terms of which the physical properties and phase diagrams are discussed. The random fields follow bimodal probability distributions and the richest criticality is found when the fields, applied in the two Ising systems, are not correlated. The phase diagrams present new interesting topologies, with a wide variety of critical points, which are expected to be useful in describing different complex phenomena.

Gauge theory at asymptotical distance and singularity

One of the fundamental problems of gravity to be an infinite range interaction is whether it has a gauge group via the internal structure, so that Einstein type theories couldn’t explain it yet. Gauge groups in Yang Mill type theories, i.e. weak [11] and strong [3, 8] interactions, are generally non Abelian Lie group via the Hermitian structure induced by complex valued smooth vector fields. A group labeled by U(1)⊗Ũ(d) obtained as the reduction of unitary Lie groups is presented as candidate a gauge group for each behavior at large scale and at small scale of a gauge field, in this context, integrated Einstein Yang Mills formalism on a unique geometrical frame is given. Also as seen that the (anti) self dual gauge theory of the group G = U(1) ⊗ Ũ(4) is indeed a G = U(1) theory in real 4 dimension, i.e the Maxwell theory for the electromagnetism to be an infinite range interaction. Mathematics Subject Classification: 22E15, 53C07, 53C80, 70S05, 70S15, 83C05

Competing interactions and traveling wave solutions in lattice differential equations

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj [1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.

Observation of a Superradiant Mott Insulator in the Dicke-Hubbard Model.

It is well known that the bosonic Hubbard model possesses a Mott insulator phase. Likewise, it is known that the Dicke model exhibits a self-organized superradiant phase. By implementing an optical lattice inside of a high-finesse optical cavity, both models are merged such that an extended Hubbard model with cavity-mediated infinite range interactions arises. In addition to a normal superfluid phase, two superradiant phases are found, one of them coherent and hence superfluid and one incoherent Mott insulating.

Bekenstein-Hawking Entropy and Strange Metals

We examine models of fermions with infinite-range interactions which realize non-Fermi liquids with a continuously variable U(1) charge density $\mathcal{Q}$, and a non-zero entropy density $\mathcal{S}$ at vanishing temperature. Real time correlators of operators carrying U(1) charge $q$ at a low temperature $T$ are characterized by a $\mathcal{Q}$-dependent frequency $\omega_{\mathcal{S}} = (q \, T/\hbar) (\partial \mathcal{S}/\partial{\mathcal{Q}})$ which determines a spectral asymmetry. We show that the correlators match precisely with those of the AdS$_2$ horizons of extremal charged black holes. On the black hole side, the matching employs $\mathcal{S}$ as the Bekenstein-Hawking entropy density, and the laws of black hole thermodynamics which relate $(\partial{\mathcal{S}}/\partial{\mathcal{Q}})/(2 \pi)$ to the electric field strength in AdS$_2$. The fermion model entropy is computed using the microscopic degrees of freedom of a UV complete theory without supersymmetry.

Termination of the Berezinskii-Kosterlitz-Thouless phase with a new critical universality in spin-crossover systems

Two-dimensional systems with $U(1)$ symmetry exhibit a peculiar phase, i.e., the Berezinskii-Kosterlitz-Thouless (BKT) phase. In particular situations, the BKT phase exists as an intermediate-temperature phase. There have been scenarios for the phase transitions at the two end points of the intermediate BKT phase; i.e., the phase transition at the low-temperature end point is a BKT transition and that at the high-temperature end point is either a BKT transition or a first-order transition. The present study gives a novel scenario, i.e., a second-order transition with a new critical universality and a BKT transition. We found that this new phase transition is realized in spin-crossover systems on a triangular lattice with an antiferromagnetic short-range interaction. At the low-temperature transition the elastic interaction plays as a ferromagnetic infinite-range interaction and encourages the breaking of ${Z}_{2}$ symmetry between high-spin-rich and low-spin-rich states.

Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism

There has been rapid development of systems that yield strong interactions between freely propagating photons in one-dimension via controlled coupling to quantum emitters. This raises interesting possibilities such as quantum information processing with photons or quantum many-body states of light, but treating such systems generally remains a difficult task theoretically. Here, we describe a novel technique in which the dynamics and correlations of a few photons can be exactly calculated, based upon knowledge of the initial photonic state and the solution of the reduced effective dynamics of the quantum emitters alone. We show that this generalized ‘input–output’ formalism allows for a straightforward numerical implementation regardless of system details, such as emitter positions, external driving, and level structure. As a specific example, we apply our technique to show how atomic systems with infinite-range interactions and under conditions of electromagnetically induced transparency enable the selective transmission of correlated multi-photon states.

Parametric estimation of pairwise Gibbs point processes with infinite range interaction

This paper is concerned with statistical inference for infinite range interaction Gibbs point processes and in particular for the large class of Ruelle superstable and lower regular pairwise interaction models. We extend classical statistical methodologies such as the pseudolikelihood and the logistic regression methods, originally defined and studied for finite range models. Then we prove that the associated estimators are strongly consistent and satisfy a central limit theorem, provided the pairwise interaction function tends sufficiently fast to zero. To this end, we introduce a new central limit theorem for almost conditionally centered triangular arrays of random fields.