Power-law Long-range Interactions Research Articles

Entanglement phases in large- N hybrid Brownian circuits with long-range couplings

We develop solvable models of large-$N$ hybrid quantum circuits on qubits and fermions with long-range power-law interactions and continuous local monitoring, which provide analytical access to the entanglement phase diagram and error-correcting properties of many-body entangled non-equilibrium states generated by such dynamics. In one dimension, the long-range coupling is irrelevant for $\alpha>3/2$, where $\alpha$ is the power-law exponent, and the models exhibit a conventional measurement-induced phase transition between volume- and area-law entangled phases. For $1/2<\alpha<3/2$ the long-range coupling becomes relevant, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for $\alpha<1$ the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases, indicating that the phase realizes a quantum error correcting code whose code distance scales as $L^{2-2\alpha}$. While the entanglement phase diagram is the same for both the interacting qubit and fermionic hybrid Brownian circuits, we find that long-range free-fermionic circuits exhibit a distinct phase diagram with two different fractal entangled phases.

Quantum chaos and ensemble inequivalence of quantum long-range Ising chains

We use large-scale exact diagonalization to study the quantum Ising chain in a transverse field with long-range power-law interactions decaying with exponent $\alpha$. We numerically study various probes for quantum chaos and eigenstate thermalization {on} the level of eigenvalues and eigenstates. The level-spacing statistics yields a clear sign towards a Wigner-Dyson distribution and therefore towards quantum chaos across all values of $\alpha>0$. Yet, for $\alpha<1$ we find that the microcanonical entropy is nonconvex. This is due to the fact that the spectrum is organized in energetically separated multiplets for $\alpha<1$. While quantum chaotic behaviour develops within the individual multiplets, many multiplets don't overlap and don't mix with each other, as we analytically and numerically argue. Our findings suggest that a small fraction of the multiplets could persist at low energies for $\alpha\ll 1$ even for large $N$, giving rise to ensemble inequivalence.

Aging in the Long-Range Ising Model.

The current understanding of aging phenomena is mainly confined to the study of systems with short-ranged interactions. Little is known about the aging of long-ranged systems. Here, the aging in the phase-ordering kinetics of the two-dimensional Ising model with power-law long-range interactions is studied via MonteCarlo simulations. The dynamical scaling of the two-time spin-spin autocorrelator is well described by simple aging for all interaction ranges studied. The autocorrelation exponents are consistent with λ=1.25 in the effectively short-range regime, while for stronger long-range interactions the data are consistent with λ=d/2=1. For very long-ranged interactions, strong finite-size effects are observed. We discuss whether such finite-size effects could be misinterpreted phenomenologically as subaging.

Quantum chaos dynamics in long-range power law interaction systems

We use out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: the number of qubits $N$ on each site (the onsite Hilbert space dimension) and the power law exponent $\alpha$. Three light cone structures of OTOC appear at $N = 1$: (1) logarithmic when $0.5<\alpha\lesssim 0.8$, (2) sublinear power law when $0.8 \lesssim \alpha \lesssim 1.5$ and (3) linear when $\alpha \gtrsim 1.5$. The OTOC scales as $\exp(\lambda t)/x^{2\alpha} $ and $t^{2 \alpha / \zeta} / x^{ 2 \alpha} $ respectively beyond the light cones in the first two cases. When $\alpha \geq 2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large $N$ limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $ \alpha \gtrsim 1.5$. This implies the locality is never fully recovered for finite $\alpha$. Our result provides a unified physical picture for the chaos dynamics in long-range power law interaction system.

On fractional and fractal formulations of gradient linear and nonlinear elasticity

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

Fermi-Pasta-Ulam chains with harmonic and anharmonic long-range interactions

We study the dynamics of Fermi-Pasta-Ulam (FPU) chains with both harmonic and anharmonic power-law long-range interactions. We show that the dynamics is described in the continuum limit by a Generalized Fractional Boussinesq differential Equation (GFBE), whose derivation is performed in full detail. We also discuss a version of the model where couplings are alternating in sign.

Comment on “Many-body localization in Ising models with random long-range interactions”

This comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power law interactions, $r^{-\alpha}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has been earlier argued [1, 2] that this model obeys the dimensional constraint suggesting the delocalization of all finite temperature states in thermodynamic limit for $\alpha \leq 2d$ in a $d$-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model in Ref. [3]. To resolve this controversy we reexamine the model of Ref. [3] and demonstrate that the infinite temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization [2] is extended to small exponents $0 \leq \alpha \leq d$. Disagreements between two works are explained by the non-standard selection of investigated states in the ordered phase and misinterpretation of the localization-delocalization transition in Ref. [3].

Dynamical phase diagram of quantum spin chains with long-range interactions

Using an infinite Matrix Product State (iMPS) technique based on the time-dependent variational principle (TDVP), we study two major types of dynamical phase transitions (DPT) in the one-dimensional transverse-field Ising model (TFIM) with long-range power-law ($\propto1/r^{\alpha}$ with $r$ inter-spin distance) interactions out of equilibrium in the thermodynamic limit -- \textit{DPT-I}: based on an order parameter in a (quasi-)steady state, and \textit{DPT-II}: based on non-analyticities (cusps) in the Loschmidt-echo return rate. We construct the corresponding rich dynamical phase diagram, whilst considering different quench initial conditions. We find a nontrivial connection between both types of DPT based on their critical lines. Moreover, and very interestingly, we detect a new DPT-II dynamical phase in a certain range of interaction exponent $\alpha$, characterized by what we call \textit{anomalous cusps} that are distinct from the \textit{regular cusps} usually associated with DPT-II. Our results provide the characterization of experimentally accessible signatures of the dynamical phases studied in this work.

Fidelity susceptibility study of quantum long-range antiferromagnetic Ising chain

We study the fidelity susceptibility of quantum antiferromagnetic Ising chain with a long-range power law interaction $1/r^{\alpha}$ using the large-scale density matrix renormalization group method. We find that the critical adiabatic dimension $\mu=2$ and the critical exponent of the correlation length $\nu=1$ for arbitrary $\alpha>0$, indicating all quantum phase transitions are second-order Ising transitions. In addition, we numerically determine the complete phase diagram for $0 < \alpha \le 3$ from the data collapse of the fidelity susceptibility and show that the critical point $h_c$ changes monotonously with respect to $\alpha$. This work will shed light on the nature of phase transitions in the quantum long-range antiferromagnetic Ising chain from a quantum information perspective.

Metastable minima of the Heisenberg spin glass in a random magnetic field

We have studied zero-temperature metastable minima in classical m-vector component spin glasses in the presence of m-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for m>2. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the m=2 and m=3 cases in the SK model the final energy reached in the quench is very close to the energy E_{c} at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states ρ(λ) of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual sqrt[λ] form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.

Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions.

The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.

Finite-size critical scaling in Ising spin glasses in the mean-field regime.

We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.

Synchronization and plateau splitting of coupled oscillators with long-range power-law interactions.

We investigate synchronization and plateau splitting of coupled oscillators on a one-dimensional lattice with long-range interactions that decay over distance as a power law. We show that in the thermodynamic limit the dynamics of systems of coupled oscillators with power-law exponent α≤1 is identical to that of the all-to-all coupling case. For α>1, oscillatory behavior of the phase coherence appears as a result of single plateau splitting into multiple plateaus. A coarse-graining method is used to investigate the onset of plateau splitting. We analyze a simple oscillatory state formed by two plateaus in detail and propose a systematic approach to predict the onset of plateau splitting. The prediction of breaking points of plateau splitting is in quantitatively good agreement with numerical simulations.

Relationships between power-law long-range interactions and fractional mechanics

We investigate the relationships between models of power-law long-range interactions and mechanics based on fractional derivatives. We present the fractional Lagrangian density which gives the Euler–Lagrange equation that serves as the equation of motion for fractional-power-law long-range interactions. We derive this equation by the fractional variational method. In addition, we derive a Noether-like current from the fractional Lagrangian density.

Ordering of the Heisenberg spin glass in four dimensions

Ordering of the Heisenberg spin glass in four dimensions (4D) with the nearest-neighbor Gaussian coupling is investigated by equilibrium Monte Carlo simulations, with particular attention to its spin and chiral orderings. It is found that the spin and the chirality order simultaneously with a common correlation-length exponent ${\ensuremath{\nu}}_{\mathrm{CG}}={\ensuremath{\nu}}_{\mathrm{SG}}\ensuremath{\simeq}1.0$ i.e., the absence of the spin-chirality decoupling in 4D. Yet, the spin-glass ordered state exhibits a nontrivial phase-space structure associated with a continuous one-step-like replica-symmetry breaking, different in nature from that of the Ising spin glass or of the mean-field spin glass. Comparison is made with the ordering of the Heisenberg spin glass in 3D, and with that of the 1D Heisenberg spin glass with a long-range power-law interaction. It is argued that 4D might be close to the marginal dimension separating the spin-chirality decoupling/coupling regimes.

Evidence of a glass transition in a ten-state non-mean-field Potts glass

Potts glasses are prototype models that have been used to understand the structural glass transition. However, in finite space dimensions a glass transition remains to be detected in the 10-state Potts glass. Using a one-dimensional model with long-range power-law interactions we present evidence that a glass transition below the upper critical dimension can exist for short-range systems at low enough temperatures. Gaining insights into the structural glass transition for short-range systems using spin models is thus potentially possible, yet difficult.

Monte Carlo Studies of the Ordering of the One-Dimensional Heisenberg Spin Glass with Long-Range Power-Law Interactions

The nature of the ordering of the one-dimensional Heisenberg spin-glass model with a long-range power-law interaction is studied by extensive Monte Carlo simulations, with particular attention to the issue of the spin-chirality decoupling/coupling. Large system sizes up to $L=4096$ are studied. With varying the exponent $\sigma$ describing the power-law interaction, we observe three distinct types of ordering regimes. For smaller $\sigma$, the spin and the chirality order at a common finite temperature with a common correlation-length exponent, exhibiting the standard spin-chirality coupling behavior. For intermediate $\sigma$, the chirality orders at a temperature higher than the spin, exhibiting the spin-chirality decoupling behavior. For larger $\sigma$, both the spin and the chirality order at zero temperature. We construct a phase diagram in the $\sigma$ versus the temperature plane, and discuss implications of the results. Critical properties associated with both the chiral-glass and the spin-glass transitions are also determined.

Spin-Chirality Decoupling in the One-Dimensional Heisenberg Spin Glass with Long-Range Power-Law Interactions

We study the issue of the spin-chirality decoupling or coupling in the ordering of the Heisenberg spin glass by performing large-scale Monte Carlo simulations on a one-dimensional Heisenberg spin-glass model with a long-range power-law interaction up to large system sizes. We find that the spin-chirality decoupling occurs for an intermediate range of the power-law exponent. Implications to the corresponding d-dimensional short-range model are discussed.

Synchronization of oscillators with long-range power law interactions

We present analytical calculations and numerical simulations for the synchronization of oscillators interacting via a long-range power law interaction on a one-dimensional lattice. We have identified the critical value of the power law exponent α(c) across which a transition from a synchronized to an unsynchronized state takes place for a sufficiently strong but finite coupling strength in the large system limit. We find α(c)=3/2. Frequency entrainment and phase ordering are discussed as a function of α≥1 . The calculations are performed using an expansion about the aligned phase state (spin-wave approximation) and a coarse graining approach. We also generalize the spin-wave results to the d -dimensional problem.

Long-range effects on superdiffusive algebraic solitons in anharmonic chains

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The position variance of the soliton shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest neighbour interactions (NNI). In contrast to the NNI case where the position variance at moderate soliton velocities has a considerable linear time-dependence (normal diffusion), the solitons with LRI are dominated by a superdiffusive mechanism where the position variance mainly depends quadratic and cubic on time. Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on a FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI). A collective variable (CV) approach in the continuum approximation of the system leads to stochastic integro-differential equations which can be reduced to Langevin-type equations for the CV position and width. We are able to derive an analytical result for the soliton diffusion which agrees well with the simulations of the discrete system. Despite of structurally similar Langevin systems for the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic $t^{1.5}$ time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of $\pi$ smaller for algebraic solitons.