Exact Numerical Techniques Research Articles

Microscopic Origin of the Quantum Mpemba Effect in Integrable Systems.

The highly complicated nature of far from equilibrium systems can lead to a complete breakdown of the physical intuition developed in equilibrium. A famous example of this is the Mpemba effect, which states that nonequilibrium states may relax faster when they are further from equilibrium or, put another way, hot water can freeze faster than warm water. Despite possessing a storied history, the precise criteria and mechanisms underpinning this phenomenon are still not known. Here, we study a quantum version of the Mpemba effect that takes place in closed many-body systems with a U(1) conserved charge: in certain cases a more asymmetric initial configuration relaxes and restores the symmetry faster than a more symmetric one. In contrast to the classical case, we establish the criteria for this to occur in arbitrary integrable quantum systems using the recently introduced entanglement asymmetry. We describe the quantum Mpemba effect in such systems and relate the properties of the initial state, specifically its charge fluctuations, to the criteria for its occurrence. These criteria are expounded using exact analytic and numerical techniques in several examples, a free fermion model, the Rule 54 cellular automaton, and the Lieb-Liniger model.

Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model

Spectral correlations are a powerful tool to study the dynamics of quantum many-body systems. For Hermitian Hamiltonians, quantum chaotic motion is related to random matrix theory spectral correlations. Based on recent progress in the application of spectral analysis to non-Hermitian quantum systems, we show that local level statistics, which probe the dynamics around the Heisenberg time, of a non-Hermitian $q$-body Sachdev-Ye-Kitev (nHSYK) model with $N$ Majorana fermions, and its chiral and complex-fermion extensions, are also well described by random matrix theory for $q > 2$, while for $q = 2$, they are given by the equivalent of Poisson statistics. For that comparison, we combine exact diagonalization numerical techniques with analytical results obtained for some of the random matrix spectral observables. Moreover, depending on $q$ and $N$, we identify $19$ out of the $38$ non-Hermitian universality classes in the nHSYK model, including those corresponding to the tenfold way. In particular, we realize explicitly $14$ out of the $15$ universality classes corresponding to non-pseudo-Hermitian Hamiltonians that involve universal bulk correlations of classes ${\rm AI}^\dagger$ and ${\rm AII}^\dagger$, beyond the Ginibre ensembles. These results provide strong evidence of striking universal features in non-unitary many-body quantum chaos, which in all cases can be captured by nHSYK models with $q > 2$.

Total yield of electron-positron pairs produced from vacuum in strong electromagnetic fields: Validity of the locally constant field approximation

The widely used locally constant field approximation (LCFA) can be utilized in order to derive a simple closed-form expression for the total number of particles produced in the presence of a strong electromagnetic field of a general spatiotemporal configuration. A usual justification for this approximate approach is the requirement that the external field vary slowly in space and time. In this investigation, we examine the validity of the LCFA by comparing its predictions to the results obtained by means of exact nonperturbative numerical techniques. To benchmark the LCFA in the regime of small field amplitudes and low frequencies, we employ a semiclassical approach. As a reference, we consider a standing electromagnetic wave oscillating both in time and space as well as two spatially uniform field configurations: Sauter pulse and oscillating electric field. Performing a thorough numerical analysis, we identify the domain of the field parameters where the approximation is well justified. In particular, it is demonstrated that the Keldysh parameter itself is not a relevant quantity governing the accuracy of the LCFA. It is also found that for field amplitudes close to the Schwinger critical value, the LCFA may overestimate the particle yield. We discuss how this effect is related to the Pauli exclusion principle and quantum interference.

Magnetic phase diagram of a spin-1/2 XXZ chain with modulated Dzyaloshinskii-Moriya interaction.

We consider the ground-state phase diagram of a one-dimensional spin-1/2 XXZ chain with a spatially modulated Dzyaloshinskii-Moriya interaction in the presence of an alternating magnetic field applied along the z[over ̂] axis. The model is studied using the continuum-limit bosonization approach and the finite system exact numerical technique. In the absence of a magnetic field, the ground-state phase diagram of the model includes, besides the ferromagnetic and gapless Luttinger-liquid phases, two gapped phases: the composite (C1) phase characterized by the coexistence of long-range-ordered (LRO) alternating dimerization and spin chirality patterns, and the composite (C2) phase characterized by, in addition to the coexisting spin dimerization and alternating chirality patterns, the presence of LRO antiferromagnetic order. In the case of mentioned composite gapped phases, and in the case of a uniform magnetic field, the commensurate-incommensurate type quantum phase transitions from a gapful phase into a gapless phase have been identified and described using the bosonization treatment and finite chain exact diagonalization studies. The upper critical magnetic field corresponding to the transition into a fully polarized state has been also determined. It has been shown that the very presence of a staggered component of the magnetic field vapes the composite (C1) in favor of the composite gapped (C2) phase.

Entanglement revivals as a probe of scrambling in finite quantum systems

The entanglement evolution after a quantum quench became one of the tools to distinguish integrable versus chaotic (non-integrable) quantum many-body dynamics. Following this line of thoughts, here we propose that the revivals in the entanglement entropy provide a finite-size diagnostic benchmark for the purpose. Indeed, integrable models display periodic revivals manifested in a dip in the block entanglement entropy in a finite system. On the other hand, in chaotic systems, initial correlations get dispersed in the global degrees of freedom (information scrambling) and such a dip is suppressed. We show that while for integrable systems the height of the dip of the entanglement of an interval of fixed length decays as a power law with the total system size, upon breaking integrability a much faster decay is observed, signalling strong scrambling. Our results are checked by exact numerical techniques in free-fermion and free-boson theories, and by time-dependent density matrix renormalisation group in interacting integrable and chaotic models.

Engineering non-equilibrium quantum phase transitions via causally gapped Hamiltonians

We introduce a phenomenological theory for many-body control of critical phenomena by engineering causally-induced gaps for quantum Hamiltonian systems. The core mechanisms are controlling information flow within and/or between clusters that are created near a quantum critical point. To this end, we construct inhomogeneous quantum phase transitions via designing spatiotemporal quantum fluctuations. We show how non-equilibrium evolution of disordered quantum systems can create new effective correlation length scales and effective dynamical critical exponents. In particular, we construct a class of causally-induced non-adiabatic quantum annealing transitions for strongly disordered quantum Ising chains leading to exponential suppression of topological defects beyond standard Kibble–Zurek predictions. Using exact numerical techniques for 1D quantum Hamiltonian systems, we demonstrate that our approach exponentially outperforms adiabatic quantum computing. Using strong-disorder renormalization group (SDRG), we demonstrate the universality of inhomogeneous quantum critical dynamics and exhibit the reconstructions of causal zones during SDRG flow. We derive a scaling relation for minimal causal gaps showing they narrow more slowly than any polynomial with increasing size of system, in contrast to stretched exponential scaling in standard adiabatic evolution. Furthermore, we demonstrate similar scaling behavior for random cluster-Ising Hamiltonians with higher order interactions.

Exploring many-body localization and thermalization using semiclassical methods

The Discrete Truncated Wigner Approximation (DTWA) is a semi-classical phase space method useful for the exploration of Many-body quantum dynamics. In this work we investigate Many-Body Localization (MBL) and thermalization using DTWA and compare its performance to exact numerical solutions. By taking as a benchmark case a 1D random field Heisenberg spin chain with short range interactions, and by comparing to numerically exact techniques, we show that DTWA is able to reproduce dynamical signatures that characterize both the thermal and the MBL phases. It exhibits the best quantitative agreement at short times deep in each of the phases and larger mismatches close to the phase transition. The DTWA captures the logarithmic growth of entanglement in the MBL phase, even though a pure classical mean-field analysis would lead to no dynamics at all. Our results suggest the DTWA can become a useful method to investigate MBL and thermalization in experimentally relevant settings intractable with exact numerical techniques, such as systems with long range interactions and/or systems in higher dimensions.

Quantum Monte Carlo study of the Bose-polaron problem in a one-dimensional gas with contact interactions

We present a theoretical study based upon quantum Monte Carlo methods of the Bose polaron in one-dimensional systems with contact interactions. In this instance of the problem of a single impurity immersed in a quantum bath, the medium is a Lieb-Liniger gas of bosons ranging from the weakly interacting to the Tonks-Girardeau regime, whereas the impurity is coupled to the bath via a different contact potential producing both repulsive and attractive interactions. Both the case of a mobile impurity, having the same mass as the particles in the medium, and of a static impurity with infinite mass are considered. We make use of exact numerical techniques that allow us to calculate the ground-state energy of the impurity, its effective mass as well as the contact parameter between the impurity and the bath. These quantities are investigated as a function of the strength of interactions between the impurity and the bath and within the bath. In particular, we find that the effective mass rapidly increases to very large values when the impurity gets strongly coupled to an otherwise weakly repulsive bath. This heavy impurity hardly moves within the medium, thereby realizing the "self-localization" regime of the Landau-Pekar polaron. Furthermore, we compare our results with predictions of perturbation theory valid for weak interactions and with exact solutions available when the bosons in the medium behave as impenetrable particles.

A Model where Anesthetic Effects on Ion Channels are Mediated by a Critical Membrane

Diverse molecules induce general anesthesia with potency strongly correlated both with their hydrophobicity and their effects on certain ion channels. We recently observed that several n-alcohol anesthetics inhibit heterogeneity in plasma membrane derived vesicles by lowering the critical temperature (Tc) for phase separation. Furthermore, rare treatments that reverse anesthetic effects, including high hydrostatic pressure, actually raise Tc, reversing anesthetic effects on membrane critical temperatures (Machta et. al., Biophys J , 111:3, (2016)). We will present a model in which ion channels are regulated by the nearly critical membrane in which they are embedded, and where anesthetics interfere with and mimic this regulation. In our model, ion channels are allosterically coupled to Ising composition modes. We use a mix of Monte-Carlo and exact numerical techniques to show that this type of regulation has several surprising features (Kimchi et. al., ArXiV 1607.06836, (2016)). Owing to diverging generalized susceptibilities, such a channel's activity becomes strongly influenced by perturbations that influence the critical temperature, in a manner that can, in principle, quantitatively account for changes in channel gating that accompany addition of anesthetics. In addition, the channel's kinetics acquire a range of time scales from its surrounding membrane, naturally leading to non-Markovian dynamics.

Shielding effectiveness analysis of an infinite cylindrical structure with a lateral aperture by modal method of moments

In this study, an efficient and exact numerical technique based on modal method of moments (MoM) is presented to evaluate shielding effectiveness of an infinite cylindrical enclosure with a lateral aperture on its wall. To this end, modal vector wave functions are used to represent the electric field and consequently the surface magnetic current on the aperture. To calculate inside and outside magnetic fields, the circular waveguide dyadic Green's functions are utilised. The unknown magnetic current on the aperture is calculated using the MoM by enforcing the continuity of tangential magnetic field on the aperture. This method is employed for different enclosure sizes when illuminated by transverse magnetic (TM) polarised wave at two different incident angles.

Electron confinement in graphene with gate-defined quantum dots

We theoretically analyse the possibility to electrostatically confine electrons in circular quantum dot arrays, impressed on contacted graphene nanoribbons by top gates. Utilising exact numerical techniques, we compute the scattering efficiency of a single dot and demonstrate that for small-sized scatterers the cross-sections are dominated by quantum effects, where resonant scattering leads to a series of quasi-bound dot states. Calculating the conductance and the local density of states for quantum dot superlattices we show that the resonant carrier transport through such graphene-based nanostructures can be easily tuned by varying the gate voltage.

Order, Criticality, and Excitations in the Extended Falicov-Kimball Model

Using exact numerical techniques, we investigate the nature of excitonic (electron-hole) bound states and the development of exciton coherence in the one-dimensional half-filled extended Falicov-Kimball model. The ground-state phase diagram of the model exhibits, besides band-insulator and staggered orbital ordered phases, an excitonic insulator (EI) with power-law correlations. The criticality of the EI state shows up in the von Neumann entropy. The anomalous spectral function and condensation amplitude provide the binding energy and coherence length of the electron-hole pairs which, on their part, point towards a Coulomb interaction driven crossover from BCS-like electron-hole pairing fluctuations to tightly bound excitons. We show that while a mass imbalance between electrons and holes does not affect the location of the BCS-BEC crossover regime, it favors staggered orbital ordering to the disadvantage of the EI. Within the Bose-Einstein condensation (BEC) regime, the quasiparticle dispersion develops a flat valence-band top, in accord with the experimental finding for Ta2NiSe5.

Translocation of a polymer through a nanopore modulated by a sticky site

Using a one-dimensional model for the translocation of a polymer through a nanopore, the effect of a "sticky site" at which the polymer binds to the pore is explored via exact numerical techniques. Results for the mean translocation time and the probability of translocation on the insertion of the first monomer in the pore are generated across a wide range of driving forces and binding potential strengths (well depths). The balance between the driving force, diffusion, and well depth yields a rich set of dynamics that depend strongly on where the sticky site is located along the polymer. For example, when the sticky site is located near the head of the polymer, the translocation time is found to be a maximum at an intermediate driving force with events at lower driving forces taking less time. Additionally, the critical well depth at which the sticky site dominates the dynamics, is found to be a non-monotonic function of the driving force when the sticky site is located at the head or tail of the polymer, but not in the middle. Modeling of the process yields good agreement with simulation results.

Universal and nonuniversal amplitude ratios for scaling corrections on Ising strips

We consider strips of Ising spins at criticality. For strips of width N sites, subdominant (additive) finite-size corrections to scaling are assumed to be of the form a(k)/N(k) for the free energy, and b(k)/N(k) for inverse correlation length, with integer values of k. We investigate the set {a(k),b(k)} (k≥2) by exact evaluation and numerical transfer-matrix diagonalization techniques, and their changes upon varying anisotropy of couplings, spin quantum number S, and (finite) interaction range, in all cases for both periodic (PBCs) and free (FBCs) boundary conditions across the strip. We find that the coefficient ratios b(k)/a(k) remain constant upon varying coupling anisotropy for S=1/2 and first-neighbor couplings, for both PBCs and FBCs (albeit at distinct values in either case). Such apparently universal behavior is not maintained upon changes in S or interaction range.

Supersolid Droplet Crystal in a Dipole-Blockaded Gas

A novel supersolid phase is predicted for an ensemble of Rydberg atoms in the dipole-blockade regime, interacting via a repulsive dipolar potential softened at short distances. Using exact numerical techniques, we study the low-temperature phase diagram of this system, and observe an intriguing phase consisting of a crystal of mesoscopic superfluid droplets. At low temperature, phase coherence throughout the whole system, and the ensuing bulk superfluidity, are established through tunnelling of identical particles between neighboring droplets.

Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic–numeric computing. Taking a tropical view of this problem leads to efficient preprocessing techniques, applying polyhedral methods to the exact exponents and numerical techniques to the approximate coefficients. With Maple we will illustrate our use of tropical algebraic geometry.

An integrated analysis of radial velocities in planet searches

We discuss a Bayesian approach to the analysis of radial velocities in planet searches. We use a combination of exact and approximate analytic and numerical techniques to efficiently evaluate chi-squared for multiple values of orbital parameters, and to carry out the marginalization integrals for a single planet including the possibility of a long term trend. The result is a robust algorithm that is rapid enough for use in real time analysis that outputs constraints on orbital parameters and false alarm probabilities for the planet and long term trend. The constraints on parameters and odds ratio that we derive compare well with previous calculations based on Markov Chain Monte Carlo methods, and we compare our results with other techniques for estimating false alarm probabilities and errors in derived orbital parameters. False alarm probabilities from the Bayesian analysis are systematically higher than frequentist false alarm probabilities, due to the different accounting of the number of trials. We show that upper limits on the velocity amplitude derived for circular orbits are a good estimate of the upper limit on the amplitude of eccentric orbits for eccentricities less than about 0.5.

Frequency selective solutions for an efficient non-ionising radiation protection in the radiofrequency and microwave ranges

The paper introduces an efficient solution for the realisation of frequency-selective devices which can shield non-ionising radiation in the radiofrequency and microwave bandwidths. The identified hardware is transparent to visible light and it can be adapted to the shielding needs, both in terms of frequency and spatial behaviours. Being transparent, it can be effectively applied on windows that normally offer the worst attenuation coefficient to outdoor versus indoor penetration. The solution exhibits enhanced performance, it can be analysed with fast but exact numerical techniques and it can be manufactured by low-cost industrial procedures. To demonstrate the validity of the approach, an example has been designed, analysed and applied to the frequency spectrum occupied by upcoming wideband access base stations.

Dipole oscillations of confined lattice bosons in one dimension

We study the dynamics of a nonintegrable system comprising interacting cold bosons trapped in an optical lattice in one dimension by means of exact time-dependent numerical density-matrix renormalization group techniques. Particles are confined by a parabolic potential, and dipole oscillations are induced by displacing the trap center of a few lattice sites. Depending on the system parameters this motion can vary from undamped to overdamped. We study the dipole oscillations as a function of the lattice displacement, the particle density, and the strength of interparticle interactions. These results explain the recent experiment [C. D. Fertig et al., Phys. Rev. Lett. 94, 120403 (2005)].

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.