Finite-size Systems Research Articles

Correlation functions of the six-vertex IRF model and its quantum spin chain

We consider the interaction-round-a-face version of the isotropic six-vertex model. The associated spin chain is made of two coupled Heisenberg spin chains with different boundary twists. The phase diagram of the model and the long distance correlations were studied in [Nucl. Phys. B, 995 (2023) 116333]. Here, we compute the short-distance correlation functions of the model in the ground state for finite system sizes via non-linear integral equations and in the thermodynamic limit. This was possible since the model satisfies the face version of the discrete quantum Knizhnik-Zamolodchikov (qKZ) equation. A suitable ansatz for the density matrix is proposed in the form of a direct sum of two Heisenberg density matrices, which allows us to obtain the discrete functional equation for the two-site function ω(λ1,λ2). Thanks to the known results on the factorization of correlation functions of the Heisenberg chain, we are able to compute the density matrix of the IRF model for up to four sites and its associated spin chain for up to three sites.

Small-amplitude synchronization in driven Potts models

We study driven q-state Potts models with thermodynamically consistent dynamics and global coupling. For a wide range of parameters, these models exhibit a dynamical phase transition from decoherent oscillations into a synchronized phase. Starting from a general microscopic dynamics for individual oscillators, we derive the normal form of the high-dimensional Hopf bifurcation that underlies the phase transition. The normal-form equations are exact in the thermodynamic limit and close to the bifurcation. Exploiting the symmetry of the model, we solve these equations and thus uncover the intricate stable synchronization patterns of driven Potts models, characterized by a rich phase diagram. Making use of thermodynamic consistency, we show that synchronization reduces dissipation in such a way that the most stable synchronized states dissipate the least entropy. Close to the phase transition, our findings condense into a linear dissipation-stability relation that connects entropy production with phase-space contraction, a stability measure. At finite system size, our findings suggest a minimum-dissipation principle for driven Potts models that holds arbitrarily far from equilibrium. Published by the American Physical Society 2024

Symmetry Restoration and Quantum Mpemba Effect in Symmetric Random Circuits.

Entanglement asymmetry, which serves as a diagnostic tool for symmetry breaking and a proxy for thermalization, has recently been proposed and studied in the context of symmetry restoration for quantum many-body systems undergoing a quench. In this Letter, we investigate symmetry restoration in various symmetric random quantum circuits, particularly focusing on the U(1) symmetry case. In contrast to nonsymmetric random circuits where the U(1) symmetry of a small subsystem can always be restored at late times, we reveal that symmetry restoration can fail in U(1)-symmetric circuits for certain weak symmetry-broken initial states in finite-size systems. In the early-time dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry is restored faster when the initial state is more asymmetric. Furthermore, we also investigate the entanglement asymmetry dynamics for SU(2) and Z_{2} symmetric circuits and identify the presence and absence of the quantum Mpemba effect for the corresponding symmetries, respectively. A unified understanding of these results is provided through the lens of quantum thermalization with conserved charges.

Finite-size effect in Kuramoto oscillators with higher-order interactions.

Finite-size systems of a Kuramoto model display intricate dynamics, especially in the presence of multi-stability where both coherent and incoherent states coexist. We investigate such a scenario in globally coupled populations of Kuramoto phase oscillators with higher-order interactions and observe that fluctuations inherent to finite-size systems drive the transition to the synchronized state before the critical point in the thermodynamic limit. Using numerical methods, we plot the first exit-time distribution of the magnitude of a complex order parameter and obtain numerical transition probabilities across various system sizes. Furthermore, we extend this study to a two-population oscillator system, and, using the velocity field of the associated order parameters, show the emergence of a new fixed point corresponding to a partially synchronized state arising due to the finite-size effect, which is absent in the thermodynamics limit.

Partition function zeros of zeta-urns

We discuss the distribution of partition function zeros for the grand-canonical ensemble of the zeta-urn model, where tuning a single parameter can give a first or any higher order condensation transition. We compute the locus of zeros for finite-size systems and test scaling relations describing the accumulation of zeros near the critical point against theoretical predictions for both the first and higher order transition regimes.

Finite-temperature Rydberg arrays: Quantum phases and entanglement characterization

As one of the most prominent platforms for analog quantum simulators, Rydberg atom arrays are a promising tool for exploring quantum phases and transitions. While the ground-state properties of one-dimensional Rydberg systems are already thoroughly examined, we extend the analysis towards the finite-temperature scenario. For this purpose, we develop a tensor network-based numerical toolbox for constructing the quantum many-body states at thermal equilibrium, which we exploit to probe classical correlations as well as entanglement monotones. We clearly observe ordered phases continuously shrinking because of thermal fluctuations at finite system sizes. Moreover, by examining the entanglement of formation and entanglement negativity of a half-system bipartition, we numerically confirm that a conformal scaling law of entanglement extends from the zero-temperature critical points into the low-temperature regime. Published by the American Physical Society 2024

Edge modes in finite-size systems with different edge terminals

Abstract We investigate the behavior of edge modes in the presence of different edge terminations and long-range (LR) hopping. Here, we mainly focus on such model crystals with two different types of structures (type I: “…–P–Q–P–Q–…” and type II: “… = P–Q = P–Q = …”), where P and Q represent crystal lines (CLs), while the symbols “–” and “=” denote the distance between the nearest neighbor (NN) CLs. Based on the lattice model Hamiltonian with LR hopping, the existence of edge modes is determined analytically by using the transfer matrix method (TMM) when different edge terminals are taken into consideration. Our findings are consistent with the numerical results obtained by the exact diagonalization method. We also notice that edge modes can exhibit different behaviors under different edge terminals. Our result is helpful in solving novel edge modes in honeycomb crystalline graphene and transition metal dichalcogenides with different edge terminals.

Jamming is a first-order transition with quenched disorder in amorphous materials sheared by cyclic quasistatic deformations

Jamming is an athermal transition between flowing and rigid states in amorphous systems such as granular matter, colloidal suspensions, complex fluids and cells. The jamming transition seems to display mixed aspects of a first-order transition, evidenced by a discontinuity in the coordination number, and a second-order transition, indicated by power-law scalings and diverging lengths. Here we demonstrate that jamming is a first-order transition with quenched disorder in cyclically sheared systems with quasistatic deformations, in two and three dimensions. Based on scaling analyses, we show that fluctuations of the jamming density in finite-sized systems have important consequences on the finite-size effects of various quantities, resulting in a square relationship between disconnected and connected susceptibilities, a key signature of the first-order transition with quenched disorder. This study puts the jamming transition into the category of a broad class of transitions in disordered systems where sample-to-sample fluctuations dominate over thermal fluctuations, suggesting that the nature and behavior of the jamming transition might be better understood within the developed theoretical framework of the athermally driven random-field Ising model.

Majorana modes in striped two-dimensional inhomogeneous topological superconductors

Majorana zero modes have gained significant interest due to their potential applications in topological quantum computing and in the realization of exotic quantum phases. These zero-energy quasiparticle excitations localize at the vortex cores of two-dimensional topological superconductors or at the ends of one-dimensional topological superconductors. Here we describe an alternative platform: a two-dimensional topological superconductor with inhomogeneous superconductivity, where Majorana modes localize at the ends of topologically nontrivial one-dimensional stripes induced by the spatial variations of the order parameter phase. In certain regimes, these Majorana modes hybridize into a single highly nonlocal state delocalized over spatially separated points, with exactly zero energy at finite system sizes and with emergent quantum-mechanical supersymmetry. We then present detailed descriptions of braiding and fusion protocols and showcase the versatility of our proposal by suggesting possible setups that can potentially lead to the realization of Yang-Lee anyons and the Sachdev-Ye-Kitaev model.

Gapless edge-modes and topology in the Qi-Wu-Zhang model: a real-space analysis

The topological phase transition in the Qi-Wu-Zhang model is studied using a real-space approach. An effective Hamiltonian for the topologically protected edge-modes in a finite-size system is developed. The topological phase transition is understood in terms of a global perturbation to the system which lifts the degeneracy of the edge-modes. The effective Hamiltonian method is also applied to a one-dimensional system with spatially varying hopping strengths to understand the impact of disorder on the edge-modes.

Reflected entropy and timelike entanglement in TT¯ -deformed CFT2s

We develop a covariant formalism to investigate the mixed state entanglement between time-dependent boosted subsystems in TT¯-deformed CFT2s through the reflected entropy. To this end we utilize a conformal perturbation theory to obtain the Rényi reflected entropy through the partition function on the replica manifold. The correction to the reflected entropy to the first order in the deformation parameter μ is then obtained in the replica limit for finite temperature and finite-sized systems. We further perform an analytic continuation of our results to obtain the reflected entropy for timelike subsystems in such TT¯-deformed CFT2s. We verify our field theoretic computations for both spacelike and timelike subsystems by obtaining the dual EWCS in the corresponding bulk cutoff AdS3 geometries and find perfect agreement between the two. Published by the American Physical Society 2024

Synchronization transitions in adaptive Kuramoto-Sakaguchi oscillators with higher-order interactions.

Coupled oscillators models help us in understanding the origin of synchronization phenomenon prevalent in both natural and artificial systems. Here, we study the coupled Kuramoto oscillator model having phase lag and adaptation in higher-order interactions. We find that the type of transition to synchronization changes from the first-order to second-order through tiered synchronization depending on the adaptation parameters. Phase lag enables this transition at a lower exponent of the adaptation parameters. Moreover, an interplay between the adaptation and phase lag parameters eliminates tiered synchronization, facilitating a direct transition from the first to second-order. In the thermodynamic limit, the Ott-Antonsen approach accurately describes all stationary and (un)stable states, with analytical results matching those obtained from numerical simulations for finite system sizes.

A model for the lattice thermal conductivity of nanocrystalline materials with arbitrary grain size distribution

Nanocrystalline materials are a popular metamaterial used to control heat transport in semiconductor industrial applications. An accurate and reliable theoretical model is urgently needed but challenging for elucidating detailed microscopic heat transfer phenomena. In this study, we develop a mathematical model to predict the lattice thermal conductivity of nanocrystalline materials with arbitrary grain size distribution. The model first obtains the effective scattering rate of grain boundaries in single-grain-size materials, and then integrates the Landauer’s system transmission concept and series transmission concept to become applicable to multiple grain sizes. It takes into account not only intrinsic and grain boundary scattering effects, but also the effects of finite system size and grain size distribution. Numerical samples of silicon nanocrystals with single grain size, dual size, constant grain size distribution, and logarithmic normal grain size distribution are constructed and full-spectrum Monte Carlo simulations are performed for comparison. The excellent agreement observed between model predictions and simulation results validates the accuracy of the proposed theoretical model. This model will be very helpful in designing micron-scale thermal manipulation systems using nanostructures.

Response theory identifies reaction coordinates and explains critical phenomena in noisy interacting systems

We consider a class of nonequilibrium systems of interacting agents with pairwise interactions and quenched disorder in the dynamics featuring, in the thermodynamic limit, phase transitions. We identify mathematical conditions on the microscopic interaction structure, namely the separability of the interaction kernel, that lead to a dimension reduction of the system in terms of a finite number of reaction coordinates (RCs). Such RCs prove to be proper nonequilibrium thermodynamic variables as they carry information on correlation, memory and resilience properties of the system. Phase transitions can be identified and quantitatively characterised as singularities of the complex valued susceptibility functions associated to the RCs. We provide analytical and numerical evidence of how the singularities affect the physical properties of finite size systems.

Boundary-induced transitions in Möbius quenches of holographic BCFT

Boundary effects play an interesting role in finite-size physical systems. In this work, we study the boundary-induced properties of 1+1-dimensional critical systems driven by inhomogeneous Möbius-like quenches. We focus on the entanglement entropy in BCFTs with a large central charge and a sparse spectrum of low-dimensional operators. We find that the choice of boundary conditions leads to different scenarios of dynamical phase transitions. We also derive these results in a holographic description in terms of intersecting branes in AdS3, and find a precise match.

Covariant phase space analysis of Lanczos-Lovelock gravity with boundaries

This work introduces a novel prescription for the expression of the thermodynamic potentials associated with the couplings of a Lanczos-Lovelock theory. These potentials emerge in theories with multiple couplings, where the ratio between them provide intrinsic length scales that break scale invariance. Our prescription, derived from the covariant phase space formalism, differs from previous approaches by enabling the construction of finite potentials without reference to any background. To do so, we consistently work with finite-size systems with Dirichlet boundary conditions and rigorously take into account boundary and corner terms: including these terms is found to be crucial for relaxing the integrability conditions for phase space quantities that were required in previous works. We apply this prescription to the first law of (extended) thermodynamics for stationary black holes, and derive a version of the Smarr formula that holds for static black holes with arbitrary asymptotic behaviour.

Symmetry, topology, and geometry: The many faces of the topological magnetoelectric effect

A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) Z2 topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular Z2 topological classification of the electronic ground state of crystalline band insulators and the associated quantization of the magnetoelectric response coefficient calculated using bulk linear-response theory but, according to standard symmetry arguments, simultaneously forbids a nonzero magnetoelectric coefficient in any physical finite-size system. This contrast between theories of magnetoelectric response in formal bulk models and in real finite-sized materials originates from the distinct approaches required to introduce notions of (electronic) polarization and orbital magnetization in these fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear-response calculations in nonmagnetic Z2 TIs that is more plainly consistent with TRS and use this interpretation to discuss the effect's observation—still absent over a decade after its prediction. Our analysis is reinforced by microscopic bulk and thin-film calculations carried out using a simplified but still realistic effective model for the well established V2VI3 [V=(Sb,Bi) and VI=(Se,Te)] family of nonmagnetic Z2 TIs. When a uniform dc magnetic field is included in this model, the anomalous n=0 Landau levels (LLs) play the central role, both in thin films and in bulk. In the former case, only the n=0 LL eigenfunctions can support a dipole moment, which vanishes if there are no magnetic surface dopants and is quantized in the thick-film limit if magnetic dopants at the top and bottom surfaces have opposite orientation. In the latter case, the Hamiltonian projected into the n=0 LL subspace is a one-dimensional Su-Schrieffer-Heeger model with ground-state polarization that is quantized in accordance with the bulk linear-response coefficient calculated for (a lattice regularization of) the starting 3D model. Motivated by analytical results, we conjecture a type of microscopic bulk-boundary correspondence: a bulk insulator with (generalized) TRS supports a magnetoelectric coefficient that is purely itinerant (which is generically related to the geometry of the ground state) if and only if magnetic surface dopants are required for the TME to manifest in finite samples thereof. We conclude that in nonmagnetic Z2 TIs the TME is activated by magnetic surface dopants, that the charge-density response to a uniform dc magnetic field is localized at the surface and specified by the configuration of those dopants, and that the TME is qualitatively less robust against disorder than the integer quantum Hall effect. Published by the American Physical Society 2024

A study of layered holographic superconductor

We have investigated a holographic model of a multi-layered superconductor in (2+1)-dimensions using the AdS/CFT correspondence. This correspondence allows us to study strongly interacting condensed matter systems through a weakly interacting gravitational theory. Our study focused on the effects of a finite system size on the superconductor’s properties. We observed significant variations in the critical temperature and critical magnetic field depending on the number of layers and the spacing between them. Notably, our results capture the transition from a two-dimensional (2D) to a three-dimensional (3D) behavior in the limit of a large number of closely spaced layers. This transition signifies a change in how the superconductivity operates within real material.

Non-Hermitian Casimir effect of magnons

There has been a growing interest in non-Hermitian quantum mechanics. The key concepts of quantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a finite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect, is one of the most striking phenomena of quantum mechanics in the sense that there are no classical analogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics, together with photonics. However, the non-Hermitian extension of the Casimir effect and the application to spintronics have not yet been investigated enough, although exploring energy sources and developing energy-efficient nanodevices are its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect into non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced as the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping constant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating behavior, including a beating one, as a function of the film thickness and is characterized by the exceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir engineering.

Exact zeros of fidelity in finite-size systems as a signature for probing quantum phase transitions.

The fidelity is widely used to detect quantum phase transitions, which is characterized by either a sharp change of fidelity or the divergence of fidelity susceptibility in the thermodynamical limit when the phase-driving parameter is across the transition point. In this work, we unveil that the occurrence of exact zeros of fidelity in finite-size systems can be applied to detect quantum phase transitions. In general, the fidelity F(γ,γ[over ̃]) always approaches zero in the thermodynamical limit, due to the Anderson orthogonality catastrophe, no matter whether the parameters of two ground states (γ and γ[over ̃]) are in the same phase or different phases, and this makes it difficult to distinguish whether an exact zero of fidelity exists by finite-size analysis. To overcome the influence of orthogonality catastrophe, we study finite-size systems with twist boundary conditions, which can be introduced by applying a magnetic flux, and demonstrate that exact zeros of fidelity can be always accessed by tuning the magnetic flux when γ and γ[over ̃] belong to different phases. On the other hand, no exact zero of fidelity can be observed if γ and γ[over ̃] are in the same phase. We demonstrate the applicability of our theoretical scheme by studying concrete examples, including the Su-Schrieffer-Heeger model, Creutz model, and Haldane model. Our work provides a practicable way to detect quantum phase transitions via the calculation of fidelity of finite-size systems.