Finite-size Scaling Research Articles

Quasiresonant diffusion of wave packets in one-dimensional disordered mosaic lattices

We investigate numerically the time evolution of wave packets incident on one-dimensional semi-infinite lattices with mosaic modulated random on-site potentials, which are characterized by the integer-valued modulation period $\ensuremath{\kappa}$ and the disorder strength $W$. For Gaussian wave packets with the central energy ${E}_{0}$ and a small spectral width, we perform extensive numerical calculations of the disorder-averaged time-dependent reflectance, $\ensuremath{\langle}R(t)\ensuremath{\rangle}$, for various values of ${E}_{0}, \ensuremath{\kappa}$, and $W$. We find that the long-time behavior of $\ensuremath{\langle}R(t)\ensuremath{\rangle}$ obeys a power law of the form ${t}^{\ensuremath{-}\ensuremath{\gamma}}$ in all cases. In the presence of the mosaic modulation, $\ensuremath{\gamma}$ is equal to 2 for almost all values of ${E}_{0}$, implying the onset of the Anderson localization, while at a finite number of discrete values of ${E}_{0}$ dependent on $\ensuremath{\kappa}, \ensuremath{\gamma}$ approaches 3/2, implying the onset of the classical diffusion. This phenomenon is independent of the disorder strength and arises in a quasiresonant manner such that $\ensuremath{\gamma}$ varies rapidly from 3/2 to 2 in a narrow energy range as ${E}_{0}$ varies away from the quasiresonance values. We deduce a simple analytical formula for the quasiresonance energies and provide an explanation of the delocalization phenomenon based on the interplay between randomness and band structure and the node structure of the wave functions. We explore the nature of the states at the quasiresonance energies using a finite-size scaling analysis of the average participation ratio and find that the states are neither extended nor exponentially localized, but critical states.

Scaling at quantum phase transitions above the upper critical dimension

The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above the upper critical dimension following the recently introduced Q-FSS formalism for thermal phase transitions. Contrary to long-standing belief, the correlation sector is affected by dangerous irrelevant variables. The presented formalism recovers a generalized hyperscaling relation and FSS form. Using this new FSS formalism, we determine the full set of critical exponents for the long-range transverse-field Ising chain in all criticality regimes ranging from the nearest-neighbor to the long-range mean-field regime. For the same model, we also explicitly confirm the effect of dangerous irrelevant variables on the characteristic length scale.

An anomalous topological phase transition in spatial random graphs

Clustering–the tendency for neighbors of nodes to be connected–quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric to non-geometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finite-size scaling behavior of clustering. Moreover, a slow decay of clustering in the non-geometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.

Surface-Dominated Finite-Size Effects in Nanoconfined Superfluid Helium.

Superfluid ^{4}He (He II) is a widely studied model system for exploring finite-size effects in strongly confined geometries. Here, we study He II confined in millimeter-scale channels of 25 and 50nm height at high pressures using a nanofluidic Helmholtz resonator. We find that the superfluid density is measurably suppressed in the confined geometry from the transition temperature down to 0.6K. Importantly, this suppression can be accounted for by rotonlike thermal excitations with an energy gap of 5K. We show that the surface-bound excitations lead to the previously unexplained lack of finite-size scaling of suppression of the superfluid density.

Quasiperiodic many-body localization transition in dimension d>1

The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension $d>1$, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension $d>1$. We argue that (i) the MBL phase is stable at strong enough quasiperiodic modulations for $d = 2$, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d \geq 3$.

Universality classes of the Anderson transitions driven by quasiperiodic potential in the three-dimensional Wigner-Dyson symmetry classes

Quasiperiodic system is an intermediate state between periodic and disordered systems with unique delocalization-localization transition driven by the quasiperiodic potential (QP). One of the intriguing questions is whether the universality class of the Anderson transition (AT) driven by QP is similar to that of the AT driven by the random potential in the same symmetry class. Here, we study the critical behavior of the ATs driven by QP in the three-dimensional (3D) Anderson model, Peierls phase model, and Ando model, which belong to the Wigner-Dyson symmetry classes. The localization length and two-terminal conductance have been calculated by the transfer matrix method, and we argue that their error estimations in statistics suffer from the correlation of QP. With the correlation under control, the critical exponents $\nu$ of the ATs driven by QP are estimated by the finite size scaling analysis of conductance, which are consistent with $\nu$'s of the ATs driven by the random potential. Moreover, the critical conductance distribution and the level spacing ratio distribution have been studied. We also find that a convolutional neural network trained by the localized/delocalized wavefunctions in a disordered system predicts the localized/delocalized wavefunctions in quasiperiodic systems. Our numerical results strongly support that the universality classes of the ATs driven by QP and random potential are similar in the 3D Wigner-Dyson symmetry classes.

Critical quantum thermometry and its feasibility in spin systems

In this work, we study temperature sensing with finite-sized strongly correlated systems exhibiting quantum phase transitions. We use the quantum Fisher information (QFI) approach to quantify the sensitivity in the temperature estimation, and apply a finite-size scaling framework to link this sensitivity to critical exponents of the system around critical points. We numerically calculate the QFI around the critical points for two experimentally-realizable systems: the spin-1 Bose-Einstein condensate and the spin-chain Heisenberg XX model in the presence of an external magnetic field. Our results confirm finite-size scaling properties of the QFI. Furthermore, we discuss experimentally-accessible observables that (nearly) saturate the QFI at the critical points for these two systems.

Universal relation between doping content and normal-state resistance in gate voltage tuned ultrathin Bi2Sr2CaCu2O8+x flakes

Gate voltage tunable ultrathin high-${T}_{c}$ cuprates supply a unique platform to investigate the electronic phase diagram and superconductor-insulator transition. One of the challenges in this field is the precise determination of the doping content in the underdoped nonsuperconducting region. Here we report the discovery of a universal relation between the doping content $p$ and the normal-state resistance at a fixed temperature $R({T}_{f})$, $p=\ensuremath{\alpha}+\ensuremath{\beta}\phantom{\rule{4pt}{0ex}}\mathrm{ln}[1/R({T}_{f})]$, in the ultrathin ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+x}$ flakes. The in-depth analysis shows that the evolution of carrier scattering probability with doping content and the change of effective mass caused by superconductor-insulator transition are two key factors leading to this logarithmic relation. Based on our finding, the more precise electronic phase diagram can be established. In addition, the superconductor-insulator transition is verified to be a quantum phase transition using a finite size scaling analysis. The scaling exponent $z\ensuremath{\nu}$ is found to have a close correlation with the disorder levels. The present result provides an important foundation to investigate the fascinating electronic states in the ultra-thin cuprates.

Density-wave-ordered phases of Rydberg atoms on a honeycomb lattice.

Rydberg atom arrays have recently emerged to be a promising platform for the exploration of exotic quantum phases of matter and quantum phenomena. In this work, we map out the ground-state phase diagram of Rydberg atoms on a honeycomb lattice as a function of the Rydberg blockade radius and the laser detuning by performing large-scale finite-size density matrix renormalization group simulations. Apart from a featureless disordered phase, we find five other intricate long-range density-wave-ordered phases within a relatively wide parameter space. The properties of these quantum phases are analyzed by calculating their Rydberg excitation profiles and static structure factors. In addition, a continuous quantum phase transition belonging to the (2+1)-dimensional Ising universality class is explored by a standard finite-size scaling analysis. Our work implies some different physics, such as the possible nontrivial quantum phase transitions and a highly degenerate string ordered phase, that a honeycomb geometry could bring to the Rydberg system and serves as a numerical guide for possible real experiments.

Transport properties in multilayer adsorption of dimers.

In this paper, we study the transport properties (percolation and conductivity) of a two-dimensional structure created by depositing dimers on a one-dimensional substrate where multilayer deposition is allowed. Specifically, we are interested in studying how the mentioned properties vary as a function of the height of the multilayer. The critical parameters of the percolation transition are calculated using finite-size scaling analysis, obtaining the scaling laws for the probability of percolation and the conductivity of the system. To calculate the electrical conductivity of the multilayer, we use the Frank-Lobb algorithm.

Holography for Ising spins on the hyperbolic plane

Motivated by the AdS/CFT correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary susceptibility which yields a scaling exponent consistent with the scaling dimension extracted from the boundary correlation function. This observation provides evidence that the connection between continuum boundary conformal symmetry and isometries of the bulk hyperbolic space survives for simple interacting field theories even when the bulk is approximated by a discrete tessellation.

Exploring bosonic and fermionic link models on (3+1)D tubes

Quantum link models (QLMs) have attracted a lot of attention in recent times as a generalization of Wilson's lattice gauge theories (LGT), and are particularly suitable for realization on quantum simulators and computers. These models are known to host new phases of matter and act as a bridge between particle and condensed matter physics. In this article, we study the Abelian $U(1)$ lattice gauge theory in $(3+1)$-d tubes using large-scale exact diagonalization (ED). We are then able to motivate the phase diagram of the model with finite size scaling techniques (FSS), and in particular propose the existence of a Coulomb phase. Furthermore, we introduce the first models involving fermionic quantum links, which generalize the gauge degrees of freedom to be of fermionic nature. We prove that while the spectra remain identical between the bosonic and the fermionic versions of the $U(1)$-symmetric quantum link models in $(2+1)$-d, they are different in $(3+1)$-d. We discuss the prospects of realizing the magnetic field interactions as correlated hopping in quantum simulator experiments.

Ising model on a 2D additive small-world network

In this article, we have employed Monte Carlo simulations to study the Ising model on a two-dimensional additive small-world network (A-SWN). The system model consists of a LxL square lattice where each site of the lattice is occupied for a spin variable that interacts with the nearest neighbor and has a certain probability p of being additionally connected at random to one of its farther neighbors. The system is in contact with a heat bath at a given temperature T and it is simulated by one-spin flip according to the Metropolis prescription. We have calculated the thermodynamic quantities of the system, such as, the magnetization per spin m, magnetic susceptibility chi, and the reduced fourth-order Binder cumulant U as a function of T for several values of lattice size L and additive probability p. We also have constructed the phase diagram for the equilibrium states of the model in the plane T versus p showing the existence of a continuous transition line between the ferromagnetic F and paramagnetic P phases. Using the finite-size scaling (FSS) theory, we have obtained the critical exponents for the system, where varying the parameter p, we have observed a change in the critical behavior from the regular square lattice Ising model to A-SWN.

Collective excitations in jammed states: ultrafast defect propagation and finite-size scaling

In crowded systems, particle currents can be mediated by propagating collective excitations which are generated as rare events, are localized, and have a finite lifetime. The theoretical description of such excitations is hampered by the problem of identifying complex many-particle transition states, calculation of their free energies, and the evaluation of propagation mechanisms and velocities. Here we show that these problems can be tackled for a highly jammed system of hard spheres in a periodic potential. We derive generation rates of collective excitations, their anomalously high velocities, and explain the occurrence of an apparent jamming transition and its strong dependence on the system size. The particle currents follow a scaling behavior, where for small systems the current is proportional to the generation rate and for large systems given by the geometric mean of the generation rate and velocity. Our theoretical approach is widely applicable to dense nonequilibrium systems in confined geometries. It provides new perspectives for studying dynamics of collective excitations in experiments.

Persistent homology analysis of a generalized Aubry-André-Harper model

Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize phases of a generalized Aubry-Andr\'e-Harper model. The persistent entropy and mean squared lifetime of features obtained using persistent homology behave similarly to conventional measures (Shannon entropy and inverse participation ratio) and can distinguish localized, extended, and critical phases. However, we find that the persistent entropy also clearly distinguishes ordered from disordered regimes of the model. The persistent homology approach can be applied to both the energy eigenstates and the wave packet propagation dynamics.

Percolation of grain boundaries and triple junctions in three-dimensions: A test of theory

Percolation is a fundamental property of grain boundary (GB) networks that has so far been analyzed through synthetic three-dimensional (3D) microstructures alone. In this work, we visualize an unprecedented 10,265 GBs in a 3D Al-Cu sample using laboratory-based X-ray diffraction tomography (LabDCT). By applying finite-size scaling with the critical exponents from standard percolation theory, we determine from our experimental data a percolation threshold of 0.236 ± 0.039 for the GBs in the thermodynamic limit. We compare our results to those from past simulations, which model the 3D microstructure using space-filling polyhedra; the thresholds show good agreement when they are normalized by the topological characteristics of the GB network. We further investigate the percolation threshold of triple junction (TJ) lines, which we show to be necessarily lower than that of the GBs. The percolation behavior of TJs is also different from that of the theoretical diamond lattice due to two factors: a surprisingly higher coordination of nodes (6.184 vs. 4) and also a spatial clustering of TJs in the microstructure. The insights obtained herein can help inform the design of failure-resistant materials via GB engineering.

Transition from metal to higher-order topological insulator driven by random flux

Random flux is commonly believed to be incapable of driving full metal-insulator transitions in non-interacting systems. Here we show that random flux can after all induce a full metal-band insulator transition in the two-dimensional Su-Schrieffer-Heeger model. Remarkably, we find that the resulting insulator can be an extrinsic higher-order topological insulator with zero-energy corner modes in proper regimes, rather than a conventional Anderson insulator. Employing both level statistics and finite-size scaling analysis, we characterize the metal-band insulator transition and numerically extract its critical exponent as $\nu=2.48\pm0.08$. To reveal the physical mechanism underlying the transition, we present an effective band structure picture based on the random flux averaged Green's function.

The location of the Fisher zeros and estimates of y T = 1/ν are found for the Baxter–Wu model

The Fisher zeros of the Baxter–Wu model are examined for the first time and for two series of finite-sized systems, with ‘spherical’ boundary conditions, their location is found to be extremely simple. They lie on the unit circle in the complex sinh[2βJ 3] plane. This is the same location as the Fisher zeros of the square lattice Ising model with nearest neighbour interactions and Brascamp–Kunz boundary conditions. The Baxter–Wu model is an Ising model with three-site interactions, J 3, on the triangle lattice. From the leading Fisher zeros, using finite-size scaling, accurate estimates of the critical exponent 1/ν are obtained and emphasis is placed on using different variables such as exp[−2βJ 3], exp[−4βJ 3], and sinh[2βJ 3] to enhance the accuracy of estimates. Furthermore, using the imaginary parts of the leading zeros versus the real part of the leading zeros, yields different results. This is similar to results of Janke and Kenna for the nearest neighbour, Ising model on the square lattice and extends this behaviour to a multisite interaction system in a different universality class than the pair-interaction cases.

Multipartite nonlocality and topological quantum phase transitions in a spinless fermion quantum wire with uniform and incommensurate potentials

Multipartite nonlocality and Bell-type inequalities are used to characterize topological quantum phase transitions (QPTs) in a spinless fermion quantum wire, where both uniform potentials and incommensurate potentials are considered. First, the nonlocality measures show clear signals at the critical points in both the uniform model and the incommensurate model. It indicates that these QPTs are accompanied by dramatic changes of multipartite quantum correlations in the ground states. Second, finite-size scaling analysis is carried out. In particular, in the incommensurate model where translation invariance is broken, with some rescaling techniques, we successfully establish the scaling formula in the large-$L$ limit. Finally, the full phase diagram of the model with mixed potentials is figured out. We find a region which is featured with strong randomness in the large-$L$ limit. The structure of this region is revealed by analyzing the energy spectrum, and an efficient approach to characterize this region is proposed.

Phase transitions above the upper critical dimension

These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension d_{uc}duc. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above d_{uc}duc. We recount how Fisher’s dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above d_{uc}duc. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.