Classification of phase transitions in small systems

Dynamical quantum phase transitions: a review

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook to major open questions as well as future directions of research.

Lee-Yang zeros and large-deviation statistics of a molecular zipper

The complex zeros of partition functions were originally investigated by Lee and Yang to explain the behavior of condensing gases. Since then, Lee-Yang zeros have become a powerful tool to describe phase transitions in interacting systems. Today, Lee-Yang zeros are no longer just a theoretical concept; they have been determined in recent experiments. In one approach, the Lee-Yang zeros are extracted from the high cumulants of thermodynamic observables at finite size. Here we employ this method to investigate a phase transition in a molecular zipper. From the energy fluctuations in small zippers, we can predict the temperature at which a phase transition occurs in the thermodynamic limit. Even when the system does not undergo a sharp transition, the Lee-Yang zeros carry important information about the large-deviation statistics and its symmetry properties. Our work suggests an interesting duality between fluctuations in small systems and their phase behavior in the thermodynamic limit. These predictions may be tested in future experiments.

Exceptional points near first- and second-order quantum phase transitions

We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.

Experimental Determination of Dynamical Lee-Yang Zeros

Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros-complex singularities of the free energy in systems of finite size-have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize nonequilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such nonequilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence

We study the orientational ordering in systems of self-propelled particles with selective interactions. To introduce the selectivity we augment the standard Vicsek model with a bounded-confidence collision rule: a given particle only aligns to neighbors who have directions quite similar to its own. Neighbors whose directions deviate more than a fixed restriction angle α are ignored. The collective dynamics of this system is studied by agent-based simulations and kinetic mean-field theory. We demonstrate that the reduction of the restriction angle leads to a critical noise amplitude decreasing monotonically with that angle, turning into a power law with exponent 3/2 for small angles. Moreover, for small system sizes we show that upon decreasing the restriction angle, the kind of the transition to polar collective motion changes from continuous to discontinuous. Thus, an apparent tricritical point with different scaling laws is identified and calculated analytically. We investigate the shifting and vanishing of this point due to the formation of density bands as the system size is increased. Agent-based simulations in small systems with large particle velocities show excellent agreement with the kinetic theory predictions. We also find that at very small interaction angles, the polar ordered phase becomes unstable with respect to the apolar phase. We derive analytical expressions for the dependence of the threshold noise on the restriction angle. We show that the mean-field kinetic theory also permits stationary nematic states below a restriction angle of 0.681π. We calculate the critical noise, at which the disordered state bifurcates to a nematic state, and find that it is always smaller than the threshold noise for the transition from disorder to polar order. The disordered-nematic transition features two tricritical points: At low and high restriction angle, the transition is discontinuous but continuous at intermediate α. We generalize our results to systems that show fragmentation into more than two groups and obtain scaling laws for the transition lines and the corresponding tricritical points. A numerical method to evaluate the nonlinear Fredholm integral equation for the stationary distribution function is also presented. This method is shown to give excellent agreement with agent-based simulations, even in strongly ordered systems at noise values close to zero.

Thermodynamic laws in isolated systems

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton β hairpin model

I compute exact partition function zeros of β hairpins, using both analytic and numerical methods, extending previous work [J. Lee, Phys. Rev. Lett. 110, 248101 (2013)] where only a restricted class of hairpins was considered. The zeros of β hairpins with an odd number of peptide bonds are computed and the difference of the distribution of zeros from those for an even number of peptide bonds is explained in terms of additional entropy of liberating the extra bond at the turn region. Upon the introduction of a hydrophobic core in the central region of the hairpin, the zeros are distributed uniformly on two concentric circles corresponding to the hydrophobic collapse and the transition to the fully folded conformation. One of the circles dissolves as the core moves toward the turn or the tip region, which is explained in terms of the similarity of the intermediate state with the folded or unfolded states. The exact partition function zeros for a hairpin with a more complex structure of native contacts, the 16 C-terminal residues of streptococcal protein G B1, are numerically computed and their loci are closely approximated by concentric circles.

Partition function zeros and phase transitions for a square-well polymer chain

The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton protein model

I compute exact partition function zeros of the Wako-Saitô-Muñoz-Eaton model for various secondary structural elements and for two proteins, 1BBL and 1I6C, by using both analytic and numerical methods. Two-state and barrierless downhill folding transitions can be distinguished by a gap in the distribution of zeros at the positive real axis.

Microcanonical entropy inflection points: key to systematic understanding of transitions in finite systems

We introduce a systematic classification method for the analogs of phase transitions in finite systems. This completely general analysis, which is applicable to any physical system and extends toward the thermodynamic limit, is based on the microcanonical entropy and its energetic derivative, the inverse caloric temperature. Inflection points of this quantity signal cooperative activity and thus serve as distinct indicators of transitions. We demonstrate the power of this method through application to the long-standing problem of liquid-solid transitions in elastic, flexible homopolymers.

Physical rationale behind the nonlinear enthalpy-entropy compensation in DNA duplex stability

The physical-chemical sense of nonlinear entropy-enthalpy compensation based upon the standard thermodynamical parameters of high-temperature melting for doublet units in DNA duplexes has been considered. We are able to show that there are three, with no other constraints equally plausible, principal levels of DNA melting/hybridization description. First, DNA structure assembly/disassembly can be seen from the viewpoint of the conventional equilibrium thermodynamics without taking special care of the heat capacity DeltaC(p) value (by simply setting it equal to zero). Second, it is possible to assume that the DeltaC(p) is finite, but independent of temperature. At this approximation level the high-temperature DNA melting cannot be described, but only some special transition between metastable states of DNA duplexes in water solutions in the vicinity of ice melting point. Third, both the latter transition and the high-temperature DNA melting can be reproduced by one and the same approach, if the DeltaC(p) is assumed to be temperature dependent. These three approximation levels are equally justified from the nonlinear entropy-enthalpy compensation standpoint and by a generalized theory of temperature effects on themodynamical stability as is outlined here. Applicability of each of the approximation levels involved is discussed.

Microcanonical versus canonical analysis of protein folding

The microcanonical analysis is shown to be a powerful tool to characterize the protein folding transition and to neatly distinguish between good and bad folders. An off-lattice model with parameter chosen to represent polymers of these two types is used to illustrate this approach. Both canonical and microcanonical ensembles are employed. The required calculations were performed using parallel tempering Monte Carlo simulations. The most revealing features of the folding transition are related to its first-order-like character, namely, the S-bend pattern in the caloric curve, which gives rise to negative microcanonical specific heats, and the bimodality of the energy distribution function at the transition temperatures. Models for a good folder are shown to be quite robust against perturbations in the interaction potential parameters.

Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions

Degeneracies near the real axis in a complex-extended parameter space of a Hermitian Hamiltonian are studied. We present a method to measure distributions of such degeneracies on the Riemann sheet of a selected level and apply it in classification of quantum phase transitions. The degeneracies are shown to behave similarly as complex zeros of a partition function.

Theoretical investigation of finite size effects at DNA melting

We investigated how the finiteness of the length of a sequence affects the phase transition that takes place at the DNA melting temperature. For this purpose, we modified the transfer integral method to adapt it to the calculation of both extensive (partition function, entropy, specific heat, etc.) and nonextensive (order parameter and average separation between paired bases) thermodynamic quantities of finite sequences with open boundary conditions, and applied the modified procedure to two different dynamical models. We characterized in some detail the three effects that take place when the length of the sequence is decreased, namely, (i) the decrease of the critical temperature, (ii) the decrease of the peak values of all quantities that diverge at the thermodynamic limit but remain finite for finite sequences, like the specific heat and the correlation length, and (iii) the broadening of the temperature range over which the transition affects the dynamics of the system. We also performed a finite size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of a homogeneous function. However, Josephson's identity is satisfied for none of the investigated models, so that the derivation of the characteristic exponents which appear, for example, in the expression of the specific heat requires some care.

Density of Yang-Lee zeros for the Ising ferromagnet

The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.

Nonanalytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: an exactly solvable one-dimensional model for evaporation

We calculate exactly both the microcanonical and canonical thermodynamic functions (TDFs) for a one-dimensional model system with piecewise constant Lennard-Jones type pair interactions. In the case of an isolated N-particle system, the microcanonical TDFs exhibit (N - 1) singular (nonanalytic) microscopic phase transitions of the formal order N/2, separating N energetically different evaporation (dissociation) states. In a suitably designed evaporation experiment, these types of phase transitions should manifest themselves in the form of pressure and temperature oscillations, indicating cooling by evaporation. In the presence of a heat bath (thermostat), such oscillations are absent, but the canonical heat capacity shows a characteristic peak, indicating the temperature-induced dissociation of the one-dimensional chain. The distribution of complex zeros of the canonical partition may be used to identify different degrees of dissociation in the canonical ensemble.

Phase Transitions and Criticality in Small Systems: Vapor−Liquid Transition in Nanoscale Spherical Cavities

Phase transformations in fluids confined to nanoscale pores, which demonstrate characteristic signatures of first-order phase transitions, have been extensively documented in experiments and molecular simulations. They are characterized by a pronounced hysteresis, which disappears above a certain temperature. A rigorous interpretation of these observations represents a fundamental problem from the point of view of statistical mechanics. Nanoscale systems are essentially small, finite volume systems, in which the concept of the thermodynamic limit is no longer valid, and the statistical ensembles are not equivalent. Here, we present a rigorous approach to the description and molecular simulations of phase transitions and criticality in small confined systems, as illustrated by the example of vapor-liquid transition (capillary condensation) in spherical cavities. The method is based on the analysis of the canonical ensemble isotherms, which can be generated by the gauge cell Monte Carlo simulation method. The method allows one to define the critical temperature of phase transition, conditions of phase equilibrium, limits of stability of metastable states, and nucleation barriers, which determine hysteretic phase transformations.

Large-N scaling behavior of the Lipkin-Meshkov-Glick model

We introduce a novel semiclassical approach to the Lipkin model. In this way the well-known phase transition arising at the critical value of the coupling is intuitively understood. New results--showing for strong couplings the existence of a threshold energy which separates deformed from undeformed states as well as the divergence of the density of states at the threshold energy--are explained straightforwardly and in quantitative terms by the appearance of a double well structure in a classical system corresponding to the Lipkin model. Previously unnoticed features of the eigenstates near the threshold energy are also predicted and found to hold.

Blocking temperature in magnetic nanoclusters

A recent study of non-extensive phase transitions in nuclei and nuclear clusters needs a probability model compatible with the appropriate Hamiltonian. For magnetic molecules a representation of the evolution by a Markov process achieves the required probability model that is used to study the probability density function (PDF) of the order parameter, i.e., the magnetization. The existence of one or more modes in this PDF is an indication for the super-paramagnetic transition of the cluster. This allows us to determine the factors that influence the blocking temperature, i.e., the temperature related to the change of the number of modes in the density. It turns out that for our model, rather than the evolution of the system implied by the Hamiltonian, the high temperature density of the magnetization is the important factor for the temperature of the transition. We find that an initial probability density function with a high entropy leads to a magnetic cluster with a high blocking temperature.

Transient backbending behavior in the Ising model with fixed magnetization

The physical origin of the backbendings in the equations of state of finite but not necessarily small systems is studied in the Ising model with fixed magnetization (IMFM) by means of the topological properties of the observable distributions and the analysis of the largest cluster with increasing lattice size. Looking at the convexity anomalies of the IMFM thermodynamic potential, it is shown that the order of the transition at the thermodynamic limit can be recognized in finite systems independently of the lattice size. General statistical mechanics arguments and analytical calculations suggest that the backbending in the caloric curve is a transient behavior which should not converge to a plateau in the thermodynamic limit, while the first-order transition (in the Ehrenfest sense) is still signaled by a discontinuity in the magnetization equation of state.

Singularities of the canonical partition functions of fluid systems with continuous interaction potentials

We observe that for finite N and V, the canonical partition function Q(N) of a fluid system of N particles is a polynomial of degree N in variable V/Nlambda(3), which has N zeros that depend only on the cluster integrals b(2)(V,T), em leader,b(N)(V,T). In the thermodynamic limit, if the zero distribution approaches the positive real axis, a phase transition arises. The behavior of phase transition is determined solely by the zero distribution near the positive real axis. Below the critical temperature, the Maxwell's equal-area rule must be used to obtain the gas-liquid coexistence regime. Several examples are given.

Numerical comparison of two approaches for the study of phase transitions in small systems

We compare two recently proposed methods for the characterization of phase transitions in small systems. The validity and usefulness of these approaches are studied for the cases of the q=4 and q=5 Potts model, i.e., systems where a thermodynamic limit and exact results exist. Guided by this analysis we then discuss the helix-coil transition in polyalanine, an example of structural transitions in biological molecules.

Deceptive signals of phase transitions in small magnetic clusters

We present an analysis of the thermodynamic properties of small transition-metal clusters and show how the commonly used indicators of phase transitions such as peaks in the specific heat or magnetic susceptibility can lead to deceptive interpretations of the underlying physics. The analysis of the distribution of zeros of the canonical partition function in the whole complex temperature plane reveals the nature of the transition. We show that signals in the magnetic susceptibility at positive temperatures have their origin at zeros lying at negative temperatures.

Investigating the phase diagram of finite extensive and nonextensive systems

Microcanonical caloric curves at constant pressure and pressure versus volume curves at constant temperature are calculated using the probability distributions of a canonical ensemble at constant pressure. It is evidenced that the constant pressure path provides a natural order parameter even for small systems. While for large systems the Coulomb interaction is destroying the liquid-gas phase transition, for smaller systems the phase transition survives. Phase diagrams of a nuclear system without Coulomb and hard-core interactions are constructed and the critical point is identified.

Topology of event distributions as a generalized definition of phase transitions in finite systems

We propose a definition of first order phase transitions in finite systems based on topology anomalies of the event distribution in the space of observations. This generalizes the definitions based on the curvature anomalies of thermodynamical potentials, provides a natural definition of order parameters, and can be related to the Yang-Lee theorem in the thermodynamical limit. It is directly operational from the experimental point of view. It allows to study phase transitions in Gibbs equilibria as well as in other ensembles such as the Tsallis ensemble.

Origins of phase transitions in small systems

The identification and classification of phases in small systems, e.g., nuclei, social and financial networks, clusters, and biological systems, where the traditional definitions of phase transitions are not applicable, is important to obtain a deeper understanding of the phenomena observed in such systems. Within a simple statistical model, we investigate the validity and applicability of different classification schemes for phase transtions in small systems. We show that the whole complex temperature plane contains necessary information in order to give a distinct classification.

Precise simulation of criticality in asymmetric fluids

Extensive grand canonical Monte Carlo simulations have been performed for the hard-core square-well fluid with interaction range b=1.5 sigma. The critical exponent for the correlation length has been estimated in an unbiased fashion as nu=0.63+/-0.03 via finite-size extrapolations of the extrema of properties measured along specially constructed, asymptotically critical loci that represent pseudosymmetry axes. The subsequent location of the critical point achieves a precision of five parts in 10(4) for Tc and about 0.3% for the critical density rhoc. The effective exponents gamma+(eff) and beta(eff) indicate Ising-type critical-point values to within 2% and 5.6%, respectively, convincingly distinguishing the universality class from the "nearby" XY and n=0 (self-avoiding walk) classes. Simulations of the heat capacity CV(T,rho) and d2psigma/dT2, where psigma is the vapor pressure below Tc, suggest a negative but small Yang-Yang anomaly, i.e., a specific-heat-like divergence in the corresponding chemical potential derivative (d2 musigma/dT2) that requires a revision of the standard asymptotic scaling description of asymmetric fluids.