Multiple Singularities of the Equilibrium Free Energy in a One-Dimensional Model of Soft Rods.

There is a misconception, widely shared among physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at nonzero temperatures, cannot show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counterexample. We consider thin rigid linear rods of equal length 2ℓ whose centers lie on a one-dimensional lattice, of lattice spacing a. The interaction between rods is a soft-core interaction, having a finite energy U per overlap of rods. We show that the equilibrium free energy per rod F[(ℓ/a),ß], at inverse temperature ß, has an infinite number of singularities, as a function of ℓ/a.

Columnar order and Ashkin-Teller criticality in mixtures of hard squares and dimers.

We show that critical exponents of the transition to columnar order in a mixture of 2×1 dimers and 2×2 hard squares on the square lattice depends on the composition of the mixture in exactly the manner predicted by the theory of Ashkin-Teller criticality, including in the hard-square limit. This result settles the question regarding the nature of the transition in the hard-square lattice gas. It also provides the first example of a polydisperse system whose critical properties depend on composition. Our ideas also lead to some interesting predictions for a class of frustrated quantum magnets that exhibit columnar ordering of the bond energies at low temperature.

Cusp singularities in the distribution of orientations of asymmetrically pivoted hard disks on a lattice.

We study a system of equal-size circular disks, each with an asymmetrically placed pivot at a fixed distance from the center. The pivots are fixed at the vertices of a regular triangular lattice. The disks can rotate freely about the pivots, with the constraint that no disks can overlap with each other. Our Monte Carlo simulations show that the one-point probability distribution of orientations has multiple cusplike singularities. We determine the exact positions and qualitative behavior of these singularities. In addition to these geometrical singularities, we also find that the system shows order-disorder transitions, with a disordered phase at large lattice spacings, a phase with spontaneously broken orientational lattice symmetry at small lattice spacings, and an intervening Berezinskii-Kosterlitz-Thouless phase in between.

Many universality classes in an interface model restricted to non-negative heights.

We present a simple one-dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site x and time t, an integer n(x,t) satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condition, so that the growing interfaces are in the Edwards-Wilkinson or in the Kardar-Parisi-Zhang universality class. In addition, there is also a constraint n(x,t)≥0. Points x where n>0 on one side and n=0 on the other are called "fronts." These fronts can be "pushed" or "pulled," depending on the control parameters. For pulled fronts, the lateral spreading is in the directed percolation (DP) universality class, while it is in a different universality class for pushed fronts, and another universality class in between. In the DP case, the activity at each active site can in general be arbitrarily large, in contrast to previous realizations of DP. Finally, we find two different types of transitions when the interface detaches from the line n=0 (with 〈n(x,t)〉→const on one side, and →∞ on the other), again with new universality classes. We also discuss a mapping of this model to the avalanche propagation in a directed Oslo rice pile model in specially prepared backgrounds.

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice.

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

Spontaneous layering and power-law order in the three-dimensional fully packed hard-plate lattice gas.

We obtain the phase diagram of fully packed hard plates on the cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities s_{µ} for "µ plates," whose normal is the µ direction (µ=x,y,z). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities s_{µ_{1}} and s_{µ_{2}} are comparable, and the third fugacity s_{µ_{3}} is much smaller, we find a spontaneously layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the µ_{3} axis. µ_{1} and µ_{2} plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. This corresponds to a twofold breaking of translation symmetry along the µ_{3} axis. In the opposite limit, with µ_{3}≫µ_{1}∼µ_{2}, we find a phase with long-range columnar order, corresponding to simultaneous twofold symmetry breaking of lattice translation symmetry in directions µ_{1} and µ_{2}. The spontaneously layered phases display critical behavior, with power-law decay of correlations in the µ_{1} and µ_{2} directions when the slabs are stacked in the µ_{3} direction, and represent examples of "floating phases" discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behavior to the constrained motion of defects in this phase, and we sketch a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.

Resonating valence bond wave functions and classical interacting dimer models.

We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for SU(g) spin systems on two-dimensional bipartite lattices to those of fully packed interacting classical dimer models on the same lattice. The interaction energy can be expressed as a sum of n-body potentials V(n), which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the n-body interaction V(n) (n>1) is of order O(g(-(n-1))) for small g(-1). The leading term is a two-body nearest-neighbor interaction V2(g) favoring two parallel dimers on elementary plaquettes. For SU(2) spins, using our calculated value of V2(g=2), we find that the long-distance behavior of the bond-energy correlation function is dominated by an oscillatory term that decays as 1/|r|α with α≈1.22. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give α≈1.20.

Continuous dissolution of structure during the unfolding of a small protein.

The unfolding kinetics of many small proteins appears to be first order, when measured by ensemble-averaging probes such as fluorescence and circular dichroism. For one such protein, monellin, it is shown here that hidden behind this deceptive simplicity is a complexity that becomes evident with the use of experimental probes that are able to discriminate between different conformations in an ensemble of structures. In this study, the unfolding of monellin has been probed by measurement of the changes in the distributions of 4 different intramolecular distances, using a multisite, time-resolved fluorescence resonance energy transfer methodology. During the course of unfolding, the protein molecules are seen to undergo slow and continuous, diffusive swelling. The swelling process can be modeled as the slow diffusive swelling of a Rouse-like chain with some additional noncovalent, intramolecular interactions. Here, we show that specific structure is lost during the swelling process gradually, and not in an all-or-none manner, during unfolding.

Phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice.

A system of hard rigid rods of length k on hypercubic lattices is known to undergo two phase transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large k, the second phase transition is a first-order transition with a discontinuity in density in all dimensions greater than 1. We show that the chemical potential at the transition is ≈kln[k/lnk] for large k, and that the density of uncovered sites drops from a value ≈(lnk)/k^{2} to a value of order exp(-ak), where a is some constant, across the transition. We conjecture that these results are asymptotically exact, in all dimensions d≥2. We also present evidence of coexistence of nematic and disordered phases from Monte Carlo simulations for rods of length 9 on the square lattice.

Sequence of phase transitions in a model of interacting rods.

In a system of interacting thin rigid rods of equal length 2ℓ on a two-dimensional grid of lattice spacing a, we show that there are multiple phase transitions as the coupling strength κ=ℓ/a and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry-breaking transition and the second belongs to less-studied phase transitions of geometrical origin. The latter class of transitions appears at fixed values of κ irrespective of the temperature, whereas the critical coupling for the spontaneous symmetry-breaking transition depends on it. By varying the temperature, the phase boundaries may cross each other, leading to a rich phase behavior with infinitely many phases. Our results are based on Monte Carlo simulations on the square lattice and a fixed-point analysis of a functional flow equation on a Bethe lattice.

Entropy of fully packed hard rigid rods on d-dimensional hypercubic lattices.

We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×k, in the limit of large k. We show that full coverage is possible only if at least one of L and M is a multiple of k, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a k×k square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large k, we show that the entropy per site S_{2}(k) tends to Ak^{-2}lnk, with A=1. We conjecture, based on a perturbative series expansion, that this large-k behavior of entropy per site is superuniversal and continues to hold on all d-dimensional hypercubic lattices, with d≥2.

Fractional Brownian motion of worms in worm algorithms for frustrated Ising magnets.

We study the distribution of lengths and other statistical properties of worms constructed by Monte Carlo worm algorithms in the power-law three-sublattice ordered phase of frustrated triangular and kagome lattice Ising antiferromagnets. Viewing each step of the worm construction as a position increment (step) of a random walker, we demonstrate that the persistence exponent θ and the dynamical exponent z of this random walk depend only on the universal power-law exponents of the underlying critical phase and not on the details of the worm algorithm or the microscopic Hamiltonian. Further, we argue that the detailed balance condition obeyed by such worm algorithms and the power-law correlations of the underlying equilibrium system together give rise to two related properties of this random walk: First, the steps of the walk are expected to be power-law correlated in time. Second, the position distribution of the walker relative to its starting point is given by the equilibrium position distribution of a particle in an attractive logarithmic central potential of strength η_{m}, where η_{m} is the universal power-law exponent of the equilibrium defect-antidefect correlation function of the underlying spin system. We derive a scaling relation, z=(2-η_{m})/(1-θ), that allows us to express the dynamical exponent z(η_{m}) of this process in terms of its persistence exponent θ(η_{m}). Our measurements of z(η_{m}) and θ(η_{m}) are consistent with this relation over a range of values of the universal equilibrium exponent η_{m} and yield subdiffusive (z>2) values of z in the entire range. Thus, we demonstrate that the worms represent a discrete-time realization of a fractional Brownian motion characterized by these properties.

Density relaxation in conserved Manna sandpiles.

We study relaxation of long-wavelength density perturbations in a one-dimensional conserved Manna sandpile. Far from criticality where correlation length ξ is finite, relaxation of density profiles having wave numbers k→0 is diffusive, with relaxation time τ_{R}∼k^{-2}/D with D being the density-dependent bulk-diffusion coefficient. Near criticality with kξ≳1, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as τ_{R}∼k^{-z}, with the dynamical exponent z=2-(1-ß)/ν_{⊥}<2, where ß is the critical order-parameter exponent and ν_{⊥} is the critical correlation-length exponent. Relaxation of initially localized density profiles on an infinite critical background exhibits a self-similar structure. In this case, the asymptotic scaling form of the time-dependent density profile is analytically calculated: we find that, at long times t, the width σ of the density perturbation grows anomalously, σ∼t^{w}, with the growth exponent ω=1/(1+ß)>1/2. In all cases, theoretical predictions are in reasonably good agreement with simulations.

Observation of Continuous Contraction and a Metastable Misfolded State during the Collapse and Folding of a Small Protein.

To obtain proper insight into how structure develops during a protein folding reaction, it is necessary to understand the nature and mechanism of the polypeptide chain collapse reaction, which marks the initiation of folding. Here, the time-resolved fluorescence resonance energy transfer technique, in which the decay of the fluorescence light intensity with time is used to determine the time evolution of the distribution of intra-molecular distances, has been utilized to study the folding of the small protein, monellin. It is seen that when folding begins, about one-third of the protein molecules collapse into a molten globule state (IMG), from which they relax by continuous further contraction to transit to the native state. The larger fraction gets trapped into a metastable misfolded state. Exit from this metastable state occurs via collapse to the lower free energy IMG state. This exit is slow, on a time scale of seconds, because of activation energy barriers. The trapped misfolded molecules as well as the IMG molecules contract continuously and slowly as structure develops. A phenomenological model of Markovian evolution of the polymer chain undergoing folding, incorporating these features, has been developed, which fits well the experimentally observed time evolution of distance distributions. The observation that the "wrong turn" to the misfolded state has not been eliminated by evolution belies the common belief that the folding of functional protein sequences is very different from that of a random heteropolymer, and the former have been selected by evolution to fold quickly.

Phase diagram of a system of hard cubes on the cubic lattice.

We study the phase diagram of a system of 2×2×2 hard cubes on a three-dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks up into parallel slabs of size 2×L×L where only a very small fraction cubes do not lie wholly within a slab. Within each slab, the cubes are disordered; translation symmetry is thus broken along exactly one principal axis. In the solidlike sublattice-ordered phase, the hard cubes preferentially occupy one of eight sublattices of the cubic lattice, breaking translational symmetry along all three principal directions. In the columnar phase, the system spontaneously breaks up into weakly interacting parallel columns of size 2×2×L, where only a very small fraction cubes do not lie wholly within a column. Within each column, the system is disordered, and thus translational symmetry is broken only along two principal directions. Using finite-size scaling, we show that the disordered-layered phase transition is continuous, while the layered-sublattice and sublattice-columnar transitions are discontinuous. We construct a Landau theory written in terms of the layering and columnar order parameters which is able to describe the different phases that are observed in the simulations and the order of the transitions. Additionally, our results near the disordered-layered transition are consistent with the O(3) universality class perturbed by cubic anisotropy as predicted by the Landau theory.

Emergence of quasiunits in the one-dimensional Zhang model.

We study the Zhang model of sandpile on a one-dimensional chain of length L , where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as L(-1/2) for large L . We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one-dimensional Abelian model with discrete heights. We argue that in the large L limit, the variance of energy at site x has a scaling form L(-1)g(x/L) , where g(xi) varies as ln(1/xi) for small xi , which agrees very well with the results from numerical simulations.

Random trimer tilings.

We study tilings of the square lattice by linear trimers. For a cylinder of circumference m , we construct a conserved functional of the base of the tilings, and use this to block diagonalize the transfer matrix. The number of blocks increases exponentially with m . The dimension of the block corresponding to the largest eigenvalue is shown to grow as (32;{13});{m} . We numerically diagonalize this block for m

General mechanism for the 1/f noise.

We consider the response of a memoryless nonlinear device that acts instantaneously, converting an input signal ξ(t) into an output η(t) at the same time t. For input Gaussian noise with power-spectrum 1/f^{α}, the nonlinearity can modify the spectral index of the output to give a spectrum that varies as 1/f^{α^{'}} with α^{'}≠α. We show that the value of α^{'} depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous 1/f noise found in nature.

Probability distribution of residence times of grains in models of rice piles.

We study the probability distribution of residence time of a grain at a site, and its total residence time inside a pile, in different rice pile models. The tails of these distributions are dominated by the grains that get deeply buried in the pile. We show that, for a pile of size L, the probabilities that the residence time at a site or the total residence time is greater than t, both decay as 1/t(ln t)x for L(omega) << t << exp(L(gamma)) where gamma is an exponent > or = 1, and values of x and omega in the two cases are different. In the Oslo rice pile model we find that the probability of the residence time T(i) at a site i being greater than or equal to t is a nonmonotonic function of L for a fixed t and does not obey simple scaling. For model in d dimensions, we show that the probability of minimum slope configuration in the steady state, for large L, varies as exp(-kappaL(d+2)) where kappa is a constant, and hence gamma=d+2.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model.

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes >10^{7}. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: ν=4/3 for the correlation length exponent, D=9/4 for the fractal dimension of avalanche clusters, and z=10/7 for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent ν. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is α_{loc}=1.