Loop Model with Generalized Fugacity in Three Dimensions

A statistical model of loops on the three-dimensional lattice is proposed and is investigated. It is O(n)-type but has loop fugacity that depends on global three-dimensional shapes of loops in a particular fashion. It is shown that, despite this non-locality and the dimensionality, a layer-to-layer transfer matrix can be constructed as a product of local vertex weights for infinitely many points in the parameter space. Using this transfer matrix, the site entropy is estimated numerically in the fully packed limit.Comment: 16pages, 4 eps figures, (v2) typos and Table 3 corrected. Refs added, (v3) an error in an explanation of fig.2 corrected. Refs added. (v4) Changes in the presentatio

Growth Algorithms for Lattice Heteropolymers at Low Temperatures

Two improved versions of the pruned-enriched-Rosenbluth method (PERM) are proposed and tested on simple models of lattice heteropolymers. Both are found to outperform not only the previous version of PERM, but also all other stochastic algorithms which have been employed on this problem, except for the core directed chain growth method (CG) of Beutler & Dill. In nearly all test cases they are faster in finding low-energy states, and in many cases they found new lowest energy states missed in previous papers. The CG method is superior to our method in some cases, but less efficient in others. On the other hand, the CG method uses heavily heuristics based on presumptions about the hydrophobic core and does not give thermodynamic properties, while the present method is a fully blind general purpose algorithm giving correct Boltzmann-Gibbs weights, and can be applied in principle to any stochastic sampling problem.Comment: 9 pages, 9 figures. J. Chem. Phys., in pres

Quantum dislocations: the fate of multiple vacancies in two dimensional solid 4He

Defects are believed to play a fundamental role in the supersolid state of 4He. We have studied solid 4He in two dimensions (2D) as function of the number of vacancies n_v, up to 30, inserted in the initial configuration at rho = 0.0765 A^-2, close to the melting density, with the exact zero temperature Shadow Path Integral Ground State method. The crystalline order is found to be stable also in presence of many vacancies and we observe two completely different regimes. For small n_v, up to about 6, vacancies form a bound state and cause a decrease of the crystalline order. At larger n_v, the formation energy of an extra vacancy at fixed density decreases by one order of magnitude to about 0.6 K. In the equilibrated state it is no more possible to recognize vacancies because they mainly transform into quantum dislocations and crystalline order is found almost independent on how many vacancies have been inserted in the initial configuration. The one--body density matrix in this latter regime shows a non decaying large distance tail: dislocations, that in 2D are point defects, turn out to be mobile, their number is fluctuating, and they are able to induce exchanges of particles across the system mainly triggered by the dislocation cores. These results indicate that the notion of incommensurate versus commensurate state loses meaning for solid 4He in 2D, because the number of lattice sites becomes ill defined when the system is not commensurate. Crystalline order is found to be stable also in 3D in presence of up to 100 vacancies

Liquids with Chiral Bond Order

I describe new phases of a chiral liquid crystal with nematic and hexatic order. I find a conical phase, similar to that of a cholesteric in an applied magnetic field for Frank elastic constants $K_2>K_3$. I discuss the role of fluctuations in the context of this phase and the possibility of satisfying the inequality for sufficiently long polymers. In addition I discuss the topological constraint relating defects in the bond order field to textures of the nematic and elucidate its physical meaning. Finally I discuss the analogy between smectic liquid crystals and chiral hexatics and propose a defect-riddled ground state, akin to the Renn-Lubensky twist grain boundary phase of chiral smectics.Comment: plain TeX, 19 Pages, four figures, uufiled and included. Minor correction and clarificatio

Rigidity of interfaces in the Falicov-Kimball model

We analyze the thermodynamic properties of interfaces in the three-dimensional Falicov Kimball model, which can be viewed as a primitive quantum lattice model of crystalline matter. In the strong coupling limit, the ionic subsystem of this model is governed by the Hamiltonian of an effective classical spin model whose leading part is the Ising Hamiltonian. We prove that the 100 interface in this model, at half-filling, is rigid, as in the three-dimensional Ising model. However, despite the above similarities with the Ising model, the thermodynamic properties of its 111 interface are very different. We prove that even though this interface is expected to be unstable for the Ising model, it is stable for the Falicov Kimball model at sufficiently low temperatures. This rigidity results from a phenomenon of "ground state selection" and is a consequence of the Fermi statistics of the electrons in the model.Comment: 79 pages, 9 figures included as ps-files, appendix added in revisio

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.Comment: Added reference

Finite-size scaling considerations on the ground state microcanonical temperature in entropic sampling simulations

In this work we discuss the behavior of the microcanonical temperature $\frac{\partial S(E)}{\partial E}$ obtained by means of numerical entropic sampling studies. It is observed that in almost all cases the slope of the logarithm of the density of states $S(E)$ is not infinite in the ground state, since as expected it should be directly related to the inverse temperature $\frac{1}{T}$. Here we show that these finite slopes are in fact due to finite-size effects and we propose an analytic expression $a\ln(bL)$ for the behavior of $\frac{\varDelta S}{\varDelta E}$ when $L\rightarrow\infty$. To test this idea we use three distinct two-dimensional square lattice models presenting second-order phase transitions. We calculated by exact means the parameters $a$ and $b$ for the two-states Ising model and for the $q=3$ and $4$ states Potts model and compared with the results obtained by entropic sampling simulations. We found an excellent agreement between exact and numerical values. We argue that this new set of parameters $a$ and $b$ represents an interesting novel issue of investigation in entropic sampling studies for different models

Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a $c=-2$ description. Using Lieb's transfer matrix and its description in terms of the Temperley-Lieb algebra $TL_n$ at $\beta = 0$, we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analysed in the scaling limit and the result for $L_0-\frac c{24}$ is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of $TL_n$ and are found to yield a $c=-2$ realisation of the Virasoro algebra, familiar from fermionic $bc$ ghost systems. In this realisation, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the $c=-2$ conformal integrals of motion. Consistent with the expression for $L_0-\frac c{24}$ obtained from the transfer matrix, we also construct higher Virasoro modes with $c=1$ and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the $c=1$ integrals of motion. Although this indicates that Lieb's transfer matrix description is incompatible with the $c=1$ interpretation, it does not rule out the existence of an alternative, $c=1$ compatible, transfer matrix description of the dimer model.Comment: 54 pages. v2: minor correction