Loop Gas Model Research Articles

Partially topological phase in a quantum loop gas model with tension and pressure

Partially topological phase in a quantum loop gas model with tension and pressure

Boundary Topological Entanglement Entropy in Two and Three Dimensions

The topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized {mathcal {S}}-matrices. We conjecture a general property of these {mathcal {S}}-matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.

Gauge-field model of superfluid turbulence in the zero-temperature limit

We present a gauge-field extension of the Bose condensate model that describes superfluid turbulence generated by the macroscopic motion of the superfluid. We first establish that the condensate model is dual to the short-range interacting loop gas model, wherein the loops represent quantum vortex lines. Vortex lines form, interact and proliferate as a result of the superfluid motion. Our extension is based on incorporating the Biot–Savart interaction between vortex lines, which is lacking in the loop gas model. We show that the extended loop gas is dual to a Ginzburg–Landau model, wherein the gauge coupling is between the macroscopic velocity field of the superfluid and the condensate. Applying the model to cylindrical and pipe flows, we describe how turbulence transitions with and without intermediate vortex flow, respectively.

Markov chain sampling of the O(n) loop models on the infinite plane.

A numerical method was recently proposed in Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)10.1103/PhysRevE.94.043322] showing a precise sampling of the infinite plane two-dimensional critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the O(n) loop gas models for n∈(1,2]. We argue that even though the Gibbs measure is nonlocal, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the O(n) models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.

Causal dynamical triangulation of a 3D tensor model

We extend the string field theory of 2D generalized causal dynamical triangulation (GCDT) with the Ishibashi-Kawai type (IK-type) interaction formulated by the matrix model to the 3D model of the surface field theory. Based on the loop gas model, we construct a tensor model for the discretized surface field and then apply it the stochastic quantization method. In the double-scaling limit, the model is characterized by two scaling dimensions $D$ and $D_N$, the power indices of the minimal length as the scaling parameter. The continuum GCDT model with the IK-type interaction is realized with a similar restriction in the $D_N$-$D$ space to the 2D model. The distinct property in the 3D model is that the quantum effect contains the IK-type interaction only, while the ordinary splitting interaction is excluded.

Causal dynamical triangulation for non-critical open-closed string field theory

We extend the 2 dimensional Causal Dynamical Triangulation (CDT) model from the usual model of closed string to the one of open-closed string. The matrix-vector model describing the loop gas model is modified so as to possess the nature of the CDT, i.e. the time foliation structure. Stochastic quantization method produces interactions of loop and line variables similar to those in the non-critical open-closed string field theories. By taking an appropriate scaling, we realize an extended model of the generalized CDT (GCDT), which keeps the causality in a broad sense.

STOCHASTIC QUANTIZATION APPROACH FOR CAUSAL DYNAMICAL TRIANGULATION STRING FIELD THEORY

We construct a two-dimensional (2D) causal dynamical triangulation (CDT) model from a matrix model which represents the loop gas model of closed string. The target-space index is reinterpreted as time or geodesic distance. We apply stochastic quantization method to the model to obtain the Generalized CDT (GCDT), which has additional interaction of creating baby universe. If we take a specific scaling in continuum limit, we realize an extended GCDT model characterized by the noncritical string field theory.

Topological phases: An expedition off lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of ‘baby universe’, here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger’s theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.

Enumeration of maps with self-avoiding loops and the model on random lattices of all topologies

We compute the generating functions of an model (loop gas model) on a random lattice of any topology. On the disc and thecylinder, the topologies were already known, and here we compute all the othertopologies. We find that they obey a slightly deformed version of the topologicalrecursion valid for the 1-Hermitian matrix models. The generating functions of genusg maps without boundaries are given by the symplectic invariantsFg of a spectral curve . This spectral curve was known before, and it is in general not algebraic.

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization (TEFR) approach that removes local entanglement and coarse grain the lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym}, T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

Breakdown of a Topological Phase: Quantum Phase Transition in a Loop Gas Model with Tension

We study the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model--the toric code--which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of "magnetic" vortices, leading to confinement of the elementary "charge" excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and relate our results to error rates for topological quantum computations.

Minimal superstrings and loop gas models

We reformulate the matrix models of minimal superstrings as loop gas models on random surfaces. In the continuum limit, this leads to the identification of minimal superstrings with certain bosonic string theories, to all orders in the genus expansion. RR vertex operators arise as operators in a Z_2 twisted sector of the matter CFT. We show how the loop gas model implements the sum over spin structures expected from the continuum RNS formulation. Open string boundary conditions are also more transparent in this language.

On Freedman's Lattice Models for Topological Phases

Freedman proposes a family of Hamiltonians which define quantum loop gas models on any celluated compact surface. We study the simplest nontrivial cases: celluations of the torus. Our numerical data support Freedman's conjecture about the ground states of the Hamiltonians. PACS: 71.10.-w; 71.35.-y

Edge excitations of an incompressible fermionic liquid in a staggered magnetic field

A model of lattice fermions in 2 + 1-dimensional space is formulated, the critical states of which are lying at the basis of such physical problems, as the 3D Ising Model(3DIM) and the edge excitations in the Hall effect. The action for these exitations coincides with the action of the so-called sign-factor model in 3DIM at one values of its parameters, and represent a model for the edge excitations, which are responsible for the plateau transitions in the Hall effect, at other values. The model can be formulated also as a loop gas models in 2D, but unlike the O( n) models, where the loop fugacity is real, here we have directed (clockwise and counterclockwise) loops and phase factors e ±2π p q i for them. The line of phase transitions in the parameter space is found and corresponding continuum limits of these models are constructed. It appears, that besides the ordinary critical line, which separates the dense and diluted phases of the models(like in ordinary O( n) models), there is a line, which corresponds to the full covering of the space by curves. The N = 2 twisted superconformal models with SU(2)/ U(1) cosec model with coupling constant k = q p − 2 describes these states.

Gas of self-avoiding loops on the brickwork lattice

An exact calculation of the phase diagram for a loop gas model on the brickwork lattice is presented. The model includes a bending energy. In the dense limit, where all the lattice sites are occupied, a phase transition occuring at an asymmetric Lifshitz tricritical point is observed as the temperature associated with the bending energy is varied. Various critical exponents are calculated. At lower densities, two lines of transitions (in the Ising universality class) are observed, terminated by a tricritical point, where there is a change in the modulation of the correlation function. To each tricritical point an associated disorder line is found.

Loop gas model for open strings

The open string with one-dimensional target space is formulated in terms of an SOS, or loop gas model on a random surface. We solve an integral equation for the loop amplitude with Dirichlet and Neumann boundary conditions imposed on different pieces of its boundary. The result is used to calculate the mean values of order and disorder operators, to construct the string propagator and find its spectrum of excitations. The latter is not sensitive either to the string tension λ or to the mass μ of the “quarks” at the ends of the string. As in the case of closed strings, the SOS formulation allows us to construct a Feynman-diagram technique for the string interaction amplitudes.

Multicritical phases of the O( n) model on a random lattice

We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/ p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known ( p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently considered independently by Saleur and Zamolodchikov. Our results may be of help in defining such models on flat honeycomb lattices; an unsolved problem in polymer theory. The phase boundaries correspond again to “topological” points with g = p/1 integer, which we study in some detail. Two qualitatively different types of critical points are discovered for each such g. For the special point g = 2 we demonstrate that the dilute phase O(−2) model does not correspond to the Parisi-Sourlas model, a result likely to hold as well for the flat case. Instead it is proven that the first multicritical O(−2) point possesses the Parisi-Sourlas supersymmetry.

Estimate of Z(2,4) for the loop-gas model on the honeycomb lattice.

By making an exact mapping of the resonant-valence-bond state to a classical O(4) model in a logarithmic potential, Shapir and Kohmoto recently obtained a mean-field estimate for the norm of the wave function from the partition function of the classical model. Their mean-field estimate is expected to be correct in high dimensions (only). We discuss the partition function of the loop-gas model, Z(x,y), on the honeycomb lattice for various values of the fugacities x and y and obtain as a special case an approximation for Z(2,4), the norm of the wave function, which should be quite reasonable for the two-dimensional case considered here.

Critical behaviour of the loop gas model on the square lattice

The critical behaviour of the two-dimensional self-avoiding loop gas model with multiplicity m=1 (singly counted non-intersecting loops) is studied numerically on a square lattice using a recently developed Monte Carlo method. The critical exponents alpha , beta , gamma and delta are evaluated in the 'critical window' between the finite-size rounding and the non-critical ('correction-to-scaling') regime using a recently calculated accurate value of Tc. Within error bars, Ising-like critical exponents of the loop gas are obtained.

Use of Microcomputers for Fast Monte Carlo Calculations

The use of microcomputers (Personal Computers, Home Computers) for sophisticated scientific calculations is exemplified. A fast Monte Carlo program is applied to simulate a whole class of statistical models on the two-dimensional square lattice in the critical region. Specifically, results on the critical behaviour of the loop gas model order parameter are presented, which show that the model lies in the Ising universality class.