# Journal of Statistical Mechanics Theory and Experiment

Published by IOP Publishing

Online ISSN: 1742-5468

Published by IOP Publishing

Online ISSN: 1742-5468

Publications

Integrative approaches to the study of complex systems demand that one knows the manner in which the parts comprising the system are connected. The structure of the complex network defining the interactions provides insight into the function and evolution of the components of the system. Unfortunately, the large size and intricacy of these networks implies that such insight is usually difficult to extract. Here, we propose a method that allows one to systematically extract and display information contained in complex networks. Specifically, we demonstrate that one can (i) find modules in complex networks and (ii) classify nodes into universal roles according to their pattern of within- and between-module connections. The method thus yields a 'cartographic representation' of complex networks.

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The Cellular Potts Model (CPM) successfully simulates drainage and shear in foams. Here we use the CPM to investigate instabilities due to the flow of a single large bubble in a dry, monodisperse two-dimensional flowing foam. As in experiments in a Hele-Shaw cell, above a threshold velocity the large bubble moves faster than the mean flow. Our simulations reproduce analytical and experimental predictions for the velocity threshold and the relative velocity of the large bubble, demonstrating the utility of the CPM in foam rheology studies.

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We suggest a fast method for finding possibly overlapping network communities of a desired size and link density. Our method is a natural generalization of the finite-T superparamagnetic Potts clustering introduced by Blatt et al (1996 Phys. Rev. Lett.76 3251) and the annealing of the Potts model with a global antiferromagnetic term recently suggested by Reichard and Bornholdt (2004 Phys. Rev. Lett.93 21870). Like in both cited works, the proposed generalization is based on ordering of the ferromagnetic Potts model; the novelty of the proposed approach lies in the adjustable dependence of the antiferromagnetic term on the population of each Potts state, which interpolates between the two previously considered cases. This adjustability allows one to empirically tune the algorithm to detect the maximum number of communities of the given size and link density. We illustrate the method by detecting protein complexes in high-throughput protein binding networks.

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Molecular spiders are synthetic biomolecular systems which have 'legs' made of short
single-stranded segments of DNA. Spiders move on a surface covered with single-stranded
DNA segments complementary to legs. Different mappings are established between various
models of spiders and simple exclusion processes. For spiders with simple gait and varying
number of legs we compute the diffusion coefficient; when the hopping is biased we also
compute their velocity.

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We investigate an idealized model of microtubule dynamics that involves: (i) attachment of guanosine triphosphate (GTP) at rate λ, (ii) conversion of GTP to guanosine diphosphate (GDP) at rate 1, and (iii) detachment of GDP at rate μ. As a function of these rates, a microtubule can grow steadily or its length can fluctuate wildly. For μ = 0, we find the exact tubule and GTP cap length distributions, and power-law length distributions of GTP and GDP islands. For μ = ∞, we argue that the time between catastrophes, where the microtubule shrinks to zero length, scales as e(λ). We also discuss the nature of the phase boundary between a growing and shrinking microtubule.

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Two recent theoretical models, Bai et al. (2004, 2007) and Tadigotla et al. (2006), formulated thermodynamic explanations of sequence-dependent transcription pausing by RNA polymerase (RNAP). The two models differ in some basic assumptions and therefore make different yet overlapping predictions for pause locations, and different predictions on pause kinetics and mechanisms. Here we present a comprehensive comparison of the two models. We show that while they have comparable predictive power of pause locations at low NTP concentrations, the Bai et al. model is more accurate than Tadigotla et al. at higher NTP concentrations. Pausing kinetics predicted by Bai et al. is also consistent with time-course transcription reactions, while Tadigotla et al. is unsuited for this type of kinetic prediction. More importantly, the two models in general predict different pausing mechanisms even for the same pausing sites, and the Bai et al. model provides an explanation more consistent with recent single molecule observations.

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All crucial features of the recently observed real-world weighted networks are obtained in a model where the weight of a link is defined with a single non-linear parameter $\alpha$ as $w_{ij}=(s_is_j)^\alpha$, $s_i$ and $s_j$ are the strengths of two end nodes of the link and $\alpha$ is a continuously tunable positive parameter. In addition the definition of strength as $s_i= \Sigma_j w_{ij}$ results a self-organizing link weight dynamics leading to a self-consistent distribution of strengths and weights on the network. Using the Barab\'asi-Albert growth dynamics all exponents of the weighted networks which are continuously tunable with $\alpha$ are obtained. It is conjectured that the weight distribution should be similar in any scale-free network. Comment: 5 pages, 4 figures

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We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings. Comment: 57 pages, in version 2: typos corrected, some sentences clarified, one appendix removed

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Through a series of exact mappings we reinterpret the Bernoulli model of sequence alignment in terms of the discrete-time totally asymmetric exclusion process with backward sequential update and step function initial condition. Using earlier results from the Bethe ansatz we obtain analytically the exact distribution of the length of the longest common subsequence of two sequences of finite lengths $X,Y$. Asymptotic analysis adapted from random matrix theory allows us to derive the thermodynamic limit directly from the finite-size result. Comment: 13 pages, 4 figures

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A lattice model of critical spanning webs is considered for the finite cylinder geometry. Due
to the presence of cycles, the model is a generalization of the known spanning
tree model which belongs to the class of logarithmic theories with central charge
c = −2. We show that in the scaling limit the universal part of the partition function for closed
boundary conditions at both edges of the cylinder coincides with the character of
symplectic fermions with periodic boundary conditions and for an open boundary at one
edge and a closed one at the other coincides with the character of symplectic fermions with
antiperiodic boundary conditions.

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We report on large scale finite-temperature Monte Carlo simulations of the
classical $120^\circ$ or $e_g$ orbital-only model on the simple cubic lattice
in three dimensions with a focus towards its critical properties. This model
displays a continuous phase transition to an orbitally ordered phase. While the
correlation length exponent $\nu\approx0.665$ is close to the 3D XY value, the
exponent $\eta \approx 0.15$ differs substantially from O(N) values. We also
introduce a discrete variant of the $e_g$ model, called $e_g$-clock model,
which is found to display the same set of exponents. Further, an emergent U(1)
symmetry is found at the critical point $T_c$, which persists for $T<T_c$ below
a crossover length scaling as $\Lambda \sim \xi^a$, with an unusually small
$a\approx1.3$.

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We study the domain wall partition function $Z_N$ for the $U_q(A_2^{(2)})$
(Izergin-Korepin) integrable $19$-vertex model on a square lattice of size $N$.
$Z_N$ is a symmetric function of two sets of parameters: horizontal
$\zeta_1,..,\zeta_N$ and vertical $z_1,..,z_N$ rapidities. For generic values
of the parameter $q$ we derive the recurrence relation for the domain wall
partition function relating $Z_{N+1}$ to $P_N Z_N$, where $P_N$ is the
proportionality factor in the recurrence, which is a polynomial symmetric in
two sets of variables $\zeta_1,..,\zeta_N$ and $z_1,..,z_N$. After setting
$q^3=-1$ the recurrence relation simplifies and we solve it in terms of a
Jacobi-Trudi-like determinant of polynomials generated by $P_N$.

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By means of an algebraic Bethe ansatz approach we study the
Zamolodchikov-Fateev and Izergin-Korepin vertex models with non-diagonal
boundaries, characterized by reflection matrices with an upper triangular form.
Generalized Bethe vectors are used to diagonalize the associated transfer
matrix. The eigenvalues as well as the Bethe equations are presented.

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Kenneth G. Wilson made deep and insightful contributions to statistical
physics, particle physics, and related fields. He was also a helpful and
thoughtful human being.

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Lars Onsager and Bruria Kaufman calculated the partition function of the
Ising model exactly in 1944 and 1949. Since then there have been many
developments in the exact solution of similar, but usually more complicated,
models. Here I shall mention a few, and show how some of the latest work seems
to be returning once again to the properties observed by Onsager and Kaufman.

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By the Bethe ansatz method we study the energy dispersion of a particle interacting by a
local interaction with fermions (or hard core bosons) of equal mass in a one-dimensional
lattice. We focus on the period of the Bloch oscillations, which turns out to be
related to the Fermi wavevector of the Fermi sea and in particular on how this
dispersion emerges as a collective effect in the thermodynamic limit. We also
discuss the adiabatic coherent collective response of the system to an applied
field.

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We study the low energy behavior of the one dimensional Hubbard model across the Mott
metal–insulator phase transition in an external magnetic field. In particular we calculate
elements of the dressed charge matrix at the critical point of the Mott transition for
arbitrary Hubbard repulsion and magnetization numerically and, in certain limiting cases,
analytically. These results are combined with a non-perturbative effective field theory
approach to reveal how the breaking of time reversal symmetry influences the Mott
transition.

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This paper gives a pedagogic derivation of the Bethe ansatz solution for 1D interacting
anyons. This includes a demonstration of the subtle role of the anyonic phases in the
Bethe ansatz arising from the anyonic commutation relations. The thermodynamic
Bethe ansatz equations defining the temperature dependent properties of the
model are also derived, from which some ground state properties are obtained.

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We discuss in details a modified variational matrix-product-state algorithm
for periodic boundary conditions, based on a recent work by P. Pippan, S.R.
White and H.G. Everts, Phys. Rev. B 81, 081103(R) (2010), which enables one to
study large systems on a ring (composed of N ~ 10^2 sites). In particular, we
introduce a couple of improvements that allow to enhance the algorithm in terms
of stability and reliability. We employ such method to compute the stiffness of
one-dimensional strongly correlated quantum lattice systems. The accuracy of
our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.

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We study one-dimensional lattice systems with pair-wise interactions of
in?nite range. We show projective convergence of Markov measures to the unique
equilibrium state. For this purpose we impose a slightly stronger condition
than summability of variations on the regularity of the interaction. With our
condition we are able to explicitly obtain stretched exponential bounds for the
rate of mixing of the equilibrium state. Finally we show convergence for the
entropy of the Markov measures to that of the equilibrium state via the
convergence of their topological pressure.

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We analyze the entanglement properties of the asymptotic steady state after a
quench from free to hard-core bosons in one dimension. The R\'enyi and von
Neumann entanglement entropies are found to be extensive, and the latter
coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble
(GGE). Computing the spectrum of the two-point function, we provide exact
analytical results both for the leading extensive parts and the subleading
terms for the entropies as well as for the cumulants of the particle number
fluctuations. We also compare the extensive part of the entanglement entropy
with the thermodynamic ones, showing that the GGE entropy equal the
entanglement one and it is the double of the diagonal entropy.

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We consider the K-body local correlations in the (repulsive) 1D Bose gas for
general K, both at finite size and in the thermodynamic limit. Concerning the
latter we develop a multiple integral formula which applies for arbitrary
states of the system with a smooth distribution of Bethe roots, including the
ground state and finite temperature Gibbs-states. In the cases K<=4 we perform
the explicit factorization of the multiple integral. In the case of K=3 we
obtain the recent result of Kormos et.al., whereas our formula for K=4 is new.
Numerical results are presented as well.

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We study thermal transport in the one dimensional classical Heisenberg model
driven by boundary heat baths in presence of a local time varying magnetic
field that acts at one end of the system. The system is studied numerically
using an energy conserving discrete-time odd even dynamics. We find that the
steady state energy current shows thermal resonance as the frequency of the
time- periodic forcing is varied. When the amplitude of the forcing field is
increased the system exhibits multiple resonance peaks instead of a single
peak. Both single and multiresonance survive in the thermodynamic limit and
their magnitudes increase as the average temperature of the system is
decreased. Finally we show that, although a reversed thermal current can be
made to flow through the bulk for a certain range of the forcing frequency, the
system fails to behave as a heat pump, thus revalidating the fact that thermal
pumping is generically absent in such force-driven lattices.

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The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated
with techniques based on the integrability of the system. Combining a knowledge of
particle quasi-momenta to exponential precision in the system size with determinant
representations of matrix elements of local operators coming from the algebraic Bethe
ansatz, we obtain rather general analytical results for the zero-temperature dynamical
correlation functions of the density and field operators. Our results thus provide
quantitative predictions for possible future experiments in atomic gases or optical
waveguides.

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We present a full identification of lattice model properties with their field
theoretical counter parts in the continuum limit for a supersymmetric model for
itinerant spinless fermions on a one dimensional chain. The continuum limit of
this model is described by an $\mathcal{N}=(2,2)$ superconformal field theory
(SCFT) with central charge c=1. We identify states and operators in the lattice
model with fields in the SCFT and we relate boundary conditions on the lattice
to sectors in the field theory. We use the dictionary we develop in this paper,
to give a pedagogical explanation of a powerful tool to study supersymmetric
models based on spectral flow. Finally, we employ the developed machinery to
explain numerically observed properties of the particle density on the open
chain presented in Beccaria et al. PRL 94:100401 (2005).

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We study the dynamics of long-wavelength fluctuations in one-dimensional (1D)
many-particle systems as described by self-consistent mode-coupling theory. The
corresponding non-linear integro-differential equations for the relevant correlators are
solved analytically and checked numerically. In particular, we find that the memory
functions exhibit a power-law decay accompanied by relatively fast oscillations.
Furthermore, the scaling behaviour and, correspondingly, the universality class depend on
the order of the leading non-linear term. In the cubic case, both viscosity and thermal
conductivity diverge in the thermodynamic limit. In the quartic case, a faster decay of the
memory functions leads to a finite viscosity, while the thermal conductivity exhibits an
even faster divergence. Finally, our analysis puts on a firmer basis the previously
conjectured connection between anomalous heat conductivity and anomalous diffusion.

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It is known that exact traveling wave solutions exist for families of (n+1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a two-dimensional representation with a generic structure. A couple of examples for n=1 and n=2 cases are presented. Comment: 12 pages, 1 figure

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We consider the Lieb-Liniger model for a gas of bosonic $\delta-$interacting
particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic
limit of the form factors of the density operator between finite entropy
eigenstates such as finite temperature states or generic non-equilibrium highly
excited states. These form factors are crucial building blocks to obtain the
thermodynamic exact dynamic correlation functions of such physically relevant
states. As a proof of principle we compute an approximated dynamic structure
factor by including only the simplest types of particle-hole excitations and
show the agreement with known results.

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We study $p$-spin glass models on regular random graphs. By analyzing the
Franz-Parisi potential with a two-body cavity field approximation under the
replica symmetric ansatz, we obtain a good approximation of the 1RSB transition
temperature for $p=3$. Our calculation method is much easier than the 1RSB
cavity method because the result is obtained by solving self-consistent
equations with Newton's method.

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The 'Cavity-Mean-Field' approximation developed for the Random Transverse
Field Ising Model on the Cayley tree [L. Ioffe and M. M\'ezard, PRL 105, 037001
(2010)] has been found to reproduce the known exact result for the surface
magnetization in $d=1$ [O. Dimitrova and M. M\'ezard, J. Stat. Mech. (2011)
P01020]. In the present paper, we propose to extend these ideas in finite
dimensions $d>1$ via a non-linear transfer approach for the surface
magnetization. In the disordered phase, the linearization of the transfer
equations correspond to the transfer matrix for a Directed Polymer in a random
medium of transverse dimension $D=d-1$, in agreement with the leading order
perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We
present numerical results of the non-linear transfer approach in dimensions
$d=2$ and $d=3$. In both cases, we find that the critical point is governed by
Infinite Disorder scaling. In particular exactly at criticality, the one-point
surface magnetization scales as $\ln m_L^{surf} \simeq - L^{\omega_c} v$, where
$\omega_c(d)$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the
corresponding Directed Polymer model, with $\omega_c(d=2)=1/3$ and
$\omega_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random
variable $v$ of order O(1) presents a power-law singularity near the origin
$P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface
magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k}
\propto L^{- x_S}$ with $x_S=\omega_c (1+a)$ independently of the order $k$.

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We present release 2.0 of the ALPS (Algorithms and Libraries for Physics
Simulations) project, an open source software project to develop libraries and
application programs for the simulation of strongly correlated quantum lattice
models such as quantum magnets, lattice bosons, and strongly correlated fermion
systems. The code development is centered on common XML and HDF5 data formats,
libraries to simplify and speed up code development, common evaluation and
plotting tools, and simulation programs. The programs enable non-experts to
start carrying out serial or parallel numerical simulations by providing basic
implementations of the important algorithms for quantum lattice models:
classical and quantum Monte Carlo (QMC) using non-local updates, extended
ensemble simulations, exact and full diagonalization (ED), the density matrix
renormalization group (DMRG) both in a static version and a dynamic
time-evolving block decimation (TEBD) code, and quantum Monte Carlo solvers for
dynamical mean field theory (DMFT). The ALPS libraries provide a powerful
framework for programers to develop their own applications, which, for
instance, greatly simplify the steps of porting a serial code onto a parallel,
distributed memory machine. Major changes in release 2.0 include the use of
HDF5 for binary data, evaluation tools in Python, support for the Windows
operating system, the use of CMake as build system and binary installation
packages for Mac OS X and Windows, and integration with the VisTrails workflow
provenance tool. The software is available from our web server at
http://alps.comp-phys.org/.

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On-site boundary conditions are often desired for lattice Boltzmann
simulations of fluid flow in complex geometries such as porous media or
microfluidic devices. The possibility to specify the exact position of the
boundary, independent of other simulation parameters, simplifies the analysis
of the system. For practical applications it should allow to freely specify the
direction of the flux, and it should be straight forward to implement in three
dimensions. Furthermore, especially for parallelized solvers it is of great
advantage if the boundary condition can be applied locally, involving only
information available on the current lattice site. We meet this need by
describing in detail how to transfer the approach suggested by Zou and He to a
D3Q19 lattice. The boundary condition acts locally, is independent of the
details of the relaxation process during collision and contains no artificial
slip. In particular, the case of an on-site no-slip boundary condition is
naturally included. We test the boundary condition in several setups and
confirm that it is capable to accurately model the velocity field up to second
order and does not contain any numerical slip.

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We study the O(n) loop model on a dynamically triangulated disk, with a new type of boundary conditions, discovered recently by Jacobsen and Saleur. The partition function of the model is that of a gas of self and mutually avoiding loops covering the disk. The Jacobsen-Saleur (JS) boundary condition prescribes that the loops that do not touch the boundary have fugacity n in [-2,2], while the loops touching at least once the boundary are given different fugacity y. The class of JS boundary conditions, labeled by the real number y, contains the Neumann (y=n) and Dirichlet (y=1) boundary conditions as particular cases. Here we consider the dense phase of the loop gas, where we compute the two-point boundary correlators of the L-leg operators with mixed Neumann-JS boundary condition. The result coincides with the boundary two-point function in Liouville theory, derived by Fateev, Zamolodchikov and Zamolodchikov. The Liouville charge of the boundary operators match, by the KPZ correspondence, with the L-leg boundary exponents conjectured by JS.

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The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables $\{h\}$ that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at $+h_{o}$ and $-h_{o}$, with the same width $\sigma$. This distribution is very general, and can recover in appropriate limits the bimodal distribution ($\sigma\to 0$) and the single gaussian one ($ho=0$). We performed Monte Carlo simulations in lattices with linear sizes in the range $L=32 - 512$. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities $h_{o}$, for some small values of the width $\sigma$. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of $\sigma$. Comment: 13 pages, 9 figures, accepted for publication in JSTAT

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We consider the critical spin-spin correlation function of the 2D Ising model
with a line defect which strength is an arbitrary function of position. By
using path-integral techniques in the continuum description of this model in
terms of fermion fields, we obtain an analytical expression for the correlator
as functional of the position dependent coupling. Thus, our result provides one
of the few analytical examples that allows to illustrate the transit of a
magnetic system from scaling to non-scaling behavior in a critical regime. We
also show that the non-scaling behavior obtained for the spin correlator along
a non-uniformly altered line of an Ising model remains unchanged in the
Ashkin-Teller model.

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We study the spectrum of Landau-Ginzburg theories in 1+1 dimensions using the
truncated conformal space approach employing a compactified boson. We study
these theories both in their broken and unbroken phases. We first demonstrate
that we can reproduce the expected spectrum of a $\Phi^2$ theory (i.e. a free
massive boson) in this framework. We then turn to $\Phi^4$ in its unbroken
phase and compare our numerical results with the predictions of two-loop
perturbation theory, finding excellent agreement. We then analyze the broken
phase of $\Phi^4$ where kink excitations together with their bound states are
present. We confirm the semiclassical predictions for this model on the number
of stable kink-antikink bound states. We also test the semiclassics in the
double well phase of $\Phi^6$ Landau-Ginzburg theory, again finding agreement.

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The two Berezinskii-Kosterlitz-Thouless phase transitions of the
two-dimensional 5-state clock model are studied on infinite strips using the
DMRG algorithm. Because of the open boundary conditions, the helicity modulus
$\Upsilon_2$ is computed by imposing twisted magnetic fields at the two
boundaries. Its scaling behavior is in good agreement with the existence of
essential singularities with $\sigma=1/2$ at the two transitions. The predicted
universal values of $\Upsilon_2$ are shown to be reached in the thermodynamic
limit. The fourth-order helicity modulus is observed to display a dip at the
low-temperature BKT transition, like the XY model. Finally, the scaling
behavior of magnetization at the low temperature transition is compatible with
$\eta=1/4$.

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The 2D Euler equations is the basic example of fluid models for which a
microcanical measure can be constructed from first principles. This measure is
defined through finite-dimensional approximations and a limiting procedure.
Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting
algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that
Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler
microcanonical measures (incorporating fixed energy and other invariants). This
is essential as microcanonical and canonical measures are known to be
inequivalent at some values of energy and vorticity distribution. Creutz's
algorithm is used to check predictions from the mean-field statistical
mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory).
We found full agreement with theory. Three different ways to compute the
temperature give consistent results. Using Creutz's algorithm, a first-order
phase transition never observed previously, and a situation of statistical
ensemble inequivalence are found and studied. Strikingly, and contrasting usual
statistical mechanics interpretations, this phase transition appears from a
disordered phase to an ordered phase (with less symmetries) when energy is
increased. We explain this paradox.

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We examine the scaling behavior of the entanglement entropy for the 2D quantum dimer model (QDM) at criticality and derive the universal finite sub-leading correction $\gamma_{QCP}$. We compute the value of $\gamma_{QCP}$ without approximation working directly with the wave function of a generalized 2D QDM at the Rokhsar-Kivelson QCP in the continuum limit. Using the replica approach, we construct the conformal boundary state corresponding to the cyclic identification of $n$-copies along the boundary of the observed region. We find that the universal finite term is $\gamma_{QCP}=\ln R$ where $R$ is the compactification radius of the bose field theory quantum Lifshitz model, the effective field theory of the 2D QDM at quantum criticality. It is shown that, at least in the cylinder geometry, no other universal contributions to the entanglement entropy are possible. We also demonstrated that the entanglement spectrum of the critical wave function on a large but finite region is described by the characters of the underlying conformal field theory. A geometric interpretation is given to the R{\'e}nyi entropy for this system at criticality, and it is shown that this is formally related to the problems of quantum Brownian motion on $n$-dimensional lattices or equivalently a system of strings interacting with a brane containing a background electromagnetic field and can be written as an expectation value of a vertex operator.

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We study scaling of the ground-state fidelity in neighborhoods of quantum
critical points in a model of interacting spinfull fermions - a BCS-like model.
Due to the exact diagonalizability of the model, in one and higher dimensions,
scaling of the ground-state fidelity can be analyzed numerically with great
accuracy, not only for small systems but also for macroscopic ones, together
with the crossover region between them. Additionally, in one-dimensional case
we have been able to derive a number of analytical formulae for fidelity and
show that they accurately fit our numerical results; these results are reported
in the article. Besides regular critical points and their neighborhoods, where
well-known scaling laws are obeyed, there is the multi-critical point and
critical points in its proximity where anomalous scaling behavior is found. We
consider also scaling of fidelity in neighborhoods of critical points where
fidelity oscillates strongly as the system size or the chemical potential is
varied. Our results for a one-dimensional version of a BCS-like model are
compared with those obtained by Rams and Damski in similar studies of a quantum
spin chain - an anisotropic XY model in transverse magnetic field.

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The distribution of internal shear stresses in a 2D dislocation system is investigated for
when external shear stress is applied. This problem serves as a natural continuation of the
previous work of Csikor and Groma (2004 Phys. Rev. B 58 2969), where an analytical
result was given for the stress distribution function at zero applied stress. First,
the internal stress distribution generated by a set of randomly positioned ideal
dislocation dipoles is studied. Analytical calculations are carried out for this case. The
theoretical predictions are checked by numerical simulations, showing perfect
agreement. It is found that for real relaxed dislocation configurations the role of
dislocation multipoles cannot be neglected, but the theory presented can still be
applied.

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We investigate and contrast, via entropic sampling based on the Wang-Landau algorithm, the effects of quenched bond randomness on the critical behavior of two Ising spin models in 2D. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple Ising model are studied and their general universality aspects are inspected by a detailed finite-size scaling (FSS) analysis. We find that, the random bond SAF model obeys weak universality, hyperscaling, and exhibits a strong saturating behavior of the specific heat due to the competing nature of interactions. On the other hand, for the random Ising model we encounter some difficulties for a definite discrimination between the two well-known scenarios of the logarithmic corrections versus the weak universality. Yet, a careful FSS analysis of our data favors the field-theoretically predicted logarithmic corrections.

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We report comprehensive inelastic neutron scattering measurements of the magnetic excitations in the 2D spin-5/2 Heisenberg antiferromagnet Rb2MnF4 as a function of temperature from deep in the Neel ordered phase up to paramagnetic, 0.13 < kBT/4JS < 1.4. Well defined spin-waves are found for wave-vectors larger than the inverse correlation length $\eta^{-1}$ for temperatures up to near the Curie-Weiss temperature, $\Theta_{CW}$. For wave-vectors smaller than $\eta^{-1}$, relaxational dynamics occurs. The observed renormalization of spin-wave energies, and evolution of excitation line-shapes, with increasing temperature are quantitatively compared with finite-temperature spin-wave theory, and computer simulations for classical spins. Random phase approximation calculations provide a good description of the low-temperature renormalisation of spin-waves. In contrast, lifetime broadening calculated using the first Born approximation shows, at best, modest agreement around the zone boundary at low temperatures. Classical dynamics simulations using an appropriate quantum-classical correspondence were found to provide a good description of the intermediate- and high-temperature regimes over all wave-vector and energy scales, and the crossover from quantum to classical dynamics observed around $\Theta_{CW}/S$, where the spin S=5/2. A characterisation of the data over the whole wave-vector/energy/temperature parameter space is given. In this, $T^2$ behaviour is found to dominate the wave-vector and temperature dependence of the line widths over a large parameter range, and no evidence of hydrodynamic behaviour or dynamical scaling behaviour found within the accuracy of the data sets.

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We present a thorough study of the static properties of 2D models of spin-ice
type on the square lattice or, in other words, the sixteen-vertex model. We use
extensive Monte Carlo simulations to determine the phase diagram and critical
properties of the finite dimensional system. We put forward a suitable
mean-field approximation, by defining the model on carefully chosen trees. We
employ the cavity (Bethe-Peierls) method to derive self-consistent equations,
the fixed points of which yield the equilibrium properties of the model on the
tree-like graph. We compare mean-field and finite dimensional results. We
discuss our findings in the context of experiments in artificial two
dimensional spin ice.

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We investigate the out of equilibrium dynamics of the two-dimensional XY
model when cooled across the Berezinskii-Kosterlitz-Thouless (BKT) phase
transition using different protocols. We focus on the evolution of the growing
correlation length and the density of topological defects (vortices). By using
Monte Carlo simulations we first determine the time and temperature dependence
of the growing correlation length after an infinitely rapid quench from above
the transition temperature to the quasi-long range order region. The functional
form is consistent with a logarithmic correction to the diffusive law and it
serves to validate dynamic scaling in this problem. This analysis clarifies the
different dynamic roles played by bound and free vortices. We then revisit the
Kibble-Zurek mechanism in thermal phase transitions in which the disordered
state is plagued with topological defects. We provide a theory of quenching
rate dependence in systems with the BKT-type transition that goes beyond the
equilibrium scaling arguments. Finally, we discuss the implications of our
results to a host of physical systems with vortex excitations including planar
ferromagnets and liquid crystals as well as the Ginzburg-Landau approach to
bidimensional freely decaying turbulence.

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Resonant activation is one of classical effects demonstrating constructive
role of noise. In resonant activation cooperative action of barrier modulation
process and noise lead to the optimal escape kinetics as measured by the mean
first passage time. Resonant activation has been observed in versatilities of
systems for various types of barrier modulation processes and noise types.
Here, we show that resonant activation is also observed in 2D and 3D systems
driven by bi-variate and tri-variate $\alpha$-stable noises. Strength of
resonant activation is sensitive to the exact value of the noise parameters. In
particular, the decrease in the stability index $\alpha$ results in the
disappearance of the resonant activation.

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We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials.
Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.

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We study the behaviour of the 2d Ising model in the symmetric high
temperature phase in presence of a small magnetic perturbation. We successfully
compare the quantum field theory predictions for the shift in the mass spectrum
of the theory with a set of high precision transfer matrix results. Our results
rule out a prediction for the same quantity obtained some years ago with strong
coupling methods.

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We study the fate of the 2d kinetic q-state Potts model after a sudden quench
to zero temperature. Both ground states and complicated static states are
reached with non-zero probabilities. These outcomes resemble those found in the
quench of the 2d Ising model; however, the variety of static states in the
q-state Potts model (with q>=3) is much richer than in the Ising model, where
static states are either ground or stripe states. Another possibility is that
the system gets trapped on a set of equal-energy blinker states where a subset
of spins can flip ad infinitum; these states are similar to those found in the
quench of the 3d Ising model. The evolution towards the final energy is also
unusual---at long times, sudden and massive energy drops may occur that are
accompanied by macroscopic reordering of the domain structure. This
indeterminacy in the zero-temperature quench of the kinetic Potts model is at
odds with basic predictions from the theory of phase-ordering kinetics. We also
propose a continuum description of coarsening with more than two equivalent
ground states. The resulting time-dependent Ginzburg-Landau equations reproduce
the complex cluster patterns that arise in the quench of the kinetic Potts
model.

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In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when
$y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the
determinantal formulae defined by conformal $(2m,1)$ models. Our approach
follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and
uses a Lax pair representation of the conformal $(2m,1)$ models (giving
Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinantal formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.

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