1,055 research outputs found
Fisher Zeros and Correlation Decay in the Ising Model
The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts in a graph, and it has also been widely studied in computer science, especially in the context of approximate counting and sampling. In this paper, we study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz\u27s self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems
Topological phases protected by symmetry can occur in gapped
and---surprisingly---in critical systems. We consider non-interacting fermions
in one dimension with spinless time-reversal symmetry. It is known that the
phases are classified by a topological invariant and a central charge
. We investigate the correlations of string operators, giving insight into
the interplay between topology and criticality. In the gapped phases, these
non-local string order parameters allow us to extract . Remarkably,
ratios of correlation lengths are universal. In the critical phases, the
scaling dimensions of these operators serve as an order parameter, encoding
and . We derive exact asymptotics of these correlation functions
using Toeplitz determinant theory. We include physical discussion, e.g.,
relating lattice operators to the conformal field theory. Moreover, we discuss
the dual spin chains. Using the aforementioned universality, the topological
invariant of the spin chain can be obtained from correlations of local
observables.Comment: 35 pages, 5 page appendi
On spinodal points and Lee-Yang edge singularities
We address a number of outstanding questions associated with the analytic
properties of the universal equation of state of the theory, which
describes the critical behavior of the Ising model and ubiquitous critical
points of the liquid-gas type. We focus on the relation between spinodal points
that limit the domain of metastability for temperatures below the critical
temperature, i.e., , and Lee-Yang edge singularities that
restrict the domain of analyticity around the point of zero magnetic field
for . The extended analyticity conjecture (due to Fonseca and
Zamolodchikov) posits that, for , the Lee-Yang edge
singularities are the closest singularities to the real axis. This has
interesting implications, in particular, that the spinodal singularities must
lie off the real axis for , in contrast to the commonly known result
of the mean-field approximation. We find that the parametric representation of
the Ising equation of state obtained in the expansion, as
well as the equation of state of the -symmetric theory at
large , are both nontrivially consistent with the conjecture. We analyze the
reason for the difficulty of addressing this issue using the
expansion. It is related to the long-standing paradox associated with the fact
that the vicinity of the Lee-Yang edge singularity is described by Fisher's
theory, which remains nonperturbative even for , where the
equation of state of the theory is expected to approach the mean-field
result. We resolve this paradox by deriving the Ginzburg criterion that
determines the size of the region around the Lee-Yang edge singularity where
mean-field theory no longer applies.Comment: 26 pages, 8 figures; v2: shortened Sec. 4.1 and streamlined
arguments/notation in Sec. 4.2, details moved to appendix, added reference 1
Characterizing correlations with full counting statistics: classical Ising and quantum XY spin chains
We propose to describe correlations in classical and quantum systems in terms
of full counting statistics of a suitably chosen discrete observable. The
method is illustrated with two exactly solvable examples: the classical
one-dimensional Ising model and the quantum spin-1/2 XY chain. For the
one-dimensional Ising model, our method results in a phase diagram with two
phases distinguishable by the long-distance behavior of the Jordan-Wigner
strings. For the quantum XY chain, the method reproduces the previously known
phase diagram.Comment: 6 pages, section on Lee-Yang zeros added, published versio
The importance of the Ising model
Understanding the relationship which integrable (solvable) models, all of
which possess very special symmetry properties, have with the generic
non-integrable models that are used to describe real experiments, which do not
have the symmetry properties, is one of the most fundamental open questions in
both statistical mechanics and quantum field theory. The importance of the
two-dimensional Ising model in a magnetic field is that it is the simplest
system where this relationship may be concretely studied. We here review the
advances made in this study, and concentrate on the magnetic susceptibility
which has revealed an unexpected natural boundary phenomenon. When this is
combined with the Fermionic representations of conformal characters, it is
suggested that the scaling theory, which smoothly connects the lattice with the
correlation length scale, may be incomplete for .Comment: 33 page
Experimental observations of dynamic critical phenomena in a lipid membrane
Near a critical point, the time scale of thermally-induced fluctuations
diverges in a manner determined by the dynamic universality class. Experiments
have verified predicted 3D dynamic critical exponents in many systems, but
similar experiments in 2D have been lacking for the case of conserved order
parameter. Here we analyze time-dependent correlation functions of a quasi-2D
lipid bilayer in water to show that its critical dynamics agree with a recently
predicted universality class. In particular, the effective dynamic exponent
crosses over from to as the correlation
length of fluctuations exceeds a hydrodynamic length set by the membrane and
bulk viscosities.Comment: 5 pages, 3 figures and 2 additional pages of supplemen
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