102 research outputs found
Spacetime Approach to Phase Transitions
In these notes, the application of Feynman's sum-over-paths approach to
thermal phase transitions is discussed. The paradigm of such a spacetime
approach to critical phenomena is provided by the high-temperature expansion of
spin models. This expansion, known as the hopping expansion in the context of
lattice field theory, yields a geometric description of the phase transition in
these models, with the thermal critical exponents being determined by the
fractal structure of the high-temperature graphs. The graphs percolate at the
thermal critical point and can be studied using purely geometrical observables
known from percolation theory. Besides the phase transition in spin models and
in the closely related theory, other transitions discussed from this
perspective include Bose-Einstein condensation, and the transitions in the
Higgs model and the pure U(1) gauge theory.Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the
National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected
typo
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
Duality and defects in rational conformal field theory
We study topological defect lines in two-dimensional rational conformal field
theory. Continuous variation of the location of such a defect does not change
the value of a correlator. Defects separating different phases of local CFTs
with the same chiral symmetry are included in our discussion. We show how the
resulting one-dimensional phase boundaries can be used to extract symmetries
and order-disorder dualities of the CFT.
The case of central charge c=4/5, for which there are two inequivalent world
sheet phases corresponding to the tetra-critical Ising model and the critical
three-states Potts model, is treated as an illustrative example.Comment: 78 pages, several figures; v2: typos corrected and some references
adde
Topological holography: Towards a unification of Landau and beyond-Landau physics
We outline a holographic framework that attempts to unify Landau and
beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a
modern understanding of symmetries as topological defects/operators, the
framework uses a topological order to organize the space of quantum systems
with a global symmetry in one lower dimension. The global symmetry naturally
serves as an input for the topological order. In particular, we holographically
construct a String Operator Algebra (SOA) which is the building block of
symmetric quantum systems with a given symmetry in one lower dimension.
This exposes a vast web of dualities which act on the space of -symmetric
quantum systems. The SOA facilitates the classification of gapped phases as
well as their corresponding order parameters and fundamental excitations, while
dualities help to navigate and predict various corners of phase diagrams and
analytically compute universality classes of phase transitions. A novelty of
the approach is that it treats conventional Landau and unconventional
topological phase transitions on an equal footing, thereby providing a
holographic unification of these seemingly-disparate domains of understanding.
We uncover a new feature of gapped phases and their multi-critical points,
which we dub fusion structure, that encodes information about which phases and
transitions can be dual to each other. Furthermore, we discover that self-dual
systems typically posses emergent non-invertible, i.e., beyond group-like
symmetries. We apply these ideas to quantum spin chains with finite
Abelian group symmetry, using topologically-ordered systems in . We
predict the phase diagrams of various concrete spin models, and analytically
compute the full conformal spectra of non-trivial quantum phase transitions,
which we then verify numerically.Comment: 95+appendices+references=148 page
Ising Model on the Affine Plane
We demonstrate that the Ising model on a general triangular graph with 3
distinct couplings corresponds to an affine transformed conformal
field theory (CFT). Full conformal invariance of the minimal CFT is
restored by introducing a metric on the lattice through the map which relates critical couplings to the ratio of the dual
hexagonal and triangular edge lengths. Applied to a 2d toroidal lattice, this
provides an exact lattice formulation in the continuum limit to the Ising CFT
as a function of the modular parameter. This example can be viewed as a quantum
generalization of the finite element method (FEM) applied to the strong
coupling CFT at a Wilson-Fisher IR fixed point and suggests a new approach to
conformal field theory on curved manifolds based on a synthesis of simplicial
geometry and projective geometry on the tangent planes
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