1,721,013 research outputs found
Conformal Invariance in the Long-Range Ising Model
No full textWe consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising model is also included and may be of independent interest.We consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d -dimensional short-range Ising model is also included and may be of independent interest.We consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising model is also included and may be of independent interest
EPFL lectures on conformal field theory in D ≥ 3 dimensions
No full textThis is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap.This primer develops Conformal Field Theory (CFT) from scratch, whereby CFT is viewed as any conformally-invariant theory that describes a fixed point of a renormalization group flow in quantum field theory. The book is divided into four lectures: Lecture 1 addresses the physical foundations of conformal invariance, while Lecture 2 examines the constraints imposed by conformal symmetry on the correlation functions of local operators, presented using the so-called projective null cone – a procedure also known as the embedding formalism. In turn, Lecture 3 focuses on the radial quantization and the operator product expansion, while Lecture 4 offers a very brief introduction to the conformal bootstrap. Derived from course-based notes, these lectures are intended as a first point of entry to this topic for Master and PhD students alike.This is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap
The dream of non-perturbative precision RG
No full textWilsonian non-perturbative RG is an unfinished business. It works qualitatively but not quantitatively. However the problem does not look unsolvable. I will give a panorama of old and new ideas about non-perturbative RG. One recent promising direction is Tensor Network RG.
TH colloquia: https://cern.zoom.us/j/67346292748?pwd=ZnRkQWh0ZXVFQTc5K256QW5NcEcrdz0
Four Lectures on the Random Field Ising Model, Parisi-Sourlas Supersymmetry, and Dimensional Reduction
Get PDFNumerical evidence suggests that the Random Field Ising Model loses
Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4
and 5 dimensions, while a related model of branched polymers retains these
features in any . These notes give a leisurely introduction to a recent
theory, developed jointly with A. Kaviraj and E. Trevisani, which aims to
explain these facts. Based on the lectures given in Cortona and at the IHES in
2022.Comment: 55 pages, 11 figures; v2 - minor changes, mentioned forthcoming work
by Fytas et a
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
Get PDFWe continue our study of rigorous renormalization group (RG) maps for tensor
networks that was begun in arXiv:2107.11464. In this paper we construct a
rigorous RG map for 2D tensor networks whose domain includes tensors that
represent the 2D Ising model at low temperatures with a magnetic field . We
prove that the RG map has two stable fixed points, corresponding to the two
ground states, and one unstable fixed point which is an example of a
discontinuity fixed point. For the Ising model at low temperatures the RG map
flows to one of the stable fixed points if , and to the discontinuity
fixed point if . In addition to the nearest neighbor and magnetic field
terms in the Hamiltonian, we can include small terms that need not be spin-flip
invariant. In this case we prove there is a critical value of the field
(which depends on these additional small interactions and the temperature) such
that the RG map flows to the discontinuity fixed point if and to one of
the stable fixed points otherwise. We use our RG map to give a new proof of
previous results on the first-order transition, namely, that the free energy is
analytic for , and the magnetization is discontinuous at .
The construction of our low temperature RG map, in particular the disentangler,
is surprisingly very similar to the construction of the map in arXiv:2107.11464
for the high temperature phase. We also give a pedagogical discussion of some
general rigorous transformations for infinite dimensional tensor networks and
an overview of the proof of stability of the high temperature fixed point for
the RG map in arXiv:2107.11464.Comment: Version 2: Diagram (6.44) has been corrected and footnote 14 has been
added. Remark 6.6 is new. A sentence was added at the end of the proof of
theorem 7.4. An additional acknowledgement has been added. Version 3 is the
version that was published by Annales Henri Poincar\'e. Changes from version
2 to version 3 are very mino
3D Ising Model: a view from the Conformal Bootstrap Island
Get PDFInternational audienceWe explain how the axioms of Conformal Field Theory are used to make predictions about critical exponents of continuous phase transitions in three dimensions, via a procedure called the conformal bootstrap. The method assumes conformal invariance of correlation functions, and imposes some relations between correlation functions of different orders. Numerical analysis shows that these conditions are incompatible unless the critical exponents take particular values, or more precisely that they must belong to a small island in the parameter space
Classifying irreducible fixed points of five scalar fields in perturbation theory
Get PDFClassifying perturbative fixed points near upper critical dimensions plays an
important role in understanding the space of conformal field theories and
critical phases of matter. In this work, we consider perturbative fixed points
of scalar bosons coupled with quartic interactions preserving an
arbitrary subgroup . We perform an exhaustive algorithmic
search over the symmetry groups which are irreducible and satisfy the
Landau condition, so that the fixed point can be reached by fine-tuning a
single mass term and there is no need to tune the cubic couplings. We also
impose stability of the RG flow in the space of quartic couplings, and reality.
We thus prove that there exist no new stable fixed points in
dimensions beyond the two known ones: namely the invariant fixed
point and the Cubic(5) fixed point. This work is a continuation of the
classification of such fixed points with scalars by Toledano, Michel,
Toledano, and Br\'ezin in 1985.Comment: 37 pages, 4 figures, references update
General Properties of Multiscalar RG Flows in
No full textFixed points of scalar field theories with quartic interactions in dimensions are considered in full generality. For suchtheories it is known that there exists a scalar function of the couplingsthrough which the leading-order beta-function can be expressed as a gradient.It is here proved that the fixed-point value of is bounded from below by asimple expression linear in the dimension of the vector order parameter, .Saturation of the bound requires a marginal deformation, and is shown to arisewhen fixed points with the same global symmetry coincide in coupling space.Several general results about scalar CFTs are discussed, and a review of knownfixed points is given.Fixed points of scalar field theories with quartic interactions in dimensions are considered in full generality. For such theories it is known that there exists a scalar function of the couplings through which the leading-order beta-function can be expressed as a gradient. It is here proved that the fixed-point value of is bounded from below by a simple expression linear in the dimension of the vector order parameter, . Saturation of the bound requires a marginal deformation, and is shown to arise when fixed points with the same global symmetry coincide in coupling space. Several general results about scalar CFTs are discussed, and a review of known fixed points is given
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