Conformal data from finite entanglement scaling (original) (raw)
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Critical finite-range scaling in scalar-field theories and Ising models
Physical Review E, 1993
We develop a critical finite-force-range scaling theory for D-dimensional scalar P field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension D"we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a P Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the XX oo quasi-one-dimensional Ising lQIDIi model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the QIDI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D =1 quartic field theory, and for the QIDI model they are those of a cylindrical Ising system above D, Results of. specialized numerical scaling techniques applied to transfer-matrix calculations for the Q I DI model with X~1024 strongly support our theoretically obtained scaling relations.
Precise determination of critical exponents and equation of state by field theory methods
Physics Reports, 2001
Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized φ 4 3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model [1,2] and of the equation of state of the 3D Ising model [3]. These results are among the most precise available probing field theory in a nonperturbative regime. Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large order behaviour of perturbation series has then allowed to develop efficient "summation" techniques, based on Borel transformation and conformal mapping [4]. We first discuss exponents and describe our recent results [2]. Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived. Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer
Scale and conformal covariance in quantum electrodynamics
Reports on Mathematical Physics, 1976
The paper consists of two independent parts. First, we review the situation in scale invariant massless QED from an axiomatic standpoint. Assuming that the t-functions (or, equivalently, the Euclidean Schwinger functions) transform covariantly under dilatations, we deduce that the current &(x) has zero n-point Wightman functions, but nonvanishing t-functions. Assuming in addition conformal invariance of the current-field vertex function, we write down a bootstrap equation similar to the one derived in [7] from the point of view of perturbation theory. Next we consider a non-Lagrangian, conformal invariant model of interacting antisymmetric tensor field Fpv (of scale dimension d) and Dirac field y (of dimension d'). The model involves two conserved currents (an "electric" and a "magnetic" one) and two effective coupling constants. We demonstrate that it is free of ultraviolet divergences in the range of dimensions 2 < d < 3, 312 < d' < 512. Contents B. Scale and y5-invariance of the massless QED. The gauge invariant 2-point functions C. Gauge dependent f-point functions D. Remarks on conformal covariance and bootstrap equations. 2. A conformal covariant model of "electromagnetic" interaction with two conserved currents A. Maxwell stress tensor with anomalous dimensions. 2 and 3-point functions. Electric and magnetic currents B. Feynman rules for skeleton diagrams. Absence of ultraviolet divergences.
Self-Consistent Scaling Theory for Logarithmic-Correction Exponents
Physical Review Letters, 2006
Multiplicative logarithmic corrections frequently characterize critical behaviour in statistical physics. Here, a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific-heat critical exponent vanishes. Also, the theory is widened to encompass the correlation function. The new relations are then confronted with results from the literature and some new predictions for logarithmic corrections in certain models are made.
Approximate scale invariance and broken- SU 3 coupling constants
1973
Under the assumption that the matrix elements of the divergences of the dilation current (~"2~= 0~) are dominated by two poles, it is shown that the present experimental data for the dilaton (~) are consistent with an unsubtractcd dispersion relation of F(q2)z(PIO~IP) (=~, K or ~), a significant SU 3 violation in ~PP couplings and no apparent singlet-octet mixing for the scalar mesons. As far as the slope I"=_l"/dq2lq,=o is concerned, in the SU3 symmetry limit we have F'_ ~ 1, while SUa-symmetry-breaking effects change this value to F '~ _ 0.5 in consistency with a general theorem given by 0kubo. By demanding maximal smoothness we obtain that the scale dimension of the scalar densities (which break ehiral SU3• 3 as well as scale symmetries in the energy density) is d~= I. However, our general relation between the dilaton mass (m~) and the dimension d~ (eq. (31)) gives m,~_~ 700 MeV, 600 MeV and 400 MeV for d~= 1, 2 and 3 respeclively. O.Jx) is the local stress energy-momentum tensor for hadrons, such theft the generators of the Poincar5 gro~lp are represented as follows: (1) P. = f O~o(X) d3x , (2) Jl,~ = f (x.O~o(x)-%0~o(x)) d3x' (*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction.
The functional renormalization group and O(4) scaling
The European Physical Journal C, 2010
The critical behavior of the chiral quark-meson model is studied within the Functional Renormalization Group (FRG). We derive the flow equation for the scale dependent thermodynamic potential at finite temperature and density in the presence of a symmetry-breaking external field. Within this scheme, the critical scaling behavior of the order parameter, its transverse and longitudinal susceptibilities as well as the correlation lengths near the chiral phase transition are computed. We focus on the scaling properties of these observables at non-vanishing external field when approaching the critical point from the symmetric as well as from the broken phase. We confront our numerical results with the Widom-Griffiths form of the magnetic equation of state, obtained by a systematic ǫ-expansion of the scaling function. Our results for the critical exponents are consistent with those recently computed within Lattice Monte-Carlo studies of the O(4) spin system.
Tests of Conformal Field Theory at the Yang-Lee Singularity
2009
This paper studies the Yang-Lee edge singularity of 2-dimensional (2D) Ising model based on a quantum spin chain and transfer matrix measurements on the cylinder. Based on finite-size scaling, the low-lying excitation spectrum is found at the Yang-Lee edge singularity. Based on transfer matrix techniques, the single structure constant is evaluated at the Yang-Lee edge singularity. The results of both types of measurements are found to be fully consistent with the predictions for the (A4;A1)minimal conformal field theory, which was previously identified with this critical point.
Scaling and universality of multipartite entanglement at criticality
Eprint Arxiv 0708 3391, 2007
Using the geometric entanglement measure, we study the scaling of multipartite entanglement in several 1D models at criticality, specifically the linear harmonic chain and the XY spin chain encompassing both the Ising and XX critical models. Our results provide convincing evidence that 1D models at criticality exhibit a universal logarithmic scaling behavior ~(c/12)log l in the multipartite entanglement per region for a partition of the system into regions of size l, where c is the central charge of the corresponding universality class in conformal field theory.
Physical Review D, 2012
A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of physical observables under both the renormalization scale-and scheme-parameter transformations, provide a convenient way for estimating the scale-and scheme-dependence of the physical process. In this paper, we present a solution for the scale-equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC) / Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all non-conformal {βi} terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. Different schemes lead to different effective PMC/BLM scales, but the final results are scheme independent. Conversely, from the requirement of scheme independence, one not only can obtain schemeindependent commensurate scale relations among different observables, but also determine the scale displacements among the PMC/BLM scales which are derived under different schemes. In principle, the PMC/BLM scales can be fixed order-by-order, and as a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicit application for determining the scale setting of R e + e − (Q) up to four loops is presented. By using the world average α M S s (MZ) = 0.1184 ± 0.0007, we obtain the asymptotic scale for the 't Hooft associated with the M S scheme, Λ ′ tH M S = 245 +9 −10 MeV, and the asymptotic scale for the conventional M S scheme, Λ M S = 213 +19 −8 MeV.
Entanglement entropy scaling of the XXZ chain
Journal of Statistical Mechanics: Theory and Experiment, 2013
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge c in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of c. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.