Critical Point Ising Correlations from Quantum Field Theory.

A relationship between the two-point correlation function of the Ising model in the critical region and the $〈sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(0)sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(\ensuremath{\rho})〉$ Green's function of a sine-Gordon field is constructed. This relationship is tested by means of a mass perturbation expansion with an infrared cutoff and shown to agree, at the leading-logarithm approximation and up to the second order, with the exact result of the Ising model.A relationship between the two-point correlation function of the Ising model in the critical region and the $〈sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(0)sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(\ensuremath{\rho})〉$ Green's function of a sine-Gordon field is constructed. This relationship is tested by means of a mass perturbation expansion with an infrared cutoff and shown to agree, at the leading-logarithm approximation and up to the second order, with the exact result of the Ising model.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMay 15, 1977133 citations

Random-Field Correlations and Quantum Critical Phenomena

Dimo I. Uzunov, E. R. Kokutcheva, YONKO T. MILLEV

De Gruyter eBooksDec 31, 19850 citations

Scaling Hypothesis for Matrix Product States

Bram Vanhecke, Jutho Haegeman, Karel Van Acoleyen +2 more

We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing $a$ , we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with $D$ the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing $a$, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with $D$ the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.

arXiv:1907.08603Physical Review LettersDec 18, 201959 citations

Correlations in the Ising-model

M. Helm

Physics Letters ADec 1, 19700 citations

Theory of finite-temperature crossovers near quantum critical points close to,or above, their upper-critical dimension

Subir Sachdev A

A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T= ${\mathrm{T}}_{\mathrm{c}}$ (t), t0, with ${\mathrm{T}}_{\mathrm{c}}$ (0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T- ${\mathrm{T}}_{\mathrm{c}}$ (t)|\ensuremath{\ll} ${\mathrm{T}}_{\mathrm{c}}$ (t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim} ${\mathrm{k}}_{\mathrm{B}}$ T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$ ${\mathrm{k}}_{\mathrm{B}}$ T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=${\mathrm{T}}_{\mathrm{c}}$(t), t0, with ${\mathrm{T}}_{\mathrm{c}}$(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-${\mathrm{T}}_{\mathrm{c}}$(t)|\ensuremath{\ll}${\mathrm{T}}_{\mathrm{c}}$(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}${\mathrm{k}}_{\mathrm{B}}$T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$${\mathrm{k}}_{\mathrm{B}}$T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.

arXiv:cond-mat/9606083Physical review. B, Condensed matterJan 1, 199797 citations

a

, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with

D

the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.

{\mathrm{T}}_{\mathrm{c}}

(t), t0, with

{\mathrm{T}}_{\mathrm{c}}

(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-

{\mathrm{T}}_{\mathrm{c}}

(t)|\ensuremath{\ll}

{\mathrm{T}}_{\mathrm{c}}

(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}

{\mathrm{k}}_{\mathrm{B}}

T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$

{\mathrm{k}}_{\mathrm{B}}

T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.