We study the quantum Ising model in $D$ dimensions with the equation-of-motion technique and the Majorana representation for spins. The decoupling scheme used for the Green's functions is based on the hierarchy of correlations in position space. To lowest order, this method reproduces the well-known mean field phase diagram and critical exponents. When correlations between spins are included, we show how the appearance of thermal fluctuations and magnons strongly affects the physical properties. In one dimension and for $B=0$ we demonstrate that, to first order in correlations, thermal fluctuations completely destroy the ordered phase. For nonvanishing transverse field we show that the model exhibits different behavior than its classical counterpart, especially near the quantum critical point. We discuss the connection with the Dyson's equation formalism and the explicit form of the self-energies.We study the quantum Ising model in $D$ dimensions with the equation-of-motion technique and the Majorana representation for spins. The decoupling scheme used for the Green's functions is based on the hierarchy of correlations in position space. To lowest order, this method reproduces the well-known mean field phase diagram and critical exponents. When correlations between spins are included, we show how the appearance of thermal fluctuations and magnons strongly affects the physical properties. In one dimension and for $B=0$ we demonstrate that, to first order in correlations, thermal fluctuations completely destroy the ordered phase. For nonvanishing transverse field we show that the model exhibits different behavior than its classical counterpart, especially near the quantum critical point. We discuss the connection with the Dyson's equation formalism and the explicit form of the self-energies.

arXiv:1705.03961Physical review. B./Physical review. BAug 21, 20173 citations

Schwinger Dyson Summation of Perturbation Expansions

Tom Banks

We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.

arXiv:2410.19683Oct 25, 20240 citations

Remarks on critical phenomena

R. Brout

PhysicaNov 1, 19707 citations

Critical properties of the random hierarchical model

Alba Theumann

The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential, $r_{l}^{}{}_{}{}^{(m)}$ and $U_{l}^{}{}_{}{}^{(m)}$ , respectively, for the mth spin at the $l$ level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only ${r}_{l}=〈r_{l}^{}{}_{}{}^{(m)}〉$ , ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and ${U}_{l}=〈U_{l}^{}{}_{}{}^{(m)}〉$ are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for ${U}_{l}$ and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$, $U>~0$ ; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential, $r_{l}^{}{}_{}{}^{(m)}$ and $U_{l}^{}{}_{}{}^{(m)}$, respectively, for the mth spin at the $l$ level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only ${r}_{l}=〈r_{l}^{}{}_{}{}^{(m)}〉$, ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and ${U}_{l}=〈U_{l}^{}{}_{}{}^{(m)}〉$ are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for ${U}_{l}$ and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$, $U>~0$; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.

Physical review. B, Condensed matterApr 1, 198013 citations

Critical Properties of a Quantum Hierarchical Model

Sergio A Albeverio, Yu. G. Kondratiev, Yuri V. Kozitsky

Letters in Mathematical PhysicsMay 1, 19974 citations

We study the quantum Ising model in

D

dimensions with the equation-of-motion technique and the Majorana representation for spins. The decoupling scheme used for the Green's functions is based on the hierarchy of correlations in position space. To lowest order, this method reproduces the well-known mean field phase diagram and critical exponents. When correlations between spins are included, we show how the appearance of thermal fluctuations and magnons strongly affects the physical properties. In one dimension and for

B=0

we demonstrate that, to first order in correlations, thermal fluctuations completely destroy the ordered phase. For nonvanishing transverse field we show that the model exhibits different behavior than its classical counterpart, especially near the quantum critical point. We discuss the connection with the Dyson's equation formalism and the explicit form of the self-energies.

r_{l}^{}{}_{}{}^{(m)}

and

U_{l}^{}{}_{}{}^{(m)}

, respectively, for the mth spin at the

l

level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only

{r}_{l}=〈r_{l}^{}{}_{}{}^{(m)}〉

, ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and

{U}_{l}=〈U_{l}^{}{}_{}{}^{(m)}〉

are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for

{U}_{l}

and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$,

U>~0

; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.