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arXiv:2206.12924 · Physical review. E

Low-temperature behavior of the O(N) models below two dimensions

Andrzej Chlebicki, Paweł Jakubczyk

Published Jan 17, 2023• 0 citations
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Abstract

We investigate the critical behavior and the nature of the low-temperature phase of the O(N) models treating the number of field components N and the dimension d as continuous variables with a focus on the d≤2 and N≤2 quadrant of the (d,N) plane. We precisely chart a region of the (d,N) plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension η. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the O(N) models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in d<2. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with d<2 and N<2. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents η(d,N) and ν^{-1}(d,N) and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in d<2.

DOI
10.1103/physreve.107.014121

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We investigate the critical behavior and the nature of the low-temperature phase of the O(N) models treating the number of field components N and the dimension d as continuous variables with a focus on the d≤2 and N≤2 quadrant of the (d,N) plane. We precisely chart a region of the (d,N) plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension η. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the O(N) models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in d<2. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with d<2 and N<2. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents η(d,N) and ν^{-1}(d,N) and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in d<2.
We study the critical behavior and phase diagram of the ddd-dimensional random field O(N)O(N)O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension dDR(N){d}_{\text{DR}}(N)dDR​(N) and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole (N,d)(N,d)(N,d) plane. Below dDR(N){d}_{\text{DR}}(N)dDR​(N), our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.
Analyticity of critical exponents of the O(N)O(N)O(N) models from nonperturbative renormalization
We employ the functional renormalization group framework at the second order in the derivative expansion to study the O(N) O ( N ) models continuously varying the number of field components N N and the spatial dimensionality d d . We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents \nu ν and \eta η across a line in the (d,N) ( d , N ) plane, which passes through the point (2,2) ( 2 , 2 ) . By direct numerical evaluation of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the (d,N) ( d , N ) plane, however no evidence of discontinuous or singular first and second derivatives of these functions for d>2 d > 2 . The computed derivatives of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) become increasingly large for d\to 2
We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy O(N)O(N)O(N) models near d=4d=4d=4, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at two-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at the two-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long-range order in d<4d<4d<4, are shown to fit into the picture predicted by our recent nonperturbative FRG approach.
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The Migdal-Kadanoff position-space renormalization-group scheme is used to study the critical behavior of the isotropic n-component-vector model in the previously unexplored region, 1<n<2, 1<d<2, of the n-d plane (d is the dimensionality of the space). We find a continuous phase transition at a finite temperature if d\ensuremath{\ge}dl{d}_{l}dl​(n). The lower critical dimension dl{d}_{l}dl​(n) increases continuously, but nonlinearly from 1 to 2 as n changes from 1 to 2. For dl{d}_{l}dl​(n)\ensuremath{\le}d2, the low-temperature phase is characterized by a power-law decay of the two-point correlation function with a temperature-independent exponent.