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Physical review. B, Condensed matter

Critical behavior of then-vector model for1<n<2

Manhot Lau, Chandan Dasgupta

Published Jan 1, 1987• 6 citations
Publisher

Abstract

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}${d}_{l}$(n). The lower critical dimension ${d}_{l}$(n) increases continuously, but nonlinearly from 1 to 2 as n changes from 1 to 2. For ${d}_{l}$(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.

DOI
10.1103/physrevb.35.329

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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.
Critical behavior associated with the helical or noncollinear ordering for isotropic n >3-component spins is studied near d =2 dimensions based on an O ( n )× O (2) nonlinear σ model and the renormalization-group ε= d -2 expansion. A chiral fixed point is found for general n characterized by the critical exponents that are consistent with the previous 1/ n expansion calculation. For n =3, it is argued that because of the nontrivial global topology of the order-parameter space, ∝ 1 ( S O (3))= Z 2 , and the possible existence of a finite-temperature topological transition at d =2, the standard ε= d -2 expansion technique is not applicable.
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.
The critical behavior of an m‐vector model with random impurities is investigated through the use of renormalization group techniques near four dimensions. For almost all values of m, this disordered model has two stable fixed points and a single nontrivialy unstable fixed point. The model exhibits sharp phase transitions with exponents which do not depend continuously on the concentration of impurities. The familiar m‐component fixed point, which is normally assumed to characterize the critical behavior of the pure m‐vector model, is unstable whenever m%4.
The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1<or=d<or=4 (m= infinity ). Scaling towards weak coupling, J'(L,m) approximately L -1 nu /, L>>L c (m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T - nu , for T to 0 but no finite-temperature transition. The characteristic length L c (m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.