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Physical review. B, Condensed matter

Low-temperature renormalization-group study of the random-field model

Mau-chung Chang, Elihu Abrahams

Published May 1, 1983• 12 citations
Publisher

Abstract

The continuous-spin random-field model is investigated by means of the low-temperature renormalization-group technique with the use of the replica trick. The Wilson-Kogut recursion method is applied. For short-range exchange, the results are in exact agreement with those of the random-axis model studied by Pelcovits. For long-range exchange varying with distance $R$ as ${R}^{\ensuremath{-}(d+\ensuremath{\sigma})}$, critical exponents are calculated to first order in $d\ensuremath{-}2\ensuremath{\sigma}$. They are identical to those in a $d\ensuremath{-}\ensuremath{\sigma}$ expansion of the nonrandom model. However, the hyperscaling law becomes $(d+{\ensuremath{\lambda}}_{T})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ (${\ensuremath{\lambda}}_{T}$ is the eigenvalue associated with the dangerous irrelevant operator $T$), and, for $m$-component spins, $\ensuremath{-}{\ensuremath{\lambda}}_{T}=\ensuremath{\sigma}+\frac{(d\ensuremath{-}2\ensuremath{\sigma})}{m}$.

DOI
10.1103/physrevb.27.5570

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Physical review. B, Condensed matter•Dec 1, 1983•13 citations
The continuous-spin random-field model is investigated by means of the low-temperature renormalization-group technique with the use of the replica trick. The Wilson-Kogut recursion method is applied. For short-range exchange, the results are in exact agreement with those of the random-axis model studied by Pelcovits. For long-range exchange varying with distance RRR as R\ensuremath−(d+\ensuremathσ){R}^{\ensuremath{-}(d+\ensuremath{\sigma})}R\ensuremath−(d+\ensuremathσ), critical exponents are calculated to first order in d\ensuremath−2\ensuremathσd\ensuremath{-}2\ensuremath{\sigma}d\ensuremath−2\ensuremathσ. They are identical to those in a d\ensuremath−\ensuremathσd\ensuremath{-}\ensuremath{\sigma}d\ensuremath−\ensuremathσ expansion of the nonrandom model. However, the hyperscaling law becomes (d+\ensuremathλT)\ensuremathν=2\ensuremath−\ensuremathα(d+{\ensuremath{\lambda}}_{T})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}(d+\ensuremathλT​)\ensuremathν=2\ensuremath−\ensuremathα (\ensuremathλT{\ensuremath{\lambda}}_{T}\ensuremathλT​ is the eigenvalue associated with the dangerous irrelevant operator TTT), and, for mmm-component spins, \ensuremath−\ensuremathλT=\ensuremathσ+(d\ensuremath−2\ensuremathσ)m\ensuremath{-}{\ensuremath{\lambda}}_{T}=\ensuremath{\sigma}+\frac{(d\ensuremath{-}2\ensuremath{\sigma})}{m}\ensuremath−\ensuremathλT​=\ensuremathσ+m(d\ensuremath−2\ensuremathσ)​.
We show that the hyperscaling law is modified whenever the relevant fixed point of the renormalization-group transformation is at zero temperature (infinite coupling). Instead of 2-\ensuremath{\alpha}=d\ensuremath{\nu}, the hyperscaling law now reads 2-\ensuremath{\alpha}=d'\ensuremath{\nu} where d' is the anomalous dimension which is equal to d-\ensuremath{\Vert}y\ensuremath{\Vert} and y is the exponent for the leading irrelevant operator that flows to zero temperature. We discuss the random-field Ising model in detail. We also perform a numerical domain-wall renormalization-group analysis for the d=3 model. For a Gaussian random-field distribution, our results are consistent with a second-order transition at low temperatures, and \ensuremath{\Vert}y\ensuremath{\Vert} is found to be 1.3--1.7. For a \ifmmode\pm\else\textpm\fi{}h distribution, the transition appears to be first order at T=0. .AE
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part WMF{W}_{\text{MF}}WMF​ coincides with previous results. The short-range part WSR{W}_{\text{SR}}WSR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc=5{d}_{c}=5dc​=5 at very low temperatures due to the dimensionality shift Dc=dc+1=6{D}_{c}={d}_{c}+1=6Dc​=dc​+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.
We consider the random-field O(N\mathrm{O}(NO(N) spin model with long-range exchange interactions which decay with distance rrr between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance rrr as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as NNN is finite.
Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on ddd and nnn but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to ddd and nnn.
The critical properties of a ddd-dimensional Bose model in the presence of a random quenched field are investigated by renormalization-group techniques via the replica trick. To first order in $\ensuremath{\epsilon}=6\ensuremath{-}d$ it is found that both at nonzero- and zero-temperature regimes the grand-canonical critical exponents are the same as those of the corresponding classical model so that thermal and quantum fluctuations appear to be irrelevant. Furthermore, at zero temperature, the scaling laws already established for the pure model are retrieved with the replacement of ddd with $d\ensuremath{-}4$, but the random critical exponents are not derivable from the pure ones with the dimensional shift $d\ensuremath{\rightarrow}d\ensuremath{-}4$.