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Physical Review Letters

Renormalization-Group Calculation of Local Magnetizations and Correlations: Random-Bond, Random-Field, and Spin-Glass Systems

Dicle Yeşilleten, A. Nihat Berker

Published Feb 24, 1997• 15 citations
Publisher

Abstract

We show that extensive statistical mechanics information can be obtained by the calculation of local densities such as local magnetizations and neighbor correlations, made possible by renormalization-group theory. This is illustrated with quenched random-bond, random-field, and spin-glass systems, in two and three dimensions. The different ordering behaviors thus become vividly and quantitatively apparent.

DOI
10.1103/physrevlett.78.1564

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We show that extensive statistical mechanics information can be obtained by the calculation of local densities such as local magnetizations and neighbor correlations, made possible by renormalization-group theory. This is illustrated with quenched random-bond, random-field, and spin-glass systems, in two and three dimensions. The different ordering behaviors thus become vividly and quantitatively apparent.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."
The tensor renormalization-group method, developed by Levin and Nave, brings systematic improvability to the position-space renormalization-group method and yields essentially exact results for phase diagrams and entire thermodynamic functions. The method, previously used on systems with no quenched randomness, is extended in this study to systems with quenched randomness. Local magnetizations and correlation functions as a function of spin separation are calculated as tensor products subject to renormalization-group transformation. Phase diagrams are extracted from the long-distance behavior of the correlation functions. The approach is illustrated with the quenched bond-diluted Ising model on the triangular lattice. An accurate phase diagram is obtained in temperature and bond-dilution probability for the entire temperature range down to the percolation threshold at zero temperature.
A highly detailed renormalization-group study is applied to the equilibrium statistical mechanics of the random-field Ising model in three dimensions, with analysis of the renormalization-group flows of over 8 500 000 quantities that determine the quenched probability distribution. We determine the strong-coupling fixed distribution that controls the finite-temperature phase boundary. This boundary is second order but has a magnetization behavior that is phenomenally close to a discontinuity, with the exponent \ensuremath{\beta}=0.0200\ifmmode\pm\else\textpm\fi{}0.0005. Magnetization curves are obtained, demonstrating that, for example, the calculated M changes from 0 to 0.407, almost half saturation, as (Tc{\mathit{T}}_{\mathit{c}}Tc​-T)/Tc{\mathit{T}}_{\mathit{c}}Tc​ is varied from 0 to 0.2947\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$. However, a rigorous argument is given showing that the magnetization M cannot be truly discontinuous at phase transitions controlled by a strong-coupling random-field fixed distribution, in renormalization-group studies that employ the Migdal-Kadanoff local recursion relations.
An Ising model including field randomness and $\ifmmode\pm\else\textpm\fi{}J$ bond randomness is studied by renormalization-group theory in spatial dimensions d=2d=2d=2 and 3. In d=3,d=3,d=3, with field randomness but no $\ifmmode\pm\else\textpm\fi{}J$ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in d=3,d=3,d=3, at sufficient $\ifmmode\pm\else\textpm\fi{}J$ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In d=2,d=2,d=2, no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in d=3,d=3,d=3, is seen here in d=2.d=2.d=2.