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Phase Diagrams for the Randomly Diluted Resistor Network and XY Model University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 10-1-1989 Phase Diagrams for the Randomly Diluted Resistor Network and XY Model A. Brooks Harris University of Pennsylvania, [email protected] Amnon Aharony Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Harris, A., & Aharony, A. (1989). Phase Diagrams for the Randomly Diluted Resistor Network and XY Model. Physical Review B, 40 (10), 7230-7238. http://dx.doi.org/10.1103/PhysRevB.40.7230 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/300 For more information, please contact [email protected]. Phase Diagrams for the Randomly Diluted Resistor Network and XY Model Abstract The randomly diluted resistor network and XY model at low temperature T are studied near the d-dimensional percolation threshold using the ɛ expansion, where ɛ=6-d. The series expansion of the inverse susceptibility in powers of T for the XY model is identical to that of the appropriate resistive −1 inverse susceptibility in powers of σ0 , where σ0 is the conductance of a bond. However, the temperature-dependent critical concentration pc(T) for the XY model has no analog in the resistor network, where pc clearly does not depend on σ0. This distinction arises from a rather subtle difference between the Fourier component representation of the Gaussian model for the resistor network and that of the bounded potential energy associated with the XY model. We introduce a family of models which provides a smooth interpolation between these two models and show that the phase boundary for the XY model satisfies certain simple self-consistency checks involving other susceptibilities. In particular we provide the first explicit calculation of the universal crossover function to finite temperature of the dilute XY model. Disciplines Physics This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/300 PHYSICAL REVIEW B VOLUME 40, NUMBER 10 1 OCTOBER 1989 Phase diagrams for the randomly diluted resistor network and XY model A. Brooks Harris Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 639-6 Amnon Aharony Beverly and Raymond Sackler Faculty ofExact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel (Received 17 June 1988) The randomly diluted resistor network and XY model at low temperature T are studied near the d-dimensional percolation threshold using the E expansion, where v=6 —d. The series expansion of the inverse susceptibility in powers of T for the XY model is identical to that of the appropriate o. o. resistive inverse susceptibility in powers of o ', where o is the conductance of a bond. However, the temperature-dependent critical concentration p, ( T) for the XY model has no analog in the resis- tor network, where p, clearly does not depend on o.o. This distinction arises from a rather subtle difference between the Fourier component representation of the Gaussian model for the resistor net- work and that of the bounded potential energy associated with the XYmodel. We introduce a fami- ly of models which provides a smooth interpolation between these two models and show that the phase boundary for the XY model satisfies certain simple self-consistency checks involving other susceptibilities. In particular we provide the first explicit calculation of the universal crossover function to finite temperature of the dilute XYmodel. I. INTRODUCTION where 8=6—d, d being the spatial dimension. Alterna- tively, we may write Eq. (l) as It has long been realized that there exists a precise (3) one-to-one correspondence between linearized spin-wave theory for continuous spins (such as the XY' model) and For the RRN we do not expect p, to depend on o.o, KirchhoA"s equations for a resistor network on the same ana- ' which is the parameter one would view as being the underlying lattice. Therefore, one expects that the low- log of J/T. The main result of this paper is to obtain temperature (T) correlation functions for the former spin phase boundaries for these two models consistent with model are exactly the same as suitable resistance correla- these observations from scaling functions whose expan- tions at least to leading order in T. Indeed, for randomly sions in 0.0 and T/J, respectively, are identical. Furth- diluted networks, these two problems have been shown to ermore we will introduce a model which interpolates be described by the same crossover exponent, P, near the smoothly between these two phase diagrams. We will percolation threshold. An important problem which has also find universal relations between the coefficient b in not been addressed so far concerns the fact that in spite Eq. (3) and other measurable amplitudes. of this seeming equivalence between the two models, their Explicitly the models we consider are the following. ob- phase diagrams differ in an important, but physically The RXY model is governed by the Hamiltonian vious way. This difference may be discussed in terms of the scaling variable, which for the randomly diluted XY H = — g J„„[S(x)S(x') —1], (4) (RXY) model is T/[J(p —p, )~], where J is the exchange (x, x') interaction associated with occupied bonds, and for the where (x,x') indicates that the sum is over pairs of randomly diluted resistor network (RRN) is nearest neighboring sites, x and x', S(x) k /[oo(p —p, )~], where A, is an adjustable parameter =—(cos0(x), sin0(x)) is an xy spin at the site x, and is 0. J„„. whose significance will be elucidated later and 0 is the a quenched random variable which assumes the values J conductance of an occupied bond. For the RXY model with probability p and 0 with probability 1 —p, corre- we will obtain the result for the concentration-dependent sponding, respectively to the bond x-x' being occupied critical temperature, T, (p), which one expects on the and vacant. The variable 0(x) is confined to the interval basis of scaling, namely 0 to 2~. For this model we consider the kth-order spin susceptibility X„' ' defined as ik (x) e—iko(x y(k)( T ) ( where a is a constant and within the c, expansion P is y [ ))' given by tempera- =1+ where ( ) z indicates a thermal average at 42 ture T and [ ]„anaverage over all configurations of 40 7230 1989 The American Physical Society PHASE DIAGRAMS FOR THE RANDOMLY DILUTED RESISTOR. 7231 occupied and vacant bonds. For the RRN the "Hamil- see the meaning of the Hamiltonian H„',consider its ex- tonian" is pansion in powers of [8(x)—8(x')]. To quadratic order 0's, H'~ ' in the is independent of X;„.The role of 1, is H/T = —, V(x) —V(x')] ;„ g (1„„[ to set the scale over which the quadratic approximation (X,X') to the cosine in Eq. (11) is valid, as shown in Fig. l. Also, where V(x) is a Gaussian variable interpreted as the volt- k;„setsthe scale of the barrier in the potential when it is o. age at site x and x „.is a quenched random variable extended periodically to all 0. Intuitively it is clear that which assumes the values cro with probability p and 0 as A, ;„approaches zero, the Hamiltonian of the GRXY with probability 1 —p, corresponding to the bond x-x' re- in Eq. (11) approaches that of the RRN of Eq. (6). For spectively being occupied and vacant. For this model we this Hamiltonian we define the susceptibility consider the generalized resistive susceptibility y(&) eire(x) —i) e(x') ~ defined as xy ~ [( ) T ]av (12) X g(A. ) i) v(x) —iz. v(x') ( ) y [( ) ] where ( . )'T indicates that the thermal average is tak- X en with respect to H„' . Here we implicitly assume that k It can be shown ' that is an integer multiple of A, ;„,to avoid any phase incom- 7' ' y(A. ) —(1/2)A. R(x,x')] patibilities. As we shall show, has the scaling repre- ( ) ~ [e RN P sentation X — x')" (13) = g g ( A, /2)", [v(x, x')R (x, ], (8) x' k „, where the function F is a universal function of both its ar- where in a given configuration R (x,x') is the resistance guments when 3& and Az are fixed by, for example, re- between the nodes at x and x' and v(x, x') is an indicator quiring, F(x,y) =OF(x,y) IBx =1 for x ~0. It should be function which is unity if the nodes x and x' are in the noted that the form in Eq. (13) is invariant with respect same cluster and is zero otherwise. We interpret vR" to to the transformation applied to Eqs. (11) and (12), A, k', be zero when v=0 and R = ~. b & ~min~b ~min, ~b and J~b J'. One We will be interested in the behaviors of these suscepti- sees that bilities for concentration, p, near the percolation thresh- ' Xa(N(p ) = A t rF( A 2i(. /( o ot ~ ), 0) (14a) old at p =p, . For instance, for the RXY model we ex- ( pect that and (14b) where t =p, —p, y is the susceptibility exponent for per- We will show that the power series expansion of F(x,y) colation, and x is the appropriate scaling variable, in powers of x is independent of y. Thus the coefficients o. ' x =Tl(jt~). If the nonuniversal constants A) and A2 of the expansion of X~N in powers of o are the same as are chosen, e.g., by setting G(0)=G'(0)=1, then the those of 7' ' in powers of T/J, independent of the value scaling function G' '(x) is universal.
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