Renormalization-group study of the replica action for the random field Ising model

Transcription

1 arxiv:cond-mat/ v1 [cond-mat.stat-mech] 8 Jun 1999 Renormalization-group study of the replica action for the random field Ising model Hisamitsu Mukaida mukaida@saitama-med.ac.jp and Yoshinori Sakamoto yossi@phys.cst.nihon-u.ac.jp Abstract cond-mat/ NUP-A We reexamine the effective action for the d-dimensional random field Ising model derived by Brézin and De Dominicis. We find a non-gaussian fixed point where the φ 4 couplings in the action have various scaling dimensions. The correlation-length exponent in d = 6 ǫ has the value consistent with the argument of the dimensional reduction in the leading order. 1 Introduction Recently, Brézin and De Dominicis derived effective scalar field theory for the Ising model in a Gaussian random field within the replica formalism[1]. They showed that the theory contains five φ 4 coupling constants as well as the standard φ 4 coupling constant. They also pointed out that we meet singular fluctuation on the critical surface below dimension eight when we take the zero-replica limit. In order to resolve this problem, they redefined the coupling constants such that the beta function for the new coupling constants does not explicitly contain n, the number of the replica components, and suggested instability of the non-trivial fixed point in dimension d, where 6 d is sufficiently small. However, their analysis has room for improvement because their beta function does not completely take the one-loop correction into account. In this letter, we reconsider the stability of the fixed point and discuss consistency with the argument of (d, d correspondence[]. Our strategy follows the renormalization-group(rg transformation in the disorder phase, where the infrared divergence discussed in ref.[1] does not occur, so Department of Physics, Saitama Medical College, 981 Kawakado, Iruma-gun, Saitama, , Japan Department of Physics, College of Science and Technology, Nihon University, , Kanda-Surugadai, Chiyodaku, Tokyo, , Japan 1

2 that we can take the zero-replica limit without any problem. We compute the recursion equation for the coupling constants including the all one-loop corrections surviving in the limit n 0. The RG transformation(rgt considered here is defined in the seven parameter space which includes not only the five φ 4 coupling constants but also t and u 0 : here t and u 0 correspond to the reduced temperature and strength of the random field, respectively. We cannot control the RG flow to stay in the perturbative region around the Gaussian fixed point by a one-parameter tuning because the both constants t and u 0 have the mass dimensions. However, we can redefine the coupling constants such that the perturbative RG analysis is applicable by one-parameter tuning if d 6. The trivial fixed point by the new coupling constants no longer corresponds to the Gaussian fixed point and the five φ 4 coupling constants get various scaling dimensions after the redefinition. In particular, the standard φ 4 coupling constant comes to have the mass dimensions 6 d and the other φ 4 couplings have less than that. Hence, the single φ 4 coupling constant becomes slightly relevant in d = 6 ǫ with small ǫ at this fixed point, which is quite similar to the case of the pure φ 4 theory in d = 4 ǫ. Further, it is found that the correlation-length exponent in d = 6 ǫ is the same as that of the pure φ 4 theory in d = 4 ǫ at least in the leading order of the ǫ expansion. Thus the result is consistent with the argument of the dimensional reduction[]. The effective action and RG The d-dimensional effective action derived by Brézin and De Dominicis is given by S = S 0 + S int n d d x S 0 S int d d x α,β=1 φ α (x { ( + tδ αβ u 0 } φβ (x ( u1 4! σ 4(x + u 3! σ 3(xσ 1 (x + u 3 8 σ (x + u 4 4 σ (xσ 1 (x + u 5 4! σ 1(x 4, (1 where α and β, which run from 1 to n, specify a replica component and n σ k (x (φ α (x k, (k = 1,, 4. ( α=1 The interaction conjugate to u 1 contains one summation over the replica index while the other interactions have more than one, which apparently shows that the other coupling constants u,, u 5 are less relevant in the replica limit. However, as we will see below, those interactions can affect the flow of the running coupling constants even in the zero-replica limit. The quadratic part S 0 defines the propagator: 1 Ĝ αβ (p = p + t δ u 0 αβ + (p + t + O(n A(pδ αβ + B(p + O(n. (3

3 Note that B(p dominates over A(p in low-momentum region with small t, while A(p can be more relevant in the replica limit because δ αβ associated with it reduces powers of n. We calculate perturbative RG transformation(rgt in the usual manner[3]. Namely, in the partition function, we first perform integration over the higher-momentum Fourier components ˆφ α (p = d d xφ α (xe ipx (4 with p belonging to K > {p L 1 Λ < p Λ}, where L > 1. Next, the following rescaling ˆφ(L 1 p L θ ˆφ(p (5 is applied if 0 p L 1 Λ. Finally, the coupling constants are redefined such that the action preserves the same form up to irrelevant interactions. At the tree level, we redefine the coupling constants as t = L t, u 0 = L u 0, u i = L 4 d u i, i = 1,, 5, (6 with θ = (d+/, the action remains unchanged: no surprises arise in the zero-th order. In addition, the φ 4 coupling constants seem irrelevant when d > 4, which is inconsistent with the rigorous result[4]. However, going to the one-loop correction, the situation drastically changes. Here First, let us calculate the one-loop corrections to t and u 0, which are extracted from V ( αβ (φ(x = 1 α,β d d xv ( αβ G αβ(x. (7 S int φ α (x φ β (x ( u1 φ α + u σ 1 φ α + u 3 σ + u 4 σ 1 δ αβ + ( u ( φ α + φ β + u3 φ α φ β + u 4 σ 1 (φ α + φ β + u 4 σ + u 5 σ 1, (8 and the propagator in the coordinate representation consists in the higher momentum components: G αβ (x Using eqs.(3 and (8, we can explicitly compute eq.(7. The result is (π dĝαβ(pe ipx. (9 δt = (A 1 + B 1 u 1 + A 1 (u + u 3 δu 0 = {(A 1 + B 1 u + B 1 u 3 + A 1 u 4 }, (10 3

4 up to O(n. Here we have defined A 1 = d d p (π da(p, B 1 = Next, let us look at the one-loop corrections to u i, which are computed from 1 4 α,β,γ,δ Lengthy but straightforward calculation gives, up to O(n, (π db(p. (11 d d xd d yv ( ( αβ (φ(xv γδ (φ(yg αγ(x yg βδ (x y. (1 [( A δu 1 = C ] u 1 + A u 1 (u + u 3, δu = 3 [( A + C u 1 u + Cu 1 u 3 + A u 1 u 4 + A u (u + u 3 [ ( B δu 3 = 4 u 1 + Cu A 1u + + C u 1 u 3 + A u 1u 4 + A [ δu 4 = B u 1 u + B u 1 u 3 + ( A + 3C u 1 u 4 + A u 1u 5 + ], ] u + A u u 3 + A u 3, ( ( A + 4C u + A + 8C u u 3 + 8A u u 4 + 4Cu 3 + 8A u 3 u 4 ], [ ( ] δu 5 = 6 B (u + u 3 + A + 6C u u 4 + A u u 5 + 6Cu 3 u 4 + A u 3 u 5 + 4A u 4, (13 where A (π da(p, B (π db(p, C (π da(pb(p. (14 Performing the rescaling (5, the coupling constants t and u µ respectively move to t and u µ (µ = 0,, 5, where t u 0 = L (t + δt + O(n = L (u 0 + δu 0 + O(n u i = L 4 d (u i + δu i + O(n, (i = 1,, 5. (15 Here we discuss that O(n terms appearing in the right-hand side are irrelevant if we study the critical phenomena from the disorder phase. Suppose that we repeat RGT k times keeping the O(n terms and that t reaches t k (n. If k is finite, t k (n is obtained as a polynomial of n with a finite degree. It means that lim n 0 t k (n = t k (0 does not have any singularity. On the other hand, we can obtain t k (0 by applying k times of RGT in which the replica limit is taken. When we compute the correlation length, RGT is repeated until t k (0 1, which will be satisfied with a finite k in the disorder phase. Hence we can ignore the terms of O(n. 4

5 The crucial point of this discussion is that k is finite. It could be possible that lim k t k (n is singular at n = 0. In this case, the order of two limiting procedures k and n 0 will not commute. If we want to compute critical exponents on the critical surface, we must repeat RGT infinitely many times. This could cause the singularity in the replica limit as pointed out by Brézin and De Dominicis[1]. In this letter, we restrict ourselves to the case of the disorder phase where we can take the replica limit in eq.(15, so that we hereafter ignore O(n in eq.(15. 3 Redefinition of the coupling constants In the pure φ 4 theory near the upper critical dimension, the RG flow is controlled to be in a perturbative region by requiring that the initial value of t should be small. On the other hand, the RG flow considered here cannot stay in the perturbative region by the one-parameter tuning because u 0 having the mass dimensions exists. Note that the φ 4 -coupling constants u 1,, u 5 also grow under the RGT through the loop integrals B and C, which are respectively proportional to u 0 and u 0. In order to resolve the difficulty, we introduce the new coupling constants g 0,, g 5 as follows: g µ u αµ 0 u µ, µ = 0,, 5, (16 where α 0 = α 5 =, α 1 = 1, α = α 3 = 0, α 4 = 1. (17 The scaling dimensions at the tree level of the new coupling constants are easily found from those of u µ s (See eq.(6: g µ = L βµ g µ, (18 where β 0 =, β 1 = 6 d, β = β 3 = 4 d, β 4 = d, β 5 = d. (19 The loop corrections to the new coupling constants are translated from δu µ by δg µ = δ ( u αµ 0 u µ = u α µ 0 δu µ + α µ u 1 0 δu 0g µ. (0 Our claim is that once we know the corrections δu µ by the formal perturbative expansion, we can obtain δg µ in terms of a power series of g µ in the replica limit. To prove it, we first look at a loop-correction term in δu j (j = 1,, 5, which generally has the form of u p 1 1 u p u p 5 5 F(u 0, t = u 5 α ip i 0 g p 1 1 g p g p 5 5 F(u 0, t, (1 where F(u 0, t contains the loop integration with the internal lines A(p and B(p, and contains the summation over the replica indices. Let #A, #B be the number of A(p and B(p in F(u 0, t 5

6 respectively. We notice that 5 #A + #B = p i. ( Since the interaction conjugate to u i has α i summations over the replica indices, the number of the sums in F(u 0, t is 5 ( α i p i. (3 It should be reduced up to α j by Kronecker s delta associated with A(p: otherwise, F(u 0, t becomes O(n and disappears in the limit n 0. Thus, we have the following restriction to #A: In other words, from eq.(, Since B(p is proportional to u 0, F(u 0, t can be written 5 #A ( α i p i ( α j. (4 ( 5 #B α i p i α j (5 F(u 0, t = u ( 5 α ip i α j 0 f(u 0, t, (6 where f(u 0, t does not contain any positive powers of u 0. Therefore, f(u 0, t = f(g 1 0, t has only non-negative powers of g 0. Inserting eq.(6 into eq.(1, we find that u α j 0 δu j in eq.(0 consists of a term g p 0 0 g p 1 1 g p 5 5 f(t (7 in the replica limit, where p µ 0, (µ = 0,, 5. Moreover, f(t is analytic around t = 0 because the loop integration is performed over the higher momentum region. A similar consideration can be applied to the two-point coupling constants t and u 0 and we find that δt and u 1 0 δu 0 can be expanded around g µ = 0. Thus the proof is completed. Switching from (t, u µ to (t, g µ, we can see the region where the perturbative RGT is available. In eq.(7, one can show that 5 p 0 p i. (8 This implies that g 0 is not neccesarrily small. Namely, the perturbative expansion will be reliable in d 6 if the initial conditions satisfy g i < 1 (i = 1,, 5, t < 1 and g i g 0 < 1. For example, Brézin and De Dominicis suggested that the initial coupling constants for the random field Ising model satisfy u 0, u 3 O(, u 1 O( 0, u, u 4 O(, u 5 O( 3 (9 6

7 with small, where denotes the variance of the Gaussian random field. Going to the new coupling constants, we get g 0 g 1 0, g i O(. (30 Since g 0 g 1 is equal to the standard φ 4 coupling constant u 1, it is expected that the perturbative expansion will be valid for the random field Ising model with small although g 0 g 1 exceeds. We emphasize that the trivial fixed point by the new constants (t, g µ = 0 does not correspond to the Gaussian fixed point (t, u µ = 0 because g 0 = 0 means that u 0 =. We can trace RG trajectories in the vicinity of the non-gaussian fixed point by the perturbative RGT using the new coupling constants. Finally, we compare the observation in ref.[1] and our result presented here. Brézin and De Dominicis studied the beta function choosing the one-loop corrections proportional to C on the critical surface t = 0. In this beta function one can absorb u 0 dependence by the redefinition u i u 0 u i for all i = 1,, 5. Hence the all redefined φ 4 coupling constants have the dimensions 6 d, which causes the instability of the non-trivial fixed point in d = 6 ǫ. On the other hand, we redefine the coupling constants as u i u α i 0 u i because the loop corrections proportional to A or B are taken into account. According to eq.(19, g 1 alone becomes slightly relevant while the other φ 4 couplings are all irrelevant in d = 6 ǫ with small ǫ. Therefore, it is expected that a non-trivial fixed point, which admits the ǫ expansion, exists in dimension 6 ǫ in accordance with the case of the pure φ 4 theory. 4 The ǫ-expansion near six dimensions Here we put ǫ 6 d and find a non-trivial fixed point with the ǫ-expansion. To begin with, let us compute the one-loop correction to the new coupling constants. When t is small, the loop integrals in eq. (11 can be expanded around t = 0. For example, Here we have defined A 1 = a 1 a t +, B 1 = a a 3 t +. (31 a j 1. (3 (π d pj Using eq.(31 in eq.(10, we get the one-loop correction δt and δg 0 written in terms of the new coupling constants: δt = ( 1 a g 1 + a 1 g a 1g 1 g 0 (a 3 g 1 + a g a g 1 g 0 t + O(t g 1 0 δg 0 = (a g 3 + a 1 g 4 + a 1 g g 0 (a 3 g 3 + a g 4 + a g g 0 t + O(t, (33 where we have used the parameter g 3 g + g 3 instead of g 3 for convenience. 7

8 The correction to the φ 4 coupling constants are also obtained employing eqs. (13, (0 and (33. The result is δg 1 = 3a 3 g 1 7a g 1 g 3 a 1 g 1 g 4 ( 3 a g 1 + a 1g 1 g g 0 δg = 3a 3 g 1 g 3 3a g 1 g 4 6a g g 3 + 3a g 3 a g 1 g g 0 δ g 3 = 5a 3 g 1 g 3 4a g 1 g 4 4a g 3 a 4 g 1 1 a (g 1 g + g 1 g 3 g 0 δg 4 = a 4 g 1 g 3 3a 3 g 1 g 4 a g 1g 5 4a 3 g 3 7a g 3 g 4 + a 1 g4 a (g 1g 4 + g g 3 g 0 + a 1 g g 4 g 0 δg 5 = 6a 4 g 3 36a 3 g 3 g 4 4a g 3 g 5 4a g4 + 4a 1g 4 g 5 ( a 1 g g 5 + 6a g g 4 g 0. (34 Here we have used the expansion eq.(31 and neglected O(t terms. As we have mentioned in the previous section, the corrections are obtained as the perturbative expansion by the new coupling constants. Let us find the non-trivial fixed point (t, gµ in the leading order of ǫ under the assumption t, gµ O(ǫ. Suppose that the initial value (t, g µ goes to (t k, g µk after applying the RGT k times. We have the following recursion equation: t k+1 = L (t k + δt k g µ k+1 = L βµ (g µ k + δg µ k, (35 where δt k and δg µ k means to replace t, g µ with t k, g µk in eqs.(33 and (34. The fixed point is determined by eq.(35 with t k+1 = t k = t, g µ k+1 = g µ k = gµ. Eqs.(33 and (34 imply that g 0 = 0, t, g 1 O(ǫ, g i = O(ǫ, (i =,, 5, (36 which leads to the following leading-order solution: g 1 = log L 3a 3 ǫ, a t = (1 L g 1, g 0 = g i = 0 (i =,, 5. (37 Next, we linearize the recursion equation (35 around the fixed point (37. Let h k be the sevencomponent vector defined by h k (t k, g µ k (t, gµ. The recursion equation is written as where M is the 7 7 matrix M ab = h k+1 a h k b h k+1 = Mh k + O( h k, (38, (a, b = 1,, 7. (39 hk =(t,gµ The eigenvalues of M can be read from the diagonal components in the leading order of ǫ. Using eq.(37, they are {L ǫ/3, L, L ǫ, L +ǫ, L ǫ/3, L 4, L 6+ǫ } + O(ǫ. (40 8

9 It indicates that the fixed point defined in eq.(37 is once unstable, as in the case of pure φ 4 theory with d = 4 ǫ. Moreover, the correlation-length exponent, which is given by the inverse of the largest exponent in eq.(40, is found to be 1 + ǫ 1 + O(ǫ. (41 It is exactly the same as the leading-order result in the d = 4 ǫ pure φ 4 theory. 5 Discussion In this letter, we have analyzed the effective field theory derived in ref.[1] near the upper critical dimension using the renormalization-group method. We have shown that the non-trivial fixed point is once-unstable in d = 6 ǫ, where ǫ is sufficiently small, and derived the correlation-length exponent. The result is totally consistent with the Parisi-Sourlas dimensional reduction in the leading-order computation. It is a nontrivial problem whether the consistency is preserved beyond the leading order, which will be reported elsewhere. The point of the present study is that we redefine the new coupling constants g µ by eq.(16 such that the RG flow can be stayed in a perturbative region by the one-parameter tuning near the upper critical dimension. It will be interesting to construct a model that describes vicinity of the non-gaussian fixed point (t, g µ =0, not relying on the replica formalism. Acknowledgment We would like to thank C. Itoi for valuable discussions and comments. References [1] E. Brézin and C. De Dominicis, Europhys. Lett. 44 ( [] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 ( ; Nucl. Phys. B06 ( [3] K. G. Wilson and J. Kogut, Phys. Rep. 1C ( [4] H. Tasaki, J. Stat. Phys. 54 (