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Real-space renormalization-group approaches for two-dimensional Gaussian Ising spin glass
21
km&r
1998 PHYSICS LETTERS A
Physics Letters A 250 ( 1998) 163-169
Real-space reno~a~za~ion-grout approaches for ~~o-~irnens~on~~~aussi~ Ising spin glass Departarnento de Fhica Te&ica e Experimental. iJniversidade Federal do Rio Grcwtdedo Norte, Campus iJniversitdrio, CJ? f641, 59072-970 Natal, RN, Bmzil
Received 23 July 1998: revised manuscript received 25 September 1998; accepted for publication 14 October I998 Communicated by A.R. Bishop
Abstract ho-dimensional king spin glass with Gaussian couplings is studied through different real-space ~no~~i~ati~n-group procedures. The stiffness exponent y, associated with the e~o~utiou of the coupling disi~butioo at zero temperature, is computed. 0.x results are discussed and compared witb other estimates available. @ 1998 ~ub~isb&d by Etsevier Scietlce
B.V. 05.5O.+q; 64.6O.p;k;75.10.Nr Keywords: Nearest-neighbor interactions; Ising spin glasses; Renormalization group;
PACS:
The spin-glass problem f I-3 ] represents nowadays one of the greatest challenges in the physics of disordered systems. Its mean-field theory, formulated in terms of the in~nite-rage-interaction model proposed by She~ington and ~rkpa~ick [4], is by now reasonably understo~. However, the validity of the mean-field results for the description of real (shop-rage-interaction} systems is a highly non~vi~ question, far from being satisfactorily answered [ 31. Very little is known, in what concerns short-range mode&; for exam@, different approximate methods [5131 all agree that the lower critical dimension di of the nearest-neighbor Ising spin glass lies in the range 2 < dr < 3 in such a way that there is a finite-temperature phase transition in three dimensions, but not in two. Curiously, such a conclusion was drawn through a Migdal-Kadanoff renormalization-group (MKRG) approach [ 51, almost a decade before other methods, like domain-wall (DW) based scaling approaches [6-g], powerful numerical simulations f lo-121 and extensive high-temperature series expansions f 131. The existence of quantities analogous to those of uniform systems which characterize the spin-glass transition [2], at a given critical temperature T’, like the correlation length (5 N (T - T,) -“), the SpiIk-ghSS su~eptibility (XsG = (T - Tc)-Y) and the correlation function (C(r) N r -(d-2f~1, at T = T,) is nowadays well accepted. Scaling relations also seem to hold for spin glasses, in such a way that if d > dt there are only two inde~ndent static critical exponents. Despite the fact that a phase transition does not occur at finite temperatures, the newest-neighbor twodimensional Ising spin gfass has been the object of a much interest [ 5,6,8,9,11,14-191. At zero temperature 0375..9601/98/$ - see front matter @ 1998 Published by Elsevier Science B.V. All rights reserved.
P1I SO375-9601(98)00796-8
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the system is in its ground state which might be either degenerate or unique (apart from time-reversed states), depending on the probability distribution for the coupling constants. For discrete distributions (e.g., the fJ case) frustration plays an important role, leading to a macroscopic ground-state entropy, whereas for continuous distributions there is no nontrivial degeneracy and the ground state is unique. In the former case the correlation function presents a power-law decay at zero temperature (G(r) m r-v), whereas for the latter, the system is completely frozen into one particular configuration, exhibiting no power decay, and so 77= 0. Consequently, if a phase transition occurs at zero temperature for nearest-neighbor two-dimensional spin glasses with continuous probability distributions for the couplings, it should be described by a single independent static critical exponent. Usually, the exponent Y, associated with the divergence of the correlation length, is the easiest one to be obtained. Some con~oversy has been raised in what concerns the critical exponent Y for the Gaussian Ising spin glass on a square lattice. Different works [6,8,9,14-191 lead to values within a wide range varying from 1.7 [ 171 up to 3.6 [ 61, with the estimates depending on the scaling assumptions used to obtain them. Curiously, the highest values are obtained if one studies the system at T = 0 [6,8,9,17], whereas the finite-temperature analyses followed by T -+0 extrapolations lead to much lower estimates [ 14-191. Since different works using the same scaling assumptions are in fairly good agreement, some authors claim that important ingredients might be missing in the scaling theory of the two-dimensional Ising spin glass [ 16,18,19]. The aim of the present work is to present T = 0 estimates of the exponent v for the Gaussian Ising spin glass, defined on five different hierarchical lattices which approach the square lattice. One possible way for estimating the exponent v is by making use of the scaling arguments of Bray and Moore (BM) [ 891, which hold for continuous and symme~ic dis~ibutions P(Jij) for the couplings among newest-neighbor spins in blocks of linear size L. At an in~nitesimal tem~rature, the effective couplings of the blocks present a root-mean-squ~e deviation J’( ~5.)that should scale as
J’(L)
N
JL'
,
(1)
where J = ($)I/*. The sign of the stiffness exponent y is directly associated with the low-temperature phase; for a positive (negative) y the system scales to strong (weak) couplings, characteristic of a spin-glass (paramagnetic) state at low temperatures. Therefore, one expects y < 0 for d < dt and y > 0 for d > dt, Since the temperature always appears in the renormalization-group equations as dimensionless ratios with the i.e., L - T'/J'. For a phase transition in the couplings, the scaling in Eq. ( I) is equivalent to a scaling T - LJ', limit T -+ 0 in the case d = 2,this may be identified with the correlation length 6 N T-' and therefore, one expects the validity of the scaling relation Y =
-l/y.
(2)
The application of this method for hierarchical lattices may turn out to be much simpler than the usual T = 0 domain-wall scaling approaches [ 6,8,9,1’7], or the finite-temperature numerical algorithms [ 14-191. As will be shown later, our estimates are in good agreement with some previous ones, but require a significantly lower amount of computational effort. Let us consider the Ising spin glass defined through the Hamiltonian
where the sum C ijf is restricted to newest-neigh~r pairs of spins on given lattices, whereas Jij are quenched random variables Iollowing a Gaussian probability distribution P(Jij)
=
&exp(-.$/ZJ’).
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i
Fig. I. The basic c&s of the hieing lattices with fmctaI diction Lk (a) Cetf I (D = 2); (b) ceil II (D % 2.32); (c) cell III (D = 2); (d) cell IV (D = 2); (e) cell V (fJ % 2.33). The solid circles denote the internal sites (to be decimated in the mnormalization process), whereas the open ones represent the external sites (connected to other cells of the lattice).
We shall study the model defined in Eq. (3) on hierarchical lattices, generated in such a way that, at each step, a bond is replaced by a cell, like the ones shown in Fig. 1, In Fig, la one has a diamond-like cell (scaling factor b = 2), characteristic of the MKRG (cell I); its fractal dimension is L) = in4/ In 2 = 2. Tfie b = 2 MKRG has already been applied to short-range Gaussian Ising spin glasses [ 5,8,20] ; it is expected to represent a reasonable approximation of Bravais lattices in low dimensions, but it becomes inadequate at high dimensions (which are obtained by increasing the number of parallel branches in Fig, la). In Fig. lb one has the Wheatstone-bridge of tw~dimension~ fe~o~g~e~ cell (b = 2) (cell II), which has been very successful as ~proxi~o~s in both pure and disordered limits [Zl] ; its fractal dimension is I) = fn5/ In2 % 2.32. In Figs. lc, Id and le one has b = 3 cells; apart from being larger cells, they preserve a pure ~tife~omagnetic system under the renormalization transformation (this does not occur in the cases of Figs. la and lb). The cell in Fig. lc
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EN. Nobre/Physics Letters A 250 (1998) 163-169
(cell III) presents a fractal dimension D = In 9/ In 3 = 2 and has been successful in several applications [ 21,221. The cells in Figs. Id (cell IV) and le (cell V) represent the b = 3 versions of cells I and II, and have fractal dimensions D = In9/In3 = 2 and D = In13/1n3 w 2.33, respectively. The B&f scaling procedure may be applied to these systems by considering L = b” for a lattice at its nth hierarchy level. The reno~alization-group procedure may now be carried bout; as usual, it works inversely to the generation of the hierarchica lattice, i.e., it transfo~s the cells in Fig. 1 into elementary bonds. At zero temperature, the recursion relations involving the effective coupling J,$ and the set of original couplings (JI,,}, may be written, for all cells in Fig. 1, as
(5a) where A ma
=
max({A;)),
Bmax= maxt (4))
(i= 1,2 ,‘.., P),
(5b)
represent the maximum values of the sets (Ai} and (&I, containing P elements each, with P = 2”, K denoting the total number of internal spins of the cell. For a finite temperature, the Ai’s would appear in the arguments of the Boitzmann’s weights associated with the states of the external spins 8; = Sj = 1, whereas the Bi’s would be those co~esponding to Si = 1 and Sj = -1. For celf I one has A max=max(At,A2,.%,A4), AI = Jil + Ji2 + Jlj+ J2j 3 A2 = Jil - Ji2 + Jij - J2j 9 AS = -Jil + JD - Jlj + J2j t A4 = -J;l - J;2 - Jf,j - Jzj ,
Bmax=max(Bl,B2,B3,Bq), BI 32 B3 B4
= = = =
Jil + Ji2 - Jlj - J2j 9 Jil - Ji2 - Jlj + J2j 9 - Jil + Ji2 + Jlj - J2j t -Jil - J;z + Jlj + J2j 1
Since A.t = -AZ, A4 = -AI, B3 = -Bz, B4 = -BE, Eq. (5a) may be rewritten in the familiar form
J~i= i[(IJil f Jrji - IJ~I-JIM/) + (~JQ +
Jzjl - /Jj2 - J2jl)].
For cell II, AI = Jit + J;2 + JI,~+ J2.i + AZ = J/l - J;2 + Jlj - J2j A3 = -J;I + JQ - Jlj + J2j A4 = -Ji, - J;2 - Jlj - J2j
Ji2 $ Jt2 3 - Jt2 9 + JIZ,
Bl = Jil + JD - JIj B2 = Jil - J;2 - Jlj B3 = -Jil + Ji2 + Jlj B4 = -Jil - Ji2 + Jlj
Jzj + J2j - J2j + J2j
Jlz,
Jt2 3 - Jt2 2 + 512 s
A similar procedure holds for the b = 3 cells, although one has to deal with more complicated functions of the couplings {Jlnt}, e.g., for cell III one has P = 24 = 16, whereas for IV and V, P = 26 = 64. The renormalization-group iterations are implemented by following numerically the probability dis~bution associated with the coupling constants [ 51. Operationahy, this probability distribution is represented by a pool of M real numbers {JI}, from which one may compute, at each renormalization step,
which should represent the moments of the distribution {Jij}. The process starts by creating an initial pool with numbers produced from a Gaussian random number generator [ 231. An iteration consists of M operations, where in each of them one picks randomly numbers from the pool (each chosen number is assigned to a bond in the cells of Fig. 1) in order to generate the effective exchange energy according to Eq. (5a); after that, one gets a new pool representing the renormalized probability dis~ibution. For a centered initial dis~bution, the
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-Ln J
-Ln J
6-
6-
167
163-169
‘0
5432l-
o
a m
0
.a fi B
a
Q 02,I,. 0
c
0
u
‘-q Tr
(b)
,a, 5
10
15
1 20
n
n Fig. 2. Plot of - In I (J = fi) versus are smaller than the symbol sires. (a) b
u
0
u
hierarchy level n for the cells in Fig. 1; in each case the slope equals -y In b. The error bars 2 cells; (b) b = 3 cells.
the =
lowest nontrivial moment is the second one, (~2, from which one may compute the root-mean-square deviation, J = @, to be used in Eq. ( 1) . Another interesting quantity is the kurtosis of the distribution, defined as
Kz2-1,
(8)
34
which is identically zero for a Gaussian probability distribution. We have used pools of size M = 500000 and our simulations were repeated for 200 samples, i.e., different initial sets of random numbers, in order to reduce the dependence of our results on the particular initial conditions. The stiffness exponent y may be obtained from the plot In.! versus n, as shown in Fig. 2. In each case we have applied the method of least squares to all the data points exhibited (hierarchy levels n = 1 to n = 20). One sees that the scaling proposed in Eq. ( 1) is confirmed to be very good, yielding precise exponent estimates for each one of the cells herein employed. Our estimates are presented in Table 1, where the results for the exponent Y were obtained assuming the validity of the scaling relation in Eq. (2). An interesting feature we have noticed in the present problem is that although at each step all even moments in Eq. (7) get reduced, after a few iterations (roughly four renormalization steps), the kurtosis of the renormalized distribution presents very small fluctuations around a fixed value (with changes in the third decimal digit only), under successive iterations, as shown in Table 1. This reflects the fact that the kurtosis is a dimensionless quantity, contrary to the even moments of the renormalized distribution (which depend on the length scale L). Such an effect suggests that the probability distribution approaches a well-defined functional dependence on the Table I The stiffness exponent y for each of the hierarchical lattices generated by the cells in Fig. 1. The correlation-length critical exponent Y is estimated from the scaling relation in Eq. (2). After a few iterations, the kuttosis K (see Eq. (8) ), associated with the renormalized coupling probability distribution, fluctuates within the error bars presented Cell
Y
I
-0.270 -0.290 -0.275 -0.278 -0.298
II 111 IV V
v f f f f f
0.002 0.002 0.001 0.002 0.002
3.70 f 3.45 f 3.64 f 3.60 f 3.36 f
K 0.03 0.02 0.02 0.03 0.02
0.116 f 0.175 f 0.174 f 0.099 rt 0.178 *
0.001 0.001 0.001 0.001 0.001
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EN. Nobre/Physics Letters A 250 (1998) 163-169
couplings {J;j}; although its moments evolve under the renormalization process, its functional form seems to remain invariant. Except for the results of cell IV, the value of the kurtosis appears to be related to the stiffness exponent y: larger kurtosis yields smaller exponents. Since each renormalization-group transformation is exact on its respective hierarchical lattice, in the case of pure systems [ 211, we expect our results to be very good approximations for the spin-glass problem on such lattices. Moreover, all five hierarchical lattices employed herein have been used in the literature as approximations in the study of two-dimensional pure and disordered systems [ 211. Therefore, one may compare our results with those of the Gaussian Ising spin glass on the square lattice. Indeed, some of our estimates are very close to those of early DW scaling approaches of McMillan [6] (y = -0.281 f 0.005; v = 3.56 f 0.06) and Bray and Moore [ 8,9] (y = -0.291 f 0.002; Y = 3.4 & O.l), as well as to the recent exact groundstate calculations of Rieger et al. [ 171 (y = -0.281 & 0.002; Y = 3.559 f 0.025). However, all our results differ significantly from the simulations using a Swendsen-Wang type of cluster dynamics of Liang [ 141 (v = 2.0 f 0.2), the exact counting of ground states of Kawashima and Suzuki [ 151 (y = -0.476 f O.OOS), the numerical transfer-matrix approach of Kawashima et al. [ 161 (y = -0.48 f O.Ol), the recent numerical simulations of Rieger et al. [ 171 (v = 1.7 f 0.2), Nifle and Young [ 181 (Y = 1.8 f 0.4)) and the transfermatrix method of Huse and Ko [ 191 (v M 2.7). Apparently, such puzzling wide range of estimates seem to follow into two distinct groups: the first [6,8,9,17] (v M 3.6), which the present results are corroborating, all consider the system at T = 0,making use of the scaling relation (2); the second [ 14-191 (v M 2.0), study the problem at finite temperatures, taking further T -+0 extrapolations. Some authors have found violations in the scaling relation (2) [ 16,171, leading to claims that the present understanding of the scaling picture for the two-dimensional Ising spin glass may not be correct [ 16,18,19]. Although the MKRG (cells I and IV) is well known to represent poor approximations for pure systems (specially in high dimensions), it yields results that are surprisingly close to those of the other cells. It should be mentioned that our results for cell I have already been obtained before by Bray and Moore [ 81 (Y M 3.7). The agreement of the MKRG results with those of more complex cells comes as an additional support for the use of the MKRG approach for short-range spin glasses; the MKRG critical-temperature estimates for the three-dimensional Ising spin glasses [ 51 (both Gaussian and fJ interactions), are also very close to those of more sophisticated numerical procedures [ lo- 131. Due to its previous successful applications in other two-dimensional disordered systems [21,22], and also due to the fact that its fractal dimension equals two, we expect the results for the hierarchical lattice generated by cell III to represent our best approximation for the Gaussian Ising spin glass on a square lattice. We suspect that the lattices generated by the Wheatstone-bridge cells (II and V) are not, for spin glasses, good approximations of two-dimensional Bravais lattices, as they happen to be in the case of pure systems. Surprisingly, they yield lower y estimates, in spite of the fact that their fractal dimension is greater than two. This is contrary to the usual expectation, i.e., higher dimensions should produce greater values of y; we have no explanation for such results. To conclude, we have studied the nearest-neighbor Gaussian Ising spin glass on five distinct hierarchical lattices which approach the square lattice. We have computed, in each case, the stiffness exponent y, characteristic of the evolution of the coupling probability distribution at zero temperature. Assuming the existence of a phase transition at zero temperature (which may be described in terms of a single independent critical exponent) and making use of the scaling relation 2, = -l/y, we have estimated the correlation-length critical exponent. Some of our estimates are very close to those of early T = 0 domain-wall scaling approaches, as well as to the ones of recent exact ground-state calculations, but fairly different from the results of finite-temperature algorithms. It is a pleasure to acknowledge S.G. Coutinho and E.M.F. Curado for fruitful conversations. Brazilian agencies CNPq and FINEP (Pronex project) for partial financial support.
We thank the
EN. Nobre/Physics
L.etters A 250 (1998) 163-169
169
References I I] K. Binder, A.P. Young, Rev. Mod. Phys. 58 ( 1986) 801. [2] K.H. Fischer, J.A. Hertz, Spin Glasses (Cambridge Univ. Press, Cambridge, 1991). [ 31 AI? Young, ed., Spin Glasses and Random Fields (World Scientific, Singapore, 1997). [4] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. 151 B.W. Southern, A.P. Young, J. Phys. C 10 (1977) 2179. 161 W.L. McMillan, Phys. Rev. B 29 (1984) 4026. [7] W.L. McMillan, Phys. Rev. B 30 (1984) 476; Phys. Rev. B 31 ( 1985) 340. [ 8 1 A.J. Bray, M.A. Moore, J. Phys. C 17 (1984) L463. 191 A.J. Bray, M.A. Moore, in: Heidelberg Colloquium on Glassy Dynamics, Lecture Notes in Physics 275, eds. J.L. van Hemmen, 1. Morgenstem (Springer, Berlin, 1986). [ 101 R.N. Bhatt, A.P. Young, Phys. Rev. L&t. 54 (1985) 924; A.T. Ogielski, I. Morgenstem, Phys. Rev. Lett. 54 (1985) 928. [ I1 1 R.N. Bhatt, A.P. Young, Phys. Rev. B 37 (1988) 5606. [ 12 ] N. Kawashima, A.P. Young, Phys. Rev. B 53 (1996) R484. [ 131 R. Singh, S. Chakravarty, Phys. Rev. Lett. 57 (1986) 245; L. Klein, J. Adler, A. Aharony, A.B. Harris, Y. Meir, Phys. Rev. B 43 ( 1991) 11249. [ 141 S. Liang, Phys. Rev. Lett. 69 (1992) 2145. [ 151 N. Kawashima, M. Suzuki, J. Phys. A 25 (1992) 1055. [ 161 N. Kawashima, N. Hatano, M. Suzuki, J. Phys. A 25 (1992) 4985. [ 17 1 H. Rieger, L. Santen. U. Blasum, M. Diehl, M. Jthtger, G. Rinaldi, J. Phys. A 29 (1996) 3939; Corrigendum, J. Phys. A 30 ( 1997) 8795. [ 18) M. Ney-Nifle. A.P. Young, J. Phys. A 30 (1997) 531 I. 1191 D. A. Huse, L.-F. Ko, Phys. Rev. B 56 (1997) 14597. 1201 M. Ney-Nifle, H.J. Hilhorst, Phys. Rev. Lett. 68 (1992) 2992; Physica A 193 (1993) 48; A 194 (1993) 462. [ 211 C. Tsallis, A.C.N. de Magalhiies, Phys. Rep. 268 ( 19%) 305. [22] A.C.N. de Magalhfies, C. Tsallis, G. Schwachheim, J. Phys. C 14 (1981) 1393; P.M.C. de Oliveira, Phys. Rev. B 25 ( 1982) 2034; P.M.C. de Oliveira, C. Tsallis, J. Phys. A 15 (1982) 2865. [ 231 W.H. Press. S.A. Teukolsky, W.T. Vetterling, B. Flannery, Numerical Recipes in Fortran, 2nd edition (Cambridge Univ. Press, Cambridge, 1992).
km&r
1998 PHYSICS LETTERS A
Physics Letters A 250 ( 1998) 163-169
Real-space reno~a~za~ion-grout approaches for ~~o-~irnens~on~~~aussi~ Ising spin glass Departarnento de Fhica Te&ica e Experimental. iJniversidade Federal do Rio Grcwtdedo Norte, Campus iJniversitdrio, CJ? f641, 59072-970 Natal, RN, Bmzil
Received 23 July 1998: revised manuscript received 25 September 1998; accepted for publication 14 October I998 Communicated by A.R. Bishop
Abstract ho-dimensional king spin glass with Gaussian couplings is studied through different real-space ~no~~i~ati~n-group procedures. The stiffness exponent y, associated with the e~o~utiou of the coupling disi~butioo at zero temperature, is computed. 0.x results are discussed and compared witb other estimates available. @ 1998 ~ub~isb&d by Etsevier Scietlce
B.V. 05.5O.+q; 64.6O.p;k;75.10.Nr Keywords: Nearest-neighbor interactions; Ising spin glasses; Renormalization group;
PACS:
The spin-glass problem f I-3 ] represents nowadays one of the greatest challenges in the physics of disordered systems. Its mean-field theory, formulated in terms of the in~nite-rage-interaction model proposed by She~ington and ~rkpa~ick [4], is by now reasonably understo~. However, the validity of the mean-field results for the description of real (shop-rage-interaction} systems is a highly non~vi~ question, far from being satisfactorily answered [ 31. Very little is known, in what concerns short-range mode&; for exam@, different approximate methods [5131 all agree that the lower critical dimension di of the nearest-neighbor Ising spin glass lies in the range 2 < dr < 3 in such a way that there is a finite-temperature phase transition in three dimensions, but not in two. Curiously, such a conclusion was drawn through a Migdal-Kadanoff renormalization-group (MKRG) approach [ 51, almost a decade before other methods, like domain-wall (DW) based scaling approaches [6-g], powerful numerical simulations f lo-121 and extensive high-temperature series expansions f 131. The existence of quantities analogous to those of uniform systems which characterize the spin-glass transition [2], at a given critical temperature T’, like the correlation length (5 N (T - T,) -“), the SpiIk-ghSS su~eptibility (XsG = (T - Tc)-Y) and the correlation function (C(r) N r -(d-2f~1, at T = T,) is nowadays well accepted. Scaling relations also seem to hold for spin glasses, in such a way that if d > dt there are only two inde~ndent static critical exponents. Despite the fact that a phase transition does not occur at finite temperatures, the newest-neighbor twodimensional Ising spin gfass has been the object of a much interest [ 5,6,8,9,11,14-191. At zero temperature 0375..9601/98/$ - see front matter @ 1998 Published by Elsevier Science B.V. All rights reserved.
P1I SO375-9601(98)00796-8
164
EN. No~~/~h~~~~s
letters
A 250 (19981 163469
the system is in its ground state which might be either degenerate or unique (apart from time-reversed states), depending on the probability distribution for the coupling constants. For discrete distributions (e.g., the fJ case) frustration plays an important role, leading to a macroscopic ground-state entropy, whereas for continuous distributions there is no nontrivial degeneracy and the ground state is unique. In the former case the correlation function presents a power-law decay at zero temperature (G(r) m r-v), whereas for the latter, the system is completely frozen into one particular configuration, exhibiting no power decay, and so 77= 0. Consequently, if a phase transition occurs at zero temperature for nearest-neighbor two-dimensional spin glasses with continuous probability distributions for the couplings, it should be described by a single independent static critical exponent. Usually, the exponent Y, associated with the divergence of the correlation length, is the easiest one to be obtained. Some con~oversy has been raised in what concerns the critical exponent Y for the Gaussian Ising spin glass on a square lattice. Different works [6,8,9,14-191 lead to values within a wide range varying from 1.7 [ 171 up to 3.6 [ 61, with the estimates depending on the scaling assumptions used to obtain them. Curiously, the highest values are obtained if one studies the system at T = 0 [6,8,9,17], whereas the finite-temperature analyses followed by T -+0 extrapolations lead to much lower estimates [ 14-191. Since different works using the same scaling assumptions are in fairly good agreement, some authors claim that important ingredients might be missing in the scaling theory of the two-dimensional Ising spin glass [ 16,18,19]. The aim of the present work is to present T = 0 estimates of the exponent v for the Gaussian Ising spin glass, defined on five different hierarchical lattices which approach the square lattice. One possible way for estimating the exponent v is by making use of the scaling arguments of Bray and Moore (BM) [ 891, which hold for continuous and symme~ic dis~ibutions P(Jij) for the couplings among newest-neighbor spins in blocks of linear size L. At an in~nitesimal tem~rature, the effective couplings of the blocks present a root-mean-squ~e deviation J’( ~5.)that should scale as
J’(L)
N
JL'
,
(1)
where J = ($)I/*. The sign of the stiffness exponent y is directly associated with the low-temperature phase; for a positive (negative) y the system scales to strong (weak) couplings, characteristic of a spin-glass (paramagnetic) state at low temperatures. Therefore, one expects y < 0 for d < dt and y > 0 for d > dt, Since the temperature always appears in the renormalization-group equations as dimensionless ratios with the i.e., L - T'/J'. For a phase transition in the couplings, the scaling in Eq. ( I) is equivalent to a scaling T - LJ', limit T -+ 0 in the case d = 2,this may be identified with the correlation length 6 N T-' and therefore, one expects the validity of the scaling relation Y =
-l/y.
(2)
The application of this method for hierarchical lattices may turn out to be much simpler than the usual T = 0 domain-wall scaling approaches [ 6,8,9,1’7], or the finite-temperature numerical algorithms [ 14-191. As will be shown later, our estimates are in good agreement with some previous ones, but require a significantly lower amount of computational effort. Let us consider the Ising spin glass defined through the Hamiltonian
where the sum C ijf is restricted to newest-neigh~r pairs of spins on given lattices, whereas Jij are quenched random variables Iollowing a Gaussian probability distribution P(Jij)
=
&exp(-.$/ZJ’).
EN.
~ob~/Ph~si~s
Letters
A 250 (19981
163469
165
A
i
Fig. I. The basic c&s of the hieing lattices with fmctaI diction Lk (a) Cetf I (D = 2); (b) ceil II (D % 2.32); (c) cell III (D = 2); (d) cell IV (D = 2); (e) cell V (fJ % 2.33). The solid circles denote the internal sites (to be decimated in the mnormalization process), whereas the open ones represent the external sites (connected to other cells of the lattice).
We shall study the model defined in Eq. (3) on hierarchical lattices, generated in such a way that, at each step, a bond is replaced by a cell, like the ones shown in Fig. 1, In Fig, la one has a diamond-like cell (scaling factor b = 2), characteristic of the MKRG (cell I); its fractal dimension is L) = in4/ In 2 = 2. Tfie b = 2 MKRG has already been applied to short-range Gaussian Ising spin glasses [ 5,8,20] ; it is expected to represent a reasonable approximation of Bravais lattices in low dimensions, but it becomes inadequate at high dimensions (which are obtained by increasing the number of parallel branches in Fig, la). In Fig. lb one has the Wheatstone-bridge of tw~dimension~ fe~o~g~e~ cell (b = 2) (cell II), which has been very successful as ~proxi~o~s in both pure and disordered limits [Zl] ; its fractal dimension is I) = fn5/ In2 % 2.32. In Figs. lc, Id and le one has b = 3 cells; apart from being larger cells, they preserve a pure ~tife~omagnetic system under the renormalization transformation (this does not occur in the cases of Figs. la and lb). The cell in Fig. lc
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(cell III) presents a fractal dimension D = In 9/ In 3 = 2 and has been successful in several applications [ 21,221. The cells in Figs. Id (cell IV) and le (cell V) represent the b = 3 versions of cells I and II, and have fractal dimensions D = In9/In3 = 2 and D = In13/1n3 w 2.33, respectively. The B&f scaling procedure may be applied to these systems by considering L = b” for a lattice at its nth hierarchy level. The reno~alization-group procedure may now be carried bout; as usual, it works inversely to the generation of the hierarchica lattice, i.e., it transfo~s the cells in Fig. 1 into elementary bonds. At zero temperature, the recursion relations involving the effective coupling J,$ and the set of original couplings (JI,,}, may be written, for all cells in Fig. 1, as
(5a) where A ma
=
max({A;)),
Bmax= maxt (4))
(i= 1,2 ,‘.., P),
(5b)
represent the maximum values of the sets (Ai} and (&I, containing P elements each, with P = 2”, K denoting the total number of internal spins of the cell. For a finite temperature, the Ai’s would appear in the arguments of the Boitzmann’s weights associated with the states of the external spins 8; = Sj = 1, whereas the Bi’s would be those co~esponding to Si = 1 and Sj = -1. For celf I one has A max=max(At,A2,.%,A4), AI = Jil + Ji2 + Jlj+ J2j 3 A2 = Jil - Ji2 + Jij - J2j 9 AS = -Jil + JD - Jlj + J2j t A4 = -J;l - J;2 - Jf,j - Jzj ,
Bmax=max(Bl,B2,B3,Bq), BI 32 B3 B4
= = = =
Jil + Ji2 - Jlj - J2j 9 Jil - Ji2 - Jlj + J2j 9 - Jil + Ji2 + Jlj - J2j t -Jil - J;z + Jlj + J2j 1
Since A.t = -AZ, A4 = -AI, B3 = -Bz, B4 = -BE, Eq. (5a) may be rewritten in the familiar form
J~i= i[(IJil f Jrji - IJ~I-JIM/) + (~JQ +
Jzjl - /Jj2 - J2jl)].
For cell II, AI = Jit + J;2 + JI,~+ J2.i + AZ = J/l - J;2 + Jlj - J2j A3 = -J;I + JQ - Jlj + J2j A4 = -Ji, - J;2 - Jlj - J2j
Ji2 $ Jt2 3 - Jt2 9 + JIZ,
Bl = Jil + JD - JIj B2 = Jil - J;2 - Jlj B3 = -Jil + Ji2 + Jlj B4 = -Jil - Ji2 + Jlj
Jzj + J2j - J2j + J2j
Jlz,
Jt2 3 - Jt2 2 + 512 s
A similar procedure holds for the b = 3 cells, although one has to deal with more complicated functions of the couplings {Jlnt}, e.g., for cell III one has P = 24 = 16, whereas for IV and V, P = 26 = 64. The renormalization-group iterations are implemented by following numerically the probability dis~bution associated with the coupling constants [ 51. Operationahy, this probability distribution is represented by a pool of M real numbers {JI}, from which one may compute, at each renormalization step,
which should represent the moments of the distribution {Jij}. The process starts by creating an initial pool with numbers produced from a Gaussian random number generator [ 231. An iteration consists of M operations, where in each of them one picks randomly numbers from the pool (each chosen number is assigned to a bond in the cells of Fig. 1) in order to generate the effective exchange energy according to Eq. (5a); after that, one gets a new pool representing the renormalized probability dis~ibution. For a centered initial dis~bution, the
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Letters A 250 (1998)
-Ln J
-Ln J
6-
6-
167
163-169
‘0
5432l-
o
a m
0
.a fi B
a
Q 02,I,. 0
c
0
u
‘-q Tr
(b)
,a, 5
10
15
1 20
n
n Fig. 2. Plot of - In I (J = fi) versus are smaller than the symbol sires. (a) b
u
0
u
hierarchy level n for the cells in Fig. 1; in each case the slope equals -y In b. The error bars 2 cells; (b) b = 3 cells.
the =
lowest nontrivial moment is the second one, (~2, from which one may compute the root-mean-square deviation, J = @, to be used in Eq. ( 1) . Another interesting quantity is the kurtosis of the distribution, defined as
Kz2-1,
(8)
34
which is identically zero for a Gaussian probability distribution. We have used pools of size M = 500000 and our simulations were repeated for 200 samples, i.e., different initial sets of random numbers, in order to reduce the dependence of our results on the particular initial conditions. The stiffness exponent y may be obtained from the plot In.! versus n, as shown in Fig. 2. In each case we have applied the method of least squares to all the data points exhibited (hierarchy levels n = 1 to n = 20). One sees that the scaling proposed in Eq. ( 1) is confirmed to be very good, yielding precise exponent estimates for each one of the cells herein employed. Our estimates are presented in Table 1, where the results for the exponent Y were obtained assuming the validity of the scaling relation in Eq. (2). An interesting feature we have noticed in the present problem is that although at each step all even moments in Eq. (7) get reduced, after a few iterations (roughly four renormalization steps), the kurtosis of the renormalized distribution presents very small fluctuations around a fixed value (with changes in the third decimal digit only), under successive iterations, as shown in Table 1. This reflects the fact that the kurtosis is a dimensionless quantity, contrary to the even moments of the renormalized distribution (which depend on the length scale L). Such an effect suggests that the probability distribution approaches a well-defined functional dependence on the Table I The stiffness exponent y for each of the hierarchical lattices generated by the cells in Fig. 1. The correlation-length critical exponent Y is estimated from the scaling relation in Eq. (2). After a few iterations, the kuttosis K (see Eq. (8) ), associated with the renormalized coupling probability distribution, fluctuates within the error bars presented Cell
Y
I
-0.270 -0.290 -0.275 -0.278 -0.298
II 111 IV V
v f f f f f
0.002 0.002 0.001 0.002 0.002
3.70 f 3.45 f 3.64 f 3.60 f 3.36 f
K 0.03 0.02 0.02 0.03 0.02
0.116 f 0.175 f 0.174 f 0.099 rt 0.178 *
0.001 0.001 0.001 0.001 0.001
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EN. Nobre/Physics Letters A 250 (1998) 163-169
couplings {J;j}; although its moments evolve under the renormalization process, its functional form seems to remain invariant. Except for the results of cell IV, the value of the kurtosis appears to be related to the stiffness exponent y: larger kurtosis yields smaller exponents. Since each renormalization-group transformation is exact on its respective hierarchical lattice, in the case of pure systems [ 211, we expect our results to be very good approximations for the spin-glass problem on such lattices. Moreover, all five hierarchical lattices employed herein have been used in the literature as approximations in the study of two-dimensional pure and disordered systems [ 211. Therefore, one may compare our results with those of the Gaussian Ising spin glass on the square lattice. Indeed, some of our estimates are very close to those of early DW scaling approaches of McMillan [6] (y = -0.281 f 0.005; v = 3.56 f 0.06) and Bray and Moore [ 8,9] (y = -0.291 f 0.002; Y = 3.4 & O.l), as well as to the recent exact groundstate calculations of Rieger et al. [ 171 (y = -0.281 & 0.002; Y = 3.559 f 0.025). However, all our results differ significantly from the simulations using a Swendsen-Wang type of cluster dynamics of Liang [ 141 (v = 2.0 f 0.2), the exact counting of ground states of Kawashima and Suzuki [ 151 (y = -0.476 f O.OOS), the numerical transfer-matrix approach of Kawashima et al. [ 161 (y = -0.48 f O.Ol), the recent numerical simulations of Rieger et al. [ 171 (v = 1.7 f 0.2), Nifle and Young [ 181 (Y = 1.8 f 0.4)) and the transfermatrix method of Huse and Ko [ 191 (v M 2.7). Apparently, such puzzling wide range of estimates seem to follow into two distinct groups: the first [6,8,9,17] (v M 3.6), which the present results are corroborating, all consider the system at T = 0,making use of the scaling relation (2); the second [ 14-191 (v M 2.0), study the problem at finite temperatures, taking further T -+0 extrapolations. Some authors have found violations in the scaling relation (2) [ 16,171, leading to claims that the present understanding of the scaling picture for the two-dimensional Ising spin glass may not be correct [ 16,18,19]. Although the MKRG (cells I and IV) is well known to represent poor approximations for pure systems (specially in high dimensions), it yields results that are surprisingly close to those of the other cells. It should be mentioned that our results for cell I have already been obtained before by Bray and Moore [ 81 (Y M 3.7). The agreement of the MKRG results with those of more complex cells comes as an additional support for the use of the MKRG approach for short-range spin glasses; the MKRG critical-temperature estimates for the three-dimensional Ising spin glasses [ 51 (both Gaussian and fJ interactions), are also very close to those of more sophisticated numerical procedures [ lo- 131. Due to its previous successful applications in other two-dimensional disordered systems [21,22], and also due to the fact that its fractal dimension equals two, we expect the results for the hierarchical lattice generated by cell III to represent our best approximation for the Gaussian Ising spin glass on a square lattice. We suspect that the lattices generated by the Wheatstone-bridge cells (II and V) are not, for spin glasses, good approximations of two-dimensional Bravais lattices, as they happen to be in the case of pure systems. Surprisingly, they yield lower y estimates, in spite of the fact that their fractal dimension is greater than two. This is contrary to the usual expectation, i.e., higher dimensions should produce greater values of y; we have no explanation for such results. To conclude, we have studied the nearest-neighbor Gaussian Ising spin glass on five distinct hierarchical lattices which approach the square lattice. We have computed, in each case, the stiffness exponent y, characteristic of the evolution of the coupling probability distribution at zero temperature. Assuming the existence of a phase transition at zero temperature (which may be described in terms of a single independent critical exponent) and making use of the scaling relation 2, = -l/y, we have estimated the correlation-length critical exponent. Some of our estimates are very close to those of early T = 0 domain-wall scaling approaches, as well as to the ones of recent exact ground-state calculations, but fairly different from the results of finite-temperature algorithms. It is a pleasure to acknowledge S.G. Coutinho and E.M.F. Curado for fruitful conversations. Brazilian agencies CNPq and FINEP (Pronex project) for partial financial support.
We thank the
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L.etters A 250 (1998) 163-169
169
References I I] K. Binder, A.P. Young, Rev. Mod. Phys. 58 ( 1986) 801. [2] K.H. Fischer, J.A. Hertz, Spin Glasses (Cambridge Univ. Press, Cambridge, 1991). [ 31 AI? Young, ed., Spin Glasses and Random Fields (World Scientific, Singapore, 1997). [4] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. 151 B.W. Southern, A.P. Young, J. Phys. C 10 (1977) 2179. 161 W.L. McMillan, Phys. Rev. B 29 (1984) 4026. [7] W.L. McMillan, Phys. Rev. B 30 (1984) 476; Phys. Rev. B 31 ( 1985) 340. [ 8 1 A.J. Bray, M.A. Moore, J. Phys. C 17 (1984) L463. 191 A.J. Bray, M.A. Moore, in: Heidelberg Colloquium on Glassy Dynamics, Lecture Notes in Physics 275, eds. J.L. van Hemmen, 1. Morgenstem (Springer, Berlin, 1986). [ 101 R.N. Bhatt, A.P. Young, Phys. Rev. L&t. 54 (1985) 924; A.T. Ogielski, I. Morgenstem, Phys. Rev. Lett. 54 (1985) 928. [ I1 1 R.N. Bhatt, A.P. Young, Phys. Rev. B 37 (1988) 5606. [ 12 ] N. Kawashima, A.P. Young, Phys. Rev. B 53 (1996) R484. [ 131 R. Singh, S. Chakravarty, Phys. Rev. Lett. 57 (1986) 245; L. Klein, J. Adler, A. Aharony, A.B. Harris, Y. Meir, Phys. Rev. B 43 ( 1991) 11249. [ 141 S. Liang, Phys. Rev. Lett. 69 (1992) 2145. [ 151 N. Kawashima, M. Suzuki, J. Phys. A 25 (1992) 1055. [ 161 N. Kawashima, N. Hatano, M. Suzuki, J. Phys. A 25 (1992) 4985. [ 17 1 H. Rieger, L. Santen. U. Blasum, M. Diehl, M. Jthtger, G. Rinaldi, J. Phys. A 29 (1996) 3939; Corrigendum, J. Phys. A 30 ( 1997) 8795. [ 18) M. Ney-Nifle. A.P. Young, J. Phys. A 30 (1997) 531 I. 1191 D. A. Huse, L.-F. Ko, Phys. Rev. B 56 (1997) 14597. 1201 M. Ney-Nifle, H.J. Hilhorst, Phys. Rev. Lett. 68 (1992) 2992; Physica A 193 (1993) 48; A 194 (1993) 462. [ 211 C. Tsallis, A.C.N. de Magalhiies, Phys. Rep. 268 ( 19%) 305. [22] A.C.N. de Magalhfies, C. Tsallis, G. Schwachheim, J. Phys. C 14 (1981) 1393; P.M.C. de Oliveira, Phys. Rev. B 25 ( 1982) 2034; P.M.C. de Oliveira, C. Tsallis, J. Phys. A 15 (1982) 2865. [ 231 W.H. Press. S.A. Teukolsky, W.T. Vetterling, B. Flannery, Numerical Recipes in Fortran, 2nd edition (Cambridge Univ. Press, Cambridge, 1992).