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Scaling at the critical temperature of a spin glass
Solid State Communications, Vol. 24, PP. 429—431, 1977.
Pergamon Press.
Printed in Great Britain
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS* J. Chalupat Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received 1 July 1977 by A.G. Chynoweth) It is argued that the critical exponent ~ for an Edwards—Anderson spin ~ass at its critical temperature can be extracted from the magnetic susceptibility in a weak uniform magnetic field. MEAN-FIELD treatments of the Edwards—Anderson spin-glass model predict [1, 2] that a spin glass undergoes a phase transition with cusps in the specific heat and magnetic susceptibility at the transition temperature T~. Experiments [3, 4] and numerical simulations [5] have yielded a cusp in the susceptibility ~ and a heat capacity C which is smooth and rounded at the transition. To resolve this qualitative discrepancy between mean-field theory and experiment is obviously of interest. Since exponent the smoothness of Cat is compatible with a critical a less than theTc mean-field value of— 1, a natural explanation is that ofcritical fluctuations round the cusp in C; such behavior has been predicted by some renormalization-group calculations [6,7]. Because the singularity in C is difficult to measure, it is important to understand how other thermodynamic quantitiesmay reveal the nature of the phase transition [8—13]. For concreteness we discuss the spin-i Ising model of reference [11] with N spins on addimensional lattice at temperature T= 1/13; the effective Hamiltonian is ,
—i3~C =
13
—~
-
t
=
f
[I dJ,~P,~(J,1) LI dH, F(H~)ln tr eC.
(2)
For Pu(Jij) = P,1(— J~~) it has been demonstrated [9] that the magnetic susceptibility ~ is simply 2 by ~ related = i3(1 to — the q); spin-glass, order parameter q = (s,) a recent measurement [4] of ~ has resulted in a value 13 = 0.8—0.9 for the order-parameter critical exponent. It is known [7] that the quantity H2 is a field thermodynamically conjugate to q, and it has been shown [10— 13] that the critical exponent y of the order-parameter susceptibility can be extracted from the behavior of a2XH/8H2 near T~. Since the exponents a and ‘y are difficult to measure directly, one might hope to obtain these quantities — a in particular — by measuring other exponents and using the scaling laws; to do so, two exponents must be known, but only 13 has been measured. The purpose of this note
~ J~s,s
1+ j3 ~ ~ (1) The exchange bonds J~ 1and the microscopic magnetic fields H, have probability distributions P,1(J,1) and P(H3. Throughout this paper we require that P,~(J,3)= P,~(—J,~), but universality indicates that the conclusions we shall draw about spin-glass critical exponents are independent condition. We= shall behavior bothofinthis uniform [P(H,) ~(H,study —H)] the andcritical in random [P(H 2) 1/2 exp (— ~H~/H2)] mag1)=a (2irH netic fields. For quantity A describing the spin glass, we denote by (A) the statistical—mechanical expectation *
value of A for a fixed configuration of bonds and fields, and by A the average of A over the bond and field distributions; we are especially interested in the average free energy
Research supported in part by NSF Grant DMR-760 1058.
is to point out thatcan thebecritical exponent 2)~, obtained from the defined magneticby q(T0) (H susceptibility ~ in a weak uniform magnetic field H. In terms of the reduced units t = I T — T~I/Ta, h, = 1311,, h = J3H and h = 1311, we have at h =0 the scaling relations for the singular 2\ part of Y(t, ~ 2) ~3a ~,
“-
~ ~‘ ~
‘
~
/Xt)
~ ~
Y(t2XtO~h),
(3b)
.
—
~(h
429
‘~(b~-tt ‘ bxhh
where b is the scale factor, and A, and X~are the cntical mdices; the scaling functions X and Y are regular for small argument formsargument such that but the for twolarge expressions in have (3b) limiting are equlvalent. At
Address after September 1, 1977: Department of Physics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
b
‘
t~tX(h2 t~h
)
t
q
=
Owe have ~
~
=
X~/(d— X~’).
(4)
430
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS
Vol. 24, No.6
But how is (S determined from the magnetic response contain quadruplets, sextuplets, etc. of identical coof the spin glass? At h = 0, ~f(t, P) formally can be ordinates besides distinct pairs, and the combinatorics expressed as the power series must allow for this possibility. But at a given order in h2 (h2/2y’ / a2\’~ nearmore Tc, the termsthan which have displayed are singular thewe corrections, as wasexplicitly illustrated ~(t, ~2) = ~ n! ~ ~h~) ~ {h 4. 1 = ~ (5) in reference [11] by explicit calculation to order h For a spin glass in a uniform magnetic field H with i~ = o, Thus we assume that when the two series in (7b) are one also can formally write summed to all orders in h and t is set to zero, the first series gives the more singular contribution to The / ~ opposite assumption leads to the situation that ~(t, h) = ~ ~(t, {h, = O}). (6) h2 couples to a scaling field which is more strongly n0 n! relevant than the scaling field thermodynamically conThe nth derivative d~in this series is a sum of n-spin correlation functions: jugate to the order parameter q; we do note that attempts [10, 13, 14] have been made to describe the / n spin glass with order parameters other than q, and that = LI ~ the role of h and ~ in such formulations is not yet fl C understood [15]. where the subscript c denotes a statistical—mechanical Comparing (3), (4), (6), and (7), we conclude that cumulant average. Performing the operations [9] 5, ~ for spin glasses described by the Edwards—Anderson — s 1 andJ,~—~— J~1for a particular spin s~,we find that formalism, the critical exponent (S can be determined by the coefficients of all odd powers of h in (6) vanish; a measuring the nonlinear behavior of the magnetic term in d2m vanishes as well unless each one of the 2m susceptibility in a weak uniform magnetic field at Tc: coordinates ~k in the setS = {~k},k = 1, 2m, is 2Th. (8) identical with one up other of the set.ofLet ~H(t = 0, h) h us continue as at ifSleast breaks intomember m distinct pairs This result is obtainable from previous scaling theories identical coordinates; we shall explain below why this of the transition [4, 10—13]. Since /3 also is found from procedure gives the part Of d2m most singular near Tc. ~H near T~,the scaling law a = 2 — /3(1 + 8) can be used A combinatorial factor expresses the number of ways to predict the behavior of the specific heat C near the that the m pairs inS may be formed, and the result is transition from the behavior of Only recently have h2~ — 1)!!( ~ a2\~ experiments [4] and numerical simulations [13] been ~(t, h) = n~:O (2n)! L performed with the intention of determining spin-glass critical exponents. The data of reference [4] permits at the crude estimate (S 5, which implies x ~(t, {h, = 0}) + less singular Part] (7a) abest —only 3. Accordingly this data seems consistent with the smooth appearance of a spin glass’s specific heat near = ~ (h2/2)’~I a2 fl ~ n! ~ ~fl2) ~(t, {h, = 0)) the phase transition. -
~.
\
.
‘I
. . . ,
~.
I(2n
h’~
+
(7b) Acknowledgements I am grateful to B. Dodson, T. Low and J. Mochel for many discussions of their experimental work, and I also thank C. Jayaprakash, S. Kirkpatrick The “less singular part” exists because the set S can and M. Wortis for helpful conversations. n0
(2n)!
(l.s.p.).
—
REFERENCES 1.
EDWARDS S.F. & ANDERSON P.W.,J. Phys. F5, 965 (1975).
2.
SHERRINGTON D. & KIRKPATRICK S.,Phys. Rev. Lett. 35, 1792 (1975); THOULESS D.J., ANDERSON P.W. &PALMERR.G.,Fhil. Mag. 35, 593 (1977).
3. 4.
WENGER L.E. & KEESOM P.H., Phys. Rev. B13, 4053 (1976). MIZOGUCHI T., MCGUIRE T.R., KIRKPATRICK S. & GAMBINO R.J.,Phys. Rev. Lett. 38,89(1977).
5.
BINDER K. & SCHRODER K.,Phys. Rev. B14, 2142 (1976).
6. 7. 8.
HARRIS A.B., LUBENSKY T.C. & CHEN J.-H.,Phys. Rev. Lett. 36, 1045 (1976). JAYAPRAKASH C., CHALUPA J. & WORTIS M.,Phys. Rev. Bl5, 1495 (1977). DOMB C.,J. Phys. A9, L17 (1976).
Vol. 24, No.6 9.
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS
FISCHERK.H.,SolidState Commun. 18, 1515 (1976).
10.
AHARONY A. & IMRY Y., Solid State Commun. 20, 899 (1976).
11. 12.
CHALU’PAJ.,SolidState Commun. 22, 315 (1977). SUZUKI M.,F~og.Theor. Phys. (to be published).
13.
BINDER K., in FestkOrperprobleme(Advances in Solid State Physics), edited by J. Treusch, Vol. XVII, pp. 5 5—84, Vieweg, Braunschweig (1977).
14. 15.
TOULOUSE G., Comm. Phys. 2, 115 (1977). FISCH R. & HARRIS A.B.,Phys. Rev. Lett. 38, 785 (1977).
431
Pergamon Press.
Printed in Great Britain
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS* J. Chalupat Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received 1 July 1977 by A.G. Chynoweth) It is argued that the critical exponent ~ for an Edwards—Anderson spin ~ass at its critical temperature can be extracted from the magnetic susceptibility in a weak uniform magnetic field. MEAN-FIELD treatments of the Edwards—Anderson spin-glass model predict [1, 2] that a spin glass undergoes a phase transition with cusps in the specific heat and magnetic susceptibility at the transition temperature T~. Experiments [3, 4] and numerical simulations [5] have yielded a cusp in the susceptibility ~ and a heat capacity C which is smooth and rounded at the transition. To resolve this qualitative discrepancy between mean-field theory and experiment is obviously of interest. Since exponent the smoothness of Cat is compatible with a critical a less than theTc mean-field value of— 1, a natural explanation is that ofcritical fluctuations round the cusp in C; such behavior has been predicted by some renormalization-group calculations [6,7]. Because the singularity in C is difficult to measure, it is important to understand how other thermodynamic quantitiesmay reveal the nature of the phase transition [8—13]. For concreteness we discuss the spin-i Ising model of reference [11] with N spins on addimensional lattice at temperature T= 1/13; the effective Hamiltonian is ,
—i3~C =
13
—~
-
t
=
f
[I dJ,~P,~(J,1) LI dH, F(H~)ln tr eC.
(2)
For Pu(Jij) = P,1(— J~~) it has been demonstrated [9] that the magnetic susceptibility ~ is simply 2 by ~ related = i3(1 to — the q); spin-glass, order parameter q = (s,) a recent measurement [4] of ~ has resulted in a value 13 = 0.8—0.9 for the order-parameter critical exponent. It is known [7] that the quantity H2 is a field thermodynamically conjugate to q, and it has been shown [10— 13] that the critical exponent y of the order-parameter susceptibility can be extracted from the behavior of a2XH/8H2 near T~. Since the exponents a and ‘y are difficult to measure directly, one might hope to obtain these quantities — a in particular — by measuring other exponents and using the scaling laws; to do so, two exponents must be known, but only 13 has been measured. The purpose of this note
~ J~s,s
1+ j3 ~ ~ (1) The exchange bonds J~ 1and the microscopic magnetic fields H, have probability distributions P,1(J,1) and P(H3. Throughout this paper we require that P,~(J,3)= P,~(—J,~), but universality indicates that the conclusions we shall draw about spin-glass critical exponents are independent condition. We= shall behavior bothofinthis uniform [P(H,) ~(H,study —H)] the andcritical in random [P(H 2) 1/2 exp (— ~H~/H2)] mag1)=a (2irH netic fields. For quantity A describing the spin glass, we denote by (A) the statistical—mechanical expectation *
value of A for a fixed configuration of bonds and fields, and by A the average of A over the bond and field distributions; we are especially interested in the average free energy
Research supported in part by NSF Grant DMR-760 1058.
is to point out thatcan thebecritical exponent 2)~, obtained from the defined magneticby q(T0) (H susceptibility ~ in a weak uniform magnetic field H. In terms of the reduced units t = I T — T~I/Ta, h, = 1311,, h = J3H and h = 1311, we have at h =0 the scaling relations for the singular 2\ part of Y(t, ~ 2) ~3a ~,
“-
~ ~‘ ~
‘
~
/Xt)
~ ~
Y(t2XtO~h),
(3b)
.
—
~(h
429
‘~(b~-tt ‘ bxhh
where b is the scale factor, and A, and X~are the cntical mdices; the scaling functions X and Y are regular for small argument formsargument such that but the for twolarge expressions in have (3b) limiting are equlvalent. At
Address after September 1, 1977: Department of Physics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
b
‘
t~tX(h2 t~h
)
t
q
=
Owe have ~
~
=
X~/(d— X~’).
(4)
430
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS
Vol. 24, No.6
But how is (S determined from the magnetic response contain quadruplets, sextuplets, etc. of identical coof the spin glass? At h = 0, ~f(t, P) formally can be ordinates besides distinct pairs, and the combinatorics expressed as the power series must allow for this possibility. But at a given order in h2 (h2/2y’ / a2\’~ nearmore Tc, the termsthan which have displayed are singular thewe corrections, as wasexplicitly illustrated ~(t, ~2) = ~ n! ~ ~h~) ~ {h 4. 1 = ~ (5) in reference [11] by explicit calculation to order h For a spin glass in a uniform magnetic field H with i~ = o, Thus we assume that when the two series in (7b) are one also can formally write summed to all orders in h and t is set to zero, the first series gives the more singular contribution to The / ~ opposite assumption leads to the situation that ~(t, h) = ~ ~(t, {h, = O}). (6) h2 couples to a scaling field which is more strongly n0 n! relevant than the scaling field thermodynamically conThe nth derivative d~in this series is a sum of n-spin correlation functions: jugate to the order parameter q; we do note that attempts [10, 13, 14] have been made to describe the / n spin glass with order parameters other than q, and that = LI ~ the role of h and ~ in such formulations is not yet fl C understood [15]. where the subscript c denotes a statistical—mechanical Comparing (3), (4), (6), and (7), we conclude that cumulant average. Performing the operations [9] 5, ~ for spin glasses described by the Edwards—Anderson — s 1 andJ,~—~— J~1for a particular spin s~,we find that formalism, the critical exponent (S can be determined by the coefficients of all odd powers of h in (6) vanish; a measuring the nonlinear behavior of the magnetic term in d2m vanishes as well unless each one of the 2m susceptibility in a weak uniform magnetic field at Tc: coordinates ~k in the setS = {~k},k = 1, 2m, is 2Th. (8) identical with one up other of the set.ofLet ~H(t = 0, h) h us continue as at ifSleast breaks intomember m distinct pairs This result is obtainable from previous scaling theories identical coordinates; we shall explain below why this of the transition [4, 10—13]. Since /3 also is found from procedure gives the part Of d2m most singular near Tc. ~H near T~,the scaling law a = 2 — /3(1 + 8) can be used A combinatorial factor expresses the number of ways to predict the behavior of the specific heat C near the that the m pairs inS may be formed, and the result is transition from the behavior of Only recently have h2~ — 1)!!( ~ a2\~ experiments [4] and numerical simulations [13] been ~(t, h) = n~:O (2n)! L performed with the intention of determining spin-glass critical exponents. The data of reference [4] permits at the crude estimate (S 5, which implies x ~(t, {h, = 0}) + less singular Part] (7a) abest —only 3. Accordingly this data seems consistent with the smooth appearance of a spin glass’s specific heat near = ~ (h2/2)’~I a2 fl ~ n! ~ ~fl2) ~(t, {h, = 0)) the phase transition. -
~.
\
.
‘I
. . . ,
~.
I(2n
h’~
+
(7b) Acknowledgements I am grateful to B. Dodson, T. Low and J. Mochel for many discussions of their experimental work, and I also thank C. Jayaprakash, S. Kirkpatrick The “less singular part” exists because the set S can and M. Wortis for helpful conversations. n0
(2n)!
(l.s.p.).
—
REFERENCES 1.
EDWARDS S.F. & ANDERSON P.W.,J. Phys. F5, 965 (1975).
2.
SHERRINGTON D. & KIRKPATRICK S.,Phys. Rev. Lett. 35, 1792 (1975); THOULESS D.J., ANDERSON P.W. &PALMERR.G.,Fhil. Mag. 35, 593 (1977).
3. 4.
WENGER L.E. & KEESOM P.H., Phys. Rev. B13, 4053 (1976). MIZOGUCHI T., MCGUIRE T.R., KIRKPATRICK S. & GAMBINO R.J.,Phys. Rev. Lett. 38,89(1977).
5.
BINDER K. & SCHRODER K.,Phys. Rev. B14, 2142 (1976).
6. 7. 8.
HARRIS A.B., LUBENSKY T.C. & CHEN J.-H.,Phys. Rev. Lett. 36, 1045 (1976). JAYAPRAKASH C., CHALUPA J. & WORTIS M.,Phys. Rev. Bl5, 1495 (1977). DOMB C.,J. Phys. A9, L17 (1976).
Vol. 24, No.6 9.
SCALING AT THE CRITICAL TEMPERATURE OF A SPIN GLASS
FISCHERK.H.,SolidState Commun. 18, 1515 (1976).
10.
AHARONY A. & IMRY Y., Solid State Commun. 20, 899 (1976).
11. 12.
CHALU’PAJ.,SolidState Commun. 22, 315 (1977). SUZUKI M.,F~og.Theor. Phys. (to be published).
13.
BINDER K., in FestkOrperprobleme(Advances in Solid State Physics), edited by J. Treusch, Vol. XVII, pp. 5 5—84, Vieweg, Braunschweig (1977).
14. 15.
TOULOUSE G., Comm. Phys. 2, 115 (1977). FISCH R. & HARRIS A.B.,Phys. Rev. Lett. 38, 785 (1977).
431