Evidence for a Genuine Ferromagnetic to Paramagnetic Reentrant Phase Transition in a Potts Spin Glass Model

5 Pages·0.16 MB·English

by  M. J. P. Gingras

Evidence for a Genuine Ferromagnetic to Paramagnetic Reentrant Phase Transition in
a Potts Spin Glass Model
Michel J.P. Gingras1 and Erik S. Sørensen2
1Department of Physics, University of Waterloo, Waterloo, Ontario, N2L-3G1 Canada
2Laboratoire de Physique Quantique, IRSAMC,Universit´e Paul Sabatier, F-31062, Toulouse, Cedex 4, France
(February 1, 2008)
Much experimentaland theoretical efforts havebeen devotedin thepast twentyyearstosearch for
8
9 agenuinethermodynamicreentrantphasetransition fromaferromagnetictoeitheraparamagnetic
9 or spin glass phase in disordered ferromagnets. So far, no real system or theoretical model of a
1 short-range spin glass system has been shown to convincingly display such a reentrant transition.
We present here results from Migdal-Kadanoff real-space renormalization-group calculations that
n
provide for the first time strong evidence for ferromagnetic to paramagnetic reentrance in Potts
a
J spin glasses on hierarchical lattices. Our results imply that there is no fundamental reason ruling
out thermodynamic reentrant phase transitions in all non mean-field randomly frustrated systems,
1
3 and may open the possibility that true reentrance might occur in some yet to be discovered real
randomly frustrated materials.

]
n
n
-
s
i
d
I. INTRODUCTION terials [7–9]. Once established at the P-F Curie tem-
.
t perature, T , ferromagnetic order remains down to zero
a c
m All real materials contain a certain amount of frozen-in temperature, though with the possibility of a transverse
- random disorder. Often, random disorder leads to ran- spin-freezing transition at T⊥ (0 < T⊥ < Tc) in XY
d domly competing, or frustrated, interactions [1]. Ran- and Heisenberg systems, which does not destroy the fer-
n dom frustration is detrimental to the type of order romagnetic order [7–9]. Above a critical disorder level,
o that an otherwise idealized pure material would dis- ferromagnetism occurs only on short length scales, and
c
[ play for zero disorder. Randomly frustrated systems are thesystemdisplays,instead,full-blownspin-glassbehav-
ubiquoutous in condensed matter physics. Examples in- ior below a glass transition temperature, Tg [2,7–9].
1 clude: magneticsystems[2],mixedmolecularcrystals[3], At the theoretical level, it is also currently believed
v superconducting Josephsonjunctions in anappliedmag- that reentrancedoes not occur in any randombond spin
3
netic field [4], liquid crystals in porous media [5], and glass model [7–21]. This is certainly the case for the in-
0
0 partially UV polymerized membranes [6]. finite range Ising, XY and Heisenberg models [2]. In
2 Oneofthemainissuesatstakeinallfrustratedsystems two and three dimensional Ising and Heisenberg models,
0 is how the low temperature phase of the pure material high-temperature series expansion [15,18], Monte Carlo
8 is affected by weak disorder, and how that state evolves simulations [2,7,9] and recent defect-wall energy calcu-
9 with increasing disorder level. In particular, one of the lations [21] find no reentrant behavior either. Recently,
t/ most intriguing questions is whether a weakly frustrated compelling renormalization-group [12,13] and quenched
a system can lose upon cooling the long-range ordered gauge symmetry arguments [16,17] have been put for-
m
phase established at higher temperature and return, or wardforabroadclassofspinglassmodels,whichinclude
d- reenter, into a thermally disordered phase, or go into a the Isingspinglass[2]andthe gaugeglassmodelfordis-
n randomly frozen glassy phase. Because of their relative orderedJosephsonjunctionarraysandvortexglassindis-
o simplicity over other systems, and because of the large ordered type-II superconductors [4,19,20,22], and which
c number of systems readily available with easily control- strongly argue against reentrance. Some of the details
v: lablelevelofdisorder,randommagnetsareidealsystems of these predictions have been quantitatively tested by
i tostudytheeffectsofweakfrustrationandtoinvestigate high-temperatureseriesexpansion[15,18],whilepossibly
X
the above question. It was originally thought that sev- the most detailed checks have been obtained from real-
r eral weakly frustrated ferromagnets, such as Eu Sr S space Migdal-Kadanoff renormalization group (MKRG)
a x 1−x
and amorphous-Fe Mn , were displaying a reentrant calculations of Ising spin glass models on so-called hi-
1−x x
transition from a ferromagnetically long-range ordered erarchical lattices [10–12]. Even in the case of the
phase to a randomly frozen spin glass phase upon cool- two-dimensionalXY modelwithrandomDzyaloshinskii-
ing and for a finite range of disorder, x [2]. However, af- Moriya couplings, which for a long-time was believed to
ter twenty years of extensive experimental research, it is be a good candidate for reentrant behavior [9,23], evi-
now generally believed that a true thermodynamic reen- dence is now rapidly accumulating that reentrance does
trant phase transition from a long-range ordered ferro- not occur in that system either [9,19,20].
magnetic (F) phase to either a spin glass (SG) or para- Summing up, it appears that the case against reen-
magnetic (P) phase does not occur in real magnetic ma- trance in randomly frustrated systems and non mean-
1
field theoretical models [24] is at this time simply over- molecular glasses, such as N -Ar and KBr-KCN, can be
2
whelming [25]. In fact, the evidence is sufficiently strong partially described by a three-dimensional 3-state Potts
that it could be interpreted as an indication that some spin glass model [3].
profound,thoughyetunknown,reason(s)formallyforbid Here we consider the situation where the bonds J in
ij
reentranceinallspinglasses,eventhosewhichdonotex- Eq.1aredistributedrandomly,andgivenbyaquenched
hibit a quenched gauge invariance [11,12,16,17]. This is biased bimodal probability distribution, P(J ):
ij
notimpossiblegiventhatourunderstandingofthenature
P(J ) = xδ(J −J)+(1−x)δ(J +J) (2)
ofthe groundstate(s) andofthe low-lyingexcitations in ij ij ij
glassy systems is still limited. In this letter we present a A bond between sites i and j has a probability x to be
counter example to this common belief, what we believe ferromagneticand of strengthJ, and a probability 1−x
is the first strong evidence for a reentrant transition in of being AF and of strength −J. We study the thermo-
a simple non mean-field spin glass model where thermal dynamicpropertiesofthissystemonhierarchicallattices
fluctuations and the question of lower-criticalspatial di- usingtheMKRGscheme[9–12,14,19,29,30]. Oneconsid-
mension play a key role [24]. Specifically, we consider ers a sequence of b J bonds in series, each we label
ij
the 3-state ferromagnetic Potts model with a concentra- J(k) (k =1,2,...b), where (b−1) spins are summed over
tion x of random antiferromagnetic bonds on hierarchi-
(we have dropped the subscript ij). The above Hamil-
callattices. We investigatethe thermodynamic behavior
tonian preserves its invariant form (apart from a spin-
of this model using the MKRG scheme, which is an ex-
independent term) under the decimation of the (b−1)
act method for hierarchical lattices [26]. We present re-
spins. This results in a new effective coupling J (l+1),
ij
sultswhichshowthatthismodelexhibitferromagneticto
at the RG decimation step
paramagnetic reentrance for a finite concentration range
ofantiferromagneticbonds. Reentranceis made possible exp{βJ(n)(l+1)} = 1+ Q
by the fact that the system prefers to lower its free en-
k=b 1+ Q −1
ergy through short range antiferromagnetic (AF) corre- k=1(cid:18) exp{βJ(k)(l)}−1 (cid:19)
lations rather than to preserve long-range ferromagnetic Q
(cid:8) (cid:9)
(3)
order. Since the lower critical dimension for antiferro-
magnetic long range order for the Q = 3 Potts model and β = 1/k T. In dimension d, b(d−1) such parallel
B
on hierarchical lattices is 4 [27], the system can be reen-
paths of b bonds in series, each with its end-to-end cou-
trant in 2 and 3 dimensions. It is interesting to note pling Jn(l+1), are then added together to give one new
that the MKRG method has in the past been used as
coupling
one of the key methods in establishing the absence of
reentrance in Ising [10–12,14], XY [9] and possibly also n=b(d−1)
in Dzyaloshinskii-MoriyaXY spin glasses [9,19,20,23]. J (l+1)= J(n)(l+1) (4)
ij
nX=1
In practice, the procedure is implemented by first cre-
II. MODEL AND METHOD
ating a large pool of N (N ≈ 106) bare couplings,
J (l = 0), distributed according to Eq. 2. Then, b cou-
ij
The Hamiltonian for the Q-state Potts model is: plings are randomly pickedout of that pool combinedto
createaserialcouplingJn(l+1)asgivenbyEq.3. Then,
H = − Jijδσi,σj (1) b(d−1) suchcouplingsJn(l+1)areaddedtogethertogive
<Xi,j> onenewcouplingJ (l+1). TheprocedureisrepeatedN
ij
times to repopulate a new pool of N couplings J (l+1)
ij
where J > 0 for ferromagnetic couplings and J < 0
ij ij at RG step (l+1).
for antiferromagneticones. δ is the Kroneckerdelta:
σi,σj The nature of the magnetic phase at a given tempera-
the spin, σ at lattice site i can take Q states, Q =
i ture,T,andconcentration,x,ofantiferromagneticbonds
0,1,2,...Q − 1. The bond energy between two spins
isdeterminedbymonitoringthel−dependenceoftheav-
is −Jij if the two spins are in the same state σi = σj, erage value, J¯(l), and the width, ∆J(l), of the distribu-
and zero otherwise. The familiar Ising model is equiva- tion of N bonds J (l). As l → ∞, J¯and ∆J evolve in
ij
lent to a Q = 2 state Potts model with a shift of total
the various phases as
energyofthesystem,andarescalingoftheexchangecou-
plingJ byafactor2. Although“lesspopular”thanthe lim J¯→ 0 , lim ∆J →0 : paramagnetic (5)
ij
Ising model, the 3-state Potts model is also important l→∞ l→∞
∆J
in modelling real condensed matter systems. For exam- lim J¯→+∞ , lim →0 : ferromagnetic (6)
ple, the two-dimensional antiferromagnetic 3-state Potts l→∞ l→∞ J¯
model on the frustrated kagom´e lattice captures some of lim J¯→−∞ , lim ∆J →0 : antiferromagnetic (7)
the essential of the low temperature thermodynamics of l→∞ l→∞ J¯
theHeisenbergantiferromagnetonthatlattice[28]. Also, J¯
lim ∆J →∞ , lim →0 : spinglass (8)
it has been suggested that the orientational freezing in
l→∞ l→∞∆J
2
To allowfor the existence of anantiferromagneticphase, boundary, one encounters overflowat iteration step l
max
we must work with odd values of b, as even values of way before J¯(l) has increased by several order of mag-
b “frustrate” the antiferromagnetic phases and maps an nitude compared to J¯(l = 0). In previous MKRG stud-
initial startup antiferromagneticallybiased P(J ) into a ies of spin glasses [9–12,14,19] the F−P or the F−SG
ij
ferromagnetic phase already at iteration step #1. Here phase boundary did not give any “peculiar” reentrant
we focus on hierarchical lattices with b=3. boundary, and the extrapolated finite temperature F−P
or F−SG boundary down to T = 0 agreed with explicit
MKRG calculations at T = 0. Consequently, there has
III. RESULTS been until now no incentive to push the limit of the nu-
merics inMKRGcalculations ofspinglassesas T →0+.
Thetemperaturevsconcentrationofantiferromagnetic However, in our case here, with this novel reentrant be-
bonds phase diagram for the three-dimensional d = 3 havior, one could be concerned that the lower reentrant
case (with b = 3) is shown in Fig. 1. The phases have portion of the phase boundary is a numerical artifact.
been determined according to the criteria given above. Specifically, it would a prioriseem possible that the flow
Firstly, there is no AF or SG phase at nonzero temper- between 0.765 and 0.855 seems to indicate a paramag-
ature in this system in the whole range 0 ≤ x ≤ 1. The netic phase according to the criterion given above, for
mostremarkablefeatureofthisphasediagramistheexis- 1 ≤ l ≤ lmax, but actually, be found to “reverse itself”
tenceofareentrantferromagnetictoparamagneticphase foravaluel>lrev withlrev >lmax,ifnumericaloverflow
transition for the range x ∈ [0.765,0.855]. The value of bounds allowed it to be seen, and such that the asymp-
0.855 obtained by extrapolating these nonzero tempera- totic large length scale behavior for 0.765 < x < 0.855
tureresultsagreeswiththeoneobtainedbyiteratingthe was ferromagnetic in the limit l → ∞. In such a sce-
MKRGequationsaboveatzerotemperatureexactly[29]. nario,the reentrantregionwouldresult froma combina-
Similar results were obtained for d=2. tion of short length scale physics added to a finite limit
tooverflowboundsimposedbythecomputerusedforthe
calculations.
To address this issue, we parametrized each of the
exp{βJ (l)}“couplingterms”viaatwo-componentvec-
ij
tor exp{βJ (l)} = {M (l),E (l)}, where M (l) and
ij ij ij ij
E (l) are the mantissa and the exponent, in base 10, of
ij
exp{βJ (l). TheMKRGcomputercodewasthenrewrit-
ij
ten in terms of direct algebraic mantissa operations and
exponentshiftingoperations. Withthisimprovedversion
of the MKRG computer code, the upper limit for over-
flowfordouble-precisioncalculationsona32-bitmachine
moves from 10308 to ≈10(10308); a tremendious improve-
ment. With this modification, the MKRG iterations be-
come forallpracticalpurposes devoidof overflowlimita-
tions. OurresultswiththisversionoftheMKRGscheme
gave an identical phase boundary to the one obtained
FIG.1. Temperature,T,concentration,x,ofantiferromag- using straighforward conventional double-precision cal-
neticbondsphasediagramfortheQ=3statePottsspinglass culations on a 32-bit machine. The results in Fig. 1 for
onad=3,b=3hierarchicallattices. Ferromagnetictopara- T/J ≤0.25 were actually obtained with the “improved”
magnetic reentrance occurs in the range 0.765 < x < 0.855. version of the MKRG scheme. We are therefore confi-
Thereisnospinglassorantiferromagneticphaseinthismodel dentthatthereentrantphasetransitiondisplayedbythe
at nonzero temperature. Q = 3 bimodal Potts spin glass model on the b = 3 hi-
erarchical lattice is a genuine one, and not an artifact
due to limitation imposed by numerical overflow at low
As observed and
temperatures.
discussed in other papers [9–12,14,19,29,30],the MKRG
The reentrance found here implies that the long range
scheme is difficult to implement for spin glass models at
ferromagnetic phase has higher entropy than the low-
low temperatures (as T →0+). The reasonfor this is as
temperature paramagnetic phase. How can we under-
follows. The occurence of a ferromagnetic phase is mon-
stand this? A first hint can be obtained by considering
itored by the criterion [J¯→ +∞,∆J/J¯→ 0]. In prac- the behavior of the flow of J¯(l) close to the upper and
tice the ferromagnetic phase is detected when numerical
lower (reentrant) F −P boundary (see Fig.2). We see
overflow occurs as l → ∞ on the ferromagnetic side of
that J(l) approaches J(l → ∞) → 0+ monotonously as
theP−Fboundary. Inaspinglassmodel,wherethe dis-
l→∞closetotheupperP−Fboundary(curveA).How-
tribution of exp{βJ (l)} is broad at low-temperature in
ij ever, J(l) swings negative for intermediate length scale
theferromagneticphaseclosetoeithertheP−ForF−SG
3
(curveB)foralltemperaturesbelowthelower(reentrant) the b = 3 hierarchical lattice is four (d = 4) [27]. Thus,
P−Fboundary before eventuallyapproachingthe trivial for a certain concentration range of random AF bonds,
paramagneticfixedpointJ(l →∞)=0. Inotherwords, reentrantbehaviorfromaferromagneticphasetoapara-
thesystemestablishesshortrangeantiferromagneticcor- magneticphasewith localantiferromagneticcorrelations
relations in the reentrant portion of the phase diagram can occur. In d = 4 (Fig. 3), the F → P reentrance
for T <0.60 and 0.755<x<0.855. dissappears and gives rise, instead, as expected from
the previous argument, to an F → AF transition upon
cooling, where here “AF” refers to the Berker-Kadanoff
phase charaterized by a nontrivial attractive fixed point
at nonzero temperature [27].
FIG. 2. Iteration number, l, dependence of the average
coupling,J¯(l)slightly intheparamagneticphaseclosetothe
the upper (curve A, x=0.80, T =0.70) and lower reentrant
(curveB, x=0.80, T =0.30) P−F phase boundary.
FIG. 3. Temperature, T, concentration, x, of antiferro-
magnetic bonds phase diagram for the Q = 3 state Potts
Interestingly,for the Potts model, a groundstate with
spin glass on a d = 4, b = 3 hierarchical lattice. The
antiferromagneticcorrelationsinpresenceofrandomfer- antiferromagnetic−“BK”phasereferstotheantiferromagnet-
romagneticbondshaslowerentropythanaferromagnetic
ically ordered phase characterized by a fixed point at finite
state with random antiferromagnetic bonds. Consider coupling J¯(l=∞)=−10.94661 [30].
three spins, σ , σ and σ with ferromagnetic bonds J
1 2 3 12
andJ . Ifoneofthebondis,instead,antiferromagnetic,
23
σ becomes an idle and entropy-carrying spin with zero
2
effective average exchange field at T =0 from ferromag- IV. CONCLUSION
neticallyalignedσ andσ . However,forσ andσ anti-
1 3 1 3
ferromagneticallyalignedviatheotherb(d−2)bonds,σ2is In conclusion, we have shown that the Q = 3 Potts
inaunique (non-idle)state foraferromagneticJ12 bond spinglassmodelontwoandthreedimensionalhierarchi-
and an antiferromagnetic J23 bond. Consequently, anti- cal lattices undergoes a ferromagnetic to paramagnetic
ferromagnetically correlated triplets of spins (σ1,σ2,σ3) reentrance upon cooling. This reentrance is due: (1) to
carrylowerentropyinpresence ofrandomferromagnetic thecombinationofantiferromagneticallycorrelatedspins
bonds than a ferromagnetic state with random antifer- atlowtemperaturesinthe phase“rich”inferromagnetic
romagnetic bonds. The antiferromagnetic state also has bondscarryinglessentropythanferromagneticallycorre-
lower energy, (E = −J), as compared to the ferromag- lated spins and, (2) to the fact that the lower criticaldi-
netic configuration(E=0). Naively, in order to minimize mension for antiferromagnetic order for the Q=3 Potts
the free energy, F = E −TS, this observation suggests antiferromagnet on hierarchical lattices is four. Conse-
that, upon cooling, local antiferromagnetic correlations quently, reentrance occurs at T > 0 in two and three
should become more and more favorablesince entropy is dimensional such lattices. The results presented here
less important at low temperatures. This makes plau- demonstrate that there is no fundamental reason forbid-
sible that the system, at low temperatures, prefers to ding a thermodynamic reentrant phase transition in all
form ferromagnetic domains that are antiferromagneti- randomly frustrated systems. Our results open the pos-
callyaligned(e.g. onintermediatelengthscales,asfound sibilitythatreentrancemightoccurinsomeyettobedis-
in Fig. 2) rather than to keep the long-range ferromag- coveredrealrandomlyfrustratedsystemswithEuclidean
neticorderestablishedathighertemperatures. However, lattices. We hope that our results will stimulate further
and this is an important point, true long-range antifer- studies in that direction.
romagneticordercannotoccursinceitis knownthatthe We thank P. Holdsworth, J. Machta, H. Nishimori,
lower critical dimension for antiferromagnetic order on Y. Ozeki, and B. Southern for useful discussions and
4
correspondence. We also thank R. Mann for generous ed. H.T. Diep, (World Scientific, Singapore, 1994), p.
CPUtimeallocationonhisDEC-Alphaworkstation. We 161.
acknowledge the NSERC of Canada and NSF DMR- [26] M. Kaufman and R. B. Griffiths, Phys. Rev. B 24, 496
9416906for financial support. (1981). R. B. Griffiths and M. Kaufman, Phys. Rev. B
26, 5022 (1982).
[27] A.N. Berker and L.P. Kadanoff, J. Phys. A 13, L259
(1980).
[28] D.A. Huse and A.D. Rutenberg, Phys. Rev. B 45, 7536
(1992).
[29] B.W. Southern et al., J. Phys.C 12, 683 (1979).
[1] G. Toulouse, Commun. Phys. 2, 115 (1977); D. [30] J.R. Banavar and A.J. Bray, Phys. Rev. B 38, 2564
Chowdhury,Spin Glasses and Other Frustrated Systems, (1988).
(Princeton Univ.Press, Princeton, NJ, 1986).
[2] K. Binder and A.P. Young, Rev. Mod. Phys., 58, 801
(1986);K.H.FischerandJ.A.Hertz,SpinGlasses,(Cam-
bridge UniversityPress, Cambridge, 1991).
[3] K.Binder and J. Reger, Adv.Phys. 41, 547 (1992).
[4] M.J.P. Gingras, Phys. Rev.B 45, 7547 (1992).
[5] W.I. Goldburg et al., Physica A 213, 61 (1995).
[6] M. Mutz et al., Phys. Rev.Lett. 67, 923 (1991).
[7] D.H. Ryan, in Recent Progess in Random Magnets, ed.
D.H.Ryan(WorldScientific,Singapore,1992),p.1.J.R.
Thompson, Phys. Rev.B 45, 3129 (1992).
[8] I.Mirebeauetal.,inRecentProgessinRandomMagnets,
ed.D.H.Ryan(World Scientific, Singapore, 1992) p.41;
I. Mirebeau et al., Phys. Rev. B, 41 (1990) 11405. B.
Hennion et al., Physica B 136, 49 (1986).
[9] M.J.P. Gingras, in Magnetic systems with competing in-
teractions, ed. H.T. Diep, (World Scientific, Singapore,
1994), p. 238; M.J.P. Gingras and E.S. Sørensen, Phys.
Rev.B 46, 3441 (1992).
[10] B.W. Southern and A.P. Young, J. Phys. C 10, 2179
(1977).
[11] A.Georges et al., J. Phys. (Paris), 46, 1827 (1985).
[12] A.Georges and P. LeDoussal (unpublished).
[13] P. LeDoussal and A.B. Harris, Phys. Rev. Lett. 61, 625
(1988); Phys.Rev. B 40, 9249 (1989).
[14] J.D.RegerandA.P.Young,J.Phys.Cond.Matt.1,915
(1989)
[15] R.R.P.Singh, Phys.Rev. Lett. 67, 899 (1991).
[16] H.Kitani, J. Phys.Soc. Jpn. 61, 4049 (1992).
[17] Y.Ozeki and H. Nishimori, J. Phys.A 26, 339 (1993).
[18] R.R.P.Singh and J. Adler, cond-mat/9603195
[19] T. Natterman et al., J. Phys. I (France) 5, 555 (1995);
S.Scheidl, Phys. Rev.B 55, 457 (1997).
[20] J. Maucourt and D.R. Grempel, Phys. Rev. B 56, 2572
(1997); J. Maucourt, Universit´e Joseph Fourier (unpub-
lished, 1997); J.M. Kosterlitz and M.V. Simkin, Phys.
Rev.Lett. 79, 1098 (1997).
[21] M. V.Simkin, Phys.Rev. B 55, 11405 (1997).
[22] D.A.Huseet al., Nature, 358, 553 (1992).
[23] M. Rubinstein et al., Phys. Rev.B 27, 1800 ( 1983).
[24] Ferromagnetictospin-glass“reentrance”hasbeenfound
in a non replica-symmetric mean-field solution of an
infinite-dimensionalcoupledferromagnet-Isingspinglass
model by H. Takayama, J. Phys. Soc. Jpn. 61, 2512
(1992).
[25] Reentrance has been found in various nonrandom uni-
formlyfrustratedspinmodels.SeeH.T.DiepandH.Gia-
comini in Magnetic systems with competing interactions,
5