Renormalization Group on hierarchical lattices in finite dimensional disordered Ising and Blume-Emery-Griffiths Models
NNoname manuscript No. (will be inserted by the editor)
F. Antenucci, , A. Crisanti , and L. Leuzzi , Critical study of hierarchicallattice renormalization group inmagnetic ordered and quencheddisordered systems: Ising andBlume-Emery-Griffiths models
June 4, 2018
Abstract
Renormalization group on hierarchical lattices is often considereda valuable tool to understand the critical behavior of more complicated statis-tical mechanical models. In presence of quenched disorder, however, in manymodel cases predictions obtained with the Migdal-Kadanoff bond removalapproach fail to quantitatively and qualitatively reproduce critical proper-ties obtained in the mean-field approximation or by numerical simulations infinite dimensions. In order to critically review this limitation we analyze thebehavior of Ising and Blume-Emery-Griffiths models on more complicatedhierarchical lattices. We find that, apart from some exceptions, the differentbehavior appears not only limited to Midgal-Kadanoff-like cells but is asso-ciated right to the hierarchization of Bravais lattices in small cells also whenin-cell loops are considered.
In this work we shall investigate the renormalization group analysis on spinsystems with quenched disorder on hierarchical lattices. We will considerboth Migdal-Kadanoff (MK) as well as more complex hierarchical latticesand we will study the critical behavior of systems with magnetic interactionsin presence of random fields and random exchange interactions.Our main aim is to investigate whether hierarchical cells more compli-cated than MK ones and more similar to the local structure of short-rangeBravais lattices can reproduce features of ferromagnets and spin-glasses so Dipartimento di Fisica, Universit`a
Sapienza , P.le Aldo Moro 2, I-00185 Roma,Italy. IPCF-CNR, UOS Roma
Kerberos , P.le Aldo Moro 2, I-00185 Roma, Italy. ISC-CNR, UOS Sapienza, P.le Aldo Moro 2, I-00185 Roma, Italy. a r X i v : . [ c ond - m a t . d i s - nn ] J a n far unobserved in position space renormalization group studies with MK lat-tices. In order to obtain a more general comprehension of the effect of bondmoving we will provide estimates for critical quantities on several hierarchi-cal lattices with different topology and compare them to known analytic andnumerical results, when available.Particular attention will be devoted to spin-glasses. Understanding thenature of the low temperature phase of spin-glasses in finite dimensional sys-tems has turned out to be an extremely difficult task. Since the resolutionof its mean-field approximation, valid above the upper critical dimension( D = 6), more than thirty years have passed without a final word about thepossible generalization of mean-field properties of spin-glasses to finite dimen-sional cases. The mean-field, else called Replica Symmetry Breaking (RSB)theory [1,2] involves a very interesting solution for the spin-glass phase andits critical properties, rich of physical (and mathematical) implications, andhas been fundamental in solving very diverse problems both in physics andin other disciplines [3,4,5]. Because of its complicated structure, to overcometechnical (maybe also conceptual) obstacles hindering the “portability” ofRSB theory predictions to short-range systems on Bravais lattice in D < ? ],a proper extension of renormalization group techniques to disordered andlocally frustrated systems is still on its way. The generalization of classicposition space renormalization methods on Bravais lattices to disordered in-teraction, such as the ones proposed for Ising spin models in the seventies [16,17], has led to controversial results. On the one hand, by means of a cumulantexpansion approach, evidence for a spin-glass phase is yielded in dimensiontwo [18,19], lower than the lower critical dimension on the Bravais lattice: D = 2 . Therefore, most of the position space renormalization group (PSRG)studies have been concentrating on hierarchical lattices for which, in theordered cases, the renormalization group flow is indeed exact (no truncationrequired). The study of these systems has brought to important results, cf.,e.g., Refs. [23,25,26,27] and references therein.However, MK lattices fail to represent short-range spin-glasses on Bravaislattices also in the mean-field approximation and are, thus, strongly limited inprobing the actual nature of the spin-glass phase [28,29]. For what concernsperturbative disorder in ferromagnetic systems, it has been found that theNishimori conjecture does not provide the exact value of the multicriticalpoint coordinate, [30,26] although recently Ohzeki, Nishimori and Berker [26, ? ] have put forward an improved conjecture for models defined on hierarchicallattices and they found that for various families of these models a noteworthyrecovery is obtained.Furthermore the lack of translational invariance on hierarchical latticesis supposed to make peculiarly difficult the study of first order transitions.In particular, for pure fixed points while the expected first order transitionis obtained in some model (see for example [32,33]), there are relevant casesin which the corresponding transition is missing on MK lattices [34,35]. Forwhat concerns disordered systems, MK lattices do not yield the first orderfixed distribution expected for the random Blume-Emery-Griffiths model [33],as obtained both in mean-field theory [36] and by numerical simulations infinite dimension [37]. We stress as in the latter case the MK lattices neithershow the expected re-entrance in the phase diagram [33,36,37].Our aim is to test whether and which of the above mentioned differ-ences are specifically due to the bond moving procedure at the basis of therenormalization group analysis. We will, thus, implement and compare theanalysis of the critical behavior of well known statistical mechanical modelswith quenched field and bond randomness on both MK hierarchical latticesand the more complex “folded hierarchical lattices”, as we will call them.The latter family consists of hierachical lattices obtained applying the tworoot reduction directly to the Bravais lattice [35] (see Figs. 1, 3, 4, 5, 11),without the bond moving specific of the Migdal-Kadanoff transformation. Sothe final lattice has no longer just a 1D topology, but retains, in a smallscale, the basic topology of the original lattice. So, unlike the MK family,in this case the original lattice is continuously reconstructed in the limit inwhich the length of the basic cell, called b in the following (see Sec. 3), goesto infinity: the original lattice is a folded hierarchical lattice with an infinitebasic cell.In the following, we will first critically revisit, in each model case, theanalyses on hierarchical lattices carried out in the literature. We will, then,compare those results to the outcome of our studies on more complex latticesin the family of folded cells.The paper is organized as follows: in Sec. 2 we recall the implementationof the PSRG in the ferromagnetic Ising model. In Sec. 3 we expose the detailsof the generalization of the PSRG in presence of generic quenched disorder.In Sec. 3.1, we investigate the random field Ising model (RFIM) and in Secs.3.2 and 3.3 we report on the Ising spin-glass, respectively below and above Fig. 1
Costruction of diamond, else called 2-dimensional Wheatstone-bridge, hi-erarchical lattice with fractal dimension d = log 5 / log 2 = 2 . . . . . the lower critical dimension. We present and compare the estimates of criti-cal parameters and discuss how they comply to known statistical mechanicalcriteria in presence of disorder (Nishimori conjecture, Harris criterion, ferro-magnetic line inversion, . . . ). In Sec. 4 we consider the Blume-Emery-Griffithsmodel on several hierarchical lattices in dimension d ≥
3. In the latter caseour analysis shows a phase diagram displaying a reentrance for strong disor-der, absent on MK lattices [33], but present in the mean-field approximation[38] and in numerical simulations on 3D cubic lattices [37,39,40].
The Position Space Renormalization Group (PSRG) approach, approximatedon realistic Bravais lattices, becomes exact when iterated on HierarchicalLattices (HL) [41,14,23,24,35]. These lattices are constructed by carryingsuccessive similar operations at each hierarchical level. E.g., at each levelone replaces bonds by well-defined unit cells. See, for example, Fig. 1 forthe diamond lattice or Fig. 2 for a MK lattice. The PSRG procedure worksthe inverse way of the lattice generation, i.e., one can implement it througha decimation of the internal sites of a given cell, leading to renormalizedquantities associated with the external sites.In the pure case, the PSRG analysis proceeds as known by finding theinteractions leaving the partition function invariant under decimation, andobtaining the critical exponents by the eigenvalues of the first derivativematrix computed on the relative fixed point [42].The well known Ising model is defined by the Hamiltonian − H ( s ) = J (cid:88) (cid:104) ij (cid:105) s i s j + h (cid:88) i s i , (1)where s i = ± (cid:104) ij (cid:105) indicates a sum over nearest-neighbor pairs. Westress that here and in the rest of the paper we include the temperature inthe definition of the couplings (reduced parameters). Fig. 2
Necklace
MK lattice. It has b = 2 and fractal dimension d = 3. Since we, eventually, want to study the critical properties of disorderedsystems, and since through the renormalization group transformation originalrandom bonds induce random fields (and vice-versa) already at the first stepof renormalization, it becomes more convenient to start using the followingHamiltonian − H ( s ) = (cid:88) (cid:104) ij (cid:105) (cid:104) J ij s i s j + h ij s i + s j h † ij s i − s j (cid:105) (2)In this way each link between two sites i and j is associated with three(possibly disordered) interactions J ij , h ij and h † ij .Decimating the inner sites { s } of the basic cell C ab of the hierarchicallattice with external sites s a and s b , while imposing the conservation of thepartition function of the cell Z C ab ≡ x s a s b = (cid:88) { s }∈C ab exp {−H [ s a , s b ; { s } ] } , (3)yields the renormalization group equations: J R = 14 log (cid:18) x ++ x −− x + − x − + (cid:19) ,h R = 12 log (cid:18) x ++ x −− (cid:19) ,h † R = 12 log (cid:18) x + − x − + (cid:19) , (4)The partition sums x s a s b , also called edge Boltzmann factors of the cell, arethe weights of the cell for fixed external spins s a and s b . The sum in Eq. (3)runs over all inner or free spins of the cell C ab . In the zero-temperature limit the relations become4 J R = max (cid:2) −H (1 , , s ) (cid:3) + max (cid:2) −H ( − , − , s ) (cid:3) − max (cid:2) −H (1 , − , s ) (cid:3) − max (cid:2) −H ( − , , s ) (cid:3) h R = max (cid:2) −H (1 , , s ) (cid:3) − max (cid:2) −H ( − , − , s ) (cid:3) h † R = max (cid:2) −H (1 , − , s ) (cid:3) − max (cid:2) −H ( − , , s ) (cid:3) (5)When the external field is missing h = h † = 0 and H ( s ) = H ( − s ), whichimplies h R = h † R = 0.2.1 The ordered ferromagnetic Ising modelFor an ordered ferromagnetic system ( J ij = J , h ij = h and h † ij = h † ) thecritical exponents can be obtained from the eigenvalues of the first derivativesmatrix ∂J R ∂J ∂J R ∂h ∂J R ∂h † ∂h R ∂J ∂h R ∂h ∂h R ∂h † ∂h † R ∂J ∂h † R ∂h ∂h † R ∂h † (6)computed on the pure fixed point corresponding to the universality class ofthe ferromagnetic transition. The derivatives are easily obtained using ∂x s a s b ∂J = (cid:88) { s }∈C ab (cid:88) (cid:104) ij (cid:105) s i s j exp[ −H ( s a , s b , { s } )] ,∂x s a s b ∂h = (cid:88) { s }∈C ab (cid:88) (cid:104) ij (cid:105) s i + s j exp[ −H ( s a , s b , { s } )] ,∂x s a s b ∂h † = (cid:88) { s }∈C ab (cid:88) (cid:104) ij (cid:105) s i − s j exp[ −H ( s a , s b , { s } )] . (7)In particular, if the fixed point is for h = h † = 0, it is easy to see thatthe matrix in Eq. (6) is diagonal and ∂h † R /∂h † ≡ c , where c is the numberof incoming (outgoing) links in the external outgoing (incoming) site. Forexample c = 4 in Fig. 3, c = 3 in Fig. 4 and c = 5 in Fig. 5. In thiscase the only relevant eigenvalues are λ T = ∂ J J R and λ h = ∂ h h R , with thecorresponding scaling exponents y T,h = log b λ T,h . Fig. 3
Wheatstone bridge (WB) hierarchical lattice obtained from a cubic lattice.It has b = 2 and Hausdorff fractal dimension d = log 12 / log 2 ≈ . The critical scaling exponents of the physical observables are related tothe scaling exponents y T,h by the scaling relations: ν = 1 y T , (8) η = d + 2 − y h , (9) α = 2 − dy T , (10) β = d − y h y T , (11) γ = 2 y h − dy T , (12) δ = y h d − y h . (13)The above PSRG scheme holds when the coupling constants J ij are allequal and the external field is ordered and homogeneous. To perform theequivalent analysis of the critical behavior in disordered systems one hasto generalize the PSRG method to probability distributions of interactionparameters. In disordered systems the PSRG transformation is described by the evolutionof a probability distribution rather than single values of coupling constants:[43,44] P (cid:48) ( J R ) = (cid:90) b d (cid:89) α =1 d J α P ( J α ) δ [ J R − R ( {J } b d )] (14) Fig. 4
Left hand side: square cells of cell spacing length b = 3. On the right side:corresponding folded square hierarchical lattices with two roots (open circles). All outcoming and incoming sites, pointed by the arrows, are put together and generate,respectively, root sites a and b . The inner sites of the square cell become the innersites of the folded square hierarchical lattice. Note that with this construction weobtain self-dual lattices with fractal dimension d = ln 9 / ln 3 = 2. Fig. 5
Folded square lattice obtained as in Fig. 4, but for b = 5. where J is the set of external parameters (couplings, fields, chemical poten-tials, . . . ), b is the length of the cell in lattice spacings (i.e., the scaling factorin the decimation procedure), d the space dimension, so that b d is the sizeof the cell in number of bonds to be decimated, and R ( {J } b d ) is the localrecursion relation for the interactions.In MK lattices, because of their 1D-like topology, see, e.g., Fig. 2, thetransformation can be divided into steps of so-called bond-moving and dec-imation, each of which involving only two bonds at a time. It is, thus, pos-sible to exactly compute the probability distribution and represent it with P ( J ) JPM fixed point 0 0.02 0.04 0.06 0.08 0.1 0.12 0 200 400 600 800 1000 1200 1400JFM fixed point 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -150 -100 -50 0 50 100 150JSG fixed point
Fig. 6
RG evolution of the probability distribution of nearest-neighbor interactionfor the spin-glass Ising model on the Wheatstone bridge lattice of Fig. 3 ( D (cid:39) . M = 10 . Paramagnetic (left), Ferromagnetic (center) and Spin-Glasss (right) phases. histograms, each bin of which characterized by a value of the interactionsand an associated probability [45].In HLs built without the MK 1D-like structure, such as the Wheatstone-bridge-like in Fig. 1, this factorization is no longer possible and we mustconsider the convolution of more than two links at a time. We are, even-tually, obliged to proceed in a statistical way. The PSRG scheme is, then,accomplished by representing the probability distribution of the couplingsby a pool of M real numbers [46] from which one can compute its asso-ciated moments, at each renormalization step. In the limit M → ∞ thesemoments should approach those of the exact renormalized probability dis-tribution. The process starts by creating a pool with M coupling constantsgenerated according to the initial distribution. A PSRG iteration consists in M operations in which one randomly picks a set of b d couplings from thepool in order to generate one renormalized coupling, which will populate therenormalized pool. Following this procedure, one creates a new pool of size M representing the renormalized probability distribution. During the PSRGprocedure the moments of the coupling distribution are of particular interestfor the identification of the phases.For example, in Ising models with quenched disorder, denoting by J theaverage of the couplings and by σ J the mean square displacement, one obtainsthe Paramagnetic (PM), Ferromagnetic (FM), and Spin Glass (SG) phases,as dominated by the attractors J → σ J →
0; PM ; J → ∞ ; σ J → ∞ ( J/σ J → ∞ ) ; FM ; J → σ J → ∞ ; SG .In Fig. 6 the typical PSRG iterations of the couplings distribution in thethree phases are shown for the random bond Ising model on the Wheatstonebridge lattice of Fig. 3.In order to reduce the dependence on a particular sequence of randomnumbers, the evolution of each distribution is analyzed over N S differentsamples. This is especially relevant when the starting pool is near a criticalpoint, and random fluctuations can lead different samples of the same dis-tribution into different attractors. In this case we will adopt the conventionthat a phase is identified if at least 80% of the N S samples flow into the same J/
Projections of the critical T = 0 fixed probability distribution for RFIMon WB 2D lattice for bimodal initial distribution on J, h (left),
J, h † (mid) and h, h † planes. attractor. This defines the error for the location of critical points, which canbe reduced by increasing the value of M .In presence of disorder it is hard to devise a general prescription for find-ing the critical exponents, like the one provided by Eqs. (6), (7) for orderedmodels. The idea is, then, to estimate the critical exponents by slightly per-turbing the system from the unstable fixed point distribution and measurehow fast it departs from it under successive PSRG iterations. We will discussthis procedure in detail in the following.3.1 Random Field Ising ModelIn this section we discuss the PSRG study of the Random Field Ising Model(RFIM) with bimodal and Gaussian distributed quenched external field onthe simple necklace MK lattice of Fig. 2, with fractal dimension d = 3, andon the Wheatstone-Bridge (WB) hierarchical lattice of Fig. 3, with fractaldimension d ≈ . .The initial distribution of couplings for the RFIM reads P ( J ij , h ij , h † ij ) = δ ( J ij − p ( h ij ) δ ( h † ij ) (15)where p ( h ij ) is either a bimodal or a Gaussian distribution: p ( h ij ) = [ δ ( h ij − h )+ δ ( h ij + h )] , √ πh exp (cid:110) − h ij h (cid:111) . (16)The initial distribution is an even function of h , and h † . This symmetry ispreserved under the PSRG transformation. To maintain this symmetry inour finite sample, we, actually, use a pool of 2 M interactions: for each of the M computed renormalized interactions ( J ij , h ij , h † ij ) we add to the pool alsothe corresponding ( J ij , − h ij , − h † ij ). In a previous paper of Nobre and Salmon[47] the exact PSRG transformationof the hierarchical lattice is not achieved, as pointed out by Berker [48].1Lattice h h /J bimod. MK 0 . . . . . . . . Table 1
Critical fixed points exponents for RFIM on necklace MK lattice in Fig.2 and WB lattice in Fig. 3 with bimodal and Gaussian random field distributions.
This trick becomes important for long PSRG iterations. As M increasesthe PSRG threshold step beyond which it becomes necessary quickly in-creases. This forced symmetrization is, thus, not crucial in determining crit-ical properties but helps decreasing finite size effects for small pools. For thepresent computation we take pools with up to M = 10 interactions, enoughto yield statistically stable results.The RFIM shows two phases identified by the behavior of the ratio be-tween the average coupling J = (cid:104) J ij (cid:105) (17)and the standard deviation of the fields h = (cid:104) h ij (cid:105) (18)along the PSRG flow: h /J → ∞ high temperature phase; h /J → h † field). At the critical point both J and h (as well as σ h † )flow to infinity, confirming that the critical behavior of RFIM is controlledby a zero-temperature fix point couplings distribution.For the MK lattice of Fig. 2 and for the WB lattice shown in Fig. 3 thecritical points are reported in Tab. 1.In Fig. 7 we show the projections of the critical fix point probability dis-tribution for the bimodal case on the WB hierarchical lattice (in the Gaussiancase they are qualitatively the same).At the zero-temperature fix point there are three independent exponents:the scaling exponent of the distribution of couplings z , the magnetic fieldscaling exponent y h and the thermal scaling exponent y T = 1 /ν [49]. Theycan be obtained following the procedure proposed by Cao and Machta [50],for both the bimodal and Gaussian case. Distribution growth scaling.
The exponent z describes the deviationof the couplings probability distribution from the unstable fix point distri-bution under the PSRG transform. In order to estimate z , once the PSRGflux gets close to the fix point distribution, we fix the ratio h /J at the crit-ical value in the subsequent PSRG iterations by shifting at each step all thecouplings { J ij } in the pool towards the ideal critical value.In particular, we compute the average J in the given iteration and wecompute the J ∗ that the system should display if J ∗ = h / [ h /J ] c . Then wetake J ∗ − J and we choose to change each one of the J ij in the distribution of the 80% of J ∗ − J . In our computation each one of these shifts turns out tobe very small, of the order of 0.1% for each J ij . This is, though, an essentialchange because of the unstable nature of the fix point.We, then, evaluate the rescaling factor λ ≡ h (cid:48) /h (cid:39) J/J (cid:48) at each PSRGstep next to the critical line and then take its average. The exponent isestimated as z = log b λ , (19)where the overbar denotes the average over 10 PSRG steps. Its values on theMK and WB cells both with bimodal and Gaussian distribution are shownin Tab. 2. We note that in all the cases it is z ≤ d/
2, that is the upper boundprovided by Berker and McKay [51].
External field scaling.
The exponent y h describes the rescaling ofan infinitesimal homogeneous field and can be obtained by averaging therelations in Eq. (6), (7) over the fix point distribution y h = log b (cid:28) ∂h R ∂h (cid:29) . (20)Its values are reported in Tab. 2 for the different cases. In all cases the valueis smaller than the fractal dimension d , which is 3 for MK and ∼ .
585 forWB, implying that the magnetization is continuous at the transition.
Correlation length scaling.
In order to estimate the exponent ν , wefirst reach a pool of renormalized couplings satisfactorily representing the fixpoint distribution. Next we take a copy of the pool and generate a slightlyperturbed couplings probability distribution by shifting every coupling { J ij } of the replicated pool by a small amount δ = 10 − J . The original and theperturbed pools are, then, simultaneously renormalized. To reduce statisticalfluctuations the couplings in the pool representing the fix point probabilitydistribution are shifted, after each PSRG step, to keep the distribution closeto the unstable fix point, in a manner similar to that used for the estimationof the exponent z . Note that in this way the only role of the shift δ is toaccelerate and make explicit the departure from the fixed point.By defining t n as the difference between the value of the ratio h /J inthe two pools after n PSRG iterations, the correlation length exponent isestimated as 1 ν = log b (cid:18) t n +1 t n (cid:19) , (21)where the overbar denotes the average over the PSRG iterations n . Notethat the argument of the logarithm is always positive, because leaving thefix point the second copy variance can either shrink or increase in its flux,but it does not oscillates between different PSRG steps. The sign of t n and t n +1 is, thus, the same.Typically we have n = 3 , . . . ,
9, for which the perturbed pool is not toofar from the unstable fix point, and the ratio h /J is stable. The result isindependent of M and it is quite stable over independent PSRG evolutions,at least for M (cid:38) .We obtain 1 /ν = 0 . ± .
23 for the bimodal distribution and 1 /ν =0 . ± .
20 for the Gaussian distribution on the MK lattice. Similarly, for z y h /ν bimod. MK 1 . . . . . . . . . . . . . . . . . . . . . Table 2
Critical exponents at critical fixed distribution for RFIM on necklace MKlattice in Fig. 2 and WB lattice in Fig. 3. The last two rows relate to simulationson Bravais 3D and 4D hyper-cubic lattices with bimodal distributed interactions.The more accurate exponents in Ref. [52] are obtained using the histograms rep-resentation for the interactions probability distribution, that is a feasible methodonly for MK cells. the WB lattice we find 1 /ν = 0 . ± .
35 for the bimodal distribution and1 /ν = 0 . ± .
31 for the gaussian distribution. As summerized in Table2. Once the exponents z , y h and ν are known, the exponents α and β areobtained from the scaling relations [49] α = 2 − ( d − z ) ν, (22) β = ( d − y h ) ν. (23)Notice that, at difference with Eq. (9-13), here the fixed point is at zero T (as 1 /J ) and the index z (cid:54) = 0.The critical exponents obtained on MK lattice for the bimodal case arecompatible with those obtained by Cao and Machta [50]. The exponents y h and z depend strongly on the fractal dimension of the lattice, and their valuefor the d = 3 MK lattice are in good agreement with the results for the threedimensional Bravais lattice, whilst for the d (cid:39) .
585 WB lattice they arecloser to that found for the four dimensional Bravais lattice, cf. Table 2.The value of the exponent ν is larger for the WB hierarchical lattice thanfor the MK lattice, as in agreement with the behavior in hyper-cubic latticesof increasing dimensions. Although the error bars for this exponent are large,we stress that in this case only the WB lattice gives a numerical estimatecompatible with the result for the cubic lattice.We, eventually, notice that the Gaussian and bimodal exponents are al-ways compatible with each other and belong to the same universality class.In the RFIM the bond are at first all equal, and the disorder is on site. Themajor reason of moving from MK lattices to more complex HL is, actually,the need of a better treatment of the bond structure. In the next sections wemove to models in which the disorder is on the bond from the outset.3.2 Random Bond Ising model in d ≤ . ± J bond distribu-tion on hierarchical lattices mimicking the topology of the square lattice. The initial probability distribution for interactions is P ( J ij , h ij , h † ij ) = [ pδ ( J ij − J ) + (1 − p ) δ ( J ij + J )] δ ( h ij ) δ (cid:16) h † ij (cid:17) (24)where J > p ∈ [0 ,
1] is the probability of a ferromagnetic bond.For low enough p , this model on regular lattice has an antiferromagneticphase. On hierarchical lattices the antiferromagnetic order is preserved underPSRG only when the rescaling factor b is odd, so that a symmetric phasediagram in p ↔ (1 − p ) is obtained, where the ferromagnetic phase is replacedby the antiferromagnetic phase and vice-versa.To capture some features of Bravais lattices, we shall focus on hierarchicallattices with elementary cell more complex than the 1D-like MK cells. Inparticular we shall consider the cell proposed by Nobre [30], Fig. 4, withrescaling factor b = 3 and its extension to b = 5, Fig. 5. Nishimori conjecture.
Even though the Random Bond Ising modelin 2D does not display any spin-glass phase [55], this model on hierarchicallattices is an excellent play ground to test Nishimori’s conjecture [56, ? ]. Herewe briefly recall what the conjecture is.The idea behind it stems from noting that the partition function Z ofnon-random Ising models on self-dual lattices is itself self-dual [58].Let us express Z in terms of the edge Boltzmann factors u ± ( J ) = e ± J (note the use of the reduced parameters). The Fourier transform of u ± ( J ), u ∗± ( J ), is the dual Boltzmann factor [59]. As a consequence, the critical pointof a self-dual model is obtained by the fix point condition u ± ( J ) = u ∗± ( J ),which yields J c = log( √ x k ( p, J ), which correspond to the configurationwith the spin connected by the bond equal to +1 in n − k replicas and − k replicas. Self-duality is now expressed by the invariance of Z n ≡ Z n under the simultaneous exchange x k ( p, J ) ↔ x ∗ k ( p, J ) for all k . Theoverbar denotes the average over quenched bond disorder. Unlike the non-random case, it is not possible to identify the critical point from the fix pointcondition of the duality relations, because the relations with k = 0 , . . . , n are not satisfied simultaneously. The Nishimori’s conjecture [60,61], then,identifies a point ( p N , J N ), called ”multicritical”, by means of a fix pointcondition for the leading k = 0 Boltzmann factor x ( p N , J N ) = x ∗ ( p N , J N ) , (25)on the Nishimori line e − J = (1 − p ) /p ,[62,56] where enhanced symmetrysimplifies the system properties significantly. This point of the Nishimoriline is called multicritical because, when a SG phase is present, it is thepoint at which PM, FM and SG phases are all in contact with each other.This does not occurs in 2D, see, e.g., Fig. 8 but it does occur in 3D, cf. Fig.13. The conjecture is proved exact for n = 1 , ∞ . In the limit n → x = x ∗ on the Nishimori line becomes H ( p N ) ≡ − p N log ( p N ) − (1 − p N ) log (1 − p N ) = 12 , (26) where the function H ( p N ) is the binary entropy.This conjecture turns out to be wrong for some systems on HL.[63,26]Ohzeki, Nishimori and Berker [26,31] noted that for HL’s a systematic ap-proximation for the multicritical point can be obtained by imposing Eq. (25)at each PSRG transformations.Note that in the two-dimensional case, even though the SG phase is ab-sent, the Nishimori point is expected to coincide with a critical point, unsta-ble along the phase boundary; so, as it is usually done, in the following weidentify the Nishimori point as the intersection between the Nishimori andthe critical lines.Here we test the original Nishimori conjecture on a Folded Square (FS)hierarchical lattice constructed through a bond-moving procedure that, atdifference with the MK bond-moving prescription, retains the local correla-tion of the bonds, see Figs. 4 and 5. In Fig. 8 we show the p, T (= 1 /J ) phase diagram obtained from the PSRGanalysis in two dimensions on the lattice shown in Fig. 5 with rescaling factor b = 5.For p = 1 we find the critical temperature is T c = 2 . T c = 2 / log(1 + √ (cid:39) . b = 3,[30] and WB with d = 2 .
32, [27] follows from the duality properties of the unit cells.[35]
Nishimori point.
The position of the multi-critical point expectedfrom Nishimori’s conjecture, Eq. (26), is p N (cid:39) . p N = 0 . H ( p N ) = 0 . T N = 0 . N S = 20 independent PSRG calculationswith a poll of M = 10 initial bond configurations, we estimate (cf. Table 3) p N = 0 . , T N = 0 . H ( p N ) = 0 . b = 3. We conclude thatthe conjecture fails also on this more complex hierarchical lattice.The folded square cell estimates with b = 3 or b = 5 turn out to be ingood agreement with each other and with the estimate p N = 0 . p N = 0 . p N = 0 . Critical slope.
A quantity usually studied is the slope of the criticalline close to p = 1:[67] s ≡ T c (1) dT c ( p ) dp (cid:12)(cid:12)(cid:12)(cid:12) p =1 . (27) p N p T =0 FS b = 3 4/[30] 0 . . b = 5 5 0 . . . . . . . . ... Table 3
Multicritical point as computed on different HLs and different estimateson the square lattice.
The Domany’s perturbative approach [68] yields s = 2 √ / [ln( √ (cid:39) .
209 for the Ising with ± J bond distribution on square lattice. The ap-proach assumes weak disorder, i.e., a qualitatively irrelevant disorder thatdoes not undermine the existence of the ferromagnetic phase at low T anddoes not change the universality class of the PM/FM transition for p <
1. Bycomputing s it is, thus, argued that one can discriminate whether quencheddisorder is a relevant perturbation, causing a change in the universality class.[69] Ohzeki and collaborators [69] suggest that Domany’s method can be ap-plied to any self-dual lattice, and then one can probe the relevance of disorderfrom the slope s also for the HL of Figs. 4 and 5.The duality approach gives s = 3 . ... [69] for the lattice in Fig. 4with b = 3. From a best fit of the points close to p = 1 with a pool of size M = 5 · with N S = 20 samples we obtain s = 3 . b = 5 case of Fig. 5,with a pool of size M = 5 · with N S = 20 samples we find s = 3 . s (cid:39) . Harris criterion.
The widely accepted form of the Harris criterion [67, ? ] is that in ferromagnetic systems with random interactions the randomnessis irrelevant if α , the specific heat exponent of the corresponding pure system,is negative, while for systems with positive α the random system exhibits dif- / J p FMPM Fig. 8
Phase diagram of Ising 2D with “folded square” HL with b = 5. Thediagram is symmetric in p → − p and the anti-ferromagnetic part is not shown.The dashed line represents the Nishimori line. P ( J ) J Fig. 9
PSRG evolution of the coupling distribution on the critical line near thepure fixed point (steps from 4 to 11 are shown for p = 0 . ferent critical behavior in presence of disorder. For the folded square latticesfor both b = 3 and b = 5 we find α < FM line reentrance.
An important feature of the p, T diagram, ac-cording to the duality requirements, is the reentrance of the transition linebelow the multicritical point: p N > p T =0 . The zero temperature transitionpoint p T =0 can be estimated by finite size scaling analyses of the ground state.Calling E p and E a the ground-state energies with, respectively, periodic andanti-periodic boundary conditions in one direction, and ∆ = E p − E a the do-main wall energy, we can determine two estimates of critical concentrationsof antiferromagnetic bonds by looking at the point where the asymptotic de-pendences [ ∆ ] and [ ∆ ] / change from increasing to decreasing, where theaverage [ . . . ] is taken on different bond samples. More explicitly, by defining[ ∆ ] ∼ L ρ and [ ∆ ] / ∼ L θ , (28)via the determination of exact ground states for large system sizes and hugesample numbers, Kawashima and Rieger give the estimates p ( ρ ) T =0 = 0 . p ( θ ) T =0 = 0 . ρ and θ respectively changesign [70]. The PSRG approach with the folded square of Fig. 4 ( b = 3)leads to p T =0 = 0 . b = 5) we find p T =0 = 0 . y T = 0 . y h = 0 . b = 3 and for the folded square lattice with b = 5 we find y T = 0 . y h = 1 . y T = 1 and y h = 1 . y T and y h the critical indexesare obtained from the usual scaling relations, cf. Eq. (13) and their numericalvalues are reported and compared on Table 4. Zero temperature stiffness.
We conclude this section by discussing theexponent ν of the zero-temperature spin-glass transition for the case of aGaussian distribution of bonds with zero mean and initial width σ J . It canbe obtained directly from scaling of σ J under PSRG : σ (cid:48) J ( b ) ∼ σ J b θ . (29)The sign of the stiffness exponent θ is directly related to the low temperaturephase: for positive (negative) θ the system scales under PSRG flow towardsstrong (weak) couplings, distinctive of a low temperature spin-glass (high T paramagnetic) phase. For continuous and symmetric probability distributions P ( J ), the temperature T appears in the PSRG equations as a dimensionlessratio between couplings, so that the scaling (29) is equivalent to T ∼ L θ , or L ∼ T /θ . In a phase transition at T → ξ ∼ T − ν implying [77]: ν = − θ . (30) α -0.7385(1) -0.6353(1) 0 β γ -0.4057(1) 0.1558(1) 1.75 δ ν η Table 4
Critical indices of pure ferromagnetic critical point for Ising model on thefolded square lattice for b = 3, Fig. 4, and for b = 5, Fig. 5.Lattice type Fig. Ref. θ MK b = 2 [46]] − . b = 3 [78] − . b = 2 1 [35] − . b = 3 [78] − . b = 3 4 [35] − . b = 5 5 − . − . − . − . Table 5
Stiffness exponent θ on different HLs and different estimates on the squarelattice. In Fig. 10 the behavior of σ J is shown as function of PSRG steps for thecase of zero-average Gaussian initial bond distribution on the folded squarecell of Fig. 5. From this we get θ = − . ν = 3 . b = 3 it was found θ = − . b = 3 or b = 5 thevalue is similar to that found for the MK lattice, whilst for the WB latticesthe values are closer to that of the regular lattice, especially the case with b = 2.As a general remark, from Table 4 we observe that in passing from b = 3 to b = 5, and thus increasing the connectivity of the lattice, for the pure modelwe obtain a slight improvement for all exponents towards the exact valuesof the square lattice, though the values are very far from the exact ones. Apossible convergence is, thus, so slow that the degree of inner correlation of afolded cell of a HL necessary to reproduce 2D RG might be so large to makea single cell comparable with a whole real Bravais lattices.We now move to models in which there is a spin-glass phase, in order totest the critical behavior prediction of HL RG in the case of strong disorder.3.3 Random bond Ising model in d ≥ ± J coupling distribution,Eq. (24) on hierarchical lattices mimicking the topology of the cubic lattice. -9-8-7-6-5-4-3-2-1 0 0 5 10 15 20 25 30 l og (cid:109) RG step
Fig. 10
Standard deviation of the Gaussian probability distribution of quencheddisorder at T = 0 vs PSRG iteration steps . Recently, Salmon, Agostini and Nobre have studied the Ising spin glass on thehierarchical Wheatstone bridge (WB) lattices [27], obtaining accurate phasediagrams and showing that on these lattices the lower critical dimension forthe spin glass phase is greater than d = ln 5 / ln 2 (cid:39) .
32, cf. the WB HLin Fig. 1. The next pattern of the WB family, cf. Fig. 3, corresponding toa lattice in dimension higher than 2 . d ≈ . d = ln 35 / ln 3 (cid:39) . b = 3 and, unlike the latter, it is able toretain a possible antiferromagnetic order, as well. As a further comparisonwe will report on the critical properties of the Ising spin-glass model on aMK lattice with b = 3 and fractal dimension d = 3, cf. Fig. 12, introduced inRefs. [82,33].The resulting phase diagram is shown in Fig. 13. An important feature ofthe phase diagram is the small reentrance in the region below the multicrit-ical point. As a consequence, by lowering the temperature one can go fromthe high temperature (disordered) paramagnetic (PM) phase to an orderedferromagnetic (FM) phase and, eventually to a low temperature disorderedspin-glass (SG) phase. The values of the critical points are reported in Table6. Fig. 11
Cubic cells of length b = 3 on the left hand side, and relative foldedcube hierarchical lattice on the right hand side. The two roots are open circles. All incoming sites and all outcoming ones, pointed by the arrows in the left figure, areput together and generate root sites a and b on the right hand side figure. Theinner sites of the cubic cell become the inner sites of the hierarchical cell. Fig. 12
MK lattice with b = d = 3.[82,63] The transition at p = 1 on the folded cube cell is obtained at T c = 5 . T c =4 . ... [83]). On the d (cid:39) .
58 WB lattice the difference was about 21%.[27] On the MK lattices the best known result is obtained for the d = 3lattice in Fig. 12, where the critical temperature of the pure transition is T c = 5 . p = 1. The critical p FMSG PM FMSG PM FMSG PM Fig. 13
Phase diagram of Ising spin glass model on the Folded Cube HL, cf. Fig.11, obtained for M = 5 · and N S = 10. The dashed line represents the Nishimoriline. The plot is symmetric in p → − p and an antiferromagnetic phase is presenta small p , in place of the FM one. pT fixed points coordinatesHL FM SG T = 0 MCFold cubeFig. 11 15 . . . . . . . . . . . . . . . . . . . . . . . Table 6
Position of typical critical points of phase diagram in Fig. 13 for thehierarchical lattice 11: ferromagnetic (FM), spin-glass (SG), zero temperature ( T =0) and multi critical point (MC). For comparison also values of the same fixed pointsare reported for other hierarchical lattices discussed in the text. exponents at this point are y T = 1 . y h = 1 . y T = 1 . y h =0 . y T = 1 . y h = 1 . α − . − . − . . β . . . . γ − . − . . . δ . . . . ν . . . . η . . . . Table 7
Critical exponents in pure ferromagnetic critical point for Ising model onthe folded cubic cell and comparison.Lattice type Fig. Ref. θ MK d = 3 12 [87] 0 . d ∼ .
58 3 [27] 0 . d ∼ .
24 11 0 . . . Table 8
Stiffness exponent as computed on different HL and different estimateson the cubic lattice. means of simulations on the cubic lattice: y T = 1 . y h = 2 . α <
0. According to the Harriscriterion [67] this would imply that disorder is irrelevant in modifying theferromagnetic critical behavior, whereas for all these HL’s one finds a frozenphase different from the ferromagnetic one in a given interval of p valuesat low temperature: besides the ”weak disorder” ferromagnetic fixed point asecond “strong disorder” spin-glass fixed point arises in presence of quenchedrandomness. For the FC HL in the totally disordered case p = 0 . T c = 1 . .
5% relatively to the Bravaiscubic lattice critical point, for which the Monte Carlo simulations provide T c = 1 . T c = 1 . T c = 1 . σ J ) under successive PSRG iterations for p = 0 . θ = 0 . θ = 0 . θ = 0 . θ = 0 . θ ≈ .
27. [87] The result are summarized in Table 8.All critical points along the PM-SG transition are attracted by an unsta-ble fixed point at p = 0 .
5. In Fig. 15 we show the unstable fix point coupling distributions for the three lattices considered so far. Critical exponents forthis fix distribution are computed by adapting the methods used for theRFIM in Sec. 3.1. A first exponent y h can be obtained generalizing the purecase definition: λ h = (cid:104) ∂ h h R (cid:105) , where the average is carried out by extractingthe interactions from the fix point distribution. Indeed, in our case this iseasy and can be exactly calculated. In fact for the fix point distribution wehave h = h † = 0, and so H ( s ) = H ( − s ). It is, then, straightforward to obtain,using Eqs. (6) and (7) that ∂ h h R = c for every choice of the J interactions,where c is the number of internal sites connected to each external site: c = 9for the folded cube cell in Fig. 11 and the MK in Fig. 12, c = 4 for the WBin Fig. 3. For all these cells the exponent y h is then y h = log b (cid:104) ∂ h h R (cid:105) = 2 (seeSec. 2.1). We note that y h < d , ensuring that the magnetization is continuousat the transition (no first order transition).To get the correlation length exponent ν we note that defining t ≡ σ J − σ ∗ J ,where σ ∗ J is the value at the critical point, we obtain the scaling law ξ ( t ) = b ξ ( t (cid:48) ) = b ξ ( b x t ) = b n ξ ( b nx t ) . (31)By taking n such that b nx t = t , where t is arbitrary but fixed, we end upwith ξ ( t ) = (cid:18) t t (cid:19) x ξ ( t ) ∼ t − x , (32) note that, as always in this paper, we are using the reduced variables βJ → J l og (cid:109) RG step
Fig. 14
Graph of ln( σ J ) under repeated applications of the PSRG transformationat zero temperature and p = 0 . p ( J ) J MKWBFC
Fig. 15
Fixed probability distribution for the spin glass - paramagnet transitionfor Ising spin glass on lattice in Fig. 12 (MK), Fig. 3 (WB) and Fig. 11 (FC). Thesedistributions are reached after about 2 − −
15 steps steps,after which they starting moving away because of statistical fluctuations. from which we get ν = 1 /x : the value of ν can be estimated by the trend of σ J near the critical point distribution.We use the method already exposed in Sec. 3.1: after obtaining an instanceof the fixed distribution, we make a copy of it and multiply each coupling ofthe copy by a quantity 1+ δ , with δ ≈ .
05. The two copies of the distributionare, then, simultaneously iterated in the PSRG transformation, with the firstcopy forced near the fixed point and the second one free to flow. We eventuallyestimate ν by means of Eq. (21), where the parameter t n is now the differencebetween the values of σ J in the two copies at RG step n .Continuing this way we find the values shown in Table 9 for different HLsand we notice that they are compatible with each other within the statisticalerror. However, when compared to the estimate for the cubic Bravais lattice,none of them is compatible with it: ν = 2 . → /ν = 0 . .
24 has a value of ν further away than the value on the MK and WB ones.This is a strong signature of the limitations of the RG approach on HL’s tothe critical behavior of systems on Bravais lattices in presence of disorder.Proceeding as for the correlation length, and using the free energy f scaling law f ( t ) = b − d f ( t (cid:48) ) = b − d f ( b x t ) = b − nd f ( b nx t ) /ν MK 12 b = 3 d = 3 0 . ± . b = 2 d ∼ .
58 0 . ± . b = 3 d ∼ .
24 0 . ± . d = 3 0 . ± . Table 9
Estimates for the exponent ν at transition SG-para for the Ising spinglass model.critical MK WB Folded cube Cubicindex Fig. 12 Fig. 3 Fig. 11 Ref. [88] α − . − . − . − . β . . . . γ . . . . ν . . . . η . . . . . . . . − . Table 10
Estimates for the physical exponents at the SG-PM transition for theIsing spin glass model. The exponent η is d − d being the fractal dimension ofthe HL. with n such that b nx t = t and t arbitrary but fixed, we have f ( t ) = (cid:18) t J t (cid:19) dx f ( t ) ∼ t dx . (33)Then from ∂ f /∂t ∼ | t | − α , we obtain the scaling relation α = 2 − dx = 2 − dν, (34)as well as β = ν (1 + η )2 ,γ = (2 − η ) ν , where η = d + 2 − y h = d − y h = 2 for all our HLs. The estimates of the physical exponents arereported in Table 10.Concluding this section, on the FC lattice in Fig. 11 the estimates forthe pure criticality are much closer to those on cubic lattice (although stillnot compatible), compared to MK in Fig. 12 and WB in Fig. 3. Also forthe stiffness exponent of SG phase a remarkable improvement is observed:its estimate is compatible with its cubic value on the FC lattice but not onother HLs. For other quantities, though, at the disordered fixed point suchimprovement is unseen. In particular, in the estimate for the ν exponent atthe SG-PM transition.We can argue that the stiffness exponent depends more on the local ge-ometrical properties of the lattice, that are substantially improves in the folded cube lattice, while critical properties depend on longer distances, thatare still dominated by the hierarchical backbone an thus far from the Bravaislattice behavior.More stringent tests can be obtained for the Blume-Emery-Griffiths model,that we will analyze in the next section. For this system the inverse first or-der transition expected from mean-field theory [36] and simulation in finitedimension [37] is absent on the MK lattice in Fig. 12. [33] We now move on to a different system, the Blume-Emery-Griffiths (BEG)model, originally devised to study the superfluidity transition and phase sep-aration in He -He mixtures [89]. The model is known to display, besides asecond order phase transition, also a first order transition, both in the or-dered case (between the PM and FM phases) and in the case with quencheddisordered interactions (between the PM and SG phases).The ordered model cases have been introduced and solved in the mean-field approximation in Refs. [90,91,89]. Finite dimensional analysis has beencarried out by different means, e.g., series extrapolation techniques [92],PSRG analysis [17], Monte Carlo simulations [93], effective-field theory [94]or two-particle cluster approximation [95].Extensions to quenched disordered models, both perturbing the orderedfixed point and in the regime of strong disorder, have been studied throughoutthe years by means of mean-field theory [96,97, ? ], PSRG analysis on Migdal-Kadanoff hierarchical lattices [45,33], and Monte Carlo numerical simulations[98,37,40,39]. The latter studies show that a critical transition line separatesthe SG and PM phases. Like in the mean-field cases, it consists of a secondorder transition terminating in a tricritical point from which a first orderinverse transition starts. [37] Furthermore, a reentrance of the first ordertransition line is present for positive, finite values of the chemical potentialof the holes, [38] yielding the so called inverse freezing phenomenon of anamorphous phase arresting itself in a blocked, solid-like state upon heating.In absence of any purely ferromagnetic contribution, the PSRG approachon MK cells apparently does not show any first order phase transitions, norany reentrance, as shown by Ozcelik and Berker [33]. In this section we willdeepen their analysis at p = 1 /
2, investigating the critical behavior on theMK lattices of Fig. 12 and Fig. 16, on the Wheatstone Bridge (WB) latticeof Fig. 3 and on the Folded Cube (FC) lattice of Fig. 11.The BEG model with generic magnetic exchange interaction is defined bythe Hamiltonian (we use reduced variables) − H ( s ) = (cid:88)
MK lattice with fractal dimension d = 3 .
58, the same as WB lattice inFig. 3. random and it is convenient to start the iteration using the more generalform −H = (cid:88) (cid:104) ij (cid:105) J ij s i s j + (cid:88) (cid:104) ij (cid:105) K ij s i s j − (cid:88) (cid:104) ij (cid:105) ∆ ij (cid:0) s i + s j (cid:1) − (cid:88) (cid:104) ij (cid:105) ∆ † ij (cid:0) s i − s j (cid:1) (36)The model is, further, defined, by the multivariable initial probability distri-bution of the interactions: P ( J ij , K ij , ∆ ij , ∆ † ij ) = δ ( J ij − J ) + δ ( J ij + J )2 δ ( K ij ) δ ( ∆ ij − ∆ ) δ ( ∆ † ij ) . We notice that if ∆ (cid:28) −
1, the values s i = 0 are suppressed and the modeltends to the Ising spin glass model analyzed in previous sections: −H ( { J ij } ; { s i } ) = (cid:88) (cid:104) ij (cid:105) J ij s i s j with s i = ± P ( J ij ) = (cid:90) dK ij d∆ ij d∆ † ij P ( J ij , K ij , ∆ ij , ∆ † ij ) . Decimating the inner sites at a given hierarchical cell with fixed outersites s a and s b , and using the up-down symmetry of the Hamiltonian, therelations for the renormalized interactions imposed by the conservation ofthe partition function, cf. Eq. (3), can be written similarly to Eqs. (3)-(4) as J R = 12 log (cid:18) x ++ x + − (cid:19) ,K R = 12 log (cid:18) x ++ x + − x x x (cid:19) ,∆ R = 12 log (cid:18) x x +0 x (cid:19) , (37) ∆ † R = 12 log (cid:18) x +0 x (cid:19) , where x s a s b are the edge Boltzmann factors, cf. Eq. (3). In the T = 0 limitthese relations become2 J R = max (cid:2) −H (1 , , s ) (cid:3) − max (cid:2) −H (1 , − , s ) (cid:3) K R = max (cid:2) −H (1 , , s )) (cid:3) + max (cid:2) −H (1 , − , s ) (cid:3) − (cid:8) max (cid:2) −H (1 , , s ) (cid:3) − max (cid:2) −H (0 , , s ) (cid:3)(cid:9) +2 max (cid:2) −H (0 , , s ) (cid:3) (38)2 ∆ R = 2 max (cid:2) −H (0 , , s ) (cid:3) − max (cid:2) −H (1 , , s ) (cid:3) − max (cid:2) −H (0 , , s ) (cid:3) ∆ † R = max (cid:2) −H (1 , , s ) (cid:3) − max (cid:2) −H (0 , , s ) (cid:3) The phase diagrams relative to the different hierarchical lattices are shownin Fig. 17. They are obtained representing the probability distributions by apool of M = 10 interaction quadruples ( J, K, ∆, ∆ † ) and the RG evolutionof each distribution is analyzed over N S = 10 different samples, i.e., startingwith ten different initial realizations of the quenched couplings. As worked outin Sec. 2, paramagnetic, ferromagnetic and spin glass phases are determinedby the analysis of the PSRG flux of J = (cid:104) J ij (cid:105) and σ J = (cid:104) J ij (cid:105) − (cid:104) J ij (cid:105) .In all cases we obtain a second order transition between paramagnetic andspin glass phases, with all points on the transition line attracted by a uniquefixed distribution at ∆ → −∞ (see Fig. 18), thus belonging to the sameuniversality class of the SG-PM transition in the Ising spin glass investigatedin the previous section.4.1 Lack of first order phase transitionStudying hierarchical lattices also much more complex than MK, the firstorder transition typical of the BEG model on Bravais lattice is not found,so we have a strong indication that this is an intrinsic limit of hierarchicallattices, and not only of the MK kind of cells.We have no clear evidence on why the first order transition is missing onthe hierarchical lattices lattices we have considered so far. Based on physicalarguments we may propose the following hypothesis. Second order transition / J (cid:54) /JMK d=3MK d=3.58WB d=3.58FC d=3.24Bravais 2 nd orderBravais 1 nd order Fig. 17
Phase diagram for the BEG model on the MK( d = 3) lattice in Fig. 12,WB lattice in Fig. 3, FC lattice in Fig. 11 and simulations [37]. The line for theMKs and WB are obtained with M = 10 , whilst for the FC we use M = 5 · .Inset: detail of the reentrance region for WB and 3D cubic lattices, compared tothe MK HL line (no reentrance). are associated with an instability, the high temperature phase becomes un-stable and a new stable phase appears. This instability can manifest itselflocally, and hence hierarchical lattices can show a second order transition.First order transitions, on the contrary, are not triggered by an instability.The high temperature phase remains stable, but a thermodynamically morefavorable phase takes over. Such a situation requires some sort of long rangestructure of the lattice, that is missing in the present hierarchical lattices,and this may explain why we do not see first order transitions. If our conjec-ture is correct we may wonder if the first order transition could still appear inhierarchical lattices with folded cells for b finite but large enough. However,from our present knowledge this value of b might be so large to make theHL approach ineffective. The analysis of the arising of a possible fist ordercritical behavior in b is left for future work.4.2 Inverse freezingAn apart feature of the BEG model arises, instead, adopting WB cells: theinverse transition between spin glass and paramagnet. it occurs in the WB -4 -2 0 2 4 0.96 0.98 1 1.02 1.04 0 0.001 0.002 0.003 0.004 0.005P(J, (cid:54) ) J (cid:54) /< (cid:54) >P(J, (cid:54) ) Fig. 18 (Projection of unstable fixed distribution P ( J, K, ∆, ∆ † ) for BEG modelat the transition SG-para on the WB lattice in Fig. 3. In the distribution ∆ isrunaway to minus infinity exponentially, while other interactions remain finite andthe distribution retains this shape. The shape of relative distribution on the MKand FC lattices is very similar. d ≈ .
58 where evidence for the reentrance is clearly obtained using the T = 0Eqs. (38). Inverse freezing is predicted in mean-field theory[38] and found in3D numerical simulations.[37]. It is not found in MK lattices, instead, neitherin fractal dimension 3 (cf. Fig. 12) nor in d ≈ .
58 (cf. Fig. 16) nor in the FClattice.
We have provided a critical review of the standard methods to develop PSRGon hierachical lattices and applied them to different models. On one side thisanalysis can be useful to test general results (e.g. Nishimori conjecture, Harriscriterion), and on the other side it is far easier and faster than Monte Carlosimulations on Bravais lattice. We, however, stress that these methods displayseveral drawbacks that we critically underlined model by model.We have investigated the Random Field Ising Model (RFIM), the Ran-dom Bond Ising Model (RBIM) and the Blume-Emery-Griffiths (BEG) modeldefined on several MK and non-MK hierachical lattices, obtaining phase di-agrams and critical exponents. The RFIM has been analyzed on a non-MK (WB in Fig. 3) and we showthat the bimodal and Gaussian disordered cases belong to the same univer-sality class.The RBIM in d < . b = 3 (Fig. 4) and b = 5 (Fig. 5), where we find that the Nishimoriconjecture fails and the disorder is irrelevant.The RBIM in d > . d = 3 (Fig. 12) and d ∼ .
58 (Fig. 16), and in all the cases the first order transition taking placein the Bravais lattice for large enough chemical potential [38,37] is absent.Our results provide a clue to the possibility of obtaining approximationsof models on regular lattice by similar models on hierachical lattice. We showthat it is possible to obtain a good picture of the actual phase diagram, butfar more difficult to yield a proper determination of critical exponents.By introducing more complex elementary cells, with some non trivial in-ternal structure, one hopes of capturing the local geometrical properties ofthe bonds. At least part of it. For pure models, although it is not a system-atic approximation, a general improvement is obtained using unit cells thatlocally mimic better the connectivity of the Bravais lattice. In the disorderedcase, instead, the situation is less definite, and no net improvement is ob-served: the WB cell (Fig. 3) proves to be the slightly most reliable (generallyquantitatively better than the more complex folded cube in Fig. 11), and inparticular shows the expected inverse transition for the BEG model, but wecannot give a general explanation for this.In particular, the fractal dimension seems to play a minor role, as thethree dimensional regular lattice is better approximated by the WB with d ∼ .
58, compared to d = 3 MK (Fig. 12) and d ∼ .
24 folded cube (Fig.11), while the d ∼ .
58 MK in Fig. 16 is the worst approximation by far.The scaling factor b has the known role to determine if the antiferromag-netic order can be preserved, as it is possible only when b is odd so thatnegative interactions in the unit cell involve negative interactions betweenthe external sites (and this leads to a phase diagram symmetric under theinversion of the bonds sign). Our results indicates that this feature does notplay a crucial role in the disordered systems (at least until the negative bondsbecome dominant). The two investigated HLs with an even b , the WB andthe d ∼ .
58 MK, indeed, appear to be, respectively, the best and the worstin approximating the phase diagram of the models on regular lattice. Theonly case in which the folded cube lattice in Fig. 11 provides a remarkableimprovement with respect to the WB lattice is in the estimate of the SGstiffness exponent. A more structured inner local connectivity could, thus,become important at low temperatures.In conclusion, our results show that the approximation on HL is partic-ularly poor for disordered systems, and strongly suggest that its limitationsare intrinsic to the hierarchical nature. The most striking case is the lack of first order transition in BEG. Indeed, using more complex HLs we may havea better treatment of short distances, i.e., short loops, but longer distancesappear definitely dominated by the hierarchical backbone. Acknowledgements
The research leading to these results has received fund-ing from the People Programme (Marie Curie Actions) of the European Union’sSeventh Framework Programme FP7/2007-2013/ under REA grant agreement n290038, NETADIS project and from the Italian MIUR under the Basic ResearchInvestigation Fund FIRB2008 program, grant No. RBFR08M3P4, and under thePRIN2010 program, grant code 2010HXAW77-008.
References
1. G. Parisi. Infinite number of order parameters for spin- glasses.
Phys. Rev.Lett. , 43:1754-1756, 1979.2. G. Parisi. A sequence of approximated solutiona to the S-K model for spinglasses.
J. Phys. A: Math. Gen. , 13:L115, 1980.3. M. M´ezard, G. Parisi, and M. Virasoro.
Spin Glass Theory and Beyond . WorldScientific (Singapore), 1987.4. D.J. Amit.
Modeling Brain Functions: The World of Attractor Neural Net-works . Cambridge University Press (Cambridge, U.K.), 1992.5. M. M´ezard and A. Montanari.
Information, physics, and computation . OxfordUniversity Press (Oxford, U.K.), 2009.6. J.H. Chen and T.C. Lubensky. Mean field and (cid:15) -expansion study of spin glasses.
Phys. Rev. B , 16:2106, 1977.7. C. De Dominicis, I. Kondor, and T. Temesvari. Beyond the Sherrington–Kirkpatrick Model. In
Directions in Condensed Matter Physics , volume 12,page 119. World Scientific, 1998.8. C. De Dominicis and I. Giardina.
Random Fields and Spin Glasses . CambridgeUniversity Press (Cambridge, U.K.), 2006.9. A.J. Bray and M.A. Moore. Disappearance of the de Almeida-Thouless line insix dimensions.
Phys. Rev. B , 83:224408, 2011.10. G. Parisi and T. Temesvari. Replica symmetry breaking in and around sixdimensions.
Nucl. Phys. B [FS] , 858:293–316, 2012.11. D.L. Stein and C.M. Newman.
Spin Glasses and Complexity . Princeton Uni-versity Press, 2012.12. D.J. Amit and V. Martin-Mayor.
Field Theory; The Renormalization Groupand Critical Phenomena . World Scientific, 2005.13. M. Le Bellac.
Quantum and Statistical Field Theory . Oxford Science Publica-tions, 1992.14. L.P. Kadanoff. Notes on Migdal’s recursion formulas.
Ann. of Phys. ,100(1):359–394, 1976.15. S.K. Ma.
Modern Theory of Critical Phenomena . Benjamin-Cummings, Read-ing, 1976.16. J.M.J. Leeuwen Th. Niemeijer. Wilson Theory for Spin Systems on TriangularLattice.
Phys. Rev. Lett. , 31(23):1411, 1973.17. A.N. Berker and M. Wortis. Blume-Emery-Griffiths-Potts model in two dimen-sions: Phase diagram and critical proprieties from a position-space renormal-ization group.
Phys. Rev. B , 14(11):4946, 1976.18. K.H. Fisher W.Kinzel. Existence of a phase transition in spin glasses?
J. Phys.C , 11:2115, 1978.19. T. Tatsumi. Renormalization-Group Approach to Spin Glass Transition of aRandom Bond Ising Model in Two- and Three-Dimensions.
Prog. Theor. Phys. ,59:405, 1978.20. S. Franz, G. Parisi, and M.A. Virasoro. Interfaces and louver critical dimensionin a spin glass model.
J. Phys. I (France) , 4:1657, 1994.421. S. Franz and F.L. Toninelli. A field-theoretical approach to the spin glasstransition: models with long but finite interaction range.
J. Stat. Mech. , pageP01008, 2005.22. S. Boettcher. Stiffness of the Edwards-Anderson Model in all Dimensions.
Phys. Rev. Lett. , 95:197205, 2005.23. A.N. Berker and S. Ostlund. Renormalisation-group calculations of finite sys-tems: order parameter and specific heat for epitaxial ordering.
J. Phys. C ,12(22):4961, 1979.24. R.B. Griffiths and M. Kaufman. Spin systems on hierarchical lattices. Intro-duction and thermodynamic limit.
Phys. Rev. B , 26:5022–5032, 1982.25. S.R. McKay, A.N. Berker, and S. Kirkpatrick. Spin-Glass Behavior in Frus-trated Ising Models with Chaotic Renormalization-Group Trajectories.
Phys.Rev. Lett. , 48:767–770, 1982.26. M. Ohzeki, H. Nishimori, and A.N. Berker. Multicritical points for spin-glassmodels on hierarchical lattices.
Phys. Rev. E , 77:061116, 2008.27. O.R. Salmon, B.T. Agostini, and F.D. Nobre. Ising spin glasses on Wheatstone–Bridge hierarchical lattices.
Phys. Lett. A , 374(15–16):1631–1635, 2010.28. M.A. Moore, H. Bokil, and B. Drossel. Evidence for the Droplet Picture ofSpin Glasses.
Phys. Rev. Lett. , 81:4252, 1998.29. F. Ricci-Tersenghi and F. Ritort. Absence of ageing in the remanent magne-tization in Migdal-Kadanoff spin glasses.
J. Phys. A: Math. Gen. , 33:3727,2000.30. F.D. Nobre. Phase diagram of the two-dimensional ± J Ising spin glass.
Phys.Rev. E , 64:046108, 2001.31. H. Nishimori and M. Ohzeki. Multicritical point of spin glasses.
Phys. A: Stat.Mech. and its Appl. , 389:2907–2910, 2010.32. D. Andelman and A.N. Berker. q-state Potts models in d dimensions: Migdal-Kadanoff approximation. J. Phys. A: Math. Gen. , 14(4):L91, 1981.33. V.O. Ozcelik and A.N. Berker. Blume-Emery-Griffiths spin glass and invertedtricritical points.
Phys. Rev. E , 78:031104, 2008.34. L.R. da Silva, C. Tsallis, and G. Schwachheim. Anisotropic cubic lattice Pottsferromagnet: renormalisation group treatment.
J. Phys. A: Math. and Gen. ,17:3209, 1984.35. C. Tsallis and A.C.N. de Magalh˜aes. Pure and random Potts-like models: real-space renormalization-group approach.
Phys. Rep. , 268(5–6):305–430, 1996.36. A. Crisanti and L. Leuzzi. Stable solution of the simplest spin model for inversefreezing.
Phys, Rev. Lett. , 95:087201, 2005.37. M. Paoluzzi, L. Leuzzi, and A. Crisanti. Thermodynamic First Order Transitionand Inverse Freezing in a 3D Spin Glass.
Phys. Rev. Lett. , 104:120602, 2010.38. A. Crisanti, L. Leuzzi, and T. Rizzo. Complexity in mean-field spin glassmodels: The Ising p-spin.
Phys. Rev. B , 71:094202, 2005.39. L. Leuzzi, M. Paoluzzi, and A. Crisanti. Random Blume-Capel model on a cubiclattice: First-order inverse freezing in a three-dimensional spin-glass system.
Phys. Rev. B , 83:014107, 2011.40. M. Paoluzzi, L. Leuzzi, and A. Crisanti. The overlap parameter across aninverse first-order phase transition in a 3D spin-glass.
Philos. Mag. , 91:1966–1976, 2011.41. A.A. Migdal. Phase transitions in gauge and spin-lattice systems.
Zh. Eksp.Teor. Fiz. , 69:1457, 1975.42. J. Cardy.
Scaling and Renormalization in Statistical Physics . Cambridge Lec-ture Notes in Physics. Cambridge University Press, 1996.43. A.B. Harris and T.C. Lubemsky. Renormalization-Group Approach to theCritical Behavior of Random-Spin Models.
Phys. Rev. Lett. , 33:1540, 1974.44. D. Andelman and A.N. Berker. Scale-invariant quenched disorder and its sta-bility criterion at random critical points.
Phys. Rev. B , 29:2630–2635, 1984.45. A. Falicov and A.N. Berker. Tricritical and Critical End-Point Phenomenaunder Random Bonds.
Phys. Rev. Lett. , 76:4380–4383, 1996.46. B.W. Southern and A.P. Young. Real space rescaling study of spin glass be-haviour in three dimensions.
J. of Phys. C , 10:2179, 1977.547. O.R. Salmon and F.D. Nobre. Spin-glass attractor on tridimensional hierar-chical lattices in the presence of an external magnetic field.
Phys. Rev. E ,79:051122, 2009.48. A.N. Berker. Comment on “Spin-glass attractor on tridimensional hierarchicallattices in the presence of an external magnetic field”.
Phys. Rev. E , 81:043101,2010.49. A.J. Bray and M.A. Moore. Scaling theory of the random-field Ising model.
J.of Phys. C: Solid State Physics , 18(28):L927, 1985.50. M.S. Cao and J. Machta. Migdal-Kadanoff study of the random-field Isingmodel.
Phys. Rev. B , 48:3177–3182, 1993.51. A.N. Berker and S.R. McKay. Modified hyperscaling relation for phase transi-tions under random fields.
Phys. Rev. B , 33:4712–4715, 1986.52. A. Falicov, A.N. Berker, and S.R. McKay. Renormalization-group theory ofthe random-field Ising model in three dimensions.
Phys. Rev. B , 51:8266–8269,1995.53. A.A. Middleton and D.S. Fisher. Three-dimensional random-field Ising magnet:Interfaces, scaling, and the nature of states.
Phys. Rev. B , 65:134411, 2002.54. A.K. Hartmann. Critical exponents of four-dimensional random-field Isingsystems.
Phys. Rev. B , 65:174427, 2002.55. M. Ohzeki and H. Nishimori. Analytical evidence for the absence of spin glasstransition on self-dual lattices.
J. Phys. A: Math. Gen. , 42:332001, 2009.56. H. Nishimori.
Statistical Physics of Spin Glasses and Information Processing:An Introduction . Oxford University Press (Oxford), 2001.57. H. Nishimori and K. Nemoto. Duality and Multicritical Point of Two-Dimensional Spin Glasses.
J. Phys. Soc. Jpn. , 71:1198, 2002.58. Franz J. Wegner. Duality in Generalized Ising Models and Phase Transitionswithout Local Order Parameters.
J. of Math. Phys. , 12(10):2259–2272, 1971.59. F.Y. Wu and Y. K. Wang. Duality transformation in a many-component spinmodel.
J. of Math. Phys. , 17(3):439–440, 1976.60. J.M. Maillard, K. Nemoto, and H. Nishimori. Symmetry, complexity and mul-ticritical point of the two-dimensional spin glass.
J. Phys. A , 36:9799, 2003.61. K. Takeda and H. Nishimori. Self-dual random-plaquette gauge model and thequantum toric code.
Nuc. Phys. B , 686(3):377–396, 2004.62. H. Nishimori. Internal Energy, Specific Heat and Correlation Function of theBond-Random Ising Model.
Prog. Theor. Phys. , 66:1169, 1981.63. M. Hinczewski and A.N. Berker. Multicritical point relations in three dualpairs of hierarchical-lattice Ising spin glasses.
Phys. Rev. B , 72:144402, 2005.64. F.D.A. Aar˜ao Reis, S.L.A. de Queiroz, and R.R. dos Santos. Universality, frus-tration, and conformal invariance in two-dimensional random Ising magnets.
Phys. Rev. B , 60:6740–6748, 1999.65. Rajiv R.P. Singh and J. Adler. High-temperature expansion study of the Nishi-mori multicritical point in two and four dimensions.
Phys. Rev. B , 54:364–367,1996.66. Y. Ozeki and N. Ito. Multicritical dynamics for the +/- J Ising Model.
J. ofPhys. A , 31:5451, 1998.67. A.B. Harris. Effect of random defects on the critical behaviour of Ising models.
J. Phys. C: Sol. St. Phys. , 7:1671, 1974.68. E. Domany. Some results for the two-dimensional Ising model with competinginteractions.
J. of Phys. C , 12:L119, 1979.69. M. Ohzeki, C. K. Thomas, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado. Lack of universality in phase boundary slopes for spin glasses on selfdual lattices.
J. of Stat. Mech. , 2011.70. N. Kawashima and H. Rieger. Finite-size scaling analysis of exact ground statesfor +/-J spin glass models in two dimensions.
Europhys. Lett. , 39:85, 1997.71. J.T. Chayes, L. Chayes, D.S. Fisher, and T. Spencer. Finite-Size Scaling andCorrelation Lengths for Disordered Systems.
Phys. Rev. Lett. , 57:2999–3002,1986.72. W. Kinzel and E. Domany. Critical properties of random Potts models.
Phys.Rev. B , 23:3421–3434, 1981.673. D. Andelman and A. Aharony. Critical behavior with axially correlated randombonds.
Phys. Rev. B , 31:4305–4312, 1985.74. B. Derrida, H. Dickinson, and J. Yeomans. On the Harris criterion for hierar-chical lattices.
J. of Phys. A , 18(1):L53, 1985.75. S. Mukherji and S.M. Bhattacharjee. Failure of the Harris criterion for directedpolymers on hierarchical lattices.
Phys. Rev. E , 52:1930–1933, 1995.76. A. Efrat. Harris criterion on hierarchical lattices: Rigorous inequalities andcounterexamples in Ising systems.
Phys. Rev. E , 63:066112, 2001.77. A.J. Bray and M.A. Moore. Lower critical dimension of Ising spin glasses: anumerical study.
J. of Phys. C , 17(18):L463, 1984.78. F.D. Nobre. Real-space renormalization-group approaches for two-dimensionalGaussian Ising spin glass.
Phys. Lett. A , 250:163, 1998.79. A.J. Bray and M.A. Moore. Heidelberg Colloquium on Glassy Dynamics. InJ.L. van Hemmen, editor,
Lecture Notes in Phys. , volume 275, pages 121–153.Springer-Verlag, Berlin, 1987.80. A.K. Hartmann, A.J. Bray, A.C. Carter, M.A. Moore, and A.P. Young. Stiff-ness exponent of two-dimensional Ising spin glasses for nonperiodic boundaryconditions using aspect-ratio scaling.
Phys. Rev. B , 66:224401, 2002.81. M. Weigel and D. Johnston. Frustration effects in antiferromagnets on planarrandom graphs.
Phys. Rev. B , 76:054408, 2007.82. A. Erbas, A. Tuncer, B. Y¨ucesoy, and A.N. Berker. Phase diagrams andcrossover in spatially anisotropic d = 3 Ising, XY magnetic, and percolationsystems: Exact renormalization-group solutions of hierarchical models. Phys.Rev. E , 72:026129, 2005.83. A.L. Talapov and H.W.J. Bl¨ote. The magnetization of the 3D Ising model.
J.of Phys. A , 29(17):5727, 1996.84. B. Nienhuis and M. Nauenberg. First-Order Phase Transitions inRenormalization-Group Theory.
Phys. Rev. Lett. , 35:477–479, 1975.85. A. Pelissetto and E. Vicari. Critical phenomena and renormalization-grouptheory.
Phys. Rep. , 368:549–727, 2002.86. H.G. Katzgraber, M. K¨orner, and A.P. Young. Universality in three-dimensional Ising spin glasses: A Monte Carlo study.
Phys. Rev. B ,73(22):224432, 2006.87. T. J¨org and H.G. Katzgraber. Evidence for Universal Scaling in the Spin-GlassPhase.
Phys. Rev. Lett. , 101(19):197205, 2008.88. M. Hasenbusch, A. Pelissetto, and E. Vicari. Critical behavior of three-dimensional Ising spin glass models.
Phys. Rev. B , 78:214205, 2008.89. M. Blume, V.J. Emery, and R.B. Griffiths. Ising Model for the λ Transitionand Phase Separation in He -He Mixtures.
Phys. Rev. A , 4:1071–1077, 1971.90. M. Blume. Theory of the First-Order Magnetic Phase Change in UO . Phys.Rev. , 141:517, 1966.91. H.W. Capel. On the possibility of first-order phase transitions in Ising systemsof triplet ions with zero-field splitting.
Physica , 32:966, 1966.92. D. M. Saul, M. Wortis, and D. Stauffer. Tricritical behavior of the Blume-Capelmodel.
Phys. Rev. B , 9:4964, 1974.93. M. Deserno. Tricriticality and the Blume-Capel model: A Monte Carlo studywithin the microcanonical ensemble.
Phys. Rev. E , 56:5204, 1997.94. K.G. Chakraborty. Effective-field model for a spin-1 Ising system with dipolarand quadrupolar interactions.
Phys. Rev. B , 29:1454–1457, 1984.95. O.R. Baran and R.R. Levitskii. Reentrant phase transitions in the Blume-Emery-Griffiths model on a simple cubic lattice: The two-particle cluster ap-proximation.
Phys. Rev. B , 65:172407, 2002.96. A. Crisanti and L. Leuzzi. First-Order Phase Transition and Phase Coexistencein a Spin-Glass Model.
Phys. Rev. Lett. , 89:237204, 2002.97. A. Crisanti and F. Ritort. Intermittency of glassy relaxation and the emergenceof a non-equilibirum spontaneous measure in the aging regime.
Europhys. Lett. ,66:253, 2004.98. I. Puha and H.T. Diep. Random-bond and random-anisotropy effects in thephase diagram of the Blume-Capel model.