Scattering Properties of Paramagnetic Ground States in the Three-Dimensional Random-Field Ising Model
Gaurav P. Shrivastav, Siddharth Krishnamoorthy, Varsha Banerjee, Sanjay Puri
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Scattering Properties of Paramagnetic Ground States in theThree-Dimensional Random-Field Ising Model
Gaurav P. Shrivastav , Siddharth Krishnamoorthy , Varsha Banerjee and Sanjay Puri School of Physical Sciences, Jawaharlal Nehru University, New Delhi – 110067, India. Department of Physics, Indian Institute of Technology,Hauz Khas, New Delhi – 110016, India.
Abstract
We study the ground-state ( T = 0) morphologies in the d = 3 random-field Ising model (RFIM)using a computationally efficient graph-cut method. We focus on paramagnetic states which arisefor disorder strengths ∆ > ∆ c , where ∆ c is the critical disorder strength at T = 0. Theseparamagnetic states consist of correlated “domains” of up and down spins which are separatedby rough, fractal interfaces. They show novel scattering properties with a cusp singularity in thecorrelation function at short distances. PACS numbers: 64.60.De - Statistical mechanics of model systems: Ising model, Monte Carlo techniques,etc.; 68.35.Rh - Phase transitions and critical phenomena; 75.60.Ch - Domain walls and domain structure random-field Ising model (RFIM) is an archetypal exam-ple of a system with quenched disorder and is described by the Hamiltonian [1, 2]: E = − J X h ij i σ i σ j − N X i =1 h i σ i , σ i = ± . (1)Here, J > { h i } are random fields, usually drawn from a Gaussian distribution whose standarddeviation ∆ is a measure of disorder. The phase diagram of the RFIM has been the subjectof much discussion. In the 2-dimensional case ( d = 2), there is no ⁀ long-range order in thepresence of disorder, no matter how small. However, in d = 3, there is a small region of( T, ∆)-values where the equilibrium phase is ferromagnetic [3, 4]. Let us focus on the casewith zero temperature ( T = 0). In that case, the system exhibits a phase transition from aferromagnetic phase (for ∆ < ∆ c ) to the paramagnetic phase (for ∆ > ∆ c ). The nature ofthis transition has received considerable attention [5–7]. An important study of the d = 3RFIM is due to Middleton and Fisher [8]. They studied a wide range of physical propertiesand convincingly demonstrated that there is a second-order phase transition at ∆ = ∆ c .At T = 0, all the information about the system is encoded in the ground-state. Further,according to the zero-temperature fixed point hypothesis , transitions at T = 0 and T = 0 arein the same universality class [9, 10]. Therefore, a study of the ground-state morphologyis important in understanding the RFIM phase diagram in d = 3. A typical method ofaccessing the ground-state is via Monte Carlo (MC) evolution (e.g., Metropolis [11], Simu-lated annealing [12], etc.) from an arbitrary initial condition. However, MC approaches fordisordered systems suffer from several drawbacks. First, the competition between exchangeinteractions and the random field introduces deep valleys in the free-energy landscape. Thesemetastable states trap the evolving system and impede the relaxation to the ground-state.The system then opts for a local minimum , which can be far removed from the global min-imum , and may not reflect any of its properties. Further, as the MC techniques involve ∼ order (1) spin-flip at a time, the possibility of escape from a local minimum to the globalminimum is small. Second, MC methods suffer from a non-polynomial (NP) divergence ofcomputation time with system size. Thus, it is computationally very demanding to reach2he global minimum for large systems with disorder. Consequently we still do not have acomplete understanding of the nature of the ground-state.To address this problem, several optimization techniques based on “max-flow/min-cut”or “graph-cuts” have been developed for a wide class of energy functions (or Hamiltonians)of binary variables [13–15]. The basic approach in a graph-cut method (GCM) is to constructa specialized graph for the energy function to be minimized such that the minimum cut onthe graph also minimizes the energy. The cut enables simultaneous relabeling of severalspin variables or nodes. As a consequence, an exponentially large portion of the phase spacecan be sampled in a single move, thereby facilitating a quick search for a global minimumor a “good-quality” local minimum. Typically, the search time in these procedures hasa polynomial dependence on the system size. An important class of energy functions arethose which are (a) quadratic and (b) satisfy a “regularity” condition. In that case, themax-flow/min-cut technique actually yields the global minimum or exact ground state ofthe energy function in polynomial time [16–18]. The Hamiltonian of the RFIM specified inEq. (1) belongs to this class [19]. We are therefore assured of reaching the exact groundstate of the RFIM if energy minimization is via graph-cuts.The literature on combinatorial optimization provides many graph-cut algorithms withdifferent polynomial complexity times. Some of the standard approaches include the Ford-Fulkerson (FF) method of augmenting paths [13], the Goldberg-Tarjan (GT) push-relabelmethod [14], and the more recent Boykov-Kolmogorov (BK) method [15]. A benchmarkingof the above algorithms on a number of typical graphs has revealed that the BK methodworks several times faster than any other. While the FF and GT algorithms exhibit an N -dependence on the system size N , the BK method is linear in N [15].In this paper, we use the BK method to study the ground-state ( T =0) morphologiesin the d = 3 RFIM. This enables us to access exact ground states for substantially largersystem sizes than in previous studies [8]. We need these large sizes to obtain smooth datafor statistical properties of the morphology, e.g., correlation function, structure factor, etc.We focus on the scattering properties of the domain structure in the paramagnetic state, i.e.,for field strengths ∆ > ∆ c . This domain morphology has several non-trivial features, whichwe highlight in this paper. The correlation function C ( r, ∆) (= h σ i σ j i − h σ i ih σ j i with r = | ~r i − ~r j | ) is a scaling function of r/ξ (∆), where the correlation length ξ diverges as ∆ → ∆ + c .At small values of r/ξ , C ( r, ∆) exhibits a cusp singularity characterized by the roughness3xponent α : C ( r, ∆) ≃ − A ( r/ξ ) α + · · · . This singularity has important consequencesfor the high-momentum behavior (“tail”) of the structure factor. A similar cusp has beenreported earlier in the context of fluctuation-dominated phase separation [20, 21], and is aconsequence of soft and ragged interfaces separating equilibrium phases. In the paramagneticphase of the RFIM, there are no coexisting equilibrium phases. Nevertheless, there existcorrelated domains of size ∼ ξ which are enriched in up or down spins. The scatteringproperties of these domain boundaries are analogous to those of fractal interfaces. We alsoprovide accurate estimates of the critical point ∆ c and the correlation length exponent ν calculated from the ground-state morphologies.Before presenting our results, we discuss the graph-cut approach, which has many po-tential applications for energy minimization in complex spin systems. This method can beapplied to energy functions of the form: E ( { s i } ) = X { ij }∈N V ij ( s i , s j ) + X i ∈S D i ( s i ) . (2)The label s i of site i ∈ S can take a value 0 or 1, and the sites are related to one another bya well-defined neighborhood N . The function D i measures the cost of assigning the label s i to the site i , and V ij ( s i , s j ) measures the penalty (or cost) of assigning labels s i and s j toadjacent sites i and j .The starting point in a GCM is to construct a specialized graph for the energy function E such that the minimum cut on the graph yields minimization of the energy. A graph G isan ordered pair of disjoint sets ( V , E ), where V is the set of vertices and E is the set of edges.An edge ij joining vertices i and j is assigned a weight V ij . A cut C is a partition of thevertices V into two sets R and Q . Any edge ij ∈ E with i ∈ R and j ∈ Q (or vice-versa) isa cut edge. The cost of the cut is defined to be the sum of the weights of the edges crossingthe cut. The minimum-cut problem is to find the cut with the smallest cost.The energy function E must satisfy the regularity condition for it to be graph-representable. Regularity is defined by the inequality V ij (0 , V ij (1 , ≤ V ij (1 , V ij (0 , σ i = ±
1) in Eq. (1) can be transformed into occupation-numbervariables ( n i = 0 ,
1) through the transformation n i = (1 + σ i ) /
2. Then, neglecting constantterms, E ( { n i } ) = − J X h ij i n i n j − N X i =1 ( h i − qJ ) n i , n i = 0 , , (3)4here q denotes the number of nearest neighbors of a lattice site. It is straightforward tocheck that the interaction term of Eq. (3) satisfies the regularity condition. Thus, the energyfunction E of Eq. (1) is graph-representable and can be minimized using a GCM to yieldthe exact ground state. Each iteration of the GCM finds an optimal subset of nodes with afixed label s i (= 0 or 1) that gives the largest decrease in energy. This computation is donevia graph-cuts on the specialized graph representing the energy function E . The algorithmrepeatedly cycles through the labels s i until the global minimum is reached.Our simulations of the T = 0 RFIM have been performed on d = 3 lattices of size L ( L ≤ L = 256. Theinitial configuration of the lattice is chosen to be a random mix of σ i = ±
1, correspondingto the paramagnetic state at ∆ = ∞ with ξ = 0. The results have been averaged over 100sets of { h i } for each value of ∆. Our studies indicate that the GCM has a 99% overlap withthe ground-state in the first iteration itself, provided the disorder strength is not too closeto the critical value ∆ c . (We do observe “critical slowing down” in the GCM as ∆ → ∆ c ,but the phenomenon is much milder than in conventional MC methods.) We also find thatthe average energy per spin in the ground-state is an order of magnitude less than thatobtained by the Metropolis algorithm. As mentioned earlier, the MC evolution invariablygets trapped in high-energy metastable states.The ground-state morphology in the paramagnetic state has the following features. Asthe disorder strength is reduced from ∆ = ∞ , there is emergence of correlated regions ordomains of size ξ , enriched in either up or down spins (see snapshots in Fig. 1 for ∆ = 2.4,2.6, 2.8). These regions grow in size with ξ → ∞ as ∆ → ∆ + c . [A similar divergence of ξ is seen for ∆ = 0, T → T + c . However, in that case, a typical MC snapshot (see Fig. 2)shows that the domain morphology is not as compact or well-defined as in Fig. 1. Fig. 2corresponds to T = 4 . T c = 4 . c ( T = 0) ≃ . ± . < ∆ c , the ground-state morphology consists of a singlelarge domain of ordered spins (up or down) with small clusters of oppositely-directed spins.The fraction of oppositely-directed spins decreases as ∆ → C ( r, ∆). The correlation5ength ξ (∆) is defined as the distance over which C ( r, ∆) decays to (say) 0 . × maximumvalue. In Fig. 3(a) we plot ξ ( δ, L ) vs. δ [where δ = (∆ − ∆ c ) / ∆ c ] for system sizes L rangingfrom 16 to 256. The correlation length diverges as ξ ∼ δ − ν when δ → + , but this is limitedby the lattice size L . The presence of these finite-size effects can be used to estimate thecritical exponent ν . The finite-size scaling ansatz ξ ( δ, L ) = δ − ν f ( Lδ ν ) results in the datacollapse seen in Fig. 3(b), yielding ν ≃ . ± . ν = 1 . ± . ν = 1 . ± .
09 (Middleton and Fisher[8]).If the system is characterized by a single length scale, the morphology of the domainsdoes not change with ∆, apart from a scale factor. In that case, the correlation functionexhibits scaling: C ( r, ∆) = g ( r/ξ ) [25]. This is verified in Fig. 4, where we plot C ( r, ∆)vs. r/ξ for different disorder amplitudes ∆ > ∆ c . The data collapse for different values of∆ is excellent, confirming that the morphologies are scale-invariant.Next, we turn our attention to the central theme of this paper, viz., the scattering prop-erties of the domain morphology in Fig. 1. The scattering of a plane wave by a rough surfacecan yield useful information about the texture of the surface [26–28]. Thus, small-angle scat-tering experiments (using X-rays, neutrons, etc.) can be used to probe the nature of domainwalls separating the components of an inhomogeneous system. These experiments yieldthe structure factor S ( k, ∆), which is the Fourier transform of the correlation function.Experimentalists are interested in the large- k (tail) behavior of S ( k, ∆), which is deter-mined by the small- r (short-distance) behavior of C ( r, ∆). In the inset of Fig. 4, we plot1 − C ( r, ∆) vs. r/ξ on a log-log scale. The small- r behavior shows a distinct cusp singularity: C ( r, ∆) ≃ − A ( r/ξ ) α + · · · with α ≃ .
5. This holds over more than a decade in r/ξ -values.A similar cusp has been reported earlier also in the context of fluctuation-dominated phase-separation (FDPS) [20, 21]. The cusp exponent α is identical to the roughness exponent of the domain boundaries. [Notice that the value we obtain for the paramagnetic phase( α para ≃ .
5) differs considerably from the roughness exponent in the ferromagnetic phase.The Middleton-Fisher value for the latter exponent is α ferro = 0 . ± .
03, which is consistentwith the theoretical result α ferro = 2 / d f = d − α . Therefore, in our present study, d f ≃ .
5, which is consistent withstudies of percolation clusters in the strong-disorder regime of the d = 3 RFIM by Seppalaet al. [31] and Ji and Robbins [32]. 6n Fig. 5, we show a schematic of a domain of size ξ , with an interface of width w . Thereis also a microscopic length scale a = 1, due to the underlying discreteness of the lattice. Thecorresponding C ( r, ∆) would show corrections to scaling, characterized by the parameter w/ξ . Systems characterized by a cusp singularity exhibit very rough interfaces with w ∼ ξ .A novel feature of the present work is that we observe this scattering phenomenology inthe paramagnetic phase of the RFIM, i.e., in the absence of interfaces between coexistingequilibrium phases. The lower frames in Fig. 1 show cross-sections of the d = 3 snapshots inthe upper frames. Notice that the domain structure is fuzzy and subject to large fluctuations,and the boundaries are ill-defined.At larger values of x = r/ξ , the correlation function is well-approximated as C ( r, ∆) ≃ − A ( r/ξ ) α − B ( r/ξ ) + · · · . (4)The linear decay in Eq. (4) is characteristic of scattering from sharp interfaces in inhomoge-neous systems, and is termed the Porod law [33]. With reference to the schematic in Fig. 5,the correlation function C ( r, ∆) exhibits (a) no systematic structure for r ∼ a ; (b) interfacialstructure or cusp singularity for w ≫ r ≫ a ; (c) Porod decay for ξ ≫ r ≫ w. The short-distance cusp singularity in C ( r, ∆) has important implications for the struc-ture factor S ( k, ∆). The scattered intensity now decays with an asymptotic power-law form[26–28] S ( k, ∆) ∼ ˜ A ( ξk ) − ( d + α ) + ˜ B ( ξk ) − ( d +1) , (5)valid for k ≪ a − . For k ∼ a − , the structure factor becomes flat, corresponding to theabsence of structure at microscopic scales. The dominant large- k behavior in Eq. (5) is S ( k ) ∼ ( ξk ) − ( d + α ) with cross-over momentum k c ∼ ξ − .In Fig. 6, we plot the structure factor for the RFIM with ∆ > ∆ c on a log-log scale.Our data is consistent with the scaling form in Eq. (5). There is a cross-over from a Porodregime (cid:2) with S ( k, ∆) ∼ k − ( d +1) (cid:3) at intermediate values of k to an asymptotic cusp regime (cid:2) with S ( k, ∆) ∼ k − ( d + α ) , α ≃ . (cid:3) . The inset of Fig. 6 shows the behavior of the cross-overmomentum, which scales as k c ∼ ξ − . We conjecture that the crossover is a generic featurein the RFIM as a consequence of interfacial roughening caused by quenched disorder [34]. Inthis context, we consider Refs. [35, 36] where the authors studied domain growth in the d = 3RFIM. They focused on the nonequilibrium evolution of the system after a quench from theparamagnetic phase (∆ = ∞ ) to the ferromagnetic phase (∆ < ∆ c ). Refs. [35, 36] observe7hat the scaling functions of the RFIM and the pure Ising system are identical, therebyexhibiting super-universality [37–39] or irrelevance of quenched randomness. However, acareful observation of the scaled correlation data (Fig. 2 in [35]) for small r/ξ reveals cleardeviations from the pure system and so from the Porod law. [Of course, we expect torecover the Porod law in the limit w/ξ → . This is possible in the domain growth problemas ξ ( t ) → ∞ as t → ∞ . ]We conclude this paper with a summary and discussion of our results. We have useda computationally efficient graph-cut method (GCM) to study the ground-state ( T = 0)properties of the RFIM in the paramagnetic state. The Boykov-Kolmogorov GCM used byus provides access to the ground-state morphology of large systems. We characterize thismorphology using correlation functions and structure factors, which contain information av-eraged over all domains and interfaces. The correlation function C ( r, ∆) is characterized bya universal scaling function for different disorder amplitudes. There are no perceptible cor-rections to scaling for different values of ∆, suggesting that the interface thickness w scaleswith the correlation length ξ . At short distances, C ( r, ∆) shows a cusp singularity reminis-cent of that seen in fluctuation-dominated phase separation [20, 21]. This is associated withscattering off rough, fractal interfaces. The corresponding structure factor S ( k, ∆) shows acrossover from a Porod regime at intermediate k values, to an asymptotic cusp regime . Theseproperties should be universal for disordered systems, which are often characterized by roughinterfaces. We believe that our results will motivate further analytical and numerical studiesof this problem.GPS and VB would like to acknowledge the support of DST Grant No. SR/S2/CMP-002/2010. We thank S.N. Maheshwari and Chetan Arora for fruitful discussions. We arealso grateful to Uma Mudenagudi and Olga Veksler for technical support in programming.8
1] T. Nattermann and J. Villain, Phase Transitions , 5 (1988).[2] T. Nattermann, in Spin Glasses and Random Fields , edited by A.P. Young (World Scientific,Singapore, 1998).[3] J.Z. Imbrie, Phys. Rev. Lett. , 1747 (1984); Commun. Math. Phys. , 145 (1985).[4] J. Bricmont and A. Kupiainen, Phys. Rev. Lett. , 1829 (1987).[5] M. Mezard and A.P. Young, Europhys. Lett. , 653 (1992); M. Mezard and R. Monasson,Phys. Rev. B , 7199 (1994).[6] H. Rieger, Phys. Rev. B , 6659 (1995).[7] J.C. Angles d’Auriac and N. Sourlas, Europhys. Lett. , 473 (1997); N. Sourlas, Comput.Phys. Commun. , 184 (1999).[8] A.A. Middleton and D.S. Fisher, Phys. Rev. B , 134411 (2002).[9] J. Villain, Phys. Rev. Lett. , 1543 (1984).[10] D.S. Fisher, Phys. Rev. Lett. , 416 (1986).[11] N. Metropolis, A.W. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. , 1087 (1953).[12] S. Kirkpatrick, C.D. Gelatt, Jr. and M.P. Vecchi, Science , 671 (1983); S. Kirkpatrick, J.Stat. Phys. , 975 (1984).[13] L. Ford and D. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962).[14] A.V. Goldberg and R.E. Tarjan, J. ACM , 921 (1988).[15] Y. Boykov and V. Kolmogorov, IEEE Transactions on PAMI , 1124 (2004).[16] C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization (Prentice-Hall, New York,1982).[17] V. Kolmogorov and R. Zabih, IEEE Transactions on PAMI , 147 (2004).[18] J.C. Picard and H.D. Ratliff, Networks , 375 (1975).[19] J.C. Angles d’Auriac, M. Preissmann and R. Rammal, J. Phys. (France) Lett. , L173 (1985).[20] D. Das and M. Barma, Phys. Rev. Lett. , 1602 (2000); M. Barma, Eur. Phys. J. B , 387(2008).[21] S. Mishra and S. Ramaswamy, Phys. Rev. Lett. , 090602 (2006).[22] M.E. Fisher, Rep. Prog. Phys. , 615 (1967).[23] K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduc- ion , Fourth Edition (Springer-Verlag, Berlin, 2002).[24] H. Rieger and A.P. Young, J. Phys. A , 5279 (1993).[25] Kinetics of Phase Transitions , edited by S. Puri and V.K. Wadhawan (Taylor and Francis,Boca Raton, 2009).[26] H.D. Bale and P.W. Schmidt, Phys. Rev. Lett. , 596 (1984); P.-Z. Wong and A.J. Bray,Phys. Rev. Lett. , 1344 (1988).[27] P.-Z. Wong, Phys. Rev. B , 7417 (1985).[28] P.-Z. Wong and A.J. Bray, Phys. Rev. B , 7751 (1988).[29] Y. Imry and S.K. Ma, Phys. Rev. Lett. , 1399 (1976).[30] T. Halpin-Healy, Phys. Rev. A , 711 (1990).[31] E.T. Seppala, A.M. Pulkkinen and M.J. Alava, Phys. Rev. B , 144403 (2002).[32] H. Ji and M.O. Robbins, Phys. Rev. B , 14519 (1992).[33] G. Porod, in Small-Angle X-Ray Scattering , edited by O. Glatter and O. Kratky (AcademicPress, New York, 1982); Y. Oono and S. Puri, Mod. Phys. Lett. B , 861 (1988).[34] D.A. Huse and C.L Henley, Phys. Rev. Lett. , 2708 (1985).[35] M. Rao and A. Chakrabarti, Phys. Rev. Lett. , 3501 (1993).[36] C. Aron, C. Chamon, L.F. Cugliandolo and M. Picco, J. Stat. Mech. P05016 (2008).[37] S. Puri, D. Chowdhury and N. Parekh, J. Phys. A , L1087 (1991); S. Puri and N. Parekh,J. Phys. A , 4127 (1992); S. Puri and N. Parekh, J. Phys. A , 2777 (1993).[38] A. J. Bray and K. Humayun, J. Phys. A , L1185 (1991).[39] R. Paul, S. Puri and H. Rieger, Europhys. Lett. , 881 (2004); Phys. Rev. E , 061109(2005). IG. 1: Ground-state morphologies of the RFIM obtained using the α -expansion GCM, for disorderstrengths ∆ = 2.4, 2.6 and 2.8. The snapshots in the top frames correspond to a 64 lattice withperiodic boundary conditions in all directions. Regions with up spins and down spins are markedblack and grey. The domains shrink in size and interfaces roughen with increasing disorder, as isevident from the cross-sections (taken at z = 32) in the bottom frames. x FIG. 2: Equilibrium morphology of the disorder-free Ising paramagnet for T = 4 . T c ≃ . latticewith periodic boundary conditions. The system is evolved from an arbitrary initial condition to itsequilibrium state, where the morphology is invariant with time. The above snapshot correspondsto a cross-section (at z = 32) of a 64 lattice. -2 -1 δ ξ ( δ , L ) L = 16L = 32L = 64L = 128L = 256 -2 -1 L δ ν -2 -1 δ ν ξ ( δ , L ) L = 16L = 32L = 64L = 128L = 256
FIG. 3: (a) Plot of correlation length ξ ( δ, L ) vs. δ , where δ = (∆ − ∆ c ) / ∆ c . We present data forcubic lattices of size L = 16, 32, 64, 128, 256 - denoted by the specified symbols. The data sets wereaveraged over 100 random-field configurations. The correlation length is defined as the distanceover which the correlation function falls to 0 . × maximum value. (b) Data collapse resultingfrom the finite-size scaling ansatz ξ ( δ, L ) = δ − ν f ( Lδ ν ), yielding the correlation length exponent ν ≃ . ξ C ( r , ∆ ) ∆ = 2.4∆ = 2.6∆ = 2.8∆ = 3.0 -1 r/ ξ -1 - C ( r , ∆ ) FIG. 4: Scaled correlation function [ C ( r, ∆) vs. r/ξ ] for specified disorder strengths. The numer-ical data has been averaged over 100 random-field configurations for a lattice of size 256 . Theinset shows the small- r/ξ behavior on a log-log scale to highlight the cusp singularity. The slopeof the solid line yields the cusp exponent α ≃ . w ξ FIG. 5: Schematic of a domain of size ξ . The characteristic interface thickness is w , and themicroscopic lattice spacing a = 1. -1 k10 S ( k , ∆ ) ∆ = 2.4∆ = 2.6∆ = 3.0 ξ −1 k c -4 -3.5 FIG. 6: Structure factor [ S ( k, ∆) vs. k ], corresponding to the correlation functions in Fig. 4. Thesolid lines denotes a Porod regime (cid:2) S ( k, ∆) ∼ k − (cid:3) at intermediate k -values, which crosses overto an asymptotic cusp regime (cid:2) S ( k, ∆) ∼ k − . (cid:3) . The inset shows the behavior of the cross-overmomentum k c vs. ξ − for several values of disorder.for several values of disorder.