,
Detection of hidden structures for arbitrary scales in complex physical systems: Supplementary Information P. Ronhovde,1 S. Chakrabarty,1 M. Sahu,1 K. K. Sahu,2 K. F. Kelton,1 N. A. Mauro,1 and Z. Nussinov∗1, 3 1
Department of Physics, Washington University in St. Louis, Campus Box 1105, 1 Brookings Drive, St. Louis, MO 63130, USA 2 Metal Physics and Technology, ETH, 8093 Zurich, Switzerland 3 Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA
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2 I.
INFORMATION MEASURES
We reproduce the information measures used in the multi-scale analysis for reference here in the SI. A randomly selected node is in community a with a probability of P (a) = na /N where na is the number of nodes in community a and N the total number of nodes. If there are q communities in a partition A. The Shannon entropy is HA = −
q X na a=1
N
log2
na . N
(1)
The mutual information I(A, B) between partitions A and B (generally distinct replicas) is I(A, B) =
qA X qB X nab a=1 b=1
N
log2
nab N . na nb
(2)
qA and qB are the number of communities in partitions A and B, nab is the number of nodes of community a of partition A that are shared with community b of partition B, and nb is the number of nodes in community b of partition B. The variation of information VI between two partitions A and B is V I(A, B) = HA + HB − 2I(A, B)
(3)
where 0 ≤ V I(A, B) ≤ log2 N . The normalized mutual information NMI is N M I(A, B) =
2I(A, B) HA + HB
(4)
where 0 ≤ N M I(A, B) ≤ 1. A high value of NMI indicates strong agreement between different replicas. VI measures the disparity between different replicas, so a lower value of VI indicates better agreement. II.
SUPPLEMENTARY METHODS: MULTIRESOLUTION ALGORITHM DETAILS
Our Potts type Hamiltonian for community detection is q 1X X H= (Vij − v) [θ (v − Vij ) + γθ (Vij − v)] . 2 a=1
(5)
i,j∈Ca
Our multiresolution replica algorithm (MRA) quantitatively assesses the strength of structures in a model graph by measuring how strongly community divisions of multiple independent replicas agree with each other. A community detection algorithm is used to solve a set of s community partitions where a unique cluster membership is assigned to each node. The algorithm is: (1) Initialize the network. Initialize the model Hamiltonian’s connection matrix Vij . Select a starting resolution specified by γ0 = 100 or larger as necessary (graphs in the current text are weighted) . (2) Solve all replicas at the current resolution. For a given replica, inititialize it to a “symmetric” state of one node per cluster. Sequentially select a node and examine its energy change as if it were placed in each neighboring cluster. Immediately move the node to the community with the lowest energy decrease as calculated by Eq. (5). Continue moving nodes until no lower energy states are found. Merge any communities that will lower the energy of the system and repeat the node dynamics until no more moves or merges are found. Repeat for p independent optimization trials and select the lowest energy configuration as the best partition for that replica. Assign any overlapping nodes (see below). Repeat for all s replicas. (3) Calculate the replica information correlations. Use Eq. (1) to calculate the information for each replica and Eqs. (2) – (4) to calculate the information correlations between all pairs of replicas. (4) Iterate to the next resolution. Calculate the next Potts model weight. The easiest method is to use a geometric decrement γi+1 = γi ∆ where ∆ = 101/20 or another desired value. Return to step (2) until the system collapses into one community or until disjoint communities are detected. (5) Identify the strongest resolutions. Select the set of Potts model weights {γj } that exhibit the strongest and most stable information correlations (generally in terms of NMI and VI). This process examines the model network over essentially all network scales. In step (5), the most strongly correlated resolutions are the best divisions of the network. Each replica and trial randomly permutes the node order in the
3 a0 a1 a2 AA * * * AB 1.92 17.4 6.09 AC 2.38 8.96 -14.9 BB * * * BC 1.88 8.00 -3.42 CC * * *
a3 a4 a5 * * * 3.05 -4.68 3.48 3.11 -3.88 4.38 * * * 2.53 -1.25 3.00 * * *
TABLE I. Fit parameters for Eq. (7) obtained from fitting configuration forces and energies to ab-initio data58 . The same-species (*) data is replaced by a suggested potential derived from generalized pseudo-potential theory.
symmetric initialization step (2). Despite this symmetric initialization, the community partitioning algorithm yields different answers depending on the particular order of the nodes in the initial state because the dynamics sample different regions of the energy landscape. We can analyze networks with a scaling often given by O(L1.3 ps) where L is the number of edges in the model network. It is natural for elements of many systems to accept multiple community memberships of a single node (e.g., social networks). These “overlapping” assignments are likewise physically realistic for identifying contiguous clusters in a complex material. Thus, we apply additional overlapping node assignments to the partitions that are determined by the community detection algorithm above. Specifically, given the partition solution in step 2, we fix the community assignments so that the original best memberships cannot be changed. We then take each node in sequence and assign it to each community for which it will lower the community energy. Alternately, we remove the node from a community if the removal will lower the community energy. The process is repeated until no new assignments or removals are applied.
III.
SUPPLEMENTARY METHODS: MOLECULAR DYNAMICS SIMULATION
The molecular dynamics (MD) simulations were carried out with the “IMD” program with pair potentials which we will describe in section III A.
A.
Empirical pair-potentials
The pair potentials were obtained by fitting forces and energies to data from experimental and first principles samples where the same pairs were found. The same-species pair potential was finally replaced by the Al-Al potential suggested by the generalized pseudopotential theory.
B.
Potential validation
A starting configuration was made randomly with 1600 atoms with 88% Al, 7% Y and 5% Fe. It was ensured that this had the correct density. Then, the IMD program was used to evolve it at a temperature 1500K for a sufficiently long time. This was then quenched to room temperature and the structure factor profile for the room temperature configuration was compared to the structure factor profile obtained from x-ray diffraction experiments done with the same system. The same process was also carried out with using first principles simulation using the VASP program. The structure factor from the two simulations were also compared.
C.
Cooling routine
We obtained twelve configurations at 1500K by running the system for a sufficiently long time at that temperature and taking 120 snapshots at fixed intervals of 2.04ps. The last of these was quenched to 300K. Before that, the lengths of the basis vectors were reduced by 1% to account for the increase in density as the system in cooled and the system was allowed to settle to the new density by running it at the high temperature for another 10.18ps. 120 configurations at 300K were obtained by running this at 300K for a long time and taking snapshots at fixed intervals of 2.04ps. It
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FIG. 1. The multiresolution analysis of a square lattice with periodic boundary conditions. Neighbors have an initial weight of Aij = 1 and non-neighbors have an initial weight of Bij = 1. We further apply a “potential” shift of φ0 = 0.97995 in order to be consistent with our previous analysis on glasses. There are three distinct plateaus in the information measures. A depiction of the system at γ = 60 for the center plateau is shown in Fig. 2.
should be noted that these configurations are very similar to each other unlike the high temperature configurations which could be treated as independent.
IV.
BENCHMARKS: CRYSTALS, CRYSTALS WITH DEFECTS, AND SPIN SYSTEMS
Before applying our method to complex systems, we verified that it yielded sensible results in simple physical cases. Among others, we tested the following systems: (i) Lattices viewed as graphs, (ii) Lattices with defects, and (iii) Defects in spin models.
A.
Multiresolution application to lattice systems
We define several uniform lattices systems for the purpose of comparing the results to the model glasses where we use relatively small systems for presentation purposes. We would normally just treat respective model networks as unweighted networks, but here we wish to maintain a consistent analysis across all systems in the paper, so we further apply a “potential” shift φ0 which corresponds to the negative of the average weight over all pairs of nodes (any non-neighbor has a weight of Bij = 1). We allow zero energy moves for the lattice solutions and perform a more strenuous optimization using s = 20 replicas and t = 40 trials. These representative networks result in “imperfect” tilings of the favored local structures due to the constraints imposed by the perfect symmetry in the Hamiltonian and by the local solution dynamics. In the depictions, different colors represent distinct clusters (best viewed in color) and edges between clusters are made partially transparent as a visualization aid. No overlapping nodes are assigned in the lattice depictions.
1.
Square lattice
We define a uniform, initially unweighted, square lattice with N = 400 nodes. Edges are assigned to each neighbor in the x and y directions with periodic boundary conditions. The “potential” shift is φ0 = 0.97995. We then perform the same multiresolution analysis on the graph as in the previous systems. In Fig. 1, we see that where are three dominant plateaus in the information measures.
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FIG. 2. A partition of a square lattice with periodic boundary conditions. The corresponding multiresolution plot is seen in Fig. 1. We use the algorithm described in the paper at γ = 60 to solve the system. To aid in visualization of the clusters, neighbor links not in the same cluster are made partially transparent. In this configuration, there were q = 120 clusters with 78 squares, 4 triads, and 38 dyads which indicates that square configuration dominates the partition, and it shows how our algorithm can naturally identify the basic unit cells of the square lattice.
Except for the plateaus, the overall multiresolution pattern resembles the LJ system in Fig. V A, except for the presence of the plateaus in the lattice plot, which suggests that the LJ system is “more ordered” than the metallic glass model. However, a purely random graph40 also shows a similar pattern, including a plateau, which indicates that there may exist an analogy between purely “random” and perfectly “ordered” systems in our analysis. However, these data are not alone sufficient to be conclusive. We select a configuration at γ = 60 corresponding to the center plateau. The lattice is depicted in Fig. 2 with q = 120 clusters of 78 squares, 4 triads, and 38 dyads. At this resolution, the square dominates the configuration which shows that our algorithm is able to isolate the basic unit cells of the lattice in a natural fashion. The plateau for γ & 100 corresponds to essentially all dyads, and the plateau for γ . 30 corresponds to a mixture of dyads, squares, and tight six-node configurations (a square plus two adjacent nodes). The lower γ plateau favors the six-node configuration in terms of the cluster energies, but the larger features are even more difficult with which to tile the lattice than for squares.
2.
Triangular lattice
Similar to the square lattice we define a uniform, initially unweighted, triangular lattice with N = 400 nodes. Edges are assigned to each triangular neighbor using periodic boundary conditions. The “potential” shift is φ0 = 0.969925. We again perform the same multiresolution analysis on the graph with the results shown in Fig. 3. There are three dominant plateaus in the information measures; and the overall multiresolution pattern resembles both the square lattice and the LJ systems except for the presence of the plateaus here. We select a configuration at γ = 50 corresponding to the “left” plateau. The lattice is depicted in Fig. 4 with q = 104 clusters of 17 triads, and 86 “diamonds”, and 1 five-node cluster. The five node cluster is a result of a preference being given for an isolated node to join a diamond as opposed to forming its own single-node cluster. At this resolution, the diamond configuration dominates. Two plateaus to the right of γ & 65 are both strongly dominated by triads of nodes. The distinction between the two is that the central plateau favors an isolated node joining a triangle, to form a rare diamond, rather than forming its own single-node cluster. Together, the different plateaus show that our algorithm is able to isolate the basic unit cells of the lattice in a natural fashion.
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FIG. 3. The multiresolution analysis of a triangular lattice with periodic boundary conditions. Neighbors have an initial weight of Aij = 1 and non-neighbors have an initial weight of Bij = 1. We further apply a “potential” shift of φ0 = 0.969925 in order to be consistent with our previous analysis on glasses. There are three distinct plateaus in the information measures, but the latter two are closely related (see text). A depiction of the system at γ = 50 for the left plateau is shown in Fig. 4.
FIG. 4. A partition of a triangular lattice with periodic boundary conditions. The corresponding multiresolution plot is seen in Fig. 3. We use the algorithm described in the paper at γ = 50 (the “left” plateau) to solve the system. To aid in visualization of the clusters, neighbor links not in the same cluster are made partially transparent. In this configuration, there were q = 104 clusters with 17 triads, 86 “diamonds,” and 1 five-node configuration (see text) which indicates that the diamond configuration dominates the partition. Our algorithm can naturally identify the different basic unit cells of the lattice. By varying φ0 (and the corresponding γ) we detect natural structures on varying scales.
3.
Cubic lattice
We further define a uniform, initially unweighted, cubic lattice with N = 1000 nodes. Edges are assigned to each neighbor in the x, y, and z directions using periodic boundary conditions. The “potential” shift is φ0 = 0.987 988. We perform the same multiresolution analysis on the graph as in the previous systems where the results are summarized in Figs. 5 and 6. Out of q = 177 clusters, we identified 66 squares, 51 six-node configurations (square plus two adjacent nodes), 45 cubes, and 15 other assorted configurations smaller than cubes. At γ ' 50, the cube is the preferred cluster in terms of the cluster energy, but they consist of only slightly more than 25% of the clusters because the large cube configuration is more difficult to identify due to the perfect network symmetry and local constraints imposed by the evolution of
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FIG. 5. The multiresolution analysis of a cubic lattice with periodic boundary conditions. Neighbors have an initial weight of Aij = 1 and non-neighbors have an initial weight of Bij = 1. We further apply a “potential” shift of φ0 = 0.987 988 in order to be consistent with our previous analysis on glasses. There are three distinct plateaus in the information measures. The leftmost short plateau is the cube preferred resolution (in terms of cluster energy) with 45 out of 177 clusters are cubic clusters (no configurations larger than cubes are found). A depiction of the system at γ = 50 for the left plateau is shown in Fig. 6.
FIG. 6. A partition of a cubic lattice with periodic boundary conditions. The corresponding multiresolution plot is seen in Fig. 5. We use the algorithm described in the paper at γ = 50 (the short “left” plateau) to solve the system. To aid in visualization of the clusters, neighbor links not in the same cluster are made partially transparent. In this configuration, there were q = 177 clusters with 66 squares, 51 six-node configurations, 45 cubes, and 15 other assorted configurations. The cubic cluster is the preferred partition (in terms of cluster energy), but it is difficult to fill the system with cubes in practice due to the perfect symmetry of the network.
the community structure during the algorithm dynamics. The “middle” plateau represents a square-dominated region (201 out of 294 clusters are squares), and the “right” plateau consists of dyads of neighbor nodes almost exclusively.
8 B.
LJ Lattice with defects
In Figs. 7 and 8, we examine a monatomic LJ system (whose ground state is a triangular lattice with inter-particle spacing given by the LJ minimum) in which defects in the form of vacancies were inserted. The system is broken into clusters such that the defects tend to congregate on boundaries between different clusters. C.
Ising spin systems
We investigated Ising systems. In Eq. (5), we set Vij to be the nearest neighbor bond energies in the Ising model. The multi-resolution analysis led to a cascade of structures similar to that in (i) up to the largest domain wall. Partitions into nearly perfect Ising domain walls sharply corresponded to VI maxima (and NMI minima). This occurs as the region near the domain walls is the one which experiences the largest fluctuations in possible assignment to the two bordering domains that it delineates. These benchmark results above are reassuring in that the systems “find” their natural structures. V.
APPLICATIONS TO COMPLEX AMORPHOUS SYSTEMS
We studied amorphous systems via different approaches starting with that of community detection In particular, in analyzing complex structures we allowed in some cases for a multiple membership of a node in different communities by further replicating each node so as to enable overlapping partitions. We used both interaction energies as well as measured pair correlations as weights in our analysis. We investigated three different cases: (i) Dynamic structures in the KA LJ glass, (ii) Static atomic structures from RMC directly applied to experimental measurements of Zr80 Pt20 , and (iii). Dynamic structures in a new model amorphous system that emulates experimental results on Al88 Y7 Fe5 (main text). The precise forms of the potentials in (i) and (iii) are less important in the broader context of the method. At low temperatures, we found larger and more pronounced compact structures than those at higher temperature. A.
Binary Kob-Andersen LJ analysis
We studied the glass-forming KA 80:20 binary liquid48,49 by MD50 to simulate an N = 2000 atom system. The pair potentials are given by σαβ 12 σαβ 6 − φαβ (r) = 4αβ (6) r r α or β designate the atomic type A or B, respectively. The dimensionless units are AA = 1.0, AB = 1.5, BB = 0.5, σAA = 1.0, σAB = 0.80, and σBB = 0.88, in agreement with the KA values. We initialized the system at a temperature T = 5 (in the units of48,49 ) and evolved it for a time that is long compared to the caging time. We saved p high temperature configurations separated by time intervals ∆t of the order of the caging time48 . The system is then rapidly quenched to T = 0.01 which is well below the glass transition temperature for the KA-LJ system. We then evolved this lower temperature and saved p configurations separated by the original time interval ∆t. Each of the p copies of the system constitutes a replica within the high/low temperature problem. We employed these replicas in our multi-resolution analysis, and the results for the low temperature system are shown in Figs. 9 and 11. Stable and compact clusters correspond to the NMI maxima. The corresponding high temperature case is shown in Fig. 10 which shows a lower correlation than the low temperature simulation. B.
Ternary Al88 Y7 Fe5 model glass
The potentials for our ternary model glass are of the form h a a1 i a2 0 V (r) = + a5 cos (a3 r + a4 ) , r r
(7)
with the parameters {ai } depending on the specific types of atom pairs (i.e., Al-Al, Al-Y, ...). See the main text for the full discussion where we solved the clusters at a community detection temperature of TCD = 0.
9 As mentioned briefly in the main text, we could solve for the partition at a temperature TCD corresponding to the simulation temperature. In this case, the cluster memberships should have an energy significantly lower than the thermal fluctuations on average. Figure 12 shows a set of sample clusters where we solved the clustering problem at TCD = 300 K. These clusters are very similar to the TCD = 0 K clusters. The TCD = 1500 K on the high temperature system are much more dispersed.
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Correspondence and requests for materials should be addressed to Z.N. (
[email protected]) Ronhovde P, et al. (2011) Detecting hidden spatial and spatio-temporal structures in glasses and complex systems by multiresolution network clustering. arXiv:cond-mat/1102.1519. Anderson PW (1995) Science 267:1610. Miracle DB, Sanders WS, Senkov ON (2003) The influence of efficient atomic packing on the constitution of metallic glasses. Philos. Mag. 83:2409–2428. Miracle DB (2004) A structural model for metallic glasses. Nature Materials 3:697–702. Schenk T, Holland-Moritz D, Simonet V, Bellissent R, Herlach DM (2002) Icosahedral short-range order in deeply undercooled metallic melts. Phys. Rev. Lett. 89:075507. Kelton KF, et al. (2003) First x-ray scattering studies on electrostatically levitated metallic liquids: Demonstrated influence of local icosahedral order on the nucleation barrier. Phys. Rev. Lett. 90:195504. Bellisario A (2009) Miracle metal: How glassy metals will define manufacturing in the 21th century. Industry Week Dec, 9. Lubchenko V, Wolynes PG (2007) Theory of structural glasses and supercooled liquids. Annu. Rev. Phys. Chem. 58:235–266. Angelani L, Parisi G, Ruocco G, Viliani G (2000) Potential energy landscape and long-time dynamics in a simple model glass. Phys. Rev. E 61:1681–1691. Parisi G (2000) The physics of the glass transition. Physica A 280:115–124. Sastry S, Debenedetti PG, Stillinger FH (1998) Signatures of distinct dynamical regimes in the energy landscape of a glass-forming liquid. Nature 393:554–557. Debenedetti PG, Stillinger FH (2001) Supercooled liquids and the glass transition. Nature 410:259–267. Lubchenko V, Wolynes PG (2004) Theory of aging in structural glasses. J. Chem. Phys. 121:2852–2865. Berthier L, Biroli G, Bouchaud JP, Jack RL (2010) Overview of different characterizations of dynamic heterogeneity. arXiv:cond-mat/1009.4665. Kirkpatrick TR, Thirumalai D, Wolynes PG (1989) Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Phys. Rev. A 40:1045–1054. Tarzia M, Moore MA (2007) Glass phenomenology from the connection to spin glasses. Phys. Rev. E 75:031502. G¨ otze W (1999) Recent tests of the mode-coupling theory for glassy dynamics. Journal of Physics: Condensed Matter 11:A1. Mayer P, Miyazaki K, Reichman DR (2006) Cooperativity beyond caging: Generalized mode-coupling theory. Phys. Rev. Lett. 97:095702. Garrahan JP (2002) Glassiness through the emergence of effective dynamical constraints in interacting systems. Journal of Physics: Condensed Matter 14:1571. Nussinov Z (2004) Avoided phase transitions and glassy dynamics in geometrically frustrated systems and non-abelian theories. Phys. Rev. B 69:014208. Tarjus G, Kivelson SA, Nussinov Z, Viot P (2005) The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessment. J Phys.: Condens. Matter 17:R1143–R1182. Falk ML, Langer JS (1998) Dynamics of viscoplastic deformation in amorphous solids. Phys. Rev. E 57:7192–7205. Kivelson D, Kivelson SA, Zhao X, Nussinov Z, Tarjus G (1995) A thermodynamic theory of supercooled liquids. Physica A: Statistical and Theoretical Physics 219:27 – 38. Nelson DR (2002) Defects and Geometry in Condensed Matter Physics (Cambridge University Press). Sadoc JF, Mosseri R (1999) Geometrical Frustration (Cambridge University Press). Ritort F, Sollich P (2003) Glassy dynamics of kinetically constrained models. Adv. Phys. 52:219–342. Aharonov E, et al. (2007) Direct identification of the glass transition: Growing length scale and the onset of plasticity. Euro. Phys. Lett. 77:56002. Boolchand P, Lucovsky G, Phillips JC, Thorpe MF (2005) Self-organization and the physics of glassy networks. Philosophical Magazine 85:3823–3838. Berthier L, Biroli G (2010) A theoretical perspective on the glass transition and nonequilibrium phenomena in disordered materials. arXiv:cond-mat/1011.2578. Chandler D, Garrahan JP (2007) Dynamics on the way to forming glass: Bubbles in space-time. Ann. Rev. Phys. Chem. 61:191–217. Montanari A, Semerjian G (2006) Rigorous inequalities between length and time scales in glassy systems. Journal of Statistical Physics 125:23–54. Tanaka H, Kawasakia T, Shintani H, Watanabe K (2010) Critical-like behaviour of glass-forming liquids. Nature Materials 9:324–331. Berthier L, et al. (2005) Direct experimental evidence of a growing length scale accompanying the glass transition. Science 310:1797–1800. Bouchaud JP, Biroli G (2004) On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses. The Journal of Chemical Physics 121:7347–7354. Kurchan J, Levine D (2009) Correlation length for amorphous systems. arXiv:cond-mat.dis-nn/0904.4850. Sheng HW, Luo WK, Alamgir FM, Bai JM, Ma E (2006) Atomic packing and short-to-medium-range order in metallic glasses. Nature 439:419–425. Honeycutt JD, Andersen HC (1987) Molecular dynamics study of melting and freezing of small Lennard-Jones clusters. J.
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Phys. Chem. 91:4950–4963. Steinhardt PJ, Nelson DR, Ronchetti M (1983) Bond-orientational order in liquids and glasses. Phys. Rev. B 28:784–805. Newman MEJ (2008) The physics of networks. Phys. Today 61:33–38. Ronhovde P, Nussinov Z (2009) Multiresolution community detection for megascale networks by information-based replica correlations. Phys. Rev. E 80:016109. Lancichinetti A, Fortunato S (2009) Community detection algorithms: a comparative analysis. Phys. Rev. E 80:056117. Ronhovde P, Nussinov Z (2010) Local resolution-limit-free Potts model for community detection. Phys. Rev. E 81:046114. Hu D, Ronhovde P, Nussinov Z (2010) Phase transition in the community detection problem: spin-glass type and dynamic perspectives. arXiv:cond-mat/1008.2699. Meil˘ a M (2007) Comparing clusterings – an information based distance. J. Multivariate Anal. 98:873–895. Danon L, D´ıaz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J. Stat. Mech.: Theory Exp. 9:P09008. Arenas A, Fern´ andez A, G´ omez S (2008) Analysis of the structure of complex networks at different resolution levels. New J. Phys. 10:053039. Fortunato S (2010) Community detection in graphs. Physics Reports 486:75 – 174. Kob W, Andersen HC (1995) Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van hove correlation function. Phys. Rev. E 51:4626–4641. Valdes LC, Affouard F, Descamps M, Habasaki J (2009) Mixing effects in glass-forming Lennard-Jones mixtures. The Journal of Chemical Physics 130:154505. Stadler J, Mikulla R, Trebin HR (1997) IMD: A software package for molecular dynamics studies on parallel computers. International Journal of Modern Physics C 8:1131–1140. Nakamura T, et al. (2002) Structural study in amorphous Zr-noble metal (Pd, Pt and Au) alloys. Journal of Non-Crystalline Solids 312-314:517 – 521. Saida J, et al. (2009) Evaluation of the local environment for nanoscale quasicrystal formation in Zr 80 Pt 20 glassy alloy using Voronoi analysis. Journal of Physics: Condensed Matter 21:375104. Sordelet D, et al. (2008) Structure of Zrx P t100−x (73 ≤ x ≤ 77) metallic glasses. Metallurgical and Materials Transactions A 39:1908–1916. Wang SY, et al. (2008) Short- and medium-range order in a Zr73 P t27 glass: Experimental and simulation studies. Phys. Rev. B 78:184204. McGreevy RL (1991) Understanding liquid structures. Journal of Physics: Condensed Matter 3:F9. Keen D, McGreevy R (1990) Structural modeling of glasses using reverse Monte Carlo simulation. Nature 344:423–425. Kim TH, Kelton KF (2007) Structural study of supercooled liquid transition metals. The Journal of Chemical Physics 126:054513. Mihalkoviˇc M, Henley CL, Widom M, Ganesh P (2008) Empirical oscillating potentials for alloys from ab-initio fits. arXiv:cond-mat.mtrl-sci/0802.2926. http://cms.mpi.univie.ac.at/vasp/. Kresse G, Hafner J (1993) Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47:558–561. Kresse G, Furthm¨ uller J (1996) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54:11169–11186. Sahu K, et al. (2010) Phase separation mediated devitrification of Al8 8Y7 Fe5 glasses. Acta Materialia 58:4199 – 4206. Perez-Garcia D, Verstraete F, Wolf MM, Cirac JI (2007) Matrix product state representations. Quantum Information and Computation 7:401–430. Amir A, Oreg Y, Imry Y (2011) Electron glass dynamics. Annual Review of Condensed Matter Physics 2:235–262. Sachdev S (2011) Quantum Phase Transitions (Cambridge University Press). Varma CM, Nussinov Z, van Saarloos W (2002) Singular or non-Fermi liquids. Physics Reports 361:267 – 417. Valla T, et al. (1999) Evidence for quantum critical behavior in the optimally doped cuprate Bi2 Sr2 CaCu2 O8+δ . Science 285:2110–2113. Marel Dvd, et al. (2003) Quantum critical behaviour in a high-Tc superconductor. Nature 425:271–274. Park T, et al. (2005) Novel dielectric anomaly in the hole-doped La2 Cu1−x Lix O4 and La2−x Srx NiO4 insulators: Signature of an electronic glassy state. Phys. Rev. Lett. 94:017002. J. Zaanen, Z. Nussinov, and S. I. Mukhin, (2004) Duality in 2+1D quantum elasticity: superconductivity and quantum nematic order Annals of Physics 310, 181–260 V. Cvetkovic, Z. Nussinov, and J. Zaanen, (2006) Topological kinematic constraints: dislocations and the glide principle Philos. Mag. 86:2995–3020
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FIG. 7. From1 . A MRA analysis of a 2D triangular LJ lattice with periodic boundary conditions. The legend for the information theory quantities is as in earlier plots. Edges are weighted according the LJ potential. There are two preferred regions, a small peak on the left (see Fig. 8) and a large plateau on the right, where the peak here corresponds to the largest possible “natural” clusters.
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FIG. 8. From1 . We use the algorithm at γ ' 31.6 (the left peak) in Fig. 7. Our method generally places defects near the boundaries of the communities in order to minimize the energy cost.
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1 .0
(a ) I
(ia ) N
8
4 6
3
I
0 .8 q
I
N
2
q
0 .6
2
0 .2
2
1
0 .0
0 0
8
4 6
3 4
2
1 0
-1
1 0
0
1 0
(b )
1 0
1
1 0
2
γ
1 0
3
1 0
4
1 0
4
I 0 .4
1 0
5
V H q
8
2
2 1 0 0
1 0
4
H
V
2
q
6
(ib ) 0 1 0
-1
1 0
0
1 0
1
1 0
γ
2
1 0
3
1 0
4
1 0
5
FIG. 9. A MRA plot for a three dimensional KA LJ system (see text) at T = 0.01 (in LJ energy units). The information overlaps show a maximum NMI (IN ) and plateaux for the other information theory measures at (i). In Fig. 11, we show the typical clusters found, and in Fig. 10, we show the corresponding plot for the high temperature system system.
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1 .0
(a ) I
N
8
4 6
3
I
0 .8 q
I
N
2
q
0 .6
2
0 .2
2
1
0 .0
0 0
8
4 6
3 4
2
1 0
-1
1 0
0
1 0
1
1 0
2
γ
1 0
3
1 0
4
1 0
5
(b )
1 0
1 0
4
I 0 .4
8
V 2
1 0 0
H 2
1 0
4
H
V
2
q
6
q 0 1 0
-1
1 0
0
1 0
1
1 0
γ
2
1 0
3
1 0
4
1 0
5
FIG. 10. A MRA plot for a high temperature three dimensional KA LJ system at T = 5 (in LJ energy units). The information overlaps show a maximum NMI (IN ) and plateau for other information theory measures at (i) that is lower than the corresponding point in Fig. 9.
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FIG. 11. The figure depicts a set of sample clusters found for the information overlaps at (i) in Fig. 9.
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FIG. 12. A set of sample clusters solved at The corresponding partition for the AlYFe system at γ = 0.001 (largest, green → Y; median size, silver → Al; smallest, red → Fe).
