Topology-driven phase transitions in the classical monomer-dimer-loop model
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Topology-driven phase transitions in the classical monomer-dimer-loop model
Sazi Li , Wei Li , , ∗ and Ziyu Chen , † Department of Physics, Beihang University, Beijing 100191, China Physics Department, Arnold Sommerfeld Center for Theoretical Physics,and Center for NanoScience, Ludwig-Maximilians-Universit¨at, 80333 Munich, Germany Key Laboratory of Micro-nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China (Dated: August 13, 2018)In this work, we investigate the classical loop models doped with monomers and dimers on a square lattice,whose partition function can be expressed as a tensor network (TN). In the thermodynamic limit, we use theboundary matrix product state technique to contract the partition function TN, and determine the thermodynamicproperties with high accuracy. In this monomer-dimer-loop model, we find a second-order phase transitionbetween a trivial monomer-condensation and a loop-condensation (LC) phases, which can not be distinguishedby any local order parameter, while nevertheless the two phases have distinct topological properties. In theLC phase, we find two degenerate dominating eigenvalues in the transfer-matrix spectrum, as well as a non-vanishing (nonlocal) string order parameter, both of which identify the topological ergodicity breaking in theLC phase and can serve as the order parameter for detecting the phase transitions.
PACS numbers: 64.60.Cn, 05.50. + q, 05.10.Cc, 64.60.F- Introduction.—
Two dimensional (2D) monomer-dimermodel has a quite venerable history in statistical mechanics[1–4]. The monomer-dimer model can be used to describe theabsorption of molecules on the surface: the molecule can oc-cupy two nearest neighboring sites and form a dimer, whilethe empty site is regarded as a monomer [1]. The monomer-dimer model can also be related to other statistical models likeIsing and height models [2, 5], etc, thus it plays the role as aquite fundamental statistical model. On a square lattice, thefully packed dimer model is found to possess algebraic decay-ing dimer-dimer correlation, however, doping the system withmonomers will drive the system out of the criticality and nophase transition occurs in a non-interacting monomer-dimermodel at finite temperatures [6–8]. On the other hand, if oneintroduces pairing interactions between the dimers, there existphase transitions between the low- T ordered phase and high- T disordered one (Kosterlitz-Thouless type for fully packedcase [6], and second-order after monomer doping [7, 8]).Loop models are also widely studied in statistical mechan-ics, which is relevant for realistic physical systems and alsoconstitutes a quite fundamental mathematical problem [9, 10].The loop structure also plays an important role in certainquantum cases, like in the ground state of toric code [11], thestring-net model [12], and the resonating A ffl eck-Kennedy-Lieb-Tasaki loop spin liquid states [13], etc. In Ref. 14,Castelnovo and Chamon couple the toric code model to a ther-mal bath, and consider the thermal superposition of all possi-ble loop coverings. They found that the concept of topologi-cal order also applies in this classical loop system, where thelow-energy phase space decompose into several distinct topo-logical sectors. The existence of distinct topological sectorsbreaks ergodicity. One needs to create / annihilate a loop withlength propositional to system size, which has huge energycost and rare probability to happen, in order to tunnel fromone sector to another, it thus leads to the topological glassbehavior [15]. The notion of topological entropy can be gen-eralized to detect such nontrivial topological order in classi- cal systems, by noticing that the topological constraint wouldalso reduce the entropy in the classical case [14]. Recently,Hermanns and Trebst have generalized this entropy character-ization to general classical string-nets and verify that there arecorresponding universal topological corrections in the Renyientropy for a number of SU(N) k anyonic theories [16].In this work, we combine the two classical models and in-troduce a monomer-dimer-loop (MDL) model on a square lat-tice. The MDL model has a rather compact tensor network(TN) representation with a small bond dimension ( D = Model and method.—
Snapshots of several classical con-figurations in di ff erent phases of MDL model are shown inFig. 1. Summing over all possible classical configurations,we have the partition function Ξ = X { c } exp [ − β ( µ N m + ν N b + uN d )] , (1)where { c } means the set of all classical monomer-dimer-loopconfigurations, N m is the total number of vertices occupied bya monomer, which has an energy of µ ; N b counts the numberof edges occupied by a loop, and ν is the energy per bond ofa loop; N d is the total number of vertices linked by a dimer(with energy per dimer as 2 u ). In the following, ν = π/ (a)(c) (b)(d) A B
FIG. 1: (Color online) Snapshots of the classical phases of the MDLmodel on a square lattice: (a) The trivial monomer-condensation,(red) dots are the monomers, A , B label the two sub-lattices; (b,c) Loop condensation in the monomer-loop and monomer-dimer-loop cases, respectively; (d) Loop condensation in the branchingmonomer-loop model (i.e., classical string-nets). T (a) contracttensors w TTT (b) M TTTT
FIG. 2: (Color online) (a) TN representation of the partition function,on a π/ M on a cylindrical geometry by contracting a column of ( w )rank-4 tensors. The dashed lines denote the contractions betweentwo tensors. lattice, which represents the partition function Z , as shownin Fig. 2(a). The partition function TN consists of tensors T s , s , s , s located at each vertex, which has four indices s i ( i ∈ { , , , } ) corresponding to the four geometric bonds.Each index has a finite bond dimension ( D = s i ∈{ , , } : s i = s i = s i . We properly initiate the tensor T , to makesure that a specific lattice site is either occupied by a loop[ T , , , = T , , , = T , , , = T , , , = T , , , = T , , , = exp ( − ν/ T )], a monomer [ T , , , = exp ( − µ/ T )], or by a dimer[ T , , , = T , , , = T , , , = T , , , = exp ( − u / T )], and therest elements are zero (forbidden).To calculate the thermodynamics of the MDL model, onehas to e ffi ciently (and accurately) contract the partition func-tion TN for calculating free energy per site f , energy per site e and other thermodynamic quantities. In this work, we definethe system on two kinds of geometries: for the infinitely large2D lattice system, we adopt the iTEBD method [21, 22] foraccurate contractions; for the cylindrical geometry with finite (small) circumferences and infinite length, we diagonalize thespectrum of the transfer matrix and evaluate properties with itsdominating eigensystems. iTEBD was initially proposed fore ffi cient simulations of the time evolution and the ground stateproperty (through imaginary-time evolution) of 1D quantumsystems, and then generalized to calculate the thermodynam-ics of 2D classical statistical models [22] and also 1D quantumlattice models [23]. In our practical simulations, we performthe contraction of MPS with transfer MPO until the prescribedconvergence criterion is reached, say, free energy per site con-verges to 10 − (almost machine precision). The total numberof iterations is around 10 ∼ , depending on the temperaturesand the physical parameters of the model. The retained bonddimension of the boundary MPS D c ≈ D c is always checked, the truncation error is less than10 − at the critical point, and reaches the machine precisionaway from the critical points. Monomer-Loop model.—
In the partition function Eq. (1),if we forbid the dimer occupation (i.e., u → ∞ ), the modelis reduced to a monomer-loop model, which can be related tothe well-known Ising model. For instance, the triangular lat-tice Ising model can be mapped to a monomer-loop model onits dual honeycomb lattice, where the loops are the magneticdomain walls separating spins which have opposite orienta-tions, and the monomers are the topological excitations ontop of that [15]. In our present model, we treat directly themonomer-loop picture, and thus can tune monomer energy µ continuously (with fixed ν = µ = , ± .
2. The specific heat C V curve is shown in Fig. 3(a),which is computed by taking first-order derivative (versus T ) of the energy per site. The latter is calculated via e = Z ∗ / Z , where Z ∗ is obtained by contracting the TN with one T tensor replaced with an impurity tensor T I = µ T , , , + ν ( T , , , + T , , , + T , , , + T , , , + T , , , + T , , , ). InFig. 3(a), divergent peaks appear at T c ≈ . , . , . µ = . , , − .
2, respectively, suggesting the presence ofsecond-order phase transitions.In addition, to confirm the existence of the phase transi-tion, we calculate the correlation length ξ of the monomer-loop model via the following formula ξ = / ln( λ λ ) , (2)where λ ( λ ) is the largest (second-largest) eigenvalue of thetransfer matrix M (in case the largest eigenvalues are n -folddegenerate, λ is the n + ξ also shows a divergent peak at T c ,confirming the criticality at the transition point.However, interestingly, we find no local order parametersfor detecting this phase transition, since both the high- andlow- T phases are disordered and have no symmetry breaking. C V D c = 20D c = 40D c = 60D c = 80 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.190500100015002000 T ξ D c = 40D c = 60D c = 80D c = 100(b) µ =0 ν =1 µ =0.2 µ =−0.2 (a) µ =0 FIG. 3: (Color online) (a) The specific heat C V of the monomer-loop model with µ = , ± . ν =
1, where T c ’s correspond to0 . , . , .
39 respectively. (b) The correlation length ξ versustemperature T of the classical loop model. The heights of the peaksat T c grow by increasing D c . We show the numerical results of bond density ( n A and n B )in Fig. 4 (a), which counts the average bonds (of the loops)per site. The results (with µ =
0) are shown in Fig. 4(a), fromwhich we can see that the low- T phase has relatively low bonddensity and thus can be regarded as monomer-condensation(MC), while the high- T region is a loop-condensation (LC)phase. Although n A and n B change from zero to nonzero val-ues when T increases, they change smoothly through the tran-sition point. In addition, the same bond densities n A = n B areobserved for all temperatures T , which suggests that the sym-metry between two sub-lattices is also intact. Therefore, weconclude that the bond density n can not serve as a local orderparameter for distinguishing two phases. Besides, in Fig. 4(a),we also show the bond density n = .
602 944 603 316 996(with 16 converged significance digits) in the T = ∞ limit,where the state is an equal-weight (classical) superposition ofall possible monomer-loop configurations. Compared to thedimer density n d = .
638 123 109 228 547 in monomer-dimermodel [8], we find that n > n d here.We also investigate the entropy of the system, including twokinds of entropies, i.e., the conventional thermodynamic en-tropy S = ( U − F ) / T and the formal “entanglement entropy” S E evaluated from the boundary MPS. The latter can be ob-tained by S E = − P i Λ i ln( Λ i ), where Λ i ’s are the Schimidtspectrum of the decomposition on a bond i . As shown inFig. 4(b), the bipartite entanglement entropy S E of the clas-sical loop model shows a clearly divergent peak at T c , indi-cating the occurrence of a phase transition. On the contrary,the conventional thermodynamic entropy S is smooth around T c . However, its first-order derivative has a divergent peak, asshown in the inset of Fig. 4(b), which is not surprising since ∂ S ∂ T = C V T . Therefore, we see that this “entanglement entropy” S E is more sensitive to the phase transition (than the thermo-dynamic entropy S ), and serves as a very useful numericaltool detecting phase transitions. Similar behaviors have al-ready been seen in our previous tensor-network study of themonomer-dimer model [8].In order to further investigate the phase transitions, we alsodefine the MDL model on cylindrical geometries. On thecylinders with finite circumferences (and infinite length), we n n A n B −8 −6 −4 −2 T n n A n B S S E S T d S / d T µ =0 ν =1 µ =0 ν =1 (a) (b) n T →∞ =1.602 944 603 316 996 FIG. 4: (Color online) (a) The mean bond occupation number n A and n B on the vertices A and B in the monomer-loop model ( µ = , ν = T = ∞ limit. The inset amplifies and shows the low temperature behaviors.(b) The entropy S and the entanglement entropy S E of the classicalloop model with µ = , ν =
1. Inset shows the first-order derivativeof entropy dS / dT . can contract the TN exactly and thus obtain the thermody-namic quantities. Specifically, we start from both ends, andcontract the boundary vectors with the transfer matrix M con-sisting of a column of rank-4 tensors [see Fig. 2(b)]. Repeat-ing the contraction process until both boundary vectors con-verge, with which we can evaluate the observables like the en-ergy expectation values. As the cylinder widths increase, theobservables should eventually converge to the thermodynamiclimit results obtained with iTEBD contractions above.We calculate the (normalized) gap of the transfer matrix δ = | λ − λ λ | for various cylinder widths w , which is shown inFig. 5(a). In the T > T c region, we observe two degeneratedominating eigenstates in the spectrum of transfer-matrix M .In particular, as shown in the inset of Fig. 5(a), δ extrapolatesto zero at critical point T = T c , in the thermodynamic limit.Therefore, δ can be taken as an order parameter detecting thephase transition between the low- and high- T phases.We are also interested in the parity of the dominating eigen-vector χ of transfer matrix M . Since the loops are closed inthe MDL model, M conserves the parity symmetry. When thecylinder is cut into two halves vertically, the number of inter-sected bonds by the cut is either even or odd, which defines theparity of eigenvector χ . For cylinders with open ends (i.e., nodangling bonds on the edges), all the allowed configurationsconstitute the even sector, and the dominating eigenvector χ e in this sector is with even parity. On the other hand, if weintroduce odd number of open strings on the cylinder, stretch-ing from the very left boundary to the rightmost side, thenall the allowed configurations constitute an odd sector, withdominating eigenvector χ o of odd parity. Fig. 5(a) shows thatthe dominating even and odd eigenvectors ( χ e and χ o ) becomedegenerate when T > T c .Furthermore, we introduce a string operator Θ windingaround the cylinder to measure the parity of eigenvectors χ .The string operator Θ = Q wi = P i is a product of the operator P = − ! (3)which lives on the horizontal edge [20]. P = −
1) if the δ w=7w=9w=11w=13 0 0.05 0.1 0.15 0.2 0.2500.050.10.150.2 1/w δ calculatedfitting Θ o w=7w=8w=11w=12 Θ e w=12 µ =0 ν =1(a) (b) µ =0 ν =1 FIG. 5: (Color online) Topological characterization of the monomer-loop model for various cylinder widths w . (a) The gap of the transfermatrix δ vanishes when T > T c , the inset shows that the extrapolated δ ∼ w = ∞ limit. (b) The odd string order parameter Θ o iszero when T ≤ T c and nonzero T > T c ; Θ e also changes its behaviorat T c . δ δ µ =0.2 µ =0.2 fitting µ =−0.2 µ =−0.2 fitting Θ o w=7w=8w=11w=12 Θ e w=12 (b) µ =0.2 µ =−0.2(a) µ =0.2 µ =−0.2 µ =−0.2 FIG. 6: (Color online) The topological properties of the monomer-loop model with µ = ± . ν = w . (a) The gap of transfer matrix δ vanishes when T > T c , the insetshows δ extrapolates to zero at T c . (b) The string order parameters Θ o and Θ e . edge is not occupied (occupied by a bond). Therefore, theexpectation value of the product of P tells whether the sys-tem is in the even or odd sector. In Fig. 5(b), we thread anopen string in the cylinder, and show the numerical resultsof Θ o = | χ o Θ M χ ∗ o / ( χ o M χ ∗ o + χ e M χ ∗ e ) | for various cylinderwidths, where the partition function χ e ( χ o ) is the even(odd)dominating eigenvectors. We observe that Θ o is a constantzero for T < T c , and becomes nonzero when T > T c . In themeantime, we also show the even Θ e = | χ e Θ M χ ∗ e / ( χ o M χ ∗ o + χ e M χ ∗ e ) | , which is a constant one in the trivial MC phase,while also changes its behavior at T c . Thus, Θ o can also betaken as an order parameter for detecting the phase transition,called nonlocal string order parameter.In addition to µ = µ = ± .
2, includingthe (normalized) gap δ in Fig. 6(a) and SOP in Fig. 6(b). Sim-ilar behaviors can be seen as in the µ = µ , collect the phasetransition points of the monomer-loop model with various pa-rameters µ , and obtain the µ − T phase diagram. When µ < T phases. T c decreases with increasing monomerenergy until µ =
1, where T c =
0, i.e., no phase transitions.Here we would like to address some remarks on the classi-cal topological order in the monomer-loop model. As a con- −8 −6 −4 −2 0 1 2024681012 µ T c second−ordertransitionLCMC FIG. 7: (Color online) The phase diagram µ − T of the monomer-loop model. The disordered monomer-condensation (MC) and topo-logically ordered loop-condensation (LC) phases are separated by asecond-order transition line when µ ∈ ( −∞ , sequence of the loop condensation [Fig. 1(b)], there exist twodegenerate eigenvectors χ in the high- T phase, meaning thatthe phase space is decomposed into two distinct topologicalsectors, which leads to a topological ergodicity breaking. Onegets exactly the same results by evaluating the thermodynamicquantities in either sector, while it is not possible to shift fromone sector to the other by changing the loop configurationsonly locally. This glassy behavior of the LC phase is due totopological reasons, therefore the LC phase can also be calleda topological glass [15], and the phase transition between MCand LC phases is thus a topology-driven transition. Monomer-Dimer-Loop model.—
In the monomer-loopmodel, the dimer occupation was not allowed (i.e., e ff ectively u = ∞ in Eq. 1). The dimer can be regarded as the “minimal”loop of length two [shown in Fig. 1 (c)]. In the following,we switch on dimer coverings, and study the full MDL modelwith u = µ = , ν = C V , the correlation length ξ , bond density n andthe entropy (the thermodynamic entropy S and the entangle-ment entropy S E ) of the system [20]. C V , ξ , S E and the first-order derivative of the thermodynamic entropy all show a di-vergent peak at T c ≈ . T c and n A = n B .Thus the bond density again can not serve as a local orderparameter.The normalized gap δ and SOP Θ o in the MDL model areshown in Fig. 8. In Fig. 8(a), we again see a twofold degen-eracy in the transfer matrix spectrum in the LC phase, indi-cating that the LC phase is also topologically ordered in theMDL model. The inset of Fig. 8(a) shows δ extrapolates tozero (in the thermodynamic limit) at T c . In Fig. 8(b), Θ o , e areshown, in which Θ o is nonvanishing in the LC phase, suggest-ing that it can also identify the classical topological order inthe MDL model. In summary, similar to the monomer-loopcase, the classical topological order exists in the LC phase ofthe general MDL model and the second-order phase transitionseparates the trivial disordered MC and topologically orderedLC phases. δ and Θ o can be used as order parameters to char-acterize the topology-driven classical phase transition well. δ w=5w=6w=7w=8 δ calculatedfitting Θ o w=5w=6w=7w=8 Θ e w=8 δ w= ∞ =0.008 (a) µ =0 ν =1 u=3 µ =0 ν =1u=3 (b) FIG. 8: (Color online) The topological properties of the MDL modelwith µ = , ν = , u = w . (a) The gapof transfer matrix δ serves as an order parameter; Inset: δ at T c forvarious cylinder widths and their extrapolation. (b) The string orderparameter Θ o and Θ e . Conclusion and outlook.—
In this work, we have system-atically studied the classical loop model with monomer anddimer doping. Using the boundary MPS contraction method,we evaluate the partition function TN and obtain the thermo-dynamic properties including the specific heat C V , correlationlength ξ and entropies. There exist second-order phase tran-sitions separating the trivial monomer-condensation and theloop-condensation phases, which can not be described by thelocal order parameters like bond density n . However, in theLC phase, we find twofold degenerate dominating eigenval-ues in the transfer matrix spectrum, one in even and the otherin odd topological sectors. The existence of two topologicalsectors actually breaks the ergodicity. The non-vanishing non-local order parameter SOP Θ o can also be used to distinguishtwo sectors and thus detect the phase transition. Therefore,these two phases can be identified by their distinct topologicalproperties, and the phase transition between them belongs toa topology-driven type.Besides the closed loop cases studied in the MDL modelabove, it is also interesting to consider the model with thebranching loops [see Fig. 1(d)], i.e., the classical string-netmodel. It is quite straightforward to generalize the tensor-network representation here to the classical string-nets, andour preliminary calculations show that there also exists asecond-order phase transition between LC and MC phases.However, owing to the existence of branching loops, thetransfer-matrix breaks the parity symmetry and no longer hasthe well-defined even and odd topological sectors as the MDLmodel has here. Our study of the classical string-nets will bepublished elsewhere. ACKNOWLEDGEMENT
W.L. acknowledges Hong-Hao Tu for helpful discussions.This work was supported in part by the National Natural Sci-ences Foundation of China (Grants No. 11274033, and No.11474015), Major Program of Instrument of the National Nat-ural Sciences Foundation of China (Grant No. 61227902),Sub Project No. XX973 (XX5XX), and the Research Fund for the Doctoral Program of Higher Education of China (GrantNo. 20131102130005 ). W.L. further acknowledges supportby the DFG through Grant No. SFB-TR12 and Cluster of Ex-cellence NIM. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] J. K. Roberts, Some properties of adsorbed films of Oxygenon Tungsten, Proc. R. Soc. London, Ser. A , 464 (1935);R. H. Fowler and G. S. Rushbrooke, An attempt to extend thestatistical theory of perfect solutions, Trans. Faraday Soc. ,1272 (1937).[2] H. N. V. Temperley and M. E. Fisher, Dimer problem in sta-tistical mechanics-an exact result, Philos. Mag. 6, 1061 (1961);M. E. 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TENSOR NETWORK REPRESENTATION
In this part, we introduce the partition function TN representation of these models and the contraction method of the nonlocalstring order parameter Θ o , e .Since each vertex is allowed to be covered by at most one loop or a “thick” dimer in the MDL model, there are elevennonzero elements in the vertex tensor T . The allowed nonzero tensor elements T s , s , s , s and their corresponding classicalloop configurations are schematically shown in Fig. S1 (a-k). T , , , = exp( − βµ ) corresponds to the absence of any loop,i.e., a monomer [Fig. S1 (a)], T , , , = T , , , = T , , , = T , , , = T , , , = T , , , = exp( − βν ) represent the one-loopconfigurations [Fig. S1 (b-g)], T , , , = T , , , = T , , , = T , , , = exp( − β u ) describe the vertex with a dimer [Fig. S1 (h-k)],and T , , , = T , , , = T , , , = T , , , = exp( − βν ) introduce the branching loop configurations [Fig. S1 (l-o)].In Fig. S2, we show the way of evaluating Θ o , e on a cylindrical geometry. Denominator is the partition function, and thenumerator measures the expectation value of Θ in the odd(even) sector. SPECIFIC HEAT, CORRELATION LENGTH, AND ENTROPY RESULTS OF A MONOMER-DIMER-LOOP MODEL
For the MDL model with µ = , ν = , u =
3, we also calculate its thermodynamic properties: the specific heat C V , the corre-lation length ξ , the bond density n and the entropy that includes the traditional thermodynamic entropy S and the entanglemententropy S E . In this part, we show the results as follows.Fig. S3 shows our computed specific heat C V of the MDL model with the energy u = µ = , ν = C V occurs at T c ≈ . ξ of the MDL model. From Fig. S4, we can also observe a divergent peak at T c , thesecond-order phase transition point.The bond densities per site n A and n B on vertexes A and B in the MDL model with µ = , ν = , u = n A and n B are changing smoothly around T c and n A is equal to n B for all temperatures, meaning that the symmetry betweentwo sub-lattices in the system is not broken.The thermodynamic entropy S and the entanglement entropy S E of the MDL model with µ = , ν = , u = S E shows a divergent peak at T c , while S is smooth around T c , and its singularity can only be seen after taking afirst-order derivative over T , which is shown in the inset of Fig. S6. T
00 00 s s s s (a)
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10 11 s s s s (m) T
11 10 s s s s (n) T
11 0 1
11 01 s s s s (o) FIG. S1: (Color online) The eleven (allowed) nonzero elements of the vertex tensor T s , s , s , s in the partition function TN representation andtheir corresponding loop configurations. In the TN, a lattice site occupied by: (a) a monomer; (b-g) a loop; (h-k) a “thick” dimer; (l-o) abranching loop. + Mχ o χ o * Mχ e χ e * Θ ο , e = Mχ o , e Θ χ o , e * FIG. S2: (Color online) The contraction method of the nonlocal string order parameter Θ o , e on a cylindrical geometry, corresponding to theequations for Θ o , e in the main text. The definition of χ o , e , M , Θ is also explained in the main text. C V D c = 20D c = 40D c = 60D c = 80 µ =0 ν =1u=3 FIG. S3: (Color online) The specific heat C V of the MDL model with the energy u = µ = ν = ξ D c = 20D c = 40D c = 60D c = 80 µ =0 ν =1u=3 FIG. S4: (Color online) The correlation length of the MDL model with µ = , ν = , u =
3. The heights of the peaks at T c grow with theincrease of D c . n n A n B 0.2 0.3 0.410 −8 −6 −4 −2 T n n A n B µ =0 ν =1u=3 FIG. S5: (Color online) The bond density per site n A and n B of the MDL model with µ = , ν = , u =
3. Inset: The amplification of n A and n B at low temperature region. S S E S d S / d T µ =0 ν =1u=3 FIG. S6: (Color online) The thermodynamic entropy S and the entanglement entropy S E of the MDL model with µ = , ν = , u =
3. Theinset shows the first-order derivative of entropy S for temperature dS / dTdT