Kosterlitz-Thouless transitions and phase diagrams of the interacting monomer-dimer model on a checkerboard lattice
KKosterlitz-Thouless transitions and phase diagrams of the interacting monomer-dimer model on acheckerboard lattice
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
Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, inwhich there are interactions (with strength v ) between the parallel dimers on one-half of the plaquettes. Forthe fully-packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer casewith finite chemical potential µ , we also find an order-disorder phase transition which is of second-order, instead.We use the boundary matrix product state approach to detect the KT and second-order phase transitions, andobtain the phase diagrams v − T and µ − T . Moreover, for the non-interacting monomer-dimer model (setting µ = ν = h ) as f = − .
662 798 972 833 746 with the dimer density n = .
638 123 109 228 547,both of 15 correct digits.
PACS numbers: 64.60.Cn, 05.50. + q, 05.10.Cc, 64.60.Fr I. INTRODUCTION
Classical monomer-dimer model in two dimensions (2D)is one of the intriguing models in statistical mechanics. Theproblem has a venerable history, it was firstly introducedin the context of the absorption of molecules on the surface. When a rigid molecule occupies two nearest neighbor (NN)sites on the (square) lattice, it can be regarded as a dimer link-ing the two NN sites, while an empty site means the presenceof a monomer. The monomer-dimer model can be related tothe Ising and the height models, playing an important role inthe statistical physics. A special case of the monomer-dimermodel, namely the fully-packed dimer model, can be ana-lytically solved, while the general monomer-dimer caseis not. Numerating all the possible configurations and cal-culating the properties of the monomer-dimer model is anNP-complete problem and thus unfortunately “intractable” incomputations. Numerically, one has to adopt some ap-proximate methods, like the Monte Carlo samplings, tostudy the monomer-dimer models.The fully-packed dimer models exhibit di ff erent proper-ties on bipartite and non-bipartite 2D planar lattices. Theformer supports a critical phase with the algebraic decay-ing dimer-dimer correlations, while the latter (say, the tri-angular and kagome lattices) have exponential dimer-dimercorrelations. Exept the 2D lattices, people have alsoinvestigated the hard-core dimer models on various 3Dlattices. Extended critical phases are found on the bipartitecubic lattice, while no critical phases are found on the non-bipartite 3D lattices. Moreover, the classical dimer modelscan be “upgraded” to the so-called quantum dimer models,by promoting the classical dimer configurations to quantumstate bases. The quantum dimer model was introduced byRokhsar and Kivelson, where the singlet formed by two ad-jacent spins plays the role of a dimer. The quantum dimermodel is one of the typical systems which exhibit nontrivialtopology and fractional excitations. Recently, F. Alet et al. introduce the interacting dimer mod-els on a square lattice, where two parallel dimers on the sameplaquette are coupled (attractive). They studied this interact-ing dimer model by Monte Carlo simulations, and found adimer order-disorder phase transition, of Kosterlitz-Thouless(KT) type, at a certain temperature. Interestingly, the in-troduction of interactions between dimers is not only of theo-retical interest in the model study, but also acquire experimen-tal realizations recently. An adsorption experiment of certainrodlike organic molecules on the graphite was reported andit is found to be relevant to the fully-packed dimer model on ahexagonal lattice, with couplings between neighboring paral-lel dimers. (a) ABT P s s s s TTPPs s (b)(c) FIG. 1: (Color online) (a) The illustration of the monomer-dimermodel on a checkerboard lattice. Interactions between the paralleldimers are introduced on the green (gray) plaquettes. The underlyingtensor network is also shown, where the plaquette tensors ( T ) onlycover one-half of the plaquettes. The square-lattice tensor networkcan be divided into two sublattices, namely A and B, denoted bydark and light green (gray) colors, respectively. Besides the plaquettetensor T , there is a local matrix P on each vertex. (b) The plaquettetensor T is shown, s i ( i ∈ { , , , } ) are the four indices of tensor T .(c) The vertex matrix P can be absorbed into a plaquette tensor T bycontracting the sharing indices s ( s ). a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r In this work, we introduce and study the interactingmonomer-dimer model on a black-white checkerboard lattice.We find and employ the compact tensor network (TN) rep-resentation of the (grand) partition function to investigate themodel. The problem of calculating the free energy is thustransformed into a problem of how to accurately contract theTN. In practice, we contract it row by row with the infi-nite time evolving block decimation (iTEBD) technique forthe matrix product state (MPS) and calculate the thermo-dynamic properties of the monomer-dimer model with highprecision. Through numerical simulations, we show that thefully-packed dimer model has a low temperature dimer or-dered phase and a high temperature critical phase, with a KTtransition separating these two phases. We observe no singu-larity in the energy and its derivative (specific heat) curves,however, we detect the KT transition by calculating the orderparameters and correlation length. On the other hand, whenmonomer doping is introduced in the model (with the chem-ical potential µ ), there is also an order-disorder phase transi-tion at certain temperature, which is instead found to be ofsecond order. Remarkably, in addition to the regular thermo-dynamic quantities including the energy derivatives and theorder parameter, etc, we also detect the (second-order) phasetransition by checking the “entanglement” properties of theboundary MPS. We even extract the corresponding conformalcentral charge of the high-T critical phase of the fully-packeddimer model, by studying the block entanglement entropiesof the boundary MPS. At last, collecting the phase transitionpoints, we show and discuss the µ − T and ν − T phase diagramsof the interacting monomer-dimer model.The rest of this paper is organized as follows. In Sec. II, weintroduce the TN representation of the partition function andthe method for accurate evaluation of the thermodynamics. InSec. III, we show the main numerical results on phase transi-tions, and the phase diagrams of the monomer-dimer model.The last section (Sec. IV) is devoted to a conclusion of thepaper. II. MODEL AND METHODS
The interacting monomer-dimer model under study is de-fined on a black-white checkerboard lattice schematically de-picted in Fig. 1 (a), where the dimers are located on the linksand occupy two lattice sites. Summing over all possible dimercoverings, we have the (grand) partition function as
Ξ = (cid:88) { c } exp [ − β ( vN d − µ N tot )] , (1)where { c } means the set of all dimer configurations, N d countsthe number of the doubly occupied green plaquettes (i.e., thereare two parallel dimers on each plaquette). ν is the couplingstrength between parallel dimers: ν < ν > ν = − N tot is the total dimer num-ber on the lattice, and µ is the chemical potential of dimers. Setting µ → ∞ , we recover the fully-packed dimer model (nomonomer doping).The partition function of the monomer-dimer model has asimple tensor-network representation on a tilted square lattice.As shown in Fig. 1, the partition function TN consists of ten-sors T s , s , s , s located at one half of the plaquettes (the greenones) and the diagonal matrices P s i , s (cid:48) i living on the vertices. T has four bond indices s i ( i ∈ { , , , } ) corresponding tothe vertices of the green plaquette. s i ∈ { , , } , meaningthe presence ( s i = ,
2) or the absence ( s i =
0) of dimerson the concerned vertex s i . In addition, we use s i = , ff erent dimers: s i = s i = T . Thenonzero tensor elements T and their corresponding classicaldimer configurations are schematically shown in Fig. 2 (a-g). T , , , = T , , , = T , , , = T , , , = T , , , = T , , , ( T , , , ) = exp( − βν )describes the plaquette with two vertical or horizontal dimers.The rest tensor elements are zero (thus forbidden to appear inthe partition function).
11 00 00 11T T s s s s s s s s s s s s T T T T
20 02 02 20 11 11 22 22 s s s s s s s s s s s s s s s s (a) (e)(d) (f) (g)(b) (c) FIG. 2: The seven nonzero tensor elements of T and the correspond-ing dimer configurations on a plaquette. (a) no dimer; (b-e) singletdimer configurations; (f, g) double occupied plaquettes (vertical orhorizontal). In the fully-packed dimer case, to ensure that every vertexis occupied by one (and only one) dimer, a 3 × P s i , s (cid:48) i is defined on every vertex, with elements P , = P , = P , = P , =
1, otherwise 0. In the monomer-dimer case, we al-low monomer doping in the model by setting P , = P , = P , = P , = P , = exp( βµ/ µ the chemicalpotential. When µ → ∞ , the monomer-dimer model recoversthe fully-packed dimer case. Networking the plaquette ten-sors T and vertex matrices P , we thus obtain a tensor networkwhich faithfully represents the (grand) partition function ofthe interacting monomer-dimer model.To calculate the thermodynamics, we adopt the infinite timeevolution block decimation (iTEBD) method for the ac-curate contraction of the partition function TN. iTEBD wasproposed for e ffi cient simulations of the time evolution andthe ground state property (through imaginary-time evolution)of 1D quantum systems, and then generalized to calculate thethermodynamics of 2D classical statistical models and 1D T C V D c =70D c =100 FIG. 3: (Color online) The specific heat C V of fully-packed dimermodel with dimer-dimer interaction ν = −
1. The results are shownto be well converged with various retained bond states D c . quantum lattice models. Within the boundary MPS frame-work, we utilize a kind of “power” method to determine thedominating eigenvector (an MPS) of the transfer operator inthe TN, which consists of a column of T tensors organizedin a matrix-product operator (MPO) form. However, unlikethe ordinary power method for matrices, in the transfer MPOcase the MPS is enlarged after each contraction step, with acomposite bond space of a direct product of MPS and MPObond bases. Thus the MPS bond dimension grows up expo-nentially with contraction steps. Therefore, one has to per-form truncations on the bond space of the enlarged boundaryMPS, and bring the bond dimension of MPS back to D c , mak-ing the contraction procedure sustainable. Performing con-traction and truncation processes iteratively until the boundaryMPS converges, we thus obtain the dominating eigenvectorsof the transfer MPO, with which we can then evaluate the ex-pectation values of the local observables including the energy,the dimer occupation numbers, and the two-point correlationfunctions like dimer-dimer correlations.In our practical calculations, we perform the contraction ofMPS with transfer MPO until the prescribed convergence cri-terion is reached, say, free energy per site converges to 10 − (in some cases even down to machine precision). The totalnumber of iterations ranges between 4000 and 10 , dependingon the temperatures and the physical parameters of the model.The retained bond dimension of the boundary MPS D c ≈ ∼ D c is always checked, the trun-cation error is less than 10 − at the critical point, and reachesthe machine precision (10 − ) away from the critical points. III. RESULTS AND DISCUSSIONSA. fully-packed dimer model
Firstly, we investigate the interacting fully-packed dimermodel on the checkerboard lattice. The specific heat C V curveis shown in Fig. 3, which is computed by taking first-order T n A D c =70B(A+B)/2A D c =100B(A+B)/2 O r de r pa r a m e t e r D c = 100 FIG. 4: (Color online) The expectation value n A ( n B ) of the dimeroccupation number on the green plaquettes A (B). The mean value n = ( n A + n B ) / | n A − n | . derivative (versus temperature T ) of the energy per site. Thelatter is obtained by contracting the TN with one T tensor re-placed with an impurity tensor T I = ν ( T , , , + T , , , ). FromFig. 3, we observe no singularities in the C V curve, suggest-ing the absence of any second-order (or lower-order) phasetransition.However, by checking the local occupation number of thedimers on the green plaquettes (i.e., dimer density) in Fig. 4,we see di ff erent dimer densities n A (cid:44) n B between the A (dark)and B (light) green plaquettes [Fig. 1 (a)] at low temperature T < T c ≈ .
4. Especially in the limit T →
0, the A plaquetteis filled with a pair of dimers ( n A =
2) while the B plaquetteis vacant ( n B = n = ( n A + n B ) / n is verified tobe a constant in the whole temperature region, this is becauseevery site is linked to one dimer in the fully-packed case, andeach green plaquette contains two sites in net. In the inset ofFig. 4, we show that the di ff erence | n A − n | is nonzero belowthe critical temperature T c , and it vanishes for T > T c . There-fore, the particle number di ff erence | n A − n | can be regarded asan order parameter detecting the phase transition between thelow-T symmetry breaking phase ( n A (cid:44) n B ) and a high-T dis-ordered phase ( n A = n B = In order to understand this KT phase transition, we calcu-lated the correlation length ξ via the following formula, ξ = / ln( λ λ ) , (2)where λ ( λ ) is the largest (second-largest) eigenvalue of thetransfer matrix M , obtained by sandwiching two boundaryMPS tensors M a , b ; a (cid:48) , b (cid:48) = (cid:88) m A ma , b ( A ∗ ) ma (cid:48) , b (cid:48) , (3)where A is the MPS tensor, with a , b the geometric indicesand m the sharing physical index. The results are shown in FIG. 5: (Color online) The correlation length ξ versus temperature T .The correlations are measured between the green plaquettes of Fig. 1,and the length unit is the distance between two nearest-neighboringplaquettes A (or B), i.e., twice the length unit of the original lattice.Upper inset shows 1 /ξ versus 1 / √ T c − T in the vicinity of T c ( T < T c ), with the solid line an exponential fitting. The lower inset shows ξ versus D c at T = . < T c and T = . > T c , the former divergeswhile the latter saturates in the large D c limit. Γ = 0 . L − . L Γ D c = 40D c = 60D c = 100D c = 150fitting η = 1 . ± . lnL l n Γ ν = 0 FIG. 6: (Color online) Log-log plot of the correlation function Γ ( L )with ν = T = ∞ limit) for the fully-packed dimercase. The fit Γ ( L ) ∼ L − η reveals the algebraic decaying behavior(with η obtained by linear fit shown in the inset). Fig. 5, where ξ converges rapidly (with increasing D c ) to afinite value for T < T c (lower inset of Fig. 5), verifying theexistence of a non-critical phase with finite correlation length.On the contrary, ξ grows almost linearly with the increase of D c for T ≥ T c (lower inset of Fig. 5). Therefore, we expectthe T ≥ T c region (shaded region in Fig. 5) is a critical phasewith divergent correlation length. The upper inset of Fig. 5is the correlation length ξ as a function of 1 / √ T c − T when T approaches T c from below ( T < T c ). It indicates that thecorrelation length ξ diverges as an exponential of 1 / √| T − T c | .In Figs. 6 and 7, we present explicitly the dimer numbercorrelation function Γ i , j = (cid:104) O i O j (cid:105) , (4)where O i = ( n A ) i − n . In Fig. 6, we set ν =
0, and the par-tition function is an equal weight superposition of all possi- lnL l n Γ D c = 150 T=1.5T=1.6T=1.8T=2T=2.2T= ∞ T=1.5 fittingT=1.6T=1.8T=2T=2.2T= ∞ η ∞ FIG. 7: (Color online) The correlation function at di ff erent temper-atures T ( > T c ) for the fully-packed dimer model. Inset: the criticalexponent η versus T , η is extracted from the fit ( Γ ∼ L − η ). ble fully-packed dimer configurations (corresponding to the T = ∞ limit). A log-log plot of Γ versus L and its algebraicfit ( Γ ( x ) ∼ x − η ) are shown in Fig. 6, where the algebraic de-cay is clearly verified. By linear fitting, the exponent η can beobtained, and is shown in Fig. 6. The deviations of Γ ( L ) fromthe algebraic behavior (due to numerical errors) can be con-tinuously corrected by increasing D c . Notice that in Ref. 13,a related exponent is determined to be η = η = η (cid:39) .
83 is obtained from dimer occupation numbercorrelations Eq. 4 (horizontal and vertical dimers are not dis-tinguished), which nevertheless well agrees with theirs.Besides the T = ∞ ( ν =
0) limit, we also studied corre-lations of other points in the critical phase with nonzero ν ,and present the results in Fig. 7, from which we can observealgebraic decaying behaviors of Γ for every T > T c . In theinset of Fig. 7, we show that the critical exponent η growsmonotonously with increasing T .Therefore, by studying the dimer-dimer correlation func-tion and correlation length, we find the phase transition oc-curring at T c is between an ordered (dimer crystal) phase anda critical (algebraic liquid) phase. This scenario also perfectlysupports the conclusion of a KT-transition at T c .To gain further insight into the underlying physics of thiscritical phase in the fully-packed dimer model, we also ex-tract the conformal central charge of this system in the criticalregime T > T c . The conformal field theory (CFT) tells usthe conformal invariance at the critical point, and sets usefulconstraints on the critical behaviors of two-dimensional clas-sical or 1D quantum systems . The universality class can becharacterized by the conformal anomaly or central charge c of the Virasoro algebra. We use MPS-based method to cal-culate the central charge, by fitting the block entanglemententropy S versus the block size L . Depending on whetherthe system is in a critical or a noncritical regime, the blockentanglement entropy has di ff erent asymptotic behaviors .In noncritical regimes, S grows monotonously with L beforesaturation; while in critical regimes, the CFT predicts a loga- S D c =40D c =60D c =80D c =100fitting 4 4.5 5 5.5 63.43.63.84 log (L) S k = 1 . c = 1 . FIG. 8: (Color online) The block entanglement entropy S as a func-tion of length L calculated by the iTEBD for the fully-packed dimermodel, with D c = , , , c (cid:39)
1. The inset: the entanglement entropy S versuslog ( L ). rithmic divergence S ≈ c ( L ) + k , (5)where L measures the site number of the block embedding inan infinite MPS, c is the central charge and k is a non-universalconstant.The entanglement entropy S is defined by S = − Tr( ρ log ρ ) , (6)where ρ is the reduced density matrix (DM) of system and canbe calculated from the converged MPS. However, notice thatfor any L , the dimension of the reduced DM supported by theMPS is D c × D c . Therefore it is not possible to capture theentanglement entropies for extremely long L ; however, by in-creasing the D c we are able to simulate the logarithmic diver-gence for su ffi ciently long L . By fitting our numerical resultsto Eq. (5), as shown in Fig. 8, we find c = B. monomer-dimer model
In this part, we study the interacting monomer-dimer model(with µ < ∞ ) on the checkerboard lattice. Fig. 9 shows thecalculated specific heat C V of the case µ =
0, where a diver-gent peak of C V occurs at T c = .
35, uncovering the existenceof a second-order phase transition.The dimer occupation numbers n A ( n B ) on the plaquettes A(B) and the average n = ( n A + n B ) / T < T c , the symmetry between the A and B plaque-ttes is broken ( n A (cid:44) n B ), while for T ≥ T c this symmetryis recovered ( n A = n B ). In contrast to the fully-packed case C V D c =40D c =60D c =80D c =100 µ =0 FIG. 9: (Color online) The specific heat C V of the monomer-dimermodel with a chemical potential µ = T n µ = 0 n A n B (n A +n B )/2 n T →∞ = 0.638 123 109 228 547 FIG. 10: (Color online) The dimer occupation number on the plaque-ttes A(B) and the average value n for µ = , ν = − ( n = n decreases with increasing temper-atures in the monomer-dimer case. The limit T → ∞ (or,equivalently µ = ν =
0) is an interesting special case, i.e., theconventional (noninteracting) monomer-dimer model. Themean value n is determined as n T →∞ = .
638 123 109 228 547,which agrees perfectly with the previous studies (0.638 12311in Ref. 9, and 0.638 1231 in Ref. 10), and provides 15 verywell converged (correct) digits. The corresponding free en-ergy per site (negative of the monomer-dimer constant h ) is f = − .
662 798 972 833 746 with 15 converged (correct) dig-its, again in perfect agreements with previous results ( h = -0.662 798 972 834 in Ref. 10, and h = ± ξ , which alsoshows a divergent peak at T c , the second-order phase transi-tion point. Notice that in the T > T c region, the correlationlength ξ is finite, in contrast to the fully-packed case where ξ diverges in the high-T disordered phase.Entropy is another interesting quantity. Actually, we referto two kinds of entropies in the calculations, i.e., the conven-tional thermodynamic entropy S = ( U − F ) / T and the for-mal “entanglement entropy” S E evaluated from the boundary ξ D c =40D c =60D c =80D c =100 µ =0 FIG. 11: (Color online) The correlation length for µ = , ν = − T c grow with the increase of D c . T S µ = 0 S E evenS E oddS d S / d T FIG. 12: (Color online) The entropy S and the entanglement entropy S E for µ =
0. Because there are A and B plaquettes, the bound-ary MPS is of period two, leading to even and odd cut entropies S E even and odd. Inset: the di ff erential of entropy S for temperature TdS / dT . MPS. Given the boundary MPS, we can take a Schmidt de-composition (once for all bonds) of the translation-invariantMPS and formally calculate its “entanglement properties”.Notice that this bipartite entanglement entropy is di ff erent theblock entanglement entropy discussed above, because the for-mer is between two half-infinite chain. As shown in Fig. 12,the bipartite entanglement entropy S E shows a clearly diver-gent peak at T c , indicating the occurrence of a phase transi-tion. This observation is quite remarkable, because the con-ventional thermodynamic entropy S is smooth around T c , andits singularity can only be seen after taking a derivative over T (inset of Fig. 12), owing to ∂ S ∂ T = C V T . Therefore, this “entan-glement entropy” S E is found to be more sensitive to the phasetransition (than the thermodynamic entropy S ), and thus canserve as an useful numerical tool detecting continuous phasetransitions.In Fig. 13, we show the semi-log plot of the correlationfunctions Γ for µ =
0. The linear fittings are performed usingthe correlation length ξ estimated from the transfer matrices −30 −25 −20 −15 −10 −5 L Γ T=0.6T=0.7T=0.8T=0.9T=1T=0.6 fittingT=0.7T=0.8T=0.9T=1
FIG. 13: (Color online) The correlation function Γ for di ff erent tem-peratures higher than T c at µ = ξ calculated directly (see details in the text). Thefitting formula is Γ ( L ) ∼ e − L /ξ . −5 0 5 10 15 20 2500.511.522.5 µ T c ∞ v = − −0.7−0.6−0.5−0.4−0.3−0.2−0.1 000.10.20.30.4 ∞µ c = −0.51 FIG. 14: (Color online) The phase diagram of the monomer-dimermodel on the checkerboard lattice with v = −
1. The (red) verticalline ( µ = ∞ ) is a critical line; while the blue curved line (with finite µ ) is a second order phase transition line. Inset: Amplification of thenegative µ region, the star denotes the terminating point µ c ≈ − . (Eq. (2)). Note that for both T > T c and < T c cases, the cor-relation functions are exponentially decaying, indicating thatthe high-T phase is non-critical under the monomer doping. C. phase diagram
As a summary of the previous studies of the phase tran-sitions, we show the phase diagrams of the monomer-dimermodel in Figs. 14 and 15. The µ − T phase diagram (withfixed v = −
1) is shown in Fig. 14. The red vertical line at µ = ∞ is a line consisting of critical points, i.e., a critical line,and the KT-transition point T c ≈ . −1 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 000.511.5 v T c fully packed−1 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 000.10.20.30.4 v T c µ = 0 disorder KT−transitionsecond−order transition (a)(b) symmetry breaking criticalsymmetry breaking
FIG. 15: (Color online) The phase diagrams of the fully-packeddimer and the monomer-dimer ( µ =
0) models on the checkerboardlattice. (a) The low-T symmetry breaking phase and the high-T criti-cal phase are separated by a KT phase transition line. (b) The low-Tsymmetry breaking and the high-T disordered phases are separatedby a second-order phase transition line. metry breaking phase and the high-T critical phase. When µ is finite, the phase boundary (blue curved) line represents con-tinuous (second-order) phase transitions, separating the low-T ordered and the high-T disordered non-critical phases. Theblue curved line terminates at µ c , which is denoted by a bluestar in the inset of Fig. 14. We estimate, by a polynomialfitting, that µ c ≈ − .
51, below which the low-T symmetrybreaking phase disappears.The ν − T phase diagrams of the fully-packed dimer and themonomer-dimer ( µ =
0) cases on the checkerboard lattice areshown in Fig. 15. The phase boundary line of the fully-packeddimer model is a KT phase transition line, which separates thelow-T ordered and the high-T critical phases. On the otherhand, the phase boundary in the monomer-dimer model with µ = T c vanishes when ν =
0, in agreement with the observation thatthe low-T symmetry breaking phase is induced by the dimer-dimer attractive interactions ν . IV. CONCLUSION AND OUTLOOK
By employing the accurate tensor network method, we havesystematically studied the interacting monomer-dimer modelon the checkerboard lattice. The specific heat C V and the or-der parameter | n A − n | show that KT phase transitions occur inthe interacting fully-packed dimer model ( µ = ∞ ), in contrastto the finite- µ case where second-order phase transitions takeplace. Collecting the phase transition points, we obtain the µ − T and ν − T phase diagrams with fixed ν = − µ = ∞ , respectively. From the phase diagrams, we find that the at-tractive interactions ν < µ = ∞ ) or the monomer-dimer case ( µ < ∞ ). Previously,people have found similar conclusions for the square-latticeinteracting dimer models . Here we show that even switchingo ff the interactions on one half of the plaquettes (thus reduc-ing to a checkerboard lattice model), there is still a symmetrybreaking dimer crystal phase at a low-T. As a consequence,the dimer crystal does not break the 90 degree lattice rota-tional symmetry on the checkerboard lattice.The e ffi cient tensor network technique enables us to cal-culate the thermodynamic properties of the monomer-dimermodels with a very high precision. For example, themonomer-dimer constant can be determined to the machineprecision. The tensor network method also provides noveltools (for example, boundary MPS entanglement entropy) fordetecting the phase transitions.Besides the square and checkerboard lattices, it calls formore investigations of this interacting monomer-dimer mod-els on other lattices, say kagome or star lattice, to explorethe dimer-dimer interaction e ff ects in more general situations.The tensor network method is also applicable for investiga-tion of these lattice dimer models and we will discuss themelsewhere. V. ACKNOWLEDGEMENT
This work was supported in part by the National NaturalSciences Foundation of China (Grants No. 11274033, andNo. 11474015), Major Program of Instrument of the NationalNatural Sciences Foundation of China (Grant No. 61227902),Sub Project No. XX973 (XX5XX), and the Research Fund forthe Doctoral Program of Higher Education of China (GrantNo. 20131102130005). W.L. acknowledges the hospital-ity of the Kavli Institute of Theoretical Physics China wherepart of this work was performed. W.L. was supported by theDFG through Grant No. SFB-TR12 and Cluster of ExcellenceNIM. ∗ Electronic address: [email protected] J. K. Roberts, Proc. R. Soc. London, Ser. A , 464 (1935); R.H. Fowler and G. S. Rushbrooke, Trans. Faraday Soc. , 1272(1937). H. N. V. Temperley and M. E. Fisher, Philos. Mag. 6, 1061 (1961);M. E. Fisher, Phys. Rev. , 1664 (1961). P. W. Kasteleyn, Physica (Amsterdam) , 1209 (1961); J. Math.Phys. (N.Y.) , 287 (1963). M. E. Fisher and J. Stephenson, Phys. Rev. , 1411 (1963). H. W. J. Bl¨ote and H. J. Hilhorst, J. Phys. A , L631 (1982); B.Nienhuis, H. J. Hilhorst, and H. W. J. Bl¨ote, ibid. , 3559 (1984). R. J. Baxter, Exactly Solved Models in Statistical Mechanics(Academic Press, 1982), Chapter 4, pp. 47-58. M. Jerrum, J. Stat. Phys., , 121 (1987). Y. Huo, H. Liang, S.-Q. Liu, and F. Bai, Phys. Rev. E , 016706(2008). R. J. Baxter, J. Math. Phys. , 650 (1968). Y. Kong, Phys. Rev. E , 061102 (2006). S. Friedland and U. N. Peled, Adv. Appl. Math. , 486 (2005). F. Alet, J.L. Jacobsen, G. Misguich, V. Pasquier, F. Mila, and M.Troyer, Phys. Rev. Lett. , 235702 (2005). F. Alet, Y. Ikhlef, J.L. Jacobsen, G. Misguich, and V. Pasquier,Phys. Rev. E , 041124 (2006). P. Fendley, R. Moessner, and S. L. Sondhi, Phys. Rev. B ,214513 (2002). W. Krauth and R. Moessner, Phys. Rev. B , 064503 (2003). G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. Lett. ,137202 (2002). D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev.Lett. , 167004 (2003). D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. , 2376 (1988). R. Moessner and S. L. Sondhi, Phys. Rev. Lett. , 1881 (2001); R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B ,024504 (2001); D. A. Ivanov, Phys. Rev. B , 094430 (2004). J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid State Phys. , 1181 (1973). M. O. Blunt et al. , Science , 1077 (2008). J. L. Jacobsen and F. Alet, Phys. Rev. Lett. , 145702 (2009). G. Vidal, Phys. Rev. Lett. , 070201 (2007). R. Or´us and G. Vidal, Phys. Rev. B , 155117 (2008). W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su,Phys. Rev. Lett. , 127202 (2011). A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, J. Stat.Phys. , 763 (1984); Nucl. Phys. B , 333 (1984); D. Friedan,Z. Qiu, and S. Shenker, Phys. Rev. Lett. , 1575 (1984). G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. ,227902 (2003). J. Eisert, M. Cramer, and M. B.Plenil, Rev. Mod. Phys. , 277(2010). C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B424