Analysis of quantum spin models on hyperbolic lattices and Bethe lattice
AAnalysis of quantum spin models on hyperboliclattices and Bethe lattice
Michal Daniˇska and Andrej Gendiar
Institute of Physics, Slovak Academy of Sciences, SK-845 11, Bratislava, SlovakiaE-mail: [email protected] and [email protected]
Abstract.
The quantum XY, Heisenberg, and transverse field Ising models onhyperbolic lattices are studied by means of the Tensor Product Variational Formulationalgorithm. The lattices are constructed by tessellation of congruent polygons withcoordination number equal to four. The calculated ground-state energies of the XYand Heisenberg models and the phase transition magnetic field of the Ising model onthe series of lattices are used to estimate the corresponding quantities of the respectivemodels on the Bethe lattice. The hyperbolic lattice geometry induces mean-field-likebehavior of the models. The ambition to obtain results on the non-Euclidean latticegeometries has been motivated by theoretical studies of the anti-de Sitter/conformalfield theory correspondence.PACS numbers: 05.30.Rt, 64.60.-i, 64.70.Tg, 68.35.Rh
Keywords : tensor product state, quantum spin systems, non-Euclidean geometry, phasetransition a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b nalysis of quantum spin models on hyperbolic lattices and Bethe lattice
1. Introduction
Many analytical and computational techniques have been developed to study quantumspin models on two-dimensional (Euclidean) lattices. Among such techniques, let usmention the corner transfer matrix approach [1], the coordinate Bethe Ansatz [2],the algebraic Bethe Ansatz [3], the vertex operator approach [4], including numericalalgorithms based on tensor product states and tensor networks [5, 6, 7, 8, 9], all of whichhave been successfully applied to the description of the energy spectrum and matrixelements of local operators in either integrable lattice models and quantum spin chainsor non-integrable quantum spin systems. However, the task of finding an appropriateapproach to analyze the quantum models on hyperbolic lattices, which belongs tochallenging problems related to the correspondence between the anti-de Sitter space andthe conformal field theory [10], still remains an open question of the quantum gravity. Aremarkable demand for an appropriate numerical tool persists. Implementation of theMonte Carlo simulations fails due to exponential increase of the number of the latticesites for models on hyperbolic lattices with respect to the expanding lattice size from thelattice center [11, 12]. Our desire is to propose a novel and sufficiently accurate numericalalgorithm, which originates from the solid state physics and inherits the typical featurescoming from widely accepted renormalization group approaches, especially based on theDensity Matrix Renormalization Group [13, 14, 15].Recently, we modified the Tensor Product Variational Formulation (TPVF) [16],which is an algorithm combining an ansatz for the ground-state in the form of theTensor product state (TPS) [5] with the Corner transfer matrix renormalization group(CTMRG) scheme [17]. This algorithm can be used to study quantum spin systems inthe thermodynamic limit on regular hyperbolic lattices of constant negative Gaussiancurvature [18]. The hyperbolic lattices are constructed by tessellation of congruent p -sided polygons (with the coordination number fixed to four). We applied the modifiedTPVF algorithm to the Euclidean square ( p = 4) and hyperbolic pentagonal ( p = 5)lattices in order to analyze the critical phenomena of the XY, Heisenberg and transversefield Ising model (TFIM). On the square lattice numerical inaccuracy varied from 1 . .
7% in TFIM at the phase transition. This observation originates inthe mean-field-like behavior induced by the TPS ansatz, which, as a consequence, cannotaccurately approximate the correct ground state of the TFIM on the two-dimensionalEuclidean lattice, which belongs to the Ising universality class. On the contrary, sincethe Hausdorff dimension of the hyperbolic lattices is infinite, spin models on theselattices belong to the mean-field universality class due to short range correlations,even though the mean-field approximation of the Hamiltonian is not applied [1]. Weconjectured that TPVF was more suitable for models on the pentagonal hyperboliclattice due to off-critical and weakly correlated characteristics [16].In this work we expand the set of hyperbolic lattices investigated by the TPVF to aseries of lattices constructed from congruent p -sided polygons, where p ∈ { , , . . . , } .In analogy to our previous studies of classical spin models on these hyperbolic lattices nalysis of quantum spin models on hyperbolic lattices and Bethe lattice p = 4 p = 5 p = 6 p = 7 p = 10 p → ∞ Figure 1.
Graphical representation of the lattices with the fixed coordination numberequal to four indexed by the lattice parameter p . The hyperbolic lattices ( p = 5 , , [19, 20, 21, 22], we expect fast convergence of the phase transition magnetic field of thequantum TFIM as well as the ground-state energies of the quantum XY and Heisenbergmodels toward the asymptotic case p → ∞ , which represents the Bethe lattice [20].Numerical results presented in the following sections are in complete agreement withthe expectations. The key feature of this work is the consequent indirect analysis ofthe quantum TFIM, XY, and Heisenberg models on the Bethe lattice with coordinationnumber four, which has not been considered yet.The article is organized as follows. In Sec. II we define the three Hamiltonianson the respective hyperbolic lattices and give a brief description of the principles ofthe TPVF algorithm, which have been discussed in [16]. An accurate analysis of thenumerical results is presented in Sec. III and we summarize them in Sec. IV.
2. The Model
We study the ground-state properties and the phase transition of the quantum TFIM,XY, and Heisenberg models in the thermodynamic limit on a series of hyperboliclattices. Each hyperbolic lattice is made from equivalent congruent p -sided polygons.The polygon vertices coincide with the lattice spin sites, where a single spin is positioned. nalysis of quantum spin models on hyperbolic lattices and Bethe lattice p ∈ { , , . . . , } .Apart from the set, we include two additional cases: p = 4 being the Euclidean squarelattice and the asymptotic case p → ∞ , which is associated to the Bethe lattice. Figure 1depicts the typical structure of the lattices. The square lattice serves as a referencelattice. The three spin models and the TPVF algorithm have been described in detailin [16], and we focus only on the substantial aspects of the models on the hyperboliclattices in the following.In general, the Hamiltonian H of any of the three models can be written as a sumof local Hamiltonians G ( p ) k of the p -sided polygonal shape, in particular, H = (cid:88) (cid:104) k (cid:105) p G ( p ) k , (1)where k labels the polygons and the sum runs over the set of all indices of the latticepolygons (cid:104) k (cid:105) p . The local Hamiltonian takes the form G ( p ) k = − p (cid:88) i =1 (cid:104) J xy (cid:16) S xk i S xk i +1 + S yk i S yk i +1 (cid:17) + J z S zk i S zk i +1 + h (cid:16) S xk i + S xk i +1 (cid:17) (cid:105) , (2)where k , k , . . . , k p label the spin positions on the k th p -sided polygon (noticing that k p +1 ≡ k ), and S xk i , S yk i , S zk i denote the corresponding Pauli spin- operators. Weconsider constant nearest-neighbor couplings J xy , J z and a uniform external magneticfield h . By setting J xy = 0 and J z = 1 we obtain the TFIM at the transversemagnetic field h , whereas the choice J xy = 1 , J z = h = 0 gives the XY model and J xy = − J z = 1 , h = 0 the Heisenberg model [16].Our task is to calculate an approximate ground-state of the system in thethermodynamic limit in the product form | Ψ p (cid:105) = lim N →∞ (cid:88) σ σ ··· σ N (cid:89) (cid:104) k (cid:105) p W p ( { σ k } ) | σ σ · · · σ N (cid:105) , (3)where N stands for the total number of the lattice spins. The basis σ j for j = 1 , . . . , N denotes a binary state, for which we use the arrow notation ↓ or ↑ in the following.The summation runs over the 2 N base spin states | σ σ · · · σ N (cid:105) , and W p ( { σ k } ) are theelements of the p -rank tensor W p depending on p spins σ k , . . . , σ k p on the k th latticepolygon. The symbol { σ k } stands for one of the 2 p base configurations of a multi-spinvariable representing the group of spins σ k , . . . , σ k p . All the tensors W p are consideredto be identical, therefore, the set of 2 p tensor elements W p ( { σ } ), where the subscript k has been omitted due to the uniformity of the tensors W p , uniquely describes the state | Ψ p (cid:105) , i.e. | Ψ p (cid:105) = | Ψ p [ W p ( { σ } )] (cid:105) . nalysis of quantum spin models on hyperbolic lattices and Bethe lattice | Ψ ∗ p (cid:105) as the best approximation of the ground-state within the class ofTPS | Ψ p (cid:105) , if the minimum of the energy normalized per bond, E ( p )0 ≡ min Ψ p lim N b →∞ N b (cid:104) Ψ p |H| Ψ p (cid:105)(cid:104) Ψ p | Ψ p (cid:105) , (4)is obtained for | Ψ ∗ p (cid:105) . Here, N b denotes the total number of bonds in the system. Theenergy E ( p )0 , due to its variational origin, serves as an upper bound of the true ground-state energy per bond E ( p )0 .Since the structure of every local Hamiltonian G ( p ) k does not depend on k (weinvestigate the system in the thermodynamic limit), the variational problem in (4) isequivalent to minimization of the local energy per bond of an arbitrary polygon in thelattice center (in order to avoid boundary effects) E ( p )0 = min Ψ p p (cid:104) Ψ p | G ( p ) (cid:96) | Ψ p (cid:105)(cid:104) Ψ p | Ψ p (cid:105) , (5)where (cid:96) is the index of the selected polygon and the normalization factor 2 /p reflectsthe fact that the p bonds of each polygon are shared with its neighbors. Moreover, if weutilize the tensor product structure of the state | Ψ p (cid:105) , we can express the denominator (cid:104) Ψ p | Ψ p (cid:105) ≡ D ( W p ( { σ } )) and the numerator (cid:104) Ψ p | G ( p ) (cid:96) | Ψ p (cid:105) ≡ N ( W p ( { σ } )) as functions ofthe tensor elements W p ( { σ } ) only. Consequently, our variational problem transformsonto a multi-dimensional minimization over 2 p tensor elements W p ( { σ } ) E ( p )0 = min W p ( { σ } ) p N ( W p ( { σ } )) D ( W p ( { σ } )) . (6)Furthermore, symmetries of the local Hamiltonian G ( p ) (cid:96) may significantly reduce thedimension of the problem. Rotational and spin-ordering symmetries are present in allthe three spin models. As a typical example, let us consider a hexagonal lattice ( p = 6)and its particular base configuration of spins on the lattice polygon { σ ∗ } = {↑↓↑↑↓↓} .Rotational symmetry requires that the tensor elements corresponding to the set ofconfigurations {↓↑↓↑↑↓} , {↓↓↑↓↑↑} , {↑↓↓↑↓↑} , {↑↑↓↓↑↓} , {↓↑↑↓↓↑} are identical to W p =6 ( { σ ∗ } ). Next, let us consider a spin-ordering operation, which reverses the orderof the polygon spins. In particular, if the spins are labelled clockwise, the operationreorders them in the anti-clockwise direction. It means that the configuration {↑↓↑↑↓↓} is equivalent to {↓↓↑↑↓↑} by the spin-ordering symmetry and to all the rotations ofthe latter configuration ( {↑↓↓↑↑↓} , {↓↑↓↓↑↑} , {↑↓↑↓↓↑} , {↑↑↓↑↓↓} , {↓↑↑↓↑↓} ) by acomposition of the spin-ordering and the rotational symmetry. As a result, the 12tensor elements W ( { σ } ) corresponding to the configuration { σ ∗ } and its 11 equivalentconfigurations are represented by a single variational parameter, as they share the samevalue.By performing a similar analysis on the set of all 2 p configurations { σ } wecan factorize it into N ( p )Ising classes of equivalence with representatives θ j , where j ∈{ , . . . , N ( p )Ising } [16]. Thus, in case of a system with the rotational and the spin-orderingsymmetry (as in the TFIM), there are only N ( p )Ising free variational parameters W p ( θ j ) nalysis of quantum spin models on hyperbolic lattices and Bethe lattice Table 1.
The numbers of the free variational parameters N ( p )Heis (for the XY and theHeisenberg models) and N ( p )Ising (for the TFIM) including the normalization parameter. p N ( p )Heis N ( p )Ising p tensor elements W p ( { σ } ). If there is no preferred spin alignment inthe system (such as in the XY model, the Heisenberg model, as well as in the TFIM atand above the phase transition magnetic field), the spin-inversion symmetry appears.For instance, if p = 4, the configuration {↑↑↑↓} is equivalent to {↓↓↓↑} , which is obtainedby flipping each spin. Such an additional symmetry results in consequent reduction ofthe set of the free variational parameters, the size of which drops to N ( p )Heis < N ( p )Ising . Thenumbers of the free variational parameters N ( p )Ising and N ( p )Heis with respect to the latticeparameter p are summarized in table 1. In addition, one more variational parametercan be eliminated from each set of the free variational parameters by setting it to 1,being the normalization condition in W p ( { σ } ) and | Ψ p (cid:105) , consequently.The free variational parameters W p ( θ j ) are optimized numerically by TPVF [16].It is based on the fact that the product structure of the state | Ψ p (cid:105) allows to calculatethe numerator N ( W p ( { σ } )) and the denominator D ( W p ( { σ } )) in (6) for the given set ofthe tensor elements W p ( { σ } ) by an appropriate modification of the CTMRG algorithm.Having applied the CTMRG as the effective and accurate numerical tool for calculationof the ratio in (6), a multi-dimensional minimizer is used for optimizing the variationalparameters W p ( θ j ) [23, 24, 25].
3. Numerical results
We study the XY and the Heisenberg models at zero magnetic field, where these modelsare known to be critical in the Euclidean space. Therefore, there is no preferreddirection (the spin alignment) in the system on the Euclidean lattice at h ≥
0, andthe spin-inversion symmetry is present. We expect that the models on hyperboliclattices also exhibit the spin-inversion symmetry. It enables to reduce the numberof the free variational parameters W p ( θ j ) within the TPVF minimization part downto N ( p )Heis as listed in table 1. Despite the significant reduction, the number of the freeparameters N ( p )Heis still grows fast with respect to the increasing lattice parameter p . Thecomputational time of the minimization algorithm is significantly prolonged due to (atleast) linear dependence on the increasing number of the free variational parameters.Also, the algorithm may possibly be trapped in a local energy minimum and thus a seriesof initial conditions has to be tested in order to obtain the global energy minimum (or, nalysis of quantum spin models on hyperbolic lattices and Bethe lattice Table 2.
The ground-state energies per bond E ( p )0 listed with respect to p for theHeisenberg and XY models. The number of block spin states [13, 17] kept was m = 20for 4 ≤ p ≤
10 and m = 10 for p = 11. The asymptotic estimate of E ( ∞ )0 correspondsto the model on the Bethe lattice. p E ( p )0 XY Heisenberg4 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . ∞ − . − . p = 11 with respect to the constraints of our computational resourcesand time.The ground-state energies E ( p )0 obtained by the TPVF algorithm for both the XYand the Heisenberg models are summarized in table 2. The energies E ( p )0 remainedidentical even if the larger set of N ( p )Ising free variational parameters W p ( θ j ) in TPVFwas used, whereby the optimal values of the parameters W ∗ p ( θ j ) coupled by spin-inversion symmetry were equal. These results witness the spin-inversion symmetry ofthe models on hyperbolic lattices. Recall that E ( p )0 represents only an upper estimateof the true ground-state energy E ( p )0 . We have shown that the energies E (4)0 of thereferencing Euclidean square lattice calculated by TPVF were higher if compared tothe Monte Carlo simulations (the relative errors for the XY and the Heisenberg models,respectively, are 1 .
2% and 2 . W , which cannot correctly reproduce the divergence of thecorrelation length in the models on the square lattice. On the other hand, any quantumspin model on hyperbolic lattice belongs to the mean-field universality class, because thehyperbolic lattices exhibit the infinite Hausdorff dimension, which significantly exceedsthe critical lattice dimension D c = 3 [1]. Because of the mean-field-like character of theTPS approximation, the TPVF algorithm is expected to be more accurate whenever ahyperbolic lattice geometry is considered [16, 20].Figure 2 illustrates a monotonically increasing and rapidly saturating curve of theenergy E ( p )0 for the XY model with respect to the lattice parameter p . The inset depictsthe tail of the curve in detail together with an exponential fit applied to the five energies nalysis of quantum spin models on hyperbolic lattices and Bethe lattice p -1.084-1.083-1.082-1.081 E ( p ) p -1.080865-1.080860-1.080855-1.080850-1.080845-1.080840-1.080835-1.080830 E ( p ) E ∞ ) = −1.08083446 + a e a p a = −1.07496644 a = −1.50635874 E ∞ ) E p ) = Figure 2.
The ground-state energy E ( p )0 of the XY model with respect to the latticeparameter p ∈ { , , . . . , } . The inset shows the zoomed-in energy including thedetails of the fitting function. p -1.310-1.305-1.300-1.295-1.290 E ( p ) p -1.29198-1.29196-1.29194-1.29192-1.29190-1.29188 E ( p ) E ∞ ) = −1.2919440 a = 0.44776132 a = −1.29952590 E ∞ ) = −1.2919443 a = −4.24906926 a = −1.47218237 + a e a p E ∞ ) E p ) = Figure 3.
The ground-state energy E ( p )0 of the Heisenberg model with respect to thelattice parameter p ∈ { , , . . . , } . The fitting function parameters are shown in theinset. E (7)0 , . . . , E (11)0 . The case p = 4, where the TPVF algorithm is not sufficiently accuratefor the reasons mentioned above, was excluded from the extrapolation analysis. Thefitting function is proposed in the form E ( p )0 = E ( ∞ )0 + a exp( a p ) , (7)where E ( ∞ )0 , a , and a are the fitting parameters, which were determined in the followingway. First we defined a function f ( E ), which returns the residual sum of squares ( RSS ) nalysis of quantum spin models on hyperbolic lattices and Bethe lattice | E − E ( p )0 | = ln | a | + a p . Then, E ( ∞ )0 was chosen as theargument, which minimizes the function f ( E ). The corresponding linear regressionln | E ( ∞ )0 − E ( p )0 | = ln | a | + a p specifies the parameters a and a . If considering anotherway, E ( ∞ )0 is such a value that the curve ln | E ( ∞ )0 − E ( p )0 | is as close as possible to a line,where the closeness is measured by the RSS . Thus obtained parameters E ( ∞ )0 , a , and a are listed in the inset of figure 2, where the dot-dashed line represents the estimateof the ground-state energy per bond of the quantum XY model on the Bethe lattice E ( ∞ )0 = lim p →∞ E ( p )0 = − . E ( p )0 of the Heisenberg model are plottedin figure 3. Again, rapid convergence of the energy to the asymptotic values isobvious from the data. Although we have not clarified the physical origin of the non-monotonic convergence (saw-like pattern) of E ( p )0 yet, a detailed analysis indicates thatthe exponential fitting function in (7) can successfully describe the data, if appliedseparately onto two sets: those with even p ∈ { , , } (the lower branch shown in theinset) and the odd p ∈ { , , , } (the upper branch). The fitting parameters of thetwo regressions are listed in the inset of figure 3. The lower and the upper branchesyield the energies E ( ∞ )0 − . E ( ∞ )0 − . E ( ∞ )0 = − . E ( p )0 yet. However, if a power-law fitting function was appliedinstead, we obtained a less accurate fitting and greater RSS . The TFIM undergoes a quantum phase transition at a nonzero magnetic field h ( p ) t > h < h ( p ) t breaks the spin-inversionsymmetry, which results in approximately twice larger set of the free variationalparameters N ( p )Ising in the TPVF algorithm if compared to N ( p )Heis in the XY and Heisenbergmodels, cf. table 1. The computational time for a particular fixed field h is, therefore,significantly prolonged. Moreover, in order to screen the vicinity of the phase transitionfield h ( p ) t , multiple calculations for a sequence of magnetic fields h had to be performed.As a consequence, in order to restrict the total computational time, we have analyzedthe TFIM on the hyperbolic lattices up to p = 10 only. (Notice that the number ofblock spins states kept was m = 20 for p ∈ { , , . . . , } , and only m = 4 for p ∈ { , } ,which was sufficient due to exponentially weak correlations caused by the hyperboliclattice geometry [21]; any further increase of the states kept m has not improved thenumerical calculations significantly).We have analyzed the phase transition of the TFIM by the expectation value of thespontaneous magnetization (cid:104) S zp (cid:105) as well as by the magnetic susceptibility χ p . Solving nalysis of quantum spin models on hyperbolic lattices and Bethe lattice Table 3.
The phase transition fields h ( p ) t of the TFIM including the estimated errors∆ ( p ) with respect to the lattice parameter p . p h ( p ) t . . . . ( p ) × − × − × − × − p ∞ h ( p ) t . . . . ( p ) × − × − × − × − the minimization problem in (6), we received the optimal tensor elements W ∗ p ( { σ } ),which uniquely define the approximative ground state | Ψ ∗ p (cid:105) via (3). Once | Ψ ∗ p (cid:105) has beenconstructed, we evaluated the spontaneous magnetization (cid:104) S zp (cid:105) = (cid:104) Ψ ∗ p | S zc | Ψ ∗ p (cid:105)(cid:104) Ψ ∗ p | Ψ ∗ p (cid:105) , (8)where c labels an arbitrary spin in the central polygon of the lattice in order to suppressboundary effects. The resulting dependence of the magnetization (cid:104) S zp (cid:105) with respect tothe magnetic field h near the phase transition field h ( p ) t is plotted in the upper graph offigure 4. The quantum phase transition of the TFIM is characterized by a non-analyticbehavior of the magnetization curve, when (cid:104) S zp (cid:105) → h → h ( p ) t from the ordered phase ( h < h ( p ) t ). The phase transition exponent β p ,which depends on the lattice geometry, describes the singularity through the scalingrelation in the ordered phase (cid:104) S zp ( h ) (cid:105) ∝ (cid:16) h ( p ) t − h (cid:17) β p . (9)Figure 4 (the lower graph) shows the squared transversal magnetization (cid:104) S zp (cid:105) , where wepoint out the linearity of the squared magnetization if approaching the phase transitionfield h ( p ) t . Such a dependence confirms the mean-field exponent β p = regardless ofthe lattice parameter p , which results in the mean-field-like behavior of the TFIM ifapproaching the phase transition.The phase transition fields h ( p ) t , calculated according to the method describedin [16], are summarized in table 3 together with their errors ∆ ( p ) . Notice that ∆ ( p ) represents only the error of the method providing that the calculated magnetization (cid:104) S zp (cid:105) is considered accurate. The data are graphically plotted in figure 5, whereas theerror bars are too small to be displayed. Using an analogous exponential fitting functionapplied on the critical magnetic fields h ( p ) t for p ∈ { , . . . , } , (cf. (7)), we calculatedthe asymptotic phase transition field of the TFIM on the Bethe lattice h ( ∞ ) t = 3 . nalysis of quantum spin models on hyperbolic lattices and Bethe lattice 〈 S z p 〉 p = 4 p = 5 p = 6 p = 7 p = 8 p = 9 p = 10 h 〈 S z p 〉 h 〈 S z p 〉 Figure 4.
The spontaneous magnetization (cid:104) S zp (cid:105) (the upper graph) and its square (cid:104) S zp (cid:105) (the lower graph) in the vicinity of the phase transitions with respect to the magneticfield h for p ∈ { , , . . . , } . The inset shows the detailed zoomed-in behavior forhigher values of p . p h t ( p ) p h t ( p ) h t ( ∞ ) = 3.29332 + a e a p a = − a = − h t ( ∞ ) h t ( p ) = Figure 5.
The phase transition field h ( p ) t of the TFIM with respect to the latticeparameter p . The horizontal dot-dashed line represents the estimated asymptotic value h ( ∞ ) t = 3 . h ( p ) t can be carried out by analyzing the magnetic susceptibility χ p = − ∂ E ( p )0 ∂h . (10) nalysis of quantum spin models on hyperbolic lattices and Bethe lattice h χ p p = 5 p = 6 p = 7 p = 8 p = 9 p = 10 Figure 6.
The magnetic susceptibility χ p of the TFIM as a function of the magneticfield h for the hyperbolic lattices with p ∈ { , . . . , } . The vertical dot-dashed linesserve as guides for the eye and correspond to the phase transitions h ( p ) t . The functional dependence of the susceptibility on the magnetic field h is shown infigure 6. A non-diverging discontinuity of χ p occurs at the identical phase transitionfields h ( p ) t , which we have determined above by the spontaneous magnetization analysisand are depicted by the vertical dot-dashed lines. The inaccuracy comes from performingthe second derivative in (10) numerically, and the additional improvement rests indecreasing the spacing interval δh , i.e, in shrinking the distance between the magneticfields, at which the ground-state energy is evaluated by TPVF. In the limit δh → h ( p ) t [16]. It is obviousthat there is no significant difference between the phase transition magnetic fields h ( p ) t obtained by the analysis of the transverse magnetization (cid:104) S zp (cid:105) and the magneticsusceptibility χ p .Except for the analysis of the phase transition by the spontaneous magnetization (cid:104) S zp (cid:105) and the magnetic susceptibility χ p , the field dependence of the set of the optimalfree variational parameters W ∗ p ( θ j ) also provides helpful information about the phasetransition h ( p ) t . The pairs of the optimal variational parameters W ∗ p ( θ j ) coupled by spin-inversion symmetry smoothly collapse onto a single curve exactly at the phase transitionfor all considered lattice geometries. This process follows the identical behavior aswe have presented in [16]. However, due to the large number of the variationalparameters N ( p )Ising , we do not plot the h -dependence of W ∗ p ( θ j ) since the behavior remainsqualitatively unchanged. nalysis of quantum spin models on hyperbolic lattices and Bethe lattice
4. Conclusions
We have investigated three quantum spin- models (Heisenberg, XY, and TFIM) ona series of hyperbolic lattices by means of the numerical algorithm Tensor ProductVariational Formulation [16]. The series of lattices is constructed by tessellation ofregular p -sided polygons with the fixed coordination number equal to four, where p ∈ { , . . . , } . The Euclidean square lattice ( p = 4) has been also considered asa reference lattice, although we have discussed in [16] that TPVF applied to the modelson the square lattice is less accurate than on the hyperbolic lattices ( p > E ( p )0 of the XY and the Heisenberg models have beenstudied in the absence of magnetic field on the series of the regular lattices with4 ≤ p ≤
11. Since no spontaneous symmetry breaking occurs in the two models at h = 0, the spin-inversion symmetry helps to accelerate the TPVF algorithm due tosignificant reduction of the number of the free variational parameters. The resultingdependence of the ground-state energy per bond E ( p )0 on the lattice geometry p differsconsiderably for the two models. While the energies E ( p )0 of the XY model form amonotonically increasing and exponentially saturated sequence with increasing p , theHeisenberg model induces a saw-like dependence containing the separated upper (odd p ) and the lower (even p ) branches, both of them converging exponentially fast to thecommon asymptotic value E ( ∞ )0 which corresponds to the ground-state energy on theBethe lattice with the coordination number four.We have analyzed the phase transition magnetic fields h ( p ) t of the TFIM by theexpectation value of the spontaneous magnetization (cid:104) S zp (cid:105) , the associated magneticexponent β p , and the magnetic susceptibility χ p . We have calculated a sequence ofthe phase transition magnetic fields h ( p ) t , which is a strictly monotonous and increasingfunction, which converges exponentially to the asymptotic value h ( ∞ ) t . This feature iscompletely analogous to a fast exponential saturation of the critical temperatures T ( p ) c wehad observed for the classical Ising model on the identical series of hyperbolic latticesin our earlier studies [19, 20]. However, we have not found physical interpretationof this phenomenon yet. The quantum spin systems (as well as the classical ones)investigated on the hyperbolic lattices belong to the mean-field universality class, sinceinfinite Hausdorff dimension of the hyperbolic lattice geometry exceeds the critical latticedimensions D c = 3 (for quantum models) and D c = 4 (for the classical ones). Thelinearity of the squared magnetization in the vicinity of the phase transition confirmsthe mean-field-like behavior, in which the associated magnetic exponents β p = .Although the set of the calculated phase transition magnetic fields h ( p ) t and theground-state energies E ( p )0 are restricted to 4 ≤ p ≤
11, which is far away from theasymptotics p → ∞ , the fast convergence and the exponential character of h ( p ) t and E ( p )0 with increasing p enables to estimate the respective quantities of the quantum spinmodels on the Bethe lattice ( p → ∞ ). In particular, we conjecture that the phasetransition field of the TFIM on the Bethe lattice is positioned at h ( ∞ ) t = 3 . nalysis of quantum spin models on hyperbolic lattices and Bethe lattice E ( ∞ )0 = − . − . Acknowledgments
The support received from the Grants QIMABOS APVV-0808-12 and VEGA-2/0130/15is acknowledged.
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