Accurate simulation of q-state clock model
AAccurate simulation of q-state clock model
Guanrong Li, Kwok Ho Pai, and Zheng-Cheng Gu ∗ Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
We accurately simulate the phase diagram and critical behavior of the q -state clock model on thesquare lattice by using the state-of-the-art loop optimization for tensor network renormalzation(loop-TNR) algorithm. The two phase transition points for q ≥ R of the compactified boson theories at both phase transition points. Inparticular, the compactification radius R at high-temperature critical point is precisely the same asthe predicted R for Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, we find that thefixed point tensors at high-temperature critical point also converge(up to numerical errors) to thesame one for large enough q and the corresponding operator product expansion(OPE) coefficient ofthe compactified boson theory can also be read out directly from the fixed point tensor. I. INTRODUCTION
Berezinskii-Kosterlitz-Thouless (BKT) transitionwas originally proposed in classical XY model with acontinuum U (1) symmetry. It is well known that sponta-neous breaking of continuum symmetry is not allowed in2D classical systems and the BKT transition provides usthe first example beyond Landau’s symmetry breakingparadigm. On the contrary, spontaneous breaking of dis-crete symmetry is generally allowed for 2D classical sys-tems and BKT transition is usually not expected for thesesystems. In recent years, people find very strong numer-ical evidence that BKT transition actually also happensin systems with discrete symmetry, e.g., the q -state clockmodel. It has been pointed out that for q ≥
5, the q -state clock model typically has two critical points . Athigh-temperature critical point, the system undergoes aBKT transition, while at low-temperature critical point,the long-range order would emerge and the usual sym-metry breaking transition happens. Theoretically, it waswell known that q -state model with q ≥ Z q deformed sine-Gordon model , and therenormalization analysis also suggests that the model willundergo two phase transitions as the temperature de-creases. Between the two phase transition points, theeffective field theory reduces to a simple compactified bo-son theory with emergent U (1) symmetry. Previously, alot of studies have been focused on how to determine thetwo critical temperatures , but how to accurately ex-tract the exact conformal data at critical points is still avery challenging problem. Tensor renormalization group(TRG) algorithm isa powerful tool to study the phase diagram of 2Dclassical statistical models and 1+1D quantum mod-els. By investigating the properties of the correspond-ing fixed point tensor, many important properties of thephase diagram can be read out directly . In recentyears, the so-called loop optimization for tensor networkrenormalzation(loop-TNR) method was proposed as areal space renormalization algorithm to accurately studycritical properties of 2D classical statistical models and1+1D quantum models. Comparing with singular value decomposition based methods, e.g., TRG and higher or-der TRG(HOTRG) , the loop-TNR algorithm has ex-tremely high accuracy and makes it possible for us toread out all the conformal data for critical systems, suchas scaling dimensions, operator product expansion(OPE)coefficient for primary fields from the corresponding fixedpoint tensor.In this paper, we use loop-TNR algorithm to study thephase transition properties of the q -state clock model.We find very strong numerical evidence that the physicsof self-dual critical points for q < q ≥ c = 1. By computing the scaling dimensions of the twophase transition points as well as the so-called self-dualpoints, we are able to determine the compactificationradius R of the corresponding compactified boson the-ory with very high accuracy. We find that the obtainedcompactification radius R perfectly agree with the fieldtheory predictions. Furthermore, we also find that forbig enough q , the corresponding fixed point tensors athigh-temperature critical point T c converge to the sameone(up to numerical errors) describing BKT transitionwith an emergent U (1) symmetry, and the correspond-ing OPE coefficient of the compactified boson theory canalso be read out directly.We stress that our method not only gives accuratecritical temperature, but also produces accurate con-formal data, especially for the cases with q = 5 and q = 6, which are very hard to be simulated by densitymatrix renormalization group(DMRG)/matirx productstate(MPS) based methods as well as Monte Carlosimulation due to the presence ofmarginal irrelevant terms . Our numerical results alsosuggest that 2D CFT could be reformulated as an in-finite dimensional fixed point tensor which encodes thecomplete conformal data, such as scaling dimensions andOPE coefficients. This might provide us an algebraic wayto reformulate and classify all 2D CFT. a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p II. q < MODELS
The q -state clock model is describe by the Hamiltonian H = − J (cid:88) (cid:104) ij (cid:105) cos ( θ i − θ j ) , (1)where θ i = 2 πn i /q, and n i ∈ { , , ...q } . We note that for q = 2 and q = 3 the model is equivalent to classical Isingmodel and 3-states Potts model. The partition functionof the q -state clock model can be expressed as a trace oflocal tensors: Z = Tr ⊗ T. (2)On square lattice, the partition function of the q -statemodel is expressed by the trace of the following elementtensor T ijkl (seen in Fig. 1): T ijkl = exp β (cos θ ij + cos θ jk + cos θ kl + cos θ li ) , (3)where θ ij = 2 π ( i − j ) /q and i, j, k, l take values { , , ...q } . i jkl FIG. 1: Tensor network representation of q -state clockmodel on square lattice.For q <
5, it is well known that the self-dual criticaltemperature reads : β c = ln (cid:0) √ (cid:1) / , q = 22 ln (cid:0) √ (cid:1) / , q = 3ln (cid:0) √ (cid:1) , q = 4 (4)We will first benchmark with these exact results to exam-ine the accuracy of our algorithm. Since the q = 2 casehas already been studied before, here we will begin withthe q = 3 and q = 4 cases. To find the critical point, wefirst calculate the gauge invariant quantity χ introducedin Ref. : χ = (cid:32)(cid:80) ij T ijij (cid:33) (cid:80) ijkl T ijkl T klij , (5)where we use the 2 by 2 block to represent the fixedpoint tensor T (composed by T A and T B on sublattices A FIG. 2: We use the 2 by 2 block to represent the fixedpoint tensor T when calculating χ , where we group theindex ( i , i ) into a single index i for tensor T (a) (b) FIG. 3: Gauge invariant quantity χ formed by T , wherethe numerator is represented by the square of part (a),and the denominator is given by part (b)and B, respectively) when calculating the gauge invariantquantity χ , as shown in Fig. 2 and Fig. 3.As seen in Fig. 4, we see that there is a sudden jumpfrom ordered phase to disordered phase. This is becausethe tensors for ordered and disordered phase would flowto different fixed points. To understand better for thegauge invariant quantity χ , we introduce matrix M h and M v : M hij = (cid:88) k T fixed-point ikjk M vij = (cid:88) k T fixed-point kikj . (6)We see that for ordered phase, the eigenvalue λ of M h and M v is: (cid:26) λ , λ , ...λ q = 1 /q others = 0 . (7)And in disordered phase, we have λ = 1, and all theothers approach 0, which shows clearly the symmetrybreaking nature of the phase transition. Here, we havealready normalized the fixed point tensor as: (cid:88) jk T fixed-point jkjk = 1 . (8)Next, we compute the central charge and scaling di-mensions for q = 3 model(here we keep D cut = 36 in ourloop-TNR algorithm). We find that the central charge c = 0 . c = 4 /
5. We see that bothcentral charge and scaling dimensions are very stable upto 20 renormalization steps, which corresponds to a totalsystem size 2 . T c = 1.4925(5) q =3 FIG. 4: The invariant quantity χ as a function oftemperature. We find that the critical temperature T c for q = 3 model is around 1.4925(5), which isintrinsically close to the prediction of the self-dualanalysis. Here we keep D cut = 36 in the loop-TNRalgorithm and system size up to 2 . q = 3( T = T c ) numerical central chargeCFT expectation FIG. 5: The scaling dimensions at the critical point of q = 3 model with D cut =36. We see that the conformaldata rapidly converges to CFT predictions during therenormalization process.Similarly, we can compute the gauge invariant quan-tity χ , central charge and scaling dimensions for the q = 4model(again, we keep D cut = 36 in our loop-TNR algo-rithm), as shown in Fig. 6 and Fig. 7. We find that c = 1 . c = 1. In fact, the critical pointof q = 4 model can be just regarded as two copies of the Ising CFT. Again, we see that both central charge andscaling dimensions are very stable up to 20 renormaliza-tion steps T c = 1.1345(5) q =4 FIG. 6: The invariant quantity χ as a function oftemperature. We find that the critical temperature T c for q = 4 model is around 1.1345(5), which isintrinsically close to the prediction of the self-dualanalysis . Here we also keep D cut = 36 in theloop-TNR algorithm and system size up to 2 . q = 4( T = T c ) numerical central chargeCFT expectation FIG. 7: The scaling dimensions at the critical point of q = 4 model with D cut =36. III. q = 5 AND q = 6 MODELS
For q ≥
5, it is conjectured that the q -state clock modelis described by Z q -deformed sine-Gordon theory S = 12 πK (cid:90) d r ( ∇ φ ) + g πα (cid:90) d r cos (cid:16) √ φ (cid:17) + g πα (cid:90) d r cos (cid:16) q √ (cid:17) , (9)where φ, Θ are compactified as φ ≡ φ + √ π , Θ ≡ Θ + √ π, and they satisfy the dual relation ∂ x φ = − K∂ y Θ , ∂ y φ = K∂ x Θ . The coupling constants
K, g , g are temperature-dependent, and α is a UV cutoff. T c T c q=5 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 FIG. 8: Invariant quantity of q = 5 model. In orderedphase, the gauge invariant quantity χ should flow to thefixed point χ = 5, while in disordered phase, χ shouldflow to χ = 1. In the middle phase, the value of χ varieswith temperature. We can read out that T c = 0 . T c = 0 . g and g become irrelevant between the two critical point T c < T < T c , the effective theory reduces to the com-pactified boson theory in the middle phase, with com-patification radius R = √ K . In addition, if g = g ,Eq. (9) is self-dual. From the scaling dimension analy-sis, the compactification radius can be computed exactlyfor both phase transition points as well as for the self-dualpoint . We have: R c = 2 √ , BKT transition point R self − dual = (cid:112) q, self-dual point R c = q/ √ , symmetry-breaking pointSimilar to the q < q = 5 model can be read out from the gauge invari-ant quantity χ . In Fig. 8, we plot χ as a function oftemperature near the critical point. Very different fromthe q < χ nearthe two phase transition points. Similar to the q < χ = 5, while in disordered phase, thefixed point tensor gives rise to χ = 1. However, in themiddle phase, The structure of fixed point tensor is very T c q=6 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 T c q=6 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 FIG. 9: Invariant quantity of q = 6 model around T c and T c . We can read that T c = 0 . T c = 0 . q = 5model.complicated and we will discuss the details later. Aninteresting feature is that the gauge invariant quantity χ becomes size independent in the middle critical phaseand this help us pin down the critical temperature forboth high-temperature and low-temperature phase tran-sitions. As seen from Fig. 8, we can read out that the lowtemperature symmetry breaking transition point T c isaround 0.908(2) while the high-temperature BKT phasetransition point is around 0.952(2). Similar analysis canbe applied to q = 6 model as well, and we can read outfrom Fig. 9 that the low-temperature critical point T c is around 0.696(2), and high-temperature phase transi-tion point T c is around 0.912(2). We note that in orderto increase the accuracy, here and below we will use the Z q symmetric loop-TNR algorithm(see Appendix B formore details) with D cut = 8 q for simulating all q -stateclock models.Since the middle phase is described by compactifiedboson model, we can further use the fixed point tensorto compute its central charge and scaling dimensions. Asseen in Fig. 10, we find c = 0 . q = 5 modelwith T = 0 . k B /J , which is intrinsically close to the q = 5( T = 0.93/ k B ) numerical R fitting FIG. 10: An example of scaling dimensions in thecritical phase for q = 5 model. T c T c q = 5Ref. q = 6Ref. - 0.9020(5)Ref. TABLE I: A comparison of T c and T c with previousresults by using other methods.theoretical prediction with c = 1. It is well known thatthe scaling dimensions of the primary fields of the com-pactified boson model can be expressed as:∆ e,m = m R + e R , (10)where R is the compactified radius and m, e are inte-gers which label the primary fields. In Fig. 10, we q =5, h = 10 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 q =6, h = 10 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 FIG. 11: Susceptibility of q = 5 and q = 6 models withexternal field h = 10 − .also plot the scaling dimension for q = 5 model with T = 0 . k B /J . We find that all the low scaling dimen-sion can be fit quite well with R = 3 . R ). We note that the deviationsfor high scaling dimensions are due to the numerical errorand we can further improve the accuracy by increasing D cut in the loop-TNR algorithm.The BKT transition point T c can also be determinedby the susceptibility peak method with extremely highaccuracy. First, by applying a very small external field,we can compute the susceptibility at different externalfield h and temperature T : χ ( h, T ) = ∂m∂h (cid:12)(cid:12)(cid:12)(cid:12) T . (11)For example, in Fig. 11, we plot the susceptibility func-tion at different system size for q = 5 and q = 6 modelswith a very small external field h = 10 − . We see thatall the susceptibility functions collapse to a single curve,which implies the thermodynamic limit has already beenachieved for physical quantities despite the fact that thegauge invariant quantity χ still has very strong size de-pendence near both critical temperatures. By plotting lgh T p e a k q =5 scaling law fitting FIG. 12: Susceptibility peak temperature versusexternal field for q = 5 model, from which we find that T c = 0 . a = 0 . b = 0 . lgh T p e a k q =6 scaling law fitting FIG. 13: Susceptibility peak temperature versusexternal field for q = 6 model, from which we find that T c = 0 . a = 0 . b = 0 . χ with different external fields, wecan read out T c by using the following formula: T peak ( h ) = T c + ah b . (12)We find that for q = 5 model, T c = 0 . a =0 . b = 0 . q = 6 model, T c = 0 . a = 0 . b = 0 . q = 5 and q = 6 models,respectively. We see that the results of T c is comparablewith what we get from the gauge invariant quantity χ .In Table I, we compare our results with all previousknown results for T c and T c using other methods. Wesee that our method gives much more accurate criticaltemperatures than HOTRG based method , and theresults are comparable with recent MPS based method and large scale Monte-Carlo results . We note thatthe small disagreement in the last digit might arise from q = 5( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 5( T = T c ) numerical R fitting FIG. 14: Scaling dimensions at the critical point T c and T c for q = 5 model, from which we can fit thecompactifiction radius R of the compactified bosontheory. We find that at T c , R = 3 . T c , R = 2 . with very highaccuracy.We further compute the scaling dimensions at T c and T c for of both q = 5 and q = 6 models. From the resultsof scaling dimension at each RG step, we can clearly ob-serve the logarithmic flow of some higher scaling dimen-sions, as seen in Fig. 14 and Fig. 15. This implies theexistence of marginal irrelevant terms for these transi-tion points, and it explains why these transition pointsare very hard to be determined accurately in previousstudies. From the scaling dimensions, we can fit the com-pactification radius R by using Eq. (10). In Table II,we list the compactification radius R at both transitionpoints and we find a perfect agreement with the field the-ory predictions. We stress that comparing with the veryrecent studies by using MPS based method , our resultsgive rise to much more accurate compactification radius R at these phase transition points.Finally, we investigate the scaling dimensions and com-pactification radius R for the so-called self-dual point. q = 6( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 6( T = T c ) numerical R fitting FIG. 15: Scaling dimensions at the critical point T c and T c for q = 6 model, from which we can fit thecompactification radius R of the compactified bosontheory. We find that at T c , R = 4 . T c , R = 2 . ,could be expressed as: (cid:101) T abcd = exp β θ a + cos θ b + cos θ c + cos θ d ) × δ mod(a+b+c+d , q) , (13)To determine the self-dual temperature, we compute themagnetization at different temperatures for both q -statemodel and its dual model. As seen in Fig. 17, thecrossing point corresponds to the dual temperature with g = g . Again, we can use the loop-TNR algorithmto compute the scaling dimensions(see in Fig. 18) andfrom the scaling dimension data, we can further fit thecompactification R . In Table II, we compare our resultswith the theoretical predictions. Again, we find a perfectagreement for both q = 5 and q = 6 models. a b cd FIG. 16: Tensor network representation of dual model,where the original lattice is shown by solid line and thedual lattice is shown by dash line. q T c T dual T c theory numerical theory numerical theory numerical5 (cid:112) / √
10 3.17354 2 √ √
18 4.23870 √
12 3.46002 2 √ TABLE II: Compactification radius R on both criticalpoints as well as self-dual point of q -state clock modelwith q = 5 and q = 6. IV. q > MODELS AND FIXED POINTTENSOR FOR BKT TRANSITIONA. Critical temperature and compactificationradius
By using the same methods for q = 5 and q = 6 models,we also studies the phase diagram for q > χ and fit-ting the susceptibility peak position under different exter-nal field, we can determine both T c and T c accurately.In Table III, we compare our results for q = 7 , , q , i.e., q > T c becomesvery close to the BKT transition value in classical XYmodel with T c = 0 . q = 5 and q = 6 models, we can also useloop-TNR method to compute the scaling dimensionsand fit the corresponding compactification radius R , seeAppendix A for more details. We find that the radius R at T c also saturates to a fixed value 2.81987 for bigenough q , which is intrinsically close to the theoreticalprediction R = 2 √
2. We can also use the same methodfor q = 5 and q = 6 models to determine the self-dualpoint and fit the corresponding compactification radius R . In Table IV, we also list the compactification radius R for T c and self-dual point T dual . Again, we find a perfectagreement with theoretical predictions. m a g n e t i z a t i o n magnetization of q=5 model q-state modeldual model0.6 0.7 0.8 0.9 1.0T0.00.20.40.60.8 m a g n e t i z a t i o n magnetization of q=6 model q-state modeldual model FIG. 17: Magnetization of q -state clock model with q = 5 , B. Fixed point tensor for BKT transition
Since the T c for q > R is also approaching the expectedvalue for BKT transition, it is natural to ask whether thecorresponding fixed point tensors in these models alsoconverge to the same one(up to numerical errors) whichcontains the complete information for BKT transitions.Below we will study the structure of fixed point tensorfor q >
1. The gauge choice of the fixed point tensor
It is well known that there exists a gauge degree of free-dom for the fixed point tensor in any TNR scheme and itis actually the major difficulty for us to understand thefull structure of fixed point tensors for critical systems.We will begin with some general discussion for the na-ture of such a gauge degree of freedom and explain why q = 5( T = T dual ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 6( T = T dual ) numerical R fitting FIG. 18: Scaling dimension on self-dual point for q = 5and q = 6 models, from which we can fit thecompactification radius R of the compactified bosontheory. We find that R = 3 . q = 5 and R = 3 . q = 6 model.it can be fixed by introducing enough symmetry condi-tions. Apparently, if we apply some invertible matriceson every legs of a tensor, the transformed tensor actuallyforms the same tensor network as before: T (cid:48) ijkl = (cid:88) i (cid:48) j (cid:48) k (cid:48) l (cid:48) T i (cid:48) j (cid:48) k (cid:48) l (cid:48) U i (cid:48) i V j (cid:48) j (cid:2) U − (cid:3) kk (cid:48) (cid:2) V − (cid:3) ll (cid:48) (14)This gives rise to great difficulty to analyze the prop-erties of the tensor components of the fixed point tensor,since they could be randomly affected by the gauge choicein numerical calculations. To get a proper gauge fixing,we have the following considerations: • The fixed point tensor(defined on the 2 by 2 plaque-tte composed by T A and T B tensors, as shown in Fig. 2)should preserve the C lattice symmetry during the loop-TNR process(see Appendix B for more details). Preserv-ing C symmetry will reduce the gauge freedom of thefixed point tensor. The gauge transformation in Eq. (14) T c T c q = 7Ref. q = 8Ref. q = 9Ref. TABLE III: A comparison of T c and T c with previousresults by using other methods. q T c T dual T c theory numerical theory numerical theory numerical7 (cid:112) / √
14 3.75035 2 √ √
32 5.67377 √
16 4.00726 2 √ (cid:112) / √
18 4.23573 2 √ TABLE IV: Compactification radius R on criticalpoints and self-dual point for q = 7 , , T (cid:48) ijkl = (cid:88) i (cid:48) j (cid:48) k (cid:48) l (cid:48) T i (cid:48) j (cid:48) k (cid:48) l (cid:48) O i (cid:48) i O j (cid:48) j O k (cid:48) k O l (cid:48) l (15)where O is an orthogonal matrix.(We assume all the ten-sors are real.) • Since the q -state clock model has a Z q internal sym-metry, we should also keep such an internal symmetryduring the whole loop-TNR process(see Appendix B formore details). By keeping the Z q symmetry, we can fur-ther reduce the gauge degrees of freedom. In fact thisis a crucial step to obtain the right fusion rule for fixedpoint tensor. It is well known that the fusion rule of com-pactifiled boson theory has a U (1) symmetry which canbe realized explicitly on XY model. However, if we onlyfocus on the leading components of primary fields anddescendant fields, Z q symmetry is a very good approxi-mation for U (1). • If we want the indices of the fixed point tensor torepresent the primary fields and their descendants forthe corresponding compactified boson theory, we needto choose a proper basis. The eigenstate of the transfermatrix is a good choice. As shown in Fig. 19 (d), weconstruct a rank-3 tensor with the building block ten-sor M ijkl in C -loop-TNR algorithm(see Appendix B formore details). This is because in usual CFT, the 3-pointcorrelation function is more fundamental and has a muchsimpler form than the 4-point correlation function. Infact, the basic renormalzation step in loop-TNR is simi- lar to the crossing symmetry for 4-point correlation func-tion. Thus, we conjecture that the rank-3 tensor con-structed here could be regarded as a 3-point correlationfunction(at least for primary fields). As illustrated inFig. 20, we construct the 2 × M ( i i )( j j ) = (cid:88) k U ( i i ) k λ k (cid:2) U − (cid:3) k ( i i ) . (16)We use eigenvectors U ( i i ) i as the basis for the fixedpoint tensor, as shown in Fig. 20 (b). As a result, thefixed point tensor is projected onto the basis representingprimary fields and their descendants. (a) (b) (c) (d) ≈ = i i j j k k M i i lk M j j k l Σ l l (e) z z z FIG. 19: From loop-TNR algorithm, a square fixedpoint tensor (a) could be approximately represented byMPS on the octagon lattice (b). Then, we decomposeoctagon MPS from (b) to (c). The rank-3 tensor in (d)is the fixed point tensor we will study here. (e) Thegeometry of the corresponding 3-point correlationfunctions. i i i r r r (a) (b) i i j j FIG. 20: (a) We choose the eigenstates of the transfermatrix in as our basis. (b) We then project the rank-3fixed point tensors on to these basis.0 T q =7 r r r T q =8 r r r T q =9 r r r T q =10 r r r r r r I I I
I α β
I β α
I γ δ
I δ γ α I β α α δ α β I α δ α β I α β α I β β γ β γ β γ I δ γ β β γ δ I δ I γ δ α α δ γ I
TABLE V: A comparison of non-zero leadingcomponents of the fixed point tensor of q-state clockmodels with q = 7 , , ,
10 at BKT critical point.
2. Operator product expansion(OPE) coefficient from thefixed point tensor
In Table V, we list the leading non-zero components ofthe fixed point tensors of different q -state clock modelsat BKT critical point. Here we normalize the largestcomponent T III = 1. We use I , α , β , γ , δ , λ and η torepresent the leading primary fields (0 , , − , , − , ,
0) and ( − , φ e ,m ] × [ φ e ,m ] = [ φ e + e ,m + m ] , (17)where [ φ e,m ] is a conformal family generatedby primary field φ e,m with conformal dimension (cid:16) ( e/R + mR/ / , ( e/R − mR/ / (cid:17) . In particular,the primary field with m = 0 is just the vertex operatorand it can be written as: φ e, ( z, z ) = e ieϕ ( z,z ) /R , (18)with ϕ ( z, z ) the free boson field. The 3-point functionhas a pretty simple form: (cid:104) φ e , ( z , z ) φ e , ( z , z ) φ e , ( z , z ) (cid:105) = C | z | ∆ +∆ − ∆ | z | ∆ +∆ − ∆ | z | ∆ +∆ − ∆ , (19)where C is the OPE coefficient, which equals 1 for e + e + e (cid:54) = 0 and vanishes for e + e + e = 0. | z | ≡ | z − z | , and the scaling dimension ∆ i = e i R .We note that in general only leading primary fields inour numerical fixed point tensor can satisfy the fusion rule since we use the Z q symmetry to approximate the U (1) symmetry in the gauge fixing procedure, and withincreasing q , more and more primary fields with correctfusion rules can be resolved numerically. (Although webelieve that the emergent U (1) must be present for allfinite q with q >
4, it is in general very hard to find theproper gauge choice for small q , especially for q = 5 and q = 6.)Next, we can try to fit our numerical fixed point tensorby using the 3-point correlation function Eq. (19). Let z = λ x , z = λ x , z = λ x . We can rewrite theright hand side of Eq. (19) as: C ( λ x ) ∆ +∆ − ∆ ( λ x ) ∆ +∆ − ∆ ( λ x ) ∆ +∆ − ∆ = C (cid:18) λ λ λ x (cid:19) ∆ (cid:18) λ λ λ x (cid:19) ∆ (cid:18) λ λ λ x (cid:19) ∆ ≡ C l ∆ l ∆ l ∆ , (20)with l = λ λ λ x , l = λ λ λ x and l = λ λ λ x , respectively.From the geometry of the square lattice, we conjecturethat our rank-3 fixed point tensor can be regarded as 3-point correlation(at least for primary fields) function onthe vertex of an isosceles right triangle on the complexplane, as seen in Fig. 19 (e). Thus we can choose λ = λ = λ / √ C l ∆ l ∆ l ∆ ≡ C l ∆ l ∆ (2 l ) ∆ . (21)where l = √ x is a fundamental inverse length scale.For q = 10 model at the temperature T c , the non-zeroleading components of fixed point tensor are given byTable VI. If we fit our data with Eq. (21), we find:∆ ( ± , = 0 . ( ± , = 0 . ( ± , = 1 . , (22)which match well with the results from our previoustransfer matrix calculation, with ∆ ( ± , = 0 . ( ± , = 0 . ( ± , = 1 . R = 2 . x = 2 . (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:80) N (cid:104)(cid:12)(cid:12)(cid:12) | T r r r | − C l ∆ l ∆ (2 l ) ∆ (cid:12)(cid:12)(cid:12) / | T r r r | (cid:105) N , (23)where N is the total number of components in our con-sideration. We find the fitting error is around 4 . × − .Thus, we conclude that the fixed point tensor can bewell described by the 3-point function(at least for pri-mary fields) and the OPE coeeficient can be read outdirectly.1 T r r r r r r T r r r r r r I I I γ I δ
I α β γ α η
I β α γ β β
I γ δ γ δ I
I δ γ γ η α
I λ η δ I γ
I η λ δ α α α I β δ β λ α α δ δ γ I α β I δ λ β α γ η λ I η α δ α λ β δ α η γ λ δ β β I α λ η I β α I η I λ β β γ η α γ β γ β η γ α β δ λ η λ I β λ δ
TABLE VI: Leading non zero components of the fixedpoint tensor of q = 10 model at BKT critical point T c . C r r r r r r C r r r r r r I I I γ I δ
I α β γ α η
I β α γ β β
I γ δ γ δ I
I δ γ γ η α
I λ η δ I γ
I η λ δ α α α I β δ β λ α α δ δ γ I α β I δ λ β α γ η λ I η α δ α λ β δ α η γ λ δ β β I α λ η I β α I η I λ β β γ η α γ β γ β η γ α β δ λ η λ I β λ δ
TABLE VII: Fitting OPE coefficients in Eq. (19) of q = 10 model at temperature T = T c , we see that theyapproach the expected value 1. C. Fixed point tensor for general cases
In fact, the above structure of fixed point tensor holdsfor the whole critical phase between T c and T c . Inthe following, we further study the fixed point tensorfor the q = 10 case at different temperatures. Table VIIIshows that all the OPE coefficients are very close to 1,as expected from the compactified boson theory. TableIX shows the comparison between the scaling dimensionsread from the fixed point tensor and from the direct cal- C t =0 . r r r C t =0 . r r r C t =0 . r r r r r r I I I
I α β
I β α
I γ δ
I δ γ α I β α α δ α β I α δ α β I α β α I β β γ β γ β γ I δ γ β β γ δ I δ I γ δ α α δ γ I
TABLE VIII: OPE coefficient in Eq. (19) fitting fromthe data of model q = 10 on different tempreature Temperature ∆ ∆ ∆ Fitting radius R From fixed point tensor0.70 0.06941 0.27824 0.62659 3.792670.72 0.07228 0.28971 0.65169 3.723120.74 0.07532 0.30182 0.67813 3.645730.76 0.07851 0.31456 0.70589 3.571060.78 0.08191 0.32813 0.73536 3.496840.80 0.08566 0.34298 0.76683 3.41513From transfer matrix0.70 0.06940 0.27795 0.62510 3.794980.72 0.07217 0.28845 0.65024 3.722900.74 0.07508 0.30057 0.67854 3.649050.76 0.07830 0.31386 0.70314 3.573250.78 0.08178 0.32827 0.73366 3.497270.80 0.08571 0.34675 0.76734 3.41494
TABLE IX: Scaling dimension of the first 3 levels readsby fitting fixed point tensor with 3-point function ofCFT and from the calculation of transfer matrix.culation of transfer matrix. We see that they also matchvery well.Therefore, we find very strong evidence that the fixedpoint tensor can be described by three-point correlationfunction for primary fields. Such a structure also explainswhy loop-TNR is a very accurate algorithm for criticalsystems since primary fields with higher scaling dimen-sions will lead to a rapid decay for the correspondingtensor components. We also find that the componentsfor descendant fields are always smaller than the corre-sponding primary field in the fixed point tensor. We be-lieve this is also because descendant fields will have biggerscaling dimensions. However, the explicit fixed point ten-sor structure for descendant fields is rather complicatedand we will leave this problem in our future study.2
V. CONCLUSIONS AND DISCUSSIONS
In summary, we use loop-TNR algorithm to study thephase transition properties of the q -state clock model.For q < q > T c and T c for both phase transitions. Byfurther computing the central charge and scaling dimen-sions at T c and T c , we can further obtain the compacti-fication radius R which also perfectly agrees with the Z q deformed sine-Gordon theory predictions. Interestingly,for big enough q , we find that the fixed point tensor at T c converges to the same one(up to numerical errors)that describes the well known BKT transitions, and thecorresponding OPE coefficient can also be read out di-rectly.For our future work, it will be of great interest to inves-tigate the explicit expression of the infinite dimensionalfixed point tensor description for the compactified bo-son theory as well as general CFT. In fact, the fixedpoint tensor provides us a purely algebraic way to de-scribe CFT which origins from a geometric perspective.Very recently, it has been shown that the p-adic CFT admits an explicit finite dimensional tensor network rep-resentation. It is somewhat not quite surprising sincep-adic CFT has no descendant fields. Since descendantfields might tell us how geometry emerges from basic al-gebraic data, it would be very important to understandthe explicit form of fixed point tensor descriptions fordescendant fields in usual CFT. ACKNOWLEDGMENTS
We are grateful to Ling-Yan Hung and Gong Chengfor very enlightening discussions for the structure of fixedpoint tensors at critical points. This work is supportedby funding from Hong Kongs Research Grants Council(GRF no.14301219) and Direct Grant no. 4053409 fromThe Chinese University of Hong Kong.
Appendix A: Transition temperatures andcompactification radius R for q > models For models with q >
6, e.g. q = 7 , , χ to determine the transitiontemperature for T c and T c , as seen in Fig. 21 and Fig.22. In Fig. 23, Fig. 24 and Fig. 25, we also use the sus-ceptibility peak method Eq. (12) to determine the BTKtransition temperature T c with very high accuracy. Re-markably, we find that for q >
6, the fitting paramters a and b are already very close to those obtained fromclassical XY model . Finally, we use the loop-TNR al-gorithm to compute the scaling dimensions at both high- temperature and low temperature critical points as wellas the self-dual point, as seen in Fig. 26, Fig. 27 andFig. 28. The corresponding compactification radius R can also be fitted by using Eq. (10). T c q=7 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 T c q =8 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 T c q=9 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 FIG. 21: Invariant quantity of for q-state clock modelwith q = 7 , , T c T c q=7 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 T c q=8 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 T c q=9 System size 2 System size 2 System size 2 System size 2 System size 2 System size 2 FIG. 22: Invariant quantity of for q-state clock modelwith q = 7 , , T c Appendix B: Imposing C rotational symmetry and Z q internal symmetry in loop-TNR algorithm In this appendix we first give a short review for theloop-TNR algorithm . Then we will discuss how to im-plement the C lattice symmetry and the internal Z q symmetry. Loop-TNR method mainly contains the fol-lowing steps, as shown in Fig. 29. In general, there will lgh T p e a k q =7 scaling law fitting FIG. 23: Susceptibility peak temperature versusexternal field. For q = 7 model, we find that T c = 0 . , a = 0 . , b = 0 . . . lgh T p e a k q =8 scaling law fitting FIG. 24: Susceptibility peak temperature versusexternal field. For q = 8 model, we find that T c = 0 . , a = 0 . b = 0 . T A and T B on sublattices A andB during the renormaliation process. • In step (a), we apply entanglement filtering to removethe corner double line(CDL) tensor. The CDL tensoronly contains local entanglement and cannot be the fixedpoint tensor describing critical systems. Ref. givesvery clear explanation on how to remove such short rangeentanglement. • In step (b), we find 8 rank-3 tensor to form a octagonmatrix product state(MPS) to approximate the squareMPS, as shown in Fig. 29 (d). We’re aiming to find theoptimal choice of those 8 rank-3 tensors S , S , ...S tominimize the cost function in Fig.29 (d), which can beexpressed as C ( S , S , ..., S ) = (cid:107) T A · T B · T A · T B − S · S · ... · S (cid:107) . (B1)Since S , S , ...S are independent variables, we can mini-4 lgh T p e a k q =9 scaling law fitting FIG. 25: Susceptibility peak temperature versusexternal field. For q = 9 model, we find that T c = 0 . , a = 0 . b = 0 . | Ψ A (cid:105) = T A · T B · T A · T B | Ψ B (cid:105) = S · S · ... · S (B2)Then, the cost function could be write down as C ( S , S , ..., S ) = (cid:104) Ψ A | Ψ A (cid:105) + (cid:104) Ψ B | Ψ B (cid:105)−(cid:104) Ψ A | Ψ B (cid:105)−(cid:104) Ψ B | Ψ A (cid:105) . (B3)Taking variation on S , we get δCδS † (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ,S ,..S = (cid:42) δ Ψ B δS † (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ B (cid:43) − (cid:42) δ Ψ B δS † (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ A (cid:43) ≡ [ N · S − W ] . (B4)The minimum of C ( S ) is given by the solution of thelinear equation: N · S = W . (B5)The cost function (B5) and N , W are illustrated in Fig.30. After optimizing S , we can go on to the next site,and if we finish the optimization from S to S , we finishone circle. By repeating this variation optimization, wecan minimize the cost function. • After minimizing the cost function, we trace the innerindices in the small circles, as shown in Fig. 29 (b), andget the coarse-grained tensor T (cid:48) A and T (cid:48) B , as in Fig. 29(c). Compared with the original tensor network, we findthe tensor network composed of the renormalized tensorelements T (cid:48) A and T (cid:48) B (a) rotates an angle of π/ • We will normalize the tensor T A and T B in every RGstep with the normalization factor as shown in Fig. 29(e). q = 7( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 8( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 9( T = T c ) numerical R fitting FIG. 26: Fitting of scaling dimensions at the criticalpoint T c for q = 7 , ,
1. loop-TNR with C lattice symmetry To keep the lattice symmetry in the renormalizationprocess, we need to find a octagon MPS with C symme-try when minimizing the cost function in Fig. 31 (d). Wecan construct this octagon MPS with the rank-4 blocktensor M ijkl , as shown in Fig. 31 (b). Then we can5 q = 7( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 8( T = T c ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 9( T = T c ) numerical R fitting FIG. 27: Fitting of scaling dimensions at the criticalpoint T c for q-state clock models with q = 7 , , C = (cid:107) T A · T B · T A · T B − M · M · M · M (cid:107) . (B6) q = 7( T = T dual ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 8( T = T dual ) numerical R fitting4 8 12 16 20loop-TNR step00.51.01.52.0 q = 9( T = T dual ) numerical R fitting FIG. 28: Fitting of scaling dimensions on self-dual pointfor q-state clock models with q = 7 , , M to build therenormalized tensor T (cid:48) A , T (cid:48) B , as shown in Fig.31 (c) T A (cid:48) ruld = (cid:88) ij M ijrd M jilu T B (cid:48) ruld = (cid:88) ij M ijld M jiru . (B7)6 (a) (b) (c) - (d) (e) FIG. 29: Loop optimization procedure, in step (a), weapply entanglement filtering, and in step (b) we find theoptimal S i to minimize cost function as shown (d).Then we trace the indices on the small square markedby the circle. (e) is gauge invariant quantity, which willbe used as the overall normalization factor. (a) (b)(c) (d) FIG. 30: Components of the cost function and itsderivative. (a) is (cid:104) Ψ A | Ψ B (cid:105) . (b) is (cid:104) Ψ B | Ψ B (cid:105) . (c) and (d)are the quantity W and N in (B5), respectively.Since the octagon network has C symmetry, the coarse-grained tensor network on the square lattice marked byblue circle has the same C symmetries.The initial value of the tensor M is very important forthe numerical accuracy. We can decompose tensor T A and T B by SVD method T Aruld ≈ (cid:88) x S ldx S rux = (cid:88) x S rux S ldx T Bruld ≈ (cid:88) x S ulx S drx = (cid:88) x S drx S ulx . (B8)Thus, the initial M is could be constructed as Fig. 31(e), with M ijkl = (cid:88) x S ixk S xjl . (B9)By keeping C lattice symmetries in each iteration step,we have partially fixed the gauge of the building block M ,which would be very important for studying the structureof the fixed point tensor. (a) (b)(d) (e) - (c) = M FIG. 31: Loop-TNR algorithm with C latticesymmetry is similar with usual loop-TNR. Notice thatthe cost function in this case is nonlinear, so we need touse nonlinear optimization algorithm, such as conjugategradient method.
2. loop-TNR with Z ( q ) symmetry in Hamiltonian As the original tensor element of q -state model T ijkl contains Z ( q ) symmetry, we can keep such a symme-try for every step in the loop-TNR algorithm. As Z ( q ) is a cyclic group, which contains group ele-ments (cid:8) I, g, g , ...g q − (cid:9) , and the generator g has the q -dimension faithful representation G q = ... ... ... ... ... ... ... ... ... ... ... . (B10)It is easy to check that: T (cid:48) ruld = (cid:88) r (cid:48) u (cid:48) l (cid:48) d (cid:48) [ G q ] rr (cid:48) [ G q ] uu (cid:48) [ G q ] ll (cid:48) [ G q ] dd (cid:48) T r (cid:48) u (cid:48) l (cid:48) d (cid:48) = T ruld . (B11)In order to find out all the irreducible representation ofthe Z q symmetry, we can just do eigenvalue decomposi-tion for G q , G q = V Λ V − , (B12)with eigenvalues Λ nn = λ n = e πin/q , n ∈ { , , , ..., q − } , and the components of the matrix V is given by: V mn = e πimn/q √ q , m, n ∈ { , , , ..., q − } . (B13)Such that: G q = V − Λ − V. (B14)Then, we define two tensors: T Aruld = (cid:88) r (cid:48) u (cid:48) l (cid:48) d (cid:48) (cid:2) V − (cid:3) rr (cid:48) (cid:2) V − (cid:3) uu (cid:48) (cid:2) V − (cid:3) ll (cid:48) (cid:2) V − (cid:3) dd (cid:48) T r (cid:48) u (cid:48) l (cid:48) d (cid:48) T Bruld = (cid:88) r (cid:48) u (cid:48) l (cid:48) d (cid:48) V rr (cid:48) V uu (cid:48) V ll (cid:48) V dd (cid:48) T r (cid:48) u (cid:48) l (cid:48) d (cid:48) . (B15)7Obviously, tensor T A and T B form the same tensor net-work with T . In the new basis tensors T A and T B satisfy: T Aruld = λ r λ u λ l λ d T Aruld T Bruld = λ − r λ − u λ − l λ − d T Bruld , (B16)which implies that T Aruld and T Bruld only have non-zerocomponents when r + u + l + d ≡ q ). Thus, tensors T A and T B are block diagonalized. 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