Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a θ term
PPrepared for submission to JHEP
UTHEP-738, UTCCS-P-125
Tensor renormalization group study of two-dimensionalU(1) lattice gauge theory with a θ term Yoshinobu Kuramashi a Yusuke Yoshimura a a Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan a r X i v : . [ h e p - l a t ] A p r ontents θ term 22.2 Gauss-Legendre quadrature method 32.3 Improved method 3 It has been argued that pure gauge theories with a θ term contain intriguing nonperturbativeaspects. Possible phase transition in the two-dimensional (2D) pure U( N ) gauge theorywas investigated at θ = 0 in the large N limit by Gross and Witten thirty years ago [1] andSeiberg discussed that it has a phase transition at θ = π in the strong coupling limit [2]. LaterWitten showed that the four-dimensional (4D) pure Yang-Mills theory yields the spontaneousCP violation at θ = π in the large N limit [3]. Recently this non-trivial phenomena was alsopredicted based on the argument of the anomaly matching between the CP symmetry andthe center symmetry [4]. Up to now, unfortunately, the numerical study with the latticeformulation has not been an efficient tool to investigate these nonperturbative phenomena.The reason is that the lattice numerical methods are based on the Monte Carlo algorithm sothat they suffer from the sign problem caused by the introduction of the θ term.In 2007 the tensor renormalization group (TRG) was proposed by Levin and Nave tostudy 2D classical spin models [5]. They pointed out that the TRG method does not sufferfrom the sign problem in principle. This is a fascinating feature to attract the attentionof the elementary particle physicists, who have been struggling with the sign problem toinvestigate the finite density QCD, the strong CP problem, the lattice supersymmetry and soon. In past several years exploratory numerical studies were performed by applying the TRGmethod to the quantum field theories in the path-integral formalism [6–19]. The authors andtheir collaborators have confirmed that the TRG method is free from the sign problem by– 1 –uccessfully demonstrating the phase structure predicted by Coleman [20] for the one-flavorSchwinger model with the θ term employing the Wilson fermion formulation [8] and theBose condensation accompanied with the Silver Blaze phenomena in the 2D complex scalar φ theory at the finite density [19].In this article we apply the TRG method to the 2D pure U(1) lattice gauge theory witha θ term. Since this is the simplest pure lattice gauge theory with a θ term and the analyticalresult for the partition function is already known [22], it is a good test case for the TRGmethod to check the feasibility to investigate the nonperturbative properties of the latticegauge theories with a θ term. In the previous studies of Schwinger model with and without the θ term [7–9], we employed the character expansion method to construct the tensor networkrepresentation following the proposal in Ref. [23]. In this work, however, we use the Gaussquadrature method with some improvement to discretize the phase in the U(1) link variable.This is motivated by the success of the Gauss quadrature method to discretize the continuousdegree of freedom in the TRG studies of the scalar field theories [16, 19].This paper is organized as follows. In Sec. 2 we explain the TRG method with the use ofthe Gauss quadrature to calculate the partition function of the 2D pure U(1) gauge theory.Numerical results for the phase transition at θ = π are presented in Sec. 3, where our resultsare compared with the exact ones which are analytically obtained. Section 4 is devoted tosummary and outlook. θ term The Euclidean action of the two-dimensional pure U(1) lattice gauge theory with a θ term isdefined by S = − β (cid:88) x cos p x − iθQ, (2.1) p x = ϕ x, + ϕ x +ˆ1 , − ϕ x +ˆ2 , − ϕ x, , (2.2) Q = 12 π (cid:88) x q x , q x = p x mod 2 π, (2.3)where ϕ x,µ ∈ [ − π, π ] is the phase of U(1) link variable at site x in µ direction. The range of q x is [ − π, π ] and it can be expressed as follows by introducing an integer n x : q x = p x + 2 πn x , n x ∈ {− , − , , , } . (2.4)For the periodic boundary condition, the topological charge Q becomes an integer: Q = (cid:88) x (cid:16) p x π + n x (cid:17) = (cid:88) x n x (2.5) See Ref. [21] for recent studies of the Schwinger model with the θ term in the Hamiltonian formalism. – 2 –he tensor may be given with continuous indices, T ( ϕ x, , ϕ x +ˆ1 , , ϕ x +ˆ2 , , ϕ x, ) = exp (cid:18) β cos p x + i θ π q x (cid:19) . (2.6)The partition function is represented as Z = (cid:32)(cid:89) x,µ (cid:90) π − π dϕ x,µ π (cid:33) (cid:89) x T ( ϕ x, , ϕ x +ˆ1 , , ϕ x +ˆ2 , , ϕ x, ) . (2.7) In order to obtain a finite dimensional tensor network, we discretize all the integrals inEq. (2.7) using a numerical quadrature. In general, an integral of a function f ( ϕ ) can beevaluated by (cid:90) dϕf ( ϕ ) ≈ K (cid:88) α =1 w α f (cid:16) ϕ ( α ) (cid:17) (2.8)where ϕ ( α ) and w α are the α -th node of the K -th polynomial and the associated weight,respectively. In this work, we use the Gauss-Legendre quadrature for discretization. Thediscretized local tensor can be expressed as T ijkl = √ w i w j w k w l (2 π ) T (cid:16) ϕ ( i ) , ϕ ( j ) , ϕ ( k ) , ϕ ( l ) (cid:17) , (2.9)and we get a finite dimensional tensor network Z ≈ (cid:88) { α } (cid:89) x T α x, α x +ˆ1 , α x +ˆ2 α x, , (2.10)where { α } represents a set of indices associated with the Gauss-Legendre quadrature . We have developed further improvement for the above method. In the singular value decom-position (SVD) procedure to prepare the initial tensor before starting the iterative TRG steps[12, 16, 19], we employ the following eigenvalue decomposition: M ijkl = √ w i w j w k w l (2 π ) (cid:90) π − π dϕ dϕ T (cid:16) ϕ ( i ) , ϕ ( j ) , ϕ , ϕ (cid:17) T ∗ (cid:16) ϕ ( k ) , ϕ ( l ) , ϕ , ϕ (cid:17) , (2.11)which is essentially equivalent to M ijkl = lim K (cid:48) →∞ K (cid:48) (cid:88) m,n =1 T ijmn T ∗ klmn . (2.12) Application of the plain Gauss-Legendre quadrature method to this model was originally proposed byYuya Shimizu. – 3 –his procedure is expected to reduce the discretization errors in M ijkl .To evaluate Eq. (2.11), we use the character expansion [24, 25]: T ( ϕ , ϕ , ϕ , ϕ ) = ∞ (cid:88) m,n = −∞ e in ( ϕ + ϕ − ϕ − ϕ ) I m ( β ) J n − m ( θ ) (2.13)where I m ( β ) is the m -th order modified Bessel function of the first kind and J n ( θ ) = ( − n θ + 2 πn sin (cid:18) θ (cid:19) . (2.14)Then, Eq. (2.11) is rewritten as M ijkl = √ w i w j w k w l (2 π ) ∞ (cid:88) n = −∞ e in ( ϕ ( i ) + ϕ ( j ) − ϕ ( k ) − ϕ ( l ) ) ∞ (cid:88) m,m (cid:48) = −∞ I m ( β ) I m (cid:48) ( β ) J n − m ( θ ) J n − m (cid:48) ( θ ) . (2.15)In the practical calculation, the sums of n, m and m (cid:48) can be truncated when the contributionsof the terms are small enough. In this work we discard the contributions of I m,m (cid:48) /I < − or J n − m,n − m (cid:48) /J < − . The partition function of Eq. (2.7) is evaluated with the TRG method at β =0.0 and 10.0 asa function of θ on a V = L × L lattice, where L is enlarged up to 1024. We choose K = 32for the polynomial order of the Gauss-Legendre quadrature in Eq. (2.8). The SVD procedurein the TRG method is truncated with D = 32. We have checked that these choices of D and K provide us sufficiently converged results for all the parameter sets employed in thiswork. Since the scaling factor of the TRG method is √
2, allowed lattice sizes for the partitionfunction are L = √ , , √ , · · · , √ , Q is quantized to be an integer. The analytic result for the partition function of Eq. (2.7) is given by [22]: Z analytic = ∞ (cid:88) Q = −∞ ( z P ( θ + 2 πQ, β )) V , (3.1) z P ( θ, β ) = (cid:90) π − π dϕ P π exp (cid:18) β cos ϕ P + i θ π ϕ P (cid:19) , (3.2)– 4 –here z P ( θ, β ) denotes the one-plaquette partition function with ϕ P ∈ [ − π, π ]. In Fig. 1 weplot the magnitude of the relative error for the free energy defined by δf = | ln Z analytic − ln Z ( K, D = 32) || ln Z analytic | (3.3)at θ = π on a 1024 × K increases even at θ = π , around which the Monte Carloapproaches do not work effectively due to large statistical errors [26]. Secondly, our methodyields more precise results than the plain Gauss-Legendre quadrature method at any valueof K . Thirdly, our choice of a parameter set of ( D, K ) = (32 ,
32) yields δf < − , whichmeans that the free energy is determined at sufficiently high precision. Hereafter we presentthe results obtained with ( D, K ) = (32 , K -14 -12 -10 -8 -6 -4 -2 δ f Plain quadratureImproved method
Figure 1 . Relative error of free energy as a function of K with D = 32 on a 1024 × K is the polynomial order of the Gauss-Legendre quadrature in Eq. (2.8). The expectation value of the topological charge (cid:104) Q (cid:105) at β = 10 . θ : (cid:104) Q (cid:105) = − i ∂ ln Z∂θ . (3.4)In Fig. 2 we show the volume dependence of (cid:104) Q (cid:105) /V around θ = π , where the analyticcalculation predicts the first order phase transition at any value of β [22]. We observe thata finite discontinuity emerges with mutual crossings of curves between different volumes at θ = π as the lattice size L is increased. This feature indicates there is a first order phasetransition at θ = π .It may be interesting to calculate the topological charge density in the strong couplinglimit β = 0 .
0, whose analytical result was obtained by Seiberg in the infinite volume limit [2]: (cid:104) Q (cid:105) V (cid:12)(cid:12)(cid:12)(cid:12) β =0 = − i (cid:18)
12 cot (cid:18) θ (cid:19) − θ (cid:19) . (3.5)– 5 – .90 0.95 1.00 1.05 1.10 θ / π -0.0050.0000.005 i < Q > / V L=8L=16L=32L=64L=128L=256
Figure 2 . Topological charge density with 8 ≤ L ≤
256 as a function of θ at β = 10 . Figure 3 compares the numerical result at β = 0 . θ = π with small lattice size of L = 4 essentially vanishes oncewe increase the lattice size up to L = 64. θ / π − i π < Q > / V L=4L=8L=64analytic result
Figure 3 . Topological charge density with 4 ≤ L ≤
64 as a function of θ at β = 0 .
0. Solid curvedenotes the analytic result of Eq. (3.5) obtained in the infinite volume limit.
We investigate the properties of the phase transition by applying the finite size scaling analysisto the topological susceptibility: χ ( L ) = − V ∂ ln Z∂θ . (3.6)Figure 4 shows the topological susceptibility as a function of θ for various lattice sizes. Thepeak structure is observed and its height χ max ( L ) grows as L increases. In order to determine– 6 –he peak position θ c ( L ) and the peak height χ max ( L ) at each L , we employ the quadraticapproximation of the topological susceptibility around the peak position: χ ( L ) ∼ χ max ( L ) + R ( θ − θ c ( L )) (3.7)with R a constant. θ / π -2 -1 χ ( L ) L=16L=32L=64L=128L=256L=512
Figure 4 . Topological susceptibility χ ( L ) as a function of θ with 16 ≤ L ≤ We expect that the peak height scales with L as χ max ( L ) ∝ L γ/ν , (3.8)where γ and ν are the critical exponents. The L dependence of the peak height χ max ( L )is plotted in Fig. 5. The solid curve represents the fit result obtained with the fit functionof χ max ( L ) = A + BL γ/ν choosing the fit range of 128 ≤ L ≤ A = − × − , B = 7 . × − and γ/ν = 1 . γ/ν = 1 . We have applied the TRG method to study the 2D pure U(1) gauge theory with a θ term.The continuous degrees of freedom are discretized with the Gauss quadrature method. Wehave confirmed that this model has a first-order phase transition at θ = π as predicted fromthe analytical calculation. The successful analysis of the model demonstrates an effectivenessof the Gauss quadrature approach to the gauge theories. It should be interesting to apply theTRG-based methods with the Gauss quadrature to higher dimensional gauge theories with θ – 7 – L -3 -2 -1 χ m a x ( L ) Numerical dataFit result
Figure 5 . Peak height of topological susceptibility χ max ( L ) as a function of L . Solid curve denotesthe fit result. term which have been hardly investigated by the Monte Carlo approach because of the signproblem. Another interesting research direction is to include fermionic degrees of freedomfollowing the Grassmann TRG method developed in Ref. [7]. This is a necessary ingredienttoward investigation of the phase structure of QCD at finite density. Acknowledgments
One of the authors (YK) thanks Yuya Shimizu for providing the results obtained by the plainGauss-Legendre quadrature method. Numerical calculation for the present work was carriedout with the Cygnus computer under the Interdisciplinary Computational Science Programof Center for Computational Sciences, University of Tsukuba. This work is supported by theMinistry of Education, Culture, Sports, Science and Technology (MEXT) as “ExploratoryChallenge on Post-K computer (Frontiers of Basic Science: Challenging the Limits)”.
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