Phase transition of four-dimensional lattice φ^4 theory with tensor renormalization group
PPrepared for submission to JHEP
UTHEP-755, UTCCS-P-136
Phase transition of four-dimensional lattice φ theorywith tensor renormalization group Shinichiro Akiyama, a Yoshinobu Kuramashi, b Yusuke Yoshimura b a Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571,Japan b Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate the phase transition of the four-dimensional single-component φ theory on the lattice using the tensor renormalization group method. We have examinedthe hopping parameter dependence of the bond energy and the vacuum condensation ofthe scalar field (cid:104) φ (cid:105) at a finite quartic coupling λ on large volumes up to V = 1024 in orderto detect the spontaneous breaking of the Z symmetry. Our results show that the systemundergoes the weak first-order phase transition at a certain critical value of the hoppingparameter. a r X i v : . [ h e p - l a t ] J a n ontents The issue of the triviality of the four-dimensional (4 d ) φ theory has been a theoreticalconcern among particle physicists, because it is related to the scalar sector in the standardmodel [1–11]. The single-component φ theory becomes equivalent to the Ising model in theinfinite limit of the quartic coupling λ = ∞ so that numerical studies of the 4 d Ising modelhave been performed as a nonperturbative test of the triviality, assuming the universality[12–18]. So far, no Monte Carlo calculation has confirmed the logarithmic correction to themean-field exponents in the scaling behavior of the specific heat, which is expected fromthe perturbative renormalization group analysis [19]. Moreover, a detailed Monte Carlostudy has found a serious finite-volume effect due to nontrivial boundary effects in the 4 d Ising model [18].Recently, the authors have investigated the phase transition of the 4 d Ising modelwith the higher-order tensor renormalization group (HOTRG) algorithm [20]. The tensorrenormalization group (TRG) method, which contains the HOTRG algorithm, has severalsuperior features over the Monte Carlo method. (i) The TRG method does not sufferfrom the sign problem as confirmed by studying various quantum field theories [25, 28–34]. (ii) Its computational cost depends on the system size only logarithmically. (iii) Itallows direct manipulation of the Grassmann variables [25–27, 35]. (iv) We can obtain thepartition function or the path-integral itself. Thanks to the above feature (ii), we have beenallowed to enlarge the lattice volume up to V = 1024 , which is essentially identified as thethermodynamic limit, and found finite jumps for the internal energy and the magnetizationas functions of temperature in the 4 d Ising model [20]. These are characteristic features ofthe first-order phase transition. Having shown that the 4 d Ising model undergoes the weakfirst-order phase transition, our interest turns to the order of the phase transition in the 4 d single-component φ theory, which has the global Z symmetry as with the Ising model. In this paper, the TRG method or the TRG approach refers to not only the original numerical algorithmproposed by Levin and Nave [21] but also its extensions [22–27]. The scenario of the weak first-order phase transition in the Ising model or the φ theory has beendiscussed phenomenologically in some recent studies [36–39]. – 1 –n this paper, we investigate the phase transition of the 4 d single-component φ theorywith the quartic coupling λ and the hopping parameter κ , employing the anisotropic TRG(ATRG) algorithm [23], which was proposed to reduce the computational cost of the TRGmethod. The ATRG has been successfully applied to analyze the 4 d complex φ theory atthe finite density with parallel computation [34]. Our main purpose is to determine theorder of the phase transition by examining the κ dependence of the bond energy and thevacuum condensation of the scalar field (cid:104) φ (cid:105) around the critical value of κ c for the fixed λ ,the latter of which is an order parameter of the phase transition caused by the spontaneous Z symmetry breaking. We do not expect that the order of phase transition may change asa function of λ so that we make a single choice of λ = 40, which avoids the weak couplingregion affected by the Gaussian fixed point at λ = 0.This paper is organized as follows. In Sec. 2 we explain the formulation of the lattice φ theory and the ATRG algorithm. We present numerical results in Sec. 3 and discussthe properties of the phase transition. Section 4 is devoted to summary and outlook. We use the following popular action for the single-component φ theory on a lattice Γ: S [ φ ] = (cid:88) n ∈ Γ (cid:34) − κ (cid:88) ν =1 ( φ n φ n +ˆ ν + φ n φ n − ˆ ν ) + φ n + λ (cid:0) φ n − (cid:1) (cid:35) , (2.1)where ˆ ν is the unit vector of the ν -direction. This formulation, which is explicit about therelation to the Ising model, is equivalent to the more conventional expression S [ ϕ ] = (cid:88) n ∈ Γ (cid:34) (cid:88) ν =1 ( ϕ n +ˆ ν − ϕ n ) + 12 m ϕ n + g ϕ n (cid:35) (2.2)with ϕ n = √ κφ n , (2.3) m = 1 − λκ − , (2.4) g = 6 λκ . (2.5)The partition function is defined by Z = (cid:90) D φ e − S [ φ ] (2.6)using the action of Eq. (2.1) with the path integral measure (cid:90) D φ = (cid:89) n ∈ Γ (cid:90) ∞−∞ d φ n . (2.7)– 2 –e express the partition function as a tensor network in the similar way to Ref. [34]. Thecontinuous variables φ n are discretized by the K -point Gauss-Hermite quadrature rule as (cid:90) ∞−∞ d φ n e − φ n f ( φ n ) (cid:39) K (cid:88) α n =1 ω α n f ( φ α n ) , (2.8)where φ α and ω α are the α -th node and its weight. The partition function is thus discretizedas Z ( K ) = (cid:88) { α } (cid:89) n,ν M α n α n +ˆ ν , (2.9)where M α n α n +ˆ ν = √ ω α n ω α n +ˆ ν exp (cid:20) κφ α n φ α n + ν − λ (cid:0) φ α n − (cid:1) − λ (cid:16) φ α n +ˆ ν − (cid:17) (cid:21) . (2.10)Each matrix M is approximated by the singular value decomposition (SVD) with a bonddimension D as M αβ (cid:39) D (cid:88) k =1 U αk σ k V βk , (2.11)where σ k is the k -th singular value sorted in the descending order, and U, V are the or-thogonal matrices composed of the singular vectors. One finally obtains a tensor networkrepresentation for Z ( K ) as Z ( K ) = (cid:88) { x,y,z,t } (cid:89) n ∈ Γ T x n y n z n t n x n − ˆ1 y n − ˆ2 z n − ˆ3 t n − ˆ4 , (2.12)where T i i i i j j j j = K (cid:88) α =1 4 (cid:89) ν =1 √ σ i ν σ j ν U αi ν V αj ν . (2.13)In this study, we employ the parallelized ATRG algorithm developed in Refs. [34, 40].We keep the bond dimension D fixed throughout the ATRG procedure. Using 2 D compu-tational processes in tensor contractions, the execution time of the 4 d parallelized ATRGscales with O ( D ). For the swapping bond parts explained in Refs. [23, 41], the random-ized SVD is applied with the choice of p = 2 D and q = 2 D , where p is the oversamplingparameter and q is the numbers of QR decomposition. The partition function of Eq. (2.12) is evaluated using the ATRG algorithm on latticeswith the volume V = L ( L = 2 m , m ∈ N ) employing the periodic boundary conditions forall the space-time directions. As explained in the previous section, there are two importantalgorithmic parameters. One is the number of nodes K in the Gauss-Hermite quadrature– 3 –ethod to discretize the scalar field. The other is the bond dimension D . We check theconvergence behavior of the free energy as a function of K and D by defining the followingquantities: δ K = (cid:12)(cid:12)(cid:12)(cid:12) ln Z ( K, D = 50) − ln Z ( K = 2000 , D = 50)ln Z ( K = 2000 , D = 50) (cid:12)(cid:12)(cid:12)(cid:12) (3.1)and δ D = (cid:12)(cid:12)(cid:12)(cid:12) ln Z ( K = 2000 , D ) − ln Z ( K = 2000 , D = 50)ln Z ( K = 2000 , D = 50) (cid:12)(cid:12)(cid:12)(cid:12) . (3.2)Figure 1 shows the K dependence of δ K with D = 50 on V = 1024 at κ = 0 . . κ =0 . κ c , as we will see below. We observe that δ K decreases monotonically as a function of K and reaches the order of 10 − around K = 1500.This shows that the Gauss-Hermite quadrature method is not affected by whether thesystem is in the symmetric or broken symmetry phase. We also plot the D dependence of δ D in Fig. 2, which shows the fluctuation of free energy is suppressed as δ D ≈ − up to D = 50. Since the double-well potential in the φ theory becomes sharper for larger λ , wetake a large number of K to achieve good convergence for δ K . In the following, numericalresults at λ = 40 are presented for K = 2000 and D = 50 which are large enough in thisstudy. K -8 -6 -4 -2 d K k = 0.0763059k = 0.0765000 Figure 1 . Convergence behavior of free energy with δ K of Eq. (3.1) at κ = 0 . . K on V = 1024 . The phase transition point κ c is determined by following the method employed in theIsing case [20]. Suppose we have obtained a coarse-grained tensor T ( m ) x n y n z n t n x n − ˆ1 y n − ˆ2 z n − ˆ3 t n − ˆ4 – 4 – D -6 -5 -4 -3 -2 d D k = 0.0763059k = 0.0765000 Figure 2 . Same as Fig. 1 for δ D of Eq. (3.2). after the m times of coarse-graining. Defining a D × D matrix as A ( m ) t n t n − ˆ4 = (cid:88) x n ,y n ,z n T ( m ) x n y n z n t n x n y n z n t n − ˆ4 , (3.3)we calculate X ( m ) = (cid:0) Tr A ( m ) (cid:1) Tr (cid:0) A ( m ) (cid:1) . (3.4)This quantity, introduced in Ref. [42], possibly counts the number of the largest singularvalue of A ( m ) . Therefore, it is expected that X ( m ) = 1 holds for the symmetric phase and X ( m ) = 2 for the broken symmetry phase. We may distinguish both phases by observingthe plateau of X ( m ) after sufficient coarse-graining iterations.In order to check the applicability of the above method to determine the value of κ c , wecalculate κ c at λ = 5 and compare it with the previous results obtained by various methodsincluding the Monte Carlo simulation [43]. Since we have found that the convergence ofthe free energy with respect to the bond dimension at λ = 5 becomes slightly slower thanthat at λ = 40, we have taken D = 55 (and K = 2000) to evaluate κ c at λ = 5. Up to D = 55, the relative error for the free energy is suppressed to O (10 − ). Figure 3 showsthe m dependence of the value of X ( m ) at κ = 0 . . κ = 7 . × − is the finest resolution across the transition point. We find X ( m ) = 1for m (cid:38)
30 at κ = 0 . X ( m ) = 2 for m (cid:38)
25 at κ = 0 . κ c = 0 . lattice,whose error bar is provided by the resolution of κ . In Fig. 4 we find that our result is– 5 – m X ( m ) k = 0.089225k = 0.089300 Figure 3 . History of X ( m ) as a function of the coarse-graining step m at κ = 0 . k c ( l = ) ATRG w/ D = 55 (This work)Monte CarloDynamical mean field theoryEffective mean field theoryKikuchi's method Figure 4 . Comparison of κ c at λ = 5 obtained by various methods. All numerical values exceptfor the ATRG result are taken from Table III in Ref. [43]. For details on the dynamical or effectivemean field theory, see Ref. [43]. For Kikuchi’s method, see Ref. [44]. – 6 –omparable to the Monte Carlo result κ c = 0 . lattice, while the previous one is on the 32 lattice.Having confirmed the validity of the method using X ( m ) , we determine κ c at λ = 40with D = 50 and K = 2000. The result is κ c = 0 . lattice,whose error bar is provided by the resolution of κ . In Fig. 5 we check the 1 /λ dependenceof κ c toward the Ising limit, where the result at λ = 100 is obtained in the same way asthe λ = 40 case with D = 50 and K = 2000. We observe that the value of κ c seemsmonotonically approaching that in the Ising case. The error bars are provided by theresolution of κ but they are all within symbols. l k c Monte CarloHOTRG w/ D = 13ATRG w/ D = 50 (This work) Figure 5 . κ c as a function of 1 /λ . 1 /λ = 0 corresponds to the Ising model. Square symbol at1 /λ = 0 denotes the result obtained by the HOTRG [20] and diamond symbol by the Monte Carloin Ref. [17]. All error bars are within symbols. We now turn to the investigation of the phase transition with the bond energy definedby E b = − ∂∂κ ln ZV (3.5)and the vacuum condensation of the scalar field (cid:104) φ (cid:105) . Both quantities are evaluated withthe impure tensor method. Figure 6 plots the bond energy as a function of κ on the 1024 lattice. The resolution of κ becomes finer toward the transition point and the finest oneis ∆ κ = 5 . × − around the transition point. The phase transition point is consistentwith κ c (gray band) determined by X ( m ) . Inset graph in Fig. 6 shows an emergence of afinite gap with mutual crossings of curves for different volumes, m ≥
23, around κ c . These– 7 –re characteristic features of the first-order phase transition as discussed in Ref. [45]. Asthe gap, we obtain ∆ E b = 0 . , (3.6)by the linear extrapolation toward the transition point both from the symmetric and brokensymmetry phases. In this extrapolation, we have used data points in [0 . , . . , . E b becomes smaller than the latent heat ∆ E = 0 . k -0.80-0.79-0.78-0.77-0.76-0.75-0.74-0.73-0.72 E b m = 23 m = 24 ( L = 64) m = 25 m = 26 m = 27 m = 28 ( L = 128) m = 40 ( L = 1024) Figure 6 . Bond energy as a function of κ . Inset graph shows it for various lattice sizes and grayband denotes κ c estimated by X ( m ) of Eq. (3.4). Another quantity to detect the phase transition is the vacuum condensation of thescalar field (cid:104) φ (cid:105) , which is the order parameter of spontaneous breaking of the Z symmetry.We calculate (cid:104) φ (cid:105) by introducing the external fields of h = 1 . × − and 2 . × − ateach κ . After taking the infinite volume limit, we extrapolate the value of (cid:104) φ (cid:105) at h = 0.Figure 7 shows the κ dependence of (cid:104) φ (cid:105) h =0 . The resolution of κ is the same as that inFig. 6. We find that the value of κ c , where the vacuum condensation sets in, is consistentwith both estimates by X ( m ) and the bond energy. A finite jump in (cid:104) φ (cid:105) h =0 at κ c is anotherindication of the first-order phase transition. We find∆ (cid:104) φ (cid:105) h =0 = 0 . , (3.7)as the value of finite jump, where we have used data points in [0 . , . . , . (cid:104) φ (cid:105) h =0 toward the transitionpoint. Note that this quantity is estimated as 0 . .0761 0.0762 0.0763 0.0764 0.0765 k < f > h = Figure 7 . Vacuum condensation (cid:104) φ (cid:105) h =0 as a function of κ on V = 1024 . Gray band in inset graphshows κ c estimated by X ( m ) of Eq. (3.4). We have investigated the phase transition of the 4 d single-component φ theory at λ = 40employing the bond energy and the vacuum condensation of the scalar field. Both quantitiesshow finite jumps at the transition point on the extremely large lattice of V = 1024 ,corresponding to the thermodynamic limit, and they indicate the weak first-order phasetransition as found in the Ising limit [20]. This means that the single-component lattice φ theory does not have a continuum limit. In the current ATRG calculation, the resultinglatent heat ∆ E b and the gap ∆ (cid:104) φ (cid:105) are smaller than those in the Ising case obtained bythe HOTRG with D = 13. As a next step, it would be interesting to investigate the phasetransition of the O(4)-symmetric φ theory, which is more relevant to the SU(2) Higgsmodel. Acknowledgments
Numerical calculation for the present work was carried out with the supercomputer Fugakuprovided by RIKEN (Project ID: hp200170) and also with the Oakforest-PACS (OFP)and the Cygnus computers under the Interdisciplinary Computational Science Program ofCenter for Computational Sciences, University of Tsukuba. This work is supported in partby Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports,Science and Technology (MEXT) (No. 20H00148).– 9 – eferences [1] K. G. Wilson and J. B. Kogut,
The Renormalization group and the ε expansion , Phys. Rept. (1974) 75–199.[2] J. Frohlich, On the Triviality of Lambda (phi**4) in D-Dimensions Theories and theApproach to the Critical Point in D > = Four-Dimensions , Nucl. Phys. B (1982)281–296.[3] R. F. Dashen and H. Neuberger,
How to Get an Upper Bound on the Higgs Mass , Phys. Rev.Lett. (1983) 1897.[4] M. Lindner, Implications of Triviality for the Standard Model , Z. Phys. C (1986) 295.[5] A. Hasenfratz, K. Jansen, C. B. Lang, T. Neuhaus and H. Yoneyama, The Triviality Boundof the Four Component phi**4 Model , Phys. Lett. B (1987) 531–535.[6] M. L¨uscher and P. Weisz,
Scaling Laws and Triviality Bounds in the Lattice φ Theory (I)One Component Model in the Symmetric Phase , Nucl. Phys.
B290 (1987) 25–60.[7] M. L¨uscher and P. Weisz,
Scaling Laws and Triviality Bounds in the Lattice φ Theory (II)One Component Model in the Phase with Spontaneous Symmetry Breaking , Nucl. Phys.
B295 (1988) 65–92.[8] M. L¨uscher and P. Weisz,
Scaling Laws and Triviality Bounds in the Lattice φ Theory (III)N Component Model , Nucl. Phys.
B318 (1989) 705–741.[9] K. Huang,
Triviality of the Higgs Field , Int. J. Mod. Phys. A4 (1989) 1037.[10] C. Frick, K. Jansen, J. Jersak, I. Montvay, P. Seuferling and G. Munster, NumericalSimulation of the Scalar Sector of the Standard Model in the Symmetric Phase , Nucl. Phys.
B331 (1990) 515–530.[11] G. D. Kribs, T. Plehn, M. Spannowsky and T. M. Tait,
Four generations and Higgs physics , Phys. Rev. D (2007) 075016, [ ].[12] H. W. J. Bl¨ote and R. H. Swendsen, Critical behavior of the four-dimensional Ising model , Phys. Rev. B (Nov, 1980) 4481–4483.[13] E. Sanchez-Velasco, A finite-size scaling study of the 4D Ising model , J. Phys.
A20 (1987)5033.[14] R. Kenna and C. B. Lang,
Renormalization group analysis of finite size scaling in the φ model , Nucl. Phys.
B393 (1993) 461–479, [ hep-lat/9210009 ].[15] E. Bittner, W. Janke and H. Markum,
Ising spins coupled to a four-dimensional discreteRegge skeleton , Phys. Rev.
D66 (2002) 024008, [ hep-lat/0205023 ].[16] R. Kenna,
Finite size scaling for O ( N ) φ -theory at the upper critical dimension , Nucl. Phys.
B691 (2004) 292–304, [ hep-lat/0405023 ].[17] P. H. Lundow and K. Markstr¨om,
Critical behavior of the Ising model on thefour-dimensional cubic lattice , Phys. Rev. E (Sep, 2009) 031104.[18] P. H. Lundow and K. Markstr¨om, Non-vanishing boundary effects and quasi-first order phasetransitions in high dimensional Ising models , Nucl. Phys.
B845 (2011) 120–139, [ ].[19] F. J. Wegner and E. K. Riedel,
Logarithmic Corrections to the Molecular-Field Behavior ofCritical and Tricritical Systems , Phys. Rev. B (Jan, 1973) 248–256. – 10 –
20] S. Akiyama, Y. Kuramashi, T. Yamashita and Y. Yoshimura,
Phase transition offour-dimensional Ising model with higher-order tensor renormalization group , Phys. Rev.
D100 (2019) 054510, [ ].[21] M. Levin and C. P. Nave,
Tensor renormalization group approach to two-dimensionalclassical lattice models , Phys. Rev. Lett. (2007) 120601, [ cond-mat/0611687 ].[22] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang and T. Xiang, Coarse-grainingrenormalization by higher-order singular value decomposition , Phys. Rev. B (Jul, 2012)045139.[23] D. Adachi, T. Okubo and S. Todo, Anisotropic Tensor Renormalization Group , Phys. Rev. B (2020) 054432, [ ].[24] D. Kadoh and K. Nakayama,
Renormalization group on a triad network , .[25] Y. Shimizu and Y. Kuramashi, Grassmann tensor renormalization group approach toone-flavor lattice Schwinger model , Phys. Rev.
D90 (2014) 014508, [ ].[26] R. Sakai, S. Takeda and Y. Yoshimura,
Higher order tensor renormalization group forrelativistic fermion systems , PTEP (2017) 063B07, [ ].[27] S. Akiyama, Y. Kuramashi, T. Yamashita and Y. Yoshimura,
Restoration of chiral symmetryin cold and dense Nambu–Jona-Lasinio model with tensor renormalization group , .[28] Y. Shimizu and Y. Kuramashi, Critical behavior of the lattice Schwinger model with atopological term at θ = π using the Grassmann tensor renormalization group , Phys. Rev.
D90 (2014) 074503, [ ].[29] Y. Shimizu and Y. Kuramashi,
Berezinskii-Kosterlitz-Thouless transition in lattice Schwingermodel with one flavor of Wilson fermion , Phys. Rev.
D97 (2018) 034502, [ ].[30] S. Takeda and Y. Yoshimura,
Grassmann tensor renormalization group for the one-flavorlattice Gross-Neveu model with finite chemical potential , PTEP (2015) 043B01,[ ].[31] D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda and Y. Yoshimura,
Tensornetwork formulation for two-dimensional lattice N = 1 Wess-Zumino model , JHEP (2018) 141, [ ].[32] D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda and Y. Yoshimura, Investigation of complex φ theory at finite density in two dimensions using TRG , JHEP (2020) 161, [ ].[33] Y. Kuramashi and Y. Yoshimura, Tensor renormalization group study of two-dimensionalU(1) lattice gauge theory with a θ term , JHEP (2020) 089, [ ].[34] S. Akiyama, D. Kadoh, Y. Kuramashi, T. Yamashita and Y. Yoshimura, Tensorrenormalization group approach to four-dimensional complex φ theory at finite density , JHEP (2020) 177, [ ].[35] Y. Yoshimura, Y. Kuramashi, Y. Nakamura, S. Takeda and R. Sakai, Calculation offermionic Green functions with Grassmann higher-order tensor renormalization group , Phys.Rev.
D97 (2018) 054511, [ ].[36] P. Cea, M. Consoli and L. Cosmai,
Two mass scales for the Higgs field? , . – 11 –
37] M. Consoli and L. Cosmai,
The mass scales of the Higgs field , Int. J. Mod. Phys. A (2020) 2050103, [ ].[38] M. Consoli and L. Cosmai, A resonance of the Higgs field at 700 GeV and a newphenomenology , .[39] M. Consoli and L. Cosmai, Spontaneous Symmetry Breaking and Its Pattern of Scales , Symmetry (2020) 2037.[40] S. Akiyama, Y. Kuramashi, T. Yamashita and Y. Yoshimura, Phase transition offour-dimensional Ising model with tensor network scheme , PoS
LATTICE2019 (2020) 138.[41] H. Oba,
Cost Reduction of Swapping Bonds Part in Anisotropic Tensor RenormalizationGroup , PTEP (2020) 013B02, [ ].[42] Z.-C. Gu and X.-G. Wen,
Tensor-entanglement-filtering renormalization approach andsymmetry-protected topological order , Phys. Rev. B (Oct, 2009) 155131.[43] O. Akerlund, P. de Forcrand, A. Georges and P. Werner, Dynamical Mean FieldApproximation Applied to Quantum Field Theory , Phys. Rev. D (2013) 125006,[ ].[44] R. Kikuchi, A theory of cooperative phenomena , Phys. Rev. (Mar, 1951) 988–1003.[45] M. Fukugita, H. Mino, M. Okawa and A. Ukawa, Finite-size scaling of the three-state Pottsmodel on a simple cubic lattice , Journal of Statistical Physics (Jun, 1990) 1397–1429.(Jun, 1990) 1397–1429.