Renormalization procedure for random tensor networks and the canonical tensor model
aa r X i v : . [ h e p - t h ] J a n YITP-15-3WITS-MITP-001
Renormalization procedure for random tensor networksand the canonical tensor model
Naoki
Sasakura a ∗ Yuki
Sato b † a Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan b National Institute for Theoretical Physics,School of Physics and Mandelstam Institute for Theoretical Physics,University of the Witwartersrand, Wits 2050, South Africa
August 12, 2018
Abstract
We discuss a renormalization procedure for random tensor networks, and showthat the corresponding renormalization-group flow is given by the Hamiltonianvector flow of the canonical tensor model, which is a discretized model of quantumgravity. The result is the generalization of the previous one concerning the relationbetween the Ising model on random networks and the canonical tensor modelwith N = 2. We also prove a general theorem which relates discontinuity of therenormalization-group flow and the phase transitions of random tensor networks. ∗ [email protected] † [email protected]
Introduction
Wilson’s renormalization group [1, 2] is an essential and pedagogical tool in modern theoreticalphysics. Once a renormalization-group flow in a parameter space is given, one can read offrelevant degrees of freedom at each step of coarse graining through change of parameters, andunderstand the phase structure in principle. Therefore, a renormalization-group flow givesus a quantitative and qualitative picture of a system concerned. The aim of this paper isto define a renormalization procedure and derive the corresponding flow equation for randomtensor networks, in particular for those proposed as Feynman-graph expressions [3, 4], throughthe use of the canonical tensor model (CTM, for short).First of all, CTM has been introduced by one of the authors as a model of quantum gravityby considering space-time as a dynamical fuzzy space [5, 6, 7]. CTM is a tensor model in thecanonical formalism, which has a canonical conjugate pair of rank-three tensors, M abc , P abc ( a, b, c = 1 , , · · · , N ), as dynamical variables. This interpretation of tensorial variables interms of a fuzzy space is different from the one made by original tensor models. Historically,tensor models have been introduced as models of simplicial quantum gravity in dimensionshigher than two [8, 9, 10]; although the original tensor models have some drawbacks, tensormodels as simplicial quantum gravity are currently in progress as colored tensor models [11, 12](See [13]-[24] for recent developments.). In CTM, N , the cardinality of the rank-three tensors,may be interpreted as the number of “points” forming a space, while physical propertiesof space-time such as dimensions, locality, etc. must emerge from the collective dynamicsof these “points.” So far, the physics of the small- N CTM is relatively well understood: theclassical dynamics of the N = 1 CTM agrees with the minisuperspace approximation of generalrelativity in arbitrary dimensions [25]; the exact physical states have been obtained for N = 2in the full theory [26, 27] and for N = 3 in an S -symmetric subsector [27]; intriguingly,physical-state wavefunctions, at least for N = 2 ,
3, have singularities where symmetries areenhanced [27]. However, similar brute-force analysis as above for
N > N dynamicsis supposed to be important. Thus, for the purpose of handling large- N behaviors of CTM,the present authors have proposed the conjecture that statistical systems on random networks[3, 4, 28], or random tensor networks , are intimately related to CTM [3]: the phase structure ofrandom tensor networks is equivalent to what is implied by considering the Hamiltonian vectorflow of CTM as the renormalization-group flow of random tensor networks. This conjecturehas been checked qualitatively for N = 2 [3]. In fact, as more or less desired, random tensornetworks turn out to be useful to find physical states of CTM with arbitrary N : some series ofexact physical states for arbitrary N have been found as integral expressions based on randomtensor networks [27].In this paper, we prove the fundamental aspect of the above conjecture: we show that theHamiltonian vector flow of CTM can be regarded as a renormalization-group flow of randomtensor networks for general N . Here the key ingredient is that the Lagrange multipliers of theHamiltonian vector flow are determined by the dynamics of random tensor networks in themanner given in this paper. This is in contrast with the previous treatment for N = 2, in1hich the Lagrange multipliers are given by a “reasonable” choice [3]. In fact, the previoustreatment turns out to have some problems for general N , as being discussed in this paper.This paper is organized as follows. In section 2, we review CTM and random tensor net-works. We argue our previous proposal [3] on the relation between CTM and random tensornetworks, and its potential problems. In Section 3, we propose a renormalization procedurefor random tensor networks based on CTM, and derive the corresponding renormalization-group flow. In Section 4, we compare our new and previous proposals with the actual phasestructures of random tensor networks for N = 2 ,
3. We find that the new proposal is con-sistent with the phase structures, while the previous one is not. In Section 5, we discuss theasymptotic behavior of the renormalization-group flow, and clarify the physical meaning ofthe renormalization parameter. In Section 6, we provide a general theorem which relates dis-continuity of the renormalization-group flow and the phase transitions of the random tensornetworks. Section 7 is devoted to summary and discussions.
In this paper, we consider a statistical system [3, 4] parameterized by a real symmetric three-index tensor P abc ( a, b, c = 1 , , . . . , N ). ∗ Its partition function is defined by † Z n ( P ) = Z dφ ( P φ ) n e − nφ , (1)where we have used the following short-hand notations, Z dφ ≡ N Y a =1 Z ∞−∞ dφ a , P φ ≡ P abc φ a φ b φ c , φ ≡ φ a φ a . (2)By using the Wick theorem, the Gaussian integration of φ in (1) can be evaluated by thesummation over the pairwise contractions of all the φ ’s in ( P φ ) n . Then the partition function(1) can graphically be represented by the summation over all the possible closed networks of n trivalent vertices. In each of such networks, every vertex is weighted by P abc , and theindices are contracted according to the connection of the network, as in Fig.1. Therefore,since the pairwise contractions are taken over all the possible ways, the statistical systemrepresented by (1) can be regarded as random tensor networks of n trivalent vertices. Ingeneral, such a network may contain disconnected sub-networks, but this probability vanishesin the thermodynamic limit n → ∞ . ‡ ∗ In this paper, the tensor variable of the statistical system is denoted by P for later convenience, insteadof M used in the previous papers [3, 4]. † For later convenience, the normalizations of the partition function and φ are taken differently from thosein the previous papers [3, 4]. This does not change the physical properties of the statistical system. ‡ This graphical property can be checked by considering a sort of “grand” partition function Z ( P ) given bya formal sum of (1) over n with n -dependent weights, and comparing explicitly the perturbative expansions in P of Z ( P ) and those of log Z ( P ) for N = 1. The former corresponds to the sums of the networks which maycontain disconnected sub-networks, while the latter to connected networks only. See [3] for details. PP PP P P PP P
Figure 1: A tensor network of n = 10 trivalent vertices.For example in N = 2, (1) gives the partition function of the Ising model on randomnetworks [3, 4], if one takes P abc = X i =1 R ai R bi R ci e Hσ i , (3)where σ i represents the spin degrees of freedom taking σ = 1 , σ = − H is a magnetic field,and R is a two-by-two matrix satisfying( R T R ) ij = e Jσ i σ j , (4)with J giving the nearest neighbor coupling of the Ising model. For a ferromagnetic case J >
0, there exists a real matrix R satisfying (4).The partition function (1) is obviously invariant under the orthogonal transformation L ∈ O ( N ), which acts on P as P ′ abc = L aa ′ L bb ′ L cc ′ P a ′ b ′ c ′ , (5)since the transformation can be absorbed by the redefinition φ ′ a = φ a ′ L a ′ a . In addition, theoverall scale transformation of P , P ′ abc = e ψ P abc , (6)with an arbitrary real number ψ , does not change the properties of the statistical system,since this merely changes the overall factor of (1). For example, for N = 2, these invariancesallow one to consider a gauge, P = 1 , P = 0 , P = x , P = x , (7)with real x i .The free energy per vertex in the thermodynamic limit can be defined by f ( P ) = − lim n →∞ log Z n ( P )2 n , (8)where we have considered only even numbers of vertices, since an odd number of trivalentvertices cannot form a closed network. The phase structure of the statistical system can be3 - - - - - Figure 2: The horizontal and vertical axes represent x and x of the gauge (7), respectively.The solid lines describe the phase transition lines of the random tensor networks with N = 2,which can exactly be obtained in the thermodynamic limit [3]. The arrows describe theHamilton vector flow (21) of CTM with N a = P abb .investigated by studying the behavior of the free energy (8). For the case of N = 2, the phasetransition lines of the free energy (8) in the gauge (7) are shown by solid lines in Fig.2 [3, 4].The transitions are first order, except at the Curie point, ( x , x ) = (0 . , f ( P ) are continuous, but the second ones are not [3, 4, 28]. In fact, for arbitrary N , the free energy can exactly be obtained by applying the Laplace method to evaluate (8)[3, 4]. The result is § f ( P ) = Min φ f ( P, φ ) = f ( P, ¯ φ ) , (9)where f ( P, φ ) = φ −
12 log h(cid:0)
P φ (cid:1) i , (10)and ¯ φ is defined so as to minimize f ( P, φ ) as a function of φ for given P , namely, ¯ φ is one ofthe solutions to the stationary condition, ∂f ( P, φ ) ∂φ a (cid:12)(cid:12)(cid:12)(cid:12) φ = ¯ φ = 2 ¯ φ a − P abc ¯ φ b ¯ φ c P ¯ φ = 0 . (11) § If P is symmetric under part of the O ( N ) transformation (5), the perturbations of the integrand of (1)around φ = ¯ φ contain some zero modes, and the application of the Laplace method will require extra treatmentto integrate over the symmetric directions. However, this integration is obviously finite, and will only generatecorrections of the free energy higher in 1 /n , which do not affect the thermodynamic limit. Thus, the exactfree energy in the thermodynamic limit is given by (9) for the whole region of P including symmetric points.Note that this argument may change, if one takes a double scaling limit accompanied with N → ∞ , which,however, is not considered in this paper. N = 2) inFig.2 can be derived from the Hamilton vector flow of CTM for N = 2, if one regards theHamilton vector flow as a renormalization-group flow, as shown in Fig.2. This is surprising,since CTM was proposed aiming for quantum gravity, and there exist no apparent reasons forCTM to be related to statistical systems on random networks. CTM is a totally constrainedsystem with a number of first-class constraints forming an algebra which resembles the Diracalgebra of the ADM formalism of general relativity [29]. In the classical case, the constraintsare given by H a = 12 P abc P bde M cde , (12) J [ ab ] = 14 ( P acd M bcd − P bcd M acd ) , (13) D = 16 P abc M abc , (14)where J and D are the kinematical symmetry generators corresponding to the SO ( N ) (5)and the scale (6) transformations, respectively, and H and J may be called Hamiltonian andmomentum constraints, respectively, in analogy with general relativity [29]. Here the bracketfor the indices of J symbolically represents the antisymmetry, J [ ab ] = −J [ ba ] , and M is thecanonical conjugate variable to P defined by { M abc , P def } = 16 X σ δ aσ ( d ) δ bσ ( e ) δ cσ ( f ) , { M abc , M def } = { P abc , P def } = 0 , (15)where { , } denotes the Poisson bracket, and the summation over σ runs over all the per-mutations of d, e, f to incorporate the symmetric property of the three-index tensors. Theconstraints form a first class constraint algebra, {H ( ξ ) , H ( ξ ) } = 16 J (cid:16) [ ˜ ξ , ˜ ξ ] (cid:17) , {J ( η ) , H ( ξ ) } = 16 H ( ηξ ) , (16) {J ( η ) , J ( η ) } = 16 J (cid:0) [ η , η ] (cid:1) , (17) {D , H ( ξ ) } = 16 H ( ξ ) , (18) {D , J ( η ) } = 0 , (19)where H ( ξ ) = ξ a H a , J ( η ) = η [ ab ] J [ ab ] and ˜ ξ ab = P abc ξ c .The Hamiltonian of CTM is given by an arbitrary linear combination of the constraints as H = N a H a + N [ ab ] J [ ab ] + N D , (20)5here N ’s are the multipliers, which may depend on P in the context of this paper, consideringa flow in the configuration space of P . Then, the Hamiltonian vector flow is given by ¶ dds P abc = { H, P abc } , (21)where s is a fictitious parameter along the flow. In the previous paper [3], which comparesCTM with the random tensor networks for N = 2, the multiplier N a is chosen to be N a = P abb , (22)based on that this is the simplest covariant choice. The other multipliers N [ ab ] and N , relatedto the symmetry generators, are chosen so that the Hamilton vector flow (21) keeps the gaugecondition (7). Indeed the flow in Fig.2 has been drawn with these choices. One can also checkthat other covariant choices such as N a = P abc P bde P cde do not change the qualitative natureof the flow and therefore the coincidence between the phase structure of the random tensornetworks and the one implied by CTM with N = 2.Though the coincidence is remarkable for N = 2, from further study generalizing gaugeconditions and values of N , we have noticed that there exist some problems in insisting thecoincidence as follows: • First of all, no physical reasons have been given for the coincidence. A primary expec-tation is that there exists a renormalization group procedure for statistical systems onrandom networks, and the procedure is described by the Hamiltonian of CTM in somemanner. However, it is unclear how one can define a renormalization group procedure forstatistical systems on random networks, which do not have regular lattice-like structures. • As will explicitly be shown later, in the case of N = 3, the phase transition lines deviatefrom the expectation from the Hamilton vector flow of CTM. What is worse is thatdifferent choices of N a , such as P abb or P abc P bde P cde , give qualitatively different Hamiltonvector flows, which ruins the predictability of the transition lines from the flow. • In Fig.2 for N = 2, on the phase transition lines, the flow goes along them, and thereexist a few fixed points of the flow on the transition lines. The fixed point at (0 ,
1) is a co-dimension two fixed point, and the associated phase transition is expected to be of secondorder rather than first order, if the flow is rigidly interpreted as a renormalization-groupflow and we follow the standard criterion [30]. This is in contradiction to the actualorder of the phase transition. This contradiction is more apparent in the diagram inanother gauge in Section 4.The purpose of the present paper is to solve all the above problems, and to show thatCTM actually gives an exact correspondence to random tensor networks. It turns out that N a should not be given by any “reasonable” choices as above, but should rather be determineddynamically as N a ∼ h φ a i to be discussed in the following sections. Then, we can show thatthe Hamiltonian of CTM actually describes a coarse-graining procedure of random tensornetworks, and that the Hamilton vector flow is in perfect agreement with the phase structureirrespective of values of N . ¶ The direction of the flow is chosen in the manner convenient for later discussions. Renormalization procedure and renormalization-groupflow
In this subsection, we discuss a renormalization group procedure of the random tensor net-works, and obtain the corresponding renormalization group flow.Let us consider an operator O which applies on Z n ( P ) as O Z n ( P ) = Z dφ { φ a H a , ( P φ ) n } e − nφ . (23)By using (12) and (15), and performing partial integrations with respect to φ , one can derive O Z n ( P ) = Z dφ { φ a H a , ( P φ ) n } e − nφ = 12 Z dφ { φ a P abc P bde M cde , ( P φ ) n } e − nφ = n Z dφ φ a P abc P bde φ c φ d φ e ( P φ ) n − e − nφ = 16 Z dφ P abc φ a φ c (cid:20) ∂∂φ b ( P φ ) n (cid:21) e − nφ = − Z dφ ( P φ ) n ∂∂φ b h P abc φ a φ c e − nφ i = 13 Z dφ (cid:2) n ( P φ ) n +1 − φ a P abb ( P φ ) n (cid:3) e − nφ = n (cid:18) n + 1 n (cid:19) n +3+ N Z n +1 ( P ) − P abb h φ a i n Z n ( P ) , (24)where h φ a i n is an expectation value defined by h φ a i n = R dφ φ a ( P φ ) n e − nφ Z n ( P ) , (25)and the numerical factor in the first term of (24) is due to the rescaling of φ for nφ → ( n +1) φ in the exponential.Here (24) and (25) must be used with a caution. If taken literally, since Z n =odd = 0, (24)and (25) do not seem useful by themselves. The reason for Z n =odd = 0 is that the contributionsat φ = ± v with arbitrary v cancel with each other in the integration of (1). To avoid thiscancellation and make (24) and (25) useful, let us consider a finite small region r ¯ φ in the spaceof φ around one of the solutions ¯ φ which minimize (10). For later convenience, we take thesign of ¯ φ so as to satisfy P ¯ φ > . (26)7his can be taken, because, if P φ = 0, f ( P, φ ) in (10) diverges and cannot be the minimum,and f ( P, φ ) = f ( P, − φ ). Especially, r ¯ φ should not contain the other minimum φ = − ¯ φ . Then,let us consider a replacement, Z n ( P ) → Z r ¯ φ dφ ( P φ ) n e − nφ . (27)In the thermodynamic limit n → ∞ , the integral (27) is dominated by the region with width∆ φ ∼ / √ n around φ = ¯ φ k . Therefore, the expression (27) approaches e − nf ( P ) in the ther-modynamic limit, irrespective of even or odd n . Moreover, since the integrand of (27) dampsexponentially in n on the boundary of r ¯ φ , the corrections generated by the partial integrationscarried out in the derivation of (24) are exponentially small. Thus, (24) is valid up to expo-nentially small corrections in n after the replacement (27). Thus, for n large enough, we cansafely use (24) and (25) as if they are meaningful irrespective of even or odd n .Taking into account the discussions above, we can put h φ a i n → ¯ φ a and Z n → e − nf ( P ) in(24) for n ≫
1. Therefore, in (24), the first term dominates over the second term, and one cansafely regard O as an operator which increases the size n of networks. To regard this operationas a flow in the space of P rather than a discrete step of increasing n , let us introduce thefollowing operator with a continuous parameter s , R ( s ) = e s O . (28)If we consider n ≫
1, we can well approximate the operation O with the first term of (24) asexplained above, and one obtains R ( s ) Z n ( P ) = ∞ X m =0 s m m ! O m Z n ( P ) ∼ ∞ X m =0 s m ( n + m − m m !( n − e − ( n + m ) f ( P )+ m . (29)By increasing s , the right-hand side is dominated by larger networks, and diverges at s = s ∞ ,which is given by s ∞ = 3 exp (cid:18) f ( P ) − (cid:19) . (30)On the other hand, in the thermodynamic limit, the left-hand side of (29) can be computedin a different manner. In the thermodynamic limit, φ a can be replaced with the mean value¯ φ a , and the operator O can be identified with a first-order partial differential operator, O → O D = ¯ φ a H a = 12 ¯ φ a P abc P bde D Pcde , (31) k If P is symmetric under part of the SO ( N ) transformation (5), an extra care will be needed as discussedin a previous footnote. However, this does not change the following argument in the thermodynaic limit. D Pabc is a partial derivative with respect to P abc with a normalization, D Pabc P def = { M abc , P def } = 16 X σ δ aσ ( d ) δ bσ ( e ) δ cσ ( f ) . (32)Here note that ¯ φ is a function of P determined through the minimization of (10). Then, theexpression in the left-hand side of (29) is obviously a solution to a first-order partial differentialequation, (cid:18) ∂∂s − O D (cid:19) R ( s ) Z n ( P ) = 0 . (33)The solution to (33) can be obtained by the method of characteristics and is given by R ( s ) Z n ( P ) = Z n ( P ( s )) , (34)where P ( s ) is a solution to a flow equation, dds P abc ( s ) = O D P abc ( s )= 16 (cid:0) ¯ φ d P dae ( s ) P ebc ( s ) + ¯ φ d P dbe ( s ) P eca ( s ) + ¯ φ d P dce ( s ) P eab ( s ) (cid:1) , (35) P abc ( s = 0) = P abc , (36)where ¯ φ must be regarded as a function of P ( s ) by substituting P with P ( s ).Here we summarize what we have obtained from the above discussions. R ( s ) Z n ( P ) canbe evaluated in two different ways. One is (29), a summation of random tensor networks, thedominant size of which increases as s increases, while P is unchanged. The other is (34), where P ( s ) changes with the flow equation (35), while the size of random networks is unchanged.This means that the change of the size of networks can be translated into the change of P .Therefore the flow of P ( s ) in (35) can be regarded as a renormalization-group flow of therandom tensor networks, where increasing s corresponds to the infrared direction.The above derivation of the renormalization group flow uses the particular form of H in (12). Since, in general, there exist various schemes for renormalization procedures forstatistical systems, one would suspect that there would be other possible forms of H whichdescribe renormalization procedures for random tensor networks. However, this is unlikely,and the form (12) would be the unique and the simplest. The reason for the uniqueness isthat, as outlined in Section 2, the algebraic consistency of H with the O ( N ) symmetry, whichis actually the symmetry of random tensor networks in the form (1), requires the unique form(12) under some physically reasonable assumptions [6]. On the other hand, the reason for thesimplest can be found by considering the diagrammatic meaning of the operation H in (23). H acts on a vertex as { φ a H a , P φ } = 12 φ a φ b P abc P cde φ d φ e , (37)9 PP Figure 3: The diagrammatic representation of the operation { φ a H a , P φ } .and hence can be regarded as an operator which inserts a vertex on an arbitrary connectionin a network (See Fig.3). This is obviously the most fundamental operation which increasesthe number of vertices of a network.Here we comment on our new proposal in comparison with the previous one [3]. Our mainclaim is that the multiplier should take N a = φ a rather than “reasonable” choices such as N a = P abb , P abc P bed P cde , etc., taken in the previous proposal. With N a = φ a , the Hamiltonianvector flow is uniquely determined by the dynamics, while “reasonable” choices are ambiguous.Even if ambiguous, there are no problems in the N = 2 case, since there are no qualitativechanges of the flow among “reasonable” choices, and the phase structure can uniquely bedetermined from the flow. However, as will be shown in Section 4, this is not true in generalfor N >
2. In fact, N = 2 is special for the following reasons. It is true that ¯ φ a can well beapproximated by ∼ const.P abb near the absorbing fixed points in Fig.2, because all of themcan be shown to be gauge-equivalent to P = 1 , others = 0. This means that at least anapproximate phase structure can be obtained even by putting N a = P abb ∼ ¯ φ a . In addition,what makes the N = 2 case very special is that the phase transition lines are the fixed pointsof the Z symmetry corresponding to reversing the sign of the magnetic field of the Ising modelon random networks. Therefore, the phase transition lines are protected by the symmetry,which stabilizes the qualitative properties of the flow under any changes of N a respecting thesymmetry.Finally, we comment on an equation which can be derived from (24) in the thermodynamiclimit. By putting Z n ( P ) ∼ e − nf ( P ) to (24), one can derive¯ φ a P abc P bde D Pcde f ( P ) = − e − f ( P )+ . (38)In fact, one can directly prove (38). By using (9) and (10), the left-hand side of (38) is givenby ¯ φ a P abc P bde D Pcde f ( P ) = ¯ φ a P abc P bde ( D Pcde ¯ φ g ) ∂f ( P, ¯ φ ) ∂ ¯ φ g − ¯ φ a P abc P bde ¯ φ c ¯ φ d ¯ φ e P ¯ φ = − P ¯ φ , (39)where we have used (11). This coincides with the right-hand side of (38), because of the choice(26) and ¯ φ = , which can be obtained by contracting (11) with ¯ φ a .10igure 4: The phase transition lines of the random tensor network with N = 2 in the gauge (40)are shown as solid lines. The horizontal and vertical axes indicate x and x , respectively. Thephase transitions are first-order except for the endpoint of a line located around ( x , x ) =(0 . , . x , x ) = (0 . ,
0) in thegauge (7). The left figure describes the Hamilton vector flow based on our former proposal, N a = P abb , while the right figure describes it based on our new proposal, N a = ¯ φ a . A locus ofgauge singularities is located at x = 0 . In this section, we will check the proposal of this paper in the cases of N = 2 , N = 2 case with a gauge, P = 1 , P = 0 . , P = x , P = x , (40)as a typical example. The difference from (7) is the gauge fixing value of P . One canobtain the phase structure in the parameter space of ( x , x ) by studying the free energy (9).Alternatively, one can apply the O (2) and scale transformations, (5) and (6), on P so that thephase structure in the gauge (7) given in Fig.2 is transformed to that in the gauge (40). Ineither way, one can determine the phase structure in the new gauge, and the result is given inFig.4. Here we draw the Hamilton vector flows for N a = P abb , based on the former proposal,and N a = ¯ φ a , based on our new proposal, in the left and right figures, respectively.The rough features of the two flows based on the different proposals seem consistent withthe phase structure: the flows depart from the transition lines, and go into the same absorptionfixed points. This was the main argument in our previous paper [3]. However, there are somephysically important differences in details between the left and the right figures. In the left11gure, on the phase transition lines, the flow is going along them. Moreover, there exist a fewfixed points of the flow on the transition lines at ( x , x ) ∼ ( − . , , ( − . , . , ( − . , − − . ,
1) is expected to be ofsecond order, rather than first order, since the points on the both sides of the transition linein its vicinity flow to the same fixed point near (0 . ,
0) without any discontinuities. On theother hand, in the right figure, the flow has discontinuity on the transition lines, except forthe Curie point at the endpoint of the transition line. Thus, the flow based on our formerproposal clearly contradicts the actual order of the phase transitions, while the one based ournew proposal is in agreement with it, i.e. first order except for the Curie point.An interesting property of the flow is that it does not vanish even on the Curie point,as can be seen in the right figure of Fig.4 and can also be checked numerically. This seemscurious, because the second derivatives of the free energy contain divergences on the point. Ina statistical system on a regular lattice, such divergences originate from an infinite correlationlength. Therefore, such a point will typically become a fixed point of a renormalization-groupflow. On the other hand, the correlation length of the Ising model on random networks isknown to be finite even on the Curie point [28]. This means that, even starting from the Curiepoint, a renormalization process will bring the system to one with a vanishing correlationlength. This implies that the Curie point cannot be a fixed point of a renormalization-groupflow, and this is correctly reflected in the fact that our flow does not vanish on the Curie point.Let us next consider the N = 3 case. There seem to exist too many parameters to treatthis case in full generality. So let us specifically consider a subspace parametrized by P iii = 1 , P ijj = x , P = x , ( i = j ) , (41)which is invariant under the S transformation permuting the index labels, 1,2,3. Throughnumerical study of the free energy (9) (and some analytic considerations), one can obtain thephase structure shown in Fig.5. In the indicated parameter region, there exist two regions ofan S symmetric phase labeled by S with ¯ φ = ¯ φ = ¯ φ . There also exist two distinct non-symmetric phases labeled by NS1 and NS2. At any point in the two regions, the minimizationof the free energy (10) has three distinct solutions of non-symmetric values, ¯ φ = ¯ φ = ¯ φ ,¯ φ = ¯ φ = ¯ φ , ¯ φ = ¯ φ = ¯ φ , and hence three distinct phases coexist in these regions. When S symmetric subspace (41) is extended to more general cases, each of NS1 and NS2 becomesthe common phase boundary of the three phases.The flow in the left figure of Fig.5 is drawn based on our previous proposal N a = P abb .There, the flow is not in good agreement with the phase structure, though it seems to capturesome rough features. We tried other possibilities such as N a = P abc P bde P cde , etc., but theflow depended on the choices, and no good agreement could be found. On the other hand,in the right figure based on our new proposal N a = ¯ φ a , the flow in the symmetric region S isconsistent with the phase structure: the flow departs from the transition lines, and, since itdoes not vanish on the lines, the order is expected to be first order. This is in agreement withthe property of the free energy except for a point ( x , x ) = (1 / , − / N = 3 in the subspace(41) are shown as solid lines. The horizontal and vertical axes indicate x and x , respectively.There exist four regions separated by the phase transition lines in the indicated parameterregion. The phases labeled by S have S symmetric expectation values ¯ φ = ¯ φ = ¯ φ . Thereare two other regions labeled by NS1 and NS2, which have non-symmetric expectation values.The phase transitions are first-order except for the meeting point of S, NS1, NS2 at ( x , x ) =(1 / , − / P is symmetric under an SO (2) transfomation [27]. In the left figure, we draw the Hamilton vector flow based on ourformer proposal N a = P abb , while the right one is based on our new proposal N a = ¯ φ a .13he flow does not vanish on the point, the correlation length is expected to be finite. This issimilar to the Curie point of the N = 2 case.In the non-symmetric phases, NS1 and NS2, the expectation values are not S symmetric.Therefore, the flow has generally directions away from the S symmetric subspace, and cannotbe drawn on the figure. To also check the consistency of the flow in these regions, it would benecessary to extend the parameter region. This would require to take a different systematicstrategy for consistency check to avoid too many parameters. In Section 6, we will take anotherway of consistency check by proving a theorem relating the renormalization group flow andthe phase transitions of the random tensor network. In Section 3, we argued that O = φ a H a provides a renormalization procedure for the randomtensor network. As can be seen from (29) and (34), P ( s ) diverges in the limit s → s ∞ in (30).On the other hand, in the numerical analysis of Section 4, P is kept normalized as (40) and(41) by appropriately tuning the multiplier N for the scale transformation D (with N ab J ab aswell). As in Fig.4 and 5, one can find fixed points of the flows in the limit ˜ s → ∞ , where ˜ s denotes the fictitious parameter parameterizing the normalized flows. In this section, we willshow that these two limits of s and ˜ s are physically equivalent.Let us first show the divergence in s → s ∞ more directly. Since ¯ φ = from (11),log P ¯ φ = − f ( P ) + . Then, by using (31) and (39), one obtains dds log P ¯ φ = 13 P ¯ φ , (42)where P is meant to be P ( s ), and hence P ¯ φ is regarded as a function of s . The solution to(42) is P ¯ φ = 1 P ¯ φ | s =0 − s , (43)which indeed diverges at s = P ¯ φ | s =0 = s ∞ . Since ¯ φ is normalized by ¯ φ = , the divergenceof (43) can be translated to the divergence of P ( s ) with a behavior, P ( s ) ∼ const.s ∞ − s , (44)or higher order in the case that some components of ¯ φ vanish in s → s ∞ .Now let us compare the two flows, unnormalized and normalized ones. For notationalsimplicity, let us denote the three indices of P abc by one index as P i . The flow equations in s s can respectively be expressed as dds P i ( s ) = ¯ φ a ( P ( s )) g ai ( P ( s )) , (45) dd ˜ s ˜ P i (˜ s ) = ¯ φ a ( ˜ P (˜ s )) g ai ( ˜ P (˜ s )) − N ˜ P i (˜ s ) , (46) P i ( s = 0) = ˜ P i (˜ s = 0) = P i , (47)where N generally depends on ˜ P i (˜ s ), and g ai ( P ) are the quadratic polynomial functions of P ,which can be read from (35). The last term of the second equation comes from N D in (20),and is assumed to be tuned to satisfy a gauge condition normalizing ˜ P (˜ s ). Here we ignore the SO ( N ) generators, J ab , for simplicity, but it is not difficult to extend the following proof toinclude them.The physical properties of the random tensor network do not depend on the overall scaleof P . So let us define the relative values of P ( s ) and ˜ P (˜ s ) as Q i ( s ) = P i ( s ) P ( s ) , ˜ Q i (˜ s ) = ˜ P i (˜ s )˜ P (˜ s ) , (48)where P ( s ) ( ˜ P (˜ s )) is taken from one of P i ( s ) (resp. ˜ P i (˜ s )), or a linear combination of them.From (11), it is obvious that ¯ φ ( P ) and ¯ φ ( ˜ P ) actually depend only on Q and ˜ Q , respectively.Then, from (45), 1 P dds Q i = 1 P dds P i P = ¯ φ a ( Q ) g ai ( P ) P − ¯ φ a ( Q ) g a ( P ) P i P = ¯ φ a ( Q ) ( g ai ( Q ) − g a ( Q ) Q i ) . (49)In the same manner, 1˜ P dds ˜ Q i = ¯ φ a ( ˜ Q ) (cid:16) g ai ( ˜ Q ) − g a ( ˜ Q ) ˜ Q i (cid:17) . (50)Note that the last term of (46) does not contribute to the flow equation of ˜ Q . Since the initialcondition (47) implies Q i ( s = 0) = ˜ Q i (˜ s = 0), and the righthand sides of (49) and (50) areidentical, the flow equations, (49) and (50), describe an identical flow with a transformationbetween the fictitious parameters, P ( s ) ds = ˜ P (˜ s ) d ˜ s. (51)As discussed above, the typical behavior of P ( s ) is (44), while ˜ P (˜ s ) is assumed to remainfinite near an absorption fixed point. In such a case, (51) implies˜ s ∼ − const. log( s ∞ − s ) . (52)15herefore, the limits of s → s ∞ and ˜ s → ∞ are physically equivalent. As can be checked easily,this physical implication does not change, even if we consider the case that P ( s ) diverges withan order higher than (44).To investigate the physical meaning of the fictitious parameter ˜ s , let us estimate (29) near s ∼ s ∞ . We obtain ∞ X m =0 s m ( n + m − m m !( n − e − ( n + m ) f ( P )+ m = e − nf ( P ) ( n − ∞ X m =0 ( m + 1)( m + 2) · · · ( n + m − (cid:18) ss ∞ (cid:19) m = const. d n − ds n − ∞ X m =0 (cid:18) ss ∞ (cid:19) m + n − = const. d n − ds n − s n − s ∞ − s ∼ const. ( s ∞ − s ) − n (53)Then the average size of networks can be estimated as h n + m i ∼ n + s dds log( s ∞ − s ) − n ∼ ns ∞ s ∞ − s . (54)Therefore ˜ s ∼ const. log (Average size) . (55)This means that ˜ s corresponds to the standard renormalization-group scale parameter oftendenoted by log Λ in field theory. In Section 4, we see that the renormalization-group flow has discontinuity on the first-orderphase transition lines in the examples of the random tensor networks. In this section, we willprove a general theorem on this aspect.By using the free energy in the thermodynamic limit (9), the stationary condition (11) andthe flow equation (35), we can prove the following theorem.
Theorem:
The following three statements are equivalent.(i) The first derivatives of f ( P ) are continuous at P .(ii) ¯ φ is continuous at P .(iii) The renormalization-group flow is continuous at P .16 roof :Let us first prove (i) ⇒ (ii). From (9), the first derivatives of f ( P ) are given by D Pabc f ( P ) = − ¯ φ a ¯ φ b ¯ φ c P ¯ φ , (56)where we have neglected the contributions from the P -dependence of ¯ φ , since ¯ φ satisfies thestationary condition (11). By contracting a pair of indices in (56), one obtains, D Paab f ( P ) = − ¯ φ a ¯ φ a ¯ φ b P ¯ φ = − φ b P ¯ φ , (57)where we have used ¯ φ = derived from (11). Here note that P ¯ φ is continuous at any P ,because the free energy f ( P ) itself in (9) is continuous at any P ∗∗ , and also ¯ φ = . Therefore,if (i) holds, (57) is continuous and hence ¯ φ is continuous; (i) ⇒ (ii) has been proven.The reverse, (ii) ⇒ (i), is obviously true from (56). Therefore, the statements (i) and (ii)are equivalent: (i) ⇔ (ii).Next, as for (ii) ⇒ (iii), it is obvious that, if ¯ φ is continuous, the renormalization groupflow (35) is also continuous.Finally, let us prove (iii) ⇒ (ii), which will complete the proof of the theorem. To prove this,we will show that there is a contradiction, if we assume both (iii) and that ¯ φ has discontinuityon P .Let us suppose that there is discontinuity of ¯ φ at a point P . Then, from the definition of¯ φ , there exist multiple distinct solutions of ¯ φ to (11) which give the same minimum of (10) at P . Let us take any two of them, ¯ φ + and ¯ φ − . As shown above, P ¯ φ is continuous at any point,which means A ≡ P ( ¯ φ + ) = P ( ¯ φ − ) , (58)where the value is denoted by A for later usage. Here note A = 0, since, otherwise, (10)diverges and cannot be the minimum. Then, since ¯ φ ± both satisfy (11), we obtain¯ φ ± a = 32 A P abc ¯ φ ± b ¯ φ ± c , (59)∆ a = 32 A P abc (cid:0) ∆ b ∆ c + 2∆ b ¯ φ − c (cid:1) , (60)where ∆ = ¯ φ + − ¯ φ − , and (60) has been obtained by considering the difference of the twoequations in (59). Note ∆ = 0, if there exist multiplicity of the solutions.On the other hand, the assumption (iii) and (35) imply dds P abc (cid:12)(cid:12)(cid:12)(cid:12) ¯ φ = ¯ φ + − dds P abc (cid:12)(cid:12)(cid:12)(cid:12) ¯ φ = ¯ φ − ∝ ∆ d P dae P ebc + ∆ d P dbe P eca + ∆ d P dce P eab = 0 . (61) ∗∗ This can be proven by using that f ( P ) is the minimum of (10), which is a continuous function of φ and P . φ + ’s or ¯ φ − ’s, and using (59), we obtain P abc ∆ a ¯ φ ± b ¯ φ ± c = 0 . (62)Finally, by contracting (60) with ∆, we obtain∆ = 32 A P abc (cid:0) ∆ a ∆ b ∆ c + 2∆ a ∆ b ¯ φ − c (cid:1) = 32 A P abc ∆ a (cid:0) ¯ φ + b ¯ φ + c − ¯ φ − b ¯ φ − c (cid:1) = 0 , (63)where we have used (62). This concludes ∆ = 0, which contradicts the initial assumption ofthe existence of discontinuity of ¯ φ . Consequently, we have proven the equivalence of (i), (ii)and (iii).By taking contrapositions, a corollary of the theorem is given by Corollary 1:
The following three statements are equivalent.(i) P is a first-order phase transition point. (Not all of the first derivatives of f ( P ) arecontinuous.)(ii) ¯ φ is not continuous at P .(iii) The renormalization-group flow is not continuous at P .Another corollary of physical interest is Corollary 2: If P is a phase transition point higher than first-order, the renormalization-group flow is continuous at the critical point.The qualitative behavior of the N = 2 renormalization-group flow shown in the right fig-ure of Fig.4 respects the theorem and corollaries as it should be: Corollary 1 is realized on thephase transition lines, and Corollary 2 on the Curie point. In the previous paper [3], it has been found that the phase structure of the Ising model onrandom networks (or random tensor networks with N = 2) can be derived from the canon-ical tensor model (CTM), if the Hamilton vector flow of the N = 2 CTM is regarded as arenormalization-group flow of the Ising model on random networks. This was a surprise, sinceCTM had been developed aiming for a model of quantum gravity in the Hamilton formalism187, 6, 5]. Considering the serious lack of real experiments on quantum gravity, the aspect thatCTM may link quantum gravity and concrete statistical systems would be encouraging.The main achievement of the present paper is to have shown that the Hamilton vectorflow of CTM with arbitrary N gives a renormalization-group flow of random tensor networks,where the N = 2 case, in particular, corresponds to the Ising model on random networks. Inthe previous paper [3], we considered the Hamiltonian of CTM, H = N a H a , with “reasonable”choices of N a . Though it was successful in the N = 2 case, we have shown in this paper thatthe previous procedure of taking H does not work for general N , and have argued that thecorrect one is given by H = φ a H a , where φ a are the integration variables for describing randomtensor networks. Here an advantage of the present procedure is that H is uniquely determinedby the dynamics of random tensor networks, but not by the ambiguous “reasonable” choices ofthe previous procedure. In fact, applied on random tensor networks, H = φ a H a is an operatorwhich randomly inserts vertices on connecting lines, and therefore it increases sizes of tensornetworks. This provides an intuitive understanding of the role of H as a renormalizationprocess. We have performed the detailed analysis of the process, and have actually derived theHamilton vector flow of CTM as a renormalization-group flow of the random tensor network.In the last section, we have proven a theorem which relates the phase transitions of the randomtensor network and discontinuity of the renormalization-group flow.The renormalization-group flow which we have obtained has discontinuities on the first-order phase transition lines. However, there is a critical argument on whether a renormalization-group flow has discontinuities on a first-order phase transition line [31]. Since the argumentbasically assumes a regular lattice-like structure of a system, it would be interesting to inves-tigate a similar argument for a system with a random network structure. The random tensornetwork would give an interesting playground to deepen the idea of the renormalization groupin a wider situation.Finally, let us comment on possible directions of further study, based on the achievementsof the present paper. One is the classification of the fixed points of the Hamilton vector flow.This will provide the classification of the phases and their transitions of the random tensornetwork. This would also be interesting from the view point of quantum gravity. As discussedin [27], the physical wave functions of CTM may have peaks at the values of P invariantunder some symmetries. In general, on such symmetric values of P , ¯ φ may have multiplesolutions, and therefore they are phase transition points. Such interplay between peaks andphase transitions may give interesting insights into quantum gravity. Another direction wouldbe to pursue possible relations between the renormalization procedure of the random tensornetwork and that of the standard field theory. In fact, the “grand” partition function [3] of therandom tensor network can be arranged to take the form of a partition function of field theoryon a lattice by considering an index set labelling lattice points and taking P so as to respectlocality. Then, the Hamilton vector flow of CTM may be regarded as a renormalization-groupflow of the standard field theory. It would be highly interesting if CTM makes a bridge betweenquantum gravity and the standard field theory.19 cknowledgment NS would like to thank the members of National Institute for Theoretical Physics, Universityof the Witwartersrand, for warm hospitality during my visit, where part of the present workhas been carried out. The visit was financially supported by the Ishizue research supportingprogram of Kyoto University. YS would like to thank Yukawa Institute for Theoretical Physicsfor comfortable hospitality and financial support, where part of this work was done. YS isgrateful to Tsunehide Kuroki, Hidehiko Shimada and Fumihiko Sugino for useful discussionsand encouragement.
A Explicit expressions of the constraints
In this appendix, we give the explicit expressions of the constraints, (12), (13), (14), in theforms used in Section 4 for N = 2 , A.1 N = 2 In a subspace, P = 1 , P = y, P = x , P = x , (64)with fixed y , which contains (40) as a special case, the constraints are given by( H , H )= 16 (cid:16) y ) ∂∂P + 3(1 + x ) y ∂∂P + (cid:0) x + 2 x + y ( x + 2 y ) (cid:1) ∂∂x + 3 x ( x + y ) ∂∂x , x ) y ∂∂P + (cid:0) x + 2 x + y ( x + 2 y ) (cid:1) ∂∂P + 3 x ( x + y ) ∂∂x + 3( x + x ) ∂∂x (cid:17) , (65) J ∝ − y ∂∂P + (1 − x ) ∂∂P + ( − x + 2 y ) ∂∂x + 3 x ∂∂x , (66) D ∝ ∂∂P + y ∂∂P + x ∂∂x + x ∂∂x . (67)A Hamilton vector flow is generated by (20), and, for a given N a , the multipliers associatedto the kinematical symmetries, N and N , can be determined so that the flow stays in thegauge (64). Then, since P and P are kept constant by such a flow, determining such aHamilton vector flow is actually equivalent to considering H = N a H a , where ∂∂P and ∂∂P are substituted by solving the linear equations, J = D = 0. Here we do not write the explicitresultant expression of H , since it is rather long but the procedure itself is elementary. Animportant issue in the procedure is that there exist exceptional points characterized by3 y + 1 − x = 0 , (68)20here the set of linear equations, J = D = 0, become singular and cannot be solved for ∂∂P , ∂∂P . On these points, N and N cannot be chosen so that the gauge be kept. Theseare the gauge singularities in Fig.4, located at x = 0 .
635 for y = 0 . A.2 N = 3 The derivation of the Hamilton vector flow in the S symmetric subspace (41) is similar tothe N = 2 case. A difference is that, in (20), the multiplier associated to J ab must be set N ab = 0 to keep the S invariance. For N a = 1, which is S symmetric, one can choose N appropriately to keep P iii = 1, and can obtain the Hamilton vector flow as H S = 2 x + x (cid:18) (1 + 3 x − x + 2 x ) ∂∂x + 6 x (1 − x ) ∂∂x (cid:19) . (69)This is used to draw the Hamilton vector flow in Fig.5. References [1] K. G. Wilson, “The Renormalization Group: Critical Phenomena and the Kondo Prob-lem,” Rev. Mod. Phys. (1975) 773.[2] K. G. Wilson and J. B. Kogut, “The Renormalization group and the epsilon expansion,”Phys. Rept. (1974) 75.[3] N. Sasakura and Y. Sato, “Ising model on random networks and the canonical tensormodel,” PTEP , no. 5, 053B03 (2014) [arXiv:1401.7806 [hep-th]].[4] N. Sasakura and Y. Sato, “Exact Free Energies of Statistical Systems on Random Net-works,” SIGMA , 087 (2014) [arXiv:1402.0740 [hep-th]].[5] N. Sasakura, “Canonical tensor models with local time,” Int. J. Mod. Phys. A (2012)1250020 [arXiv:1111.2790 [hep-th]].[6] N. Sasakura, “Uniqueness of canonical tensor model with local time,” Int. J. Mod. Phys.A (2012) 1250096 [arXiv:1203.0421 [hep-th]].[7] N. Sasakura, “A canonical rank-three tensor model with a scaling constraint,” Int. J.Mod. Phys. A (2013) 1 [arXiv:1302.1656 [hep-th]].[8] J. Ambjorn, B. Durhuus and T. Jonsson, “Three-Dimensional Simplicial Quantum Grav-ity And Generalized Matrix Models,” Mod. Phys. Lett. A , 1133 (1991).[9] N. Sasakura, “Tensor Model For Gravity And Orientability Of Manifold,” Mod. Phys.Lett. A , 2613 (1991). 2110] N. Godfrey and M. Gross, “Simplicial Quantum Gravity In More Than Two-Dimensions,”Phys. Rev. D , 1749 (1991).[11] R. Gurau, “Colored Group Field Theory,” Commun. Math. Phys. (2011) 69[arXiv:0907.2582 [hep-th]].[12] R. Gurau and J. P. Ryan, “Colored Tensor Models - a review,” SIGMA , 020 (2012)[arXiv:1109.4812 [hep-th]].[13] R. Gurau, “The 1/N Expansion of Tensor Models Beyond Perturbation Theory,” Com-mun. Math. Phys. , 973 (2014) [arXiv:1304.2666 [math-ph]].[14] V. Rivasseau, “The Tensor Track, III,” Fortsch. Phys. , 81 (2014) [arXiv:1311.1461[hep-th]].[15] J. Ben Geloun, “Renormalizable Models in Rank d ≥ , 117 (2014) [arXiv:1306.1201 [hep-th]].[16] D. Oriti, “Group Field Theory and Loop Quantum Gravity,” arXiv:1408.7112 [gr-qc].[17] D. Ousmane Samary, “Beta functions of U (1) d gauge invariant just renormalizable tensormodels,” Phys. Rev. D , no. 10, 105003 (2013) [arXiv:1303.7256 [hep-th]].[18] A. Eichhorn and T. Koslowski, “Continuum limit in matrix models for quantum grav-ity from the Functional Renormalization Group,” Phys. Rev. D , 084016 (2013)[arXiv:1309.1690 [gr-qc]].[19] M. Raasakka and A. Tanasa, “Next-to-leading order in the large N expansion of themulti-orientable random tensor model,” arXiv:1310.3132 [hep-th].[20] S. Carrozza, “Discrete Renormalization Group for SU(2) Tensorial Group Field Theory,”arXiv:1407.4615 [hep-th].[21] J. Ben Geloun and R. Toriumi, “Parametric Representation of Rank d Tensorial GroupField Theory: Abelian Models with Kinetic Term P s | p s | + µ ,” arXiv:1409.0398 [hep-th].[22] V. A. Nguyen, S. Dartois and B. Eynard, “An analysis of the intermediate field theoryof T tensor model,” arXiv:1409.5751 [math-ph].[23] D. Benedetti, J. Ben Geloun and D. Oriti, “Functional Renormalisation Group Approachfor Tensorial Group Field Theory: a Rank-3 Model,” arXiv:1411.3180 [hep-th].[24] T. Delepouve and V. Rivasseau, “Constructive Tensor Field Theory: The T Model,”arXiv:1412.5091 [math-ph].[25] N. Sasakura and Y. Sato, “Interpreting canonical tensor model in minisuperspace,” Phys.Lett. B , 32 (2014) [arXiv:1401.2062 [hep-th]].[26] N. Sasakura, “Quantum canonical tensor model and an exact wave function,” Int. J. Mod.Phys. A (2013) 1350111 [arXiv:1305.6389 [hep-th]].2227] G. Narain, N. Sasakura and Y. Sato, “Physical states in the canonical tensor model fromthe perspective of random tensor networks,” JHEP , 010 (2015) [arXiv:1410.2683[hep-th]].[28] See, S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, “Critical phenomena in complexnetworks,” Rev. Mod. Phys. , 1275 (2008) [arXiv:0705.0010 [cond-mat]], and referencestherein for the dynamics of statistical systems on random networks.[29] R. L. Arnowitt, S. Deser and C. W. Misner, “Canonical variables for general relativity,”Phys. Rev. , 1595 (1960); R. L. Arnowitt, S. Deser and C. W. Misner, “The Dynamicsof general relativity,” arXiv:gr-qc/0405109.[30] For instance, see N. Goldenfeld, “Lectures on Phase Transitions and the RenormalizationGroup,” ISBN-020155408955408.[31] For instance, see A. D. Sokal, A. C. D. van Enter and R. Fernandez, “Renormalizationtransformations in the vicinity of first-order phase transitions: What can and cannot gowrong,” Phys. Rev. Lett.66