A Proof of lemma 2

Phase Transition in Tensor Models

Abstract

Generalizing matrix models, tensor models generate dynamical triangulations in any dimension and support a expansion. Using the intermediate field representation we explicitly rewrite a quartic tensor model as a field theory for a fluctuation field around a vacuum state corresponding to the resummation of the entire leading order in (a resummation of the melonic family). We then prove that the critical regime in which the continuum limit in the sense of dynamical triangulations is reached is precisely a phase transition in the field theory sense for the fluctuation field.

1 Introduction

Matrix models [1] generate dynamical triangulations and dynamical tessellations in two dimensions. Tensor models [2, 3, 4, 5, 6] generalize matrix models and generate dynamical triangulations [7, 8] in any dimension (recently a matrix model generating dynamical triangulations is dimension three has been proposed [9]). The continuum limit of dynamical triangulations is reached when tunning the coupling constants to some critical values [10, 11, 12, 13]. In this regime the number of simplices in the triangulation diverges. Sending simultaneously the volume of the individual simplices to zero one obtains a continuous phase of infinitely refined random spaces [1, 8].

Tensor and matrix models can be regarded as field theories with no kinetic term. The fields are a pair of complex conjugated tensors (where ) transforming under the external tensor product of fundamental representations of the unitary group :

The action of tensor models is a function of and which is invariant under unitary transformations. Like matrix models, tensor models are known to possess a expansion [14, 15, 16, 17, 18]. The leading order in the expansion for matrices is given by the planar triangulations. For tensor models, the leading order in is given by a class of triangulations called melonic [12]. The melonic triangulations represent topological spheres in any dimensions, but one should emphasize that only a subclass of triangulations of the sphere are melonic. In particular the family of melonic triangulations is exponentially bounded [12]. It is not yet known whether the family of general triangulations of the sphere in dimension higher than three is exponentially bounded or not [19].

In this paper we consider a tensor model with a quartic interaction. We first rewrite this model, using the intermediate field representation [20, 17], as a coupled multi matrix model. Examining the vacua of this multi matrix model we find for small coupling a unique vacuum state invariant under unitary transformations which we call the melonic vacuum. We derive the effective theory for the fluctuations around the melonic vacuum. The translation to the melonic vacuum is not a phase transition as no symmetry gets broken (the melonic vacuum possesses the full invariance of the theory). We show that this translation completely resums the leading order in : the effective theory for the perturbation fields around the melonic vacuum contributes only to lower orders in . We show furthermore that, at each order in , only a finite number of graphs of the effective theory contribute. Finally, we show that the criticality of the leading order contribution in , which is the dynamical triangulation continuum limit, corresponds precisely to a zero eigenvalue in the effective mass matrix for the fluctuation field.

We therefore prove that the continuum limit in dynamical triangulations is a phase transition in tensor models. This phase transition presumably corresponds to the spontaneous breaking of the unitary invariance of the theory. In order to establish this rigorously, one needs in the future to identify the appropriate non unitary invariant vacuum state and study perturbations around it.

The situation is quite different for matrix models. A procedure similar to the one we use here for tensors can not be used for matrices: so far at least, the action of no matrix model could be translated to an explicit “planar vacuum state” such that the effective theory around it contributes only a finite number of graphs at each order in .

A very interesting question is whether similar results hold for group field theories [21, 22, 23] and tensor field theories [24, 25, 26, 27, 28]. Such theories have a genuine renormalization group flow [29, 30] and it would be very interesting to test whether the phase transition we identify in this paper is associated to an infrared fixed point of this flow in which case tensor field theories and group field theories would naturally generate bound states of extended geometry.

This paper is organized as follows. As our results are somewhat technical, we start by discussing in section 2 their interpretation. In section 3 we introduce the quartic tensor model and in section 4 we discuss the effective theory around an invariant field configuration. In section 5 we introduce the class of Feynman graphs (combinatorial maps) relevant for the Feynman expansion of the effective theory which we detail in section 6. Finally in section 7 we consider the translation to the invariant vacuum state and detail the expansion of the fluctuation field.

2 Discussion

The partition functions of tensor and matrix models are generating functions for dynamical triangulations [1, 7, 8]. Denoting a (strongly) connected dimensional triangulation and the number of simplices in , the logarithm of the partition function (the free energy) writes as:

 W=1NDlnZ=∑ΔgTop(Δ)N−2(D−1)!ω(Δ),

where denotes the size of the matrices (or tensors) and is a non negative integer. In the case of matrices , is the genus of the triangulations [1], while in higher dimensions [14, 16] is the degree. While the degree is not a topological invariant, it serves to organize the series defining as a series in .

In the large limit only a subclass of triangulations contribute: the planar triangulations in dimension two and the melonic triangulations in higher dimension. Both families are exponentially bounded. Similar results hold order by order in and it can be shown that [1, 31, 32] summing the families of triangulations at fixed order in , the free energy becomes (up to non universal analytic pieces in which can be disposed of by a suitable number of derivatives):

 W∼d0(gc−g)ν0+∑q≥11Nqdq(gc−g)νq.

While the critical value depends on the details of the model, all the series at fixed order in become critical at the same critical value . The (non integer) critical exponents are, to a large extent, universal for a given dimension.

Considering equilateral simplices of volume , the physical volume of a triangulation is proportional to the number of simplices, and the average physical volume is:

 Vol=VD\BraketTop(Δ)=VDg∂glnW∼g→gcVDg−gc.

The continuum limit of dynamical triangulation is reached by tuning and sending at the same time while keeping the average physical volume fixed. In this regime one obtains a phase of random infinitely refined geometries. One can study more involved matrix and tensor models, adding for instance coupling constants for lower dimensional simplices. The various continuum phases of the model will then be spanned by these new couplings.

Dynamical triangulations can also be studied [8, 33] with no reference to matrix and tensor models. In this case one restricts the family of triangulations by some prescription. For instance one can chose to restrict to triangulations of the sphere, or to foliated triangulations [33]. Of course one should check that the restricted family of triangulations is exponentially bounded (in order to be able to obtain a continuum limit) but, due to technical difficulties, this is never done in practice. Still, one can learn numerous lessons from numerical studies.

We show in this paper that a certain tensor model can be recast as a coupled multi matrix model for matrices , with . We then show that, in this new formulation, the vacuum state of the model is not , but it is given at small coupling constant by the invariant configuration:

 Hc=a01,a0=1−√1−4Dg22Dg.

We translate to the vacuum state and write the partition function of the model as some explicit factor multiplying the partition function of the effective theory of the perturbation fields around the vacuum state:

 Z(4) =e−ND(Da202+ln(1−gDa0))(1−a20)D−12∫[dMc]e−S(M),\default@crS(M) =12ND−1(1−a20)D∑c=1Trc[McMc]−12ND−2D−1Da20(D∑c=1Trc[Mc])2−Q(M). (1)

We prove on the one hand that the integral over the perturbation fields is subleading in , namely:

 −W=−1NDlnZ(4)=D2a20+ln(1−gDa0)+O(1N),

and on the other that the mass matrix of the effective theory for the perturbation , has eigenvalues with degeneracy and with degeneracy .

Dynamical triangulations continuum limit. The dynamical triangulations continuum limit is obtained from the explicit factor in Eq. (1). The leading order free energy writes in terms of as:

 −WLO=D2a20+ln(1−gDa0)=−12+1−√1−4Dg24Dg2−ln(1−√1−4Dg22Dg2),

and becomes critical for the critical constant:

 gc=12√D.

In the critical regime one obtains:

 −WLO=−12+1−√ϵ1−ϵ−ln(21−√ϵ1−ϵ)=12+ln2+12ϵ−23ϵ3/2+O(ϵ2),

hence the critical exponent .

Field theory phase transition. The effective field theory for the perturbation field (which is subleading in ) undergoes a phase transition when the mass matrix has a zero eigenvalue, that is for . At this point the invariant vacuum is no longer stable and the symmetry of the theory is broken. This occurs for the coupling constant solution of:

 1√D=1−√1−4Dg2p.t.2Dgp.t.⇒√1−4Dg2p.t.=(1−4Dg2p.t.)⇒gp.t.=12√D.

We conclude that, in tensor models, the dynamical triangulations continuum limit is a phase transition associated to a breaking of the symmetry (unitary symmetry and color permutation) of the model.

We stress that the phase transition we identify in this paper is of an entirely different nature form the phase transitions studied so far in the context of dynamical triangulations and matrix models:

• Dynamical triangulations. In dynamical triangulations [8, 33] one takes the continuum limit and obtains various continuum phases. In Euclidean dynamical triangulations [8] one typically obtains two continuum phases (crumpled and branched polymer), while in causal dynamical triangulations [33] one obtains three continuum phases. One is then interested in the transition between the continuum phases. These transitions have nothing to do with the phase transition from discrete to continuum we identify in this paper.

• Matrix models. In matrix models [34] one studies the transition from a one cut resolvent to a multi cut resolvent. The phase transition we identify in this paper is very different, and occurs in a completely different range of parameters. In matrix models the continuum limit is a change of behavior of the spectral density of a one cut solution (one passes from a square root law to a power law for the distribution of eigenvalues). It is this change of behavior which we show in this paper is a genuine phase transition in the case of tensor models.

3 The quartic melonic model

A detailed introduction to the general framework of tensor models can be found in [13] or [35], and an introduction to the quartic melonic model in [20] and [17]. An important notion for tensor models is the notion of -colored graph [5]. Such graphs are dual to -dimensional triangulations [5].

Definition 1.

A bipartite edge -colored graph (-colored graph for short) is a graph with vertex set and edge set such that:

• the vertex set is the disjoint union of the set of white vertices and the set of black vertices, with .

• the edge set is the disjoint union of sets of edges of color , with such that any edge connects a black and a white vertex.

• all the vertices have degree and all the edges incident to a vertex have distinct colors.

The fields in a tensor model are a pair of complex conjugated rank tensors , with . We call the position of an index its color and denote the set of colors . We denote . The tensors and transform under the external tensor product of fundamental representations of the unitary group :

Starting from and one can build a class of invariant polynomials called trace invariants. They are in bijection with bipartite edge -colored graphs. The trace invariant associated to is obtained by associating to every white vertex of a tensor , to every black vertex of a complex conjugated tensor and to every edge of color of , a contraction of the indices of color of the two tensors associated to its end vertices and :

The melonic quartic tensor model the probability measure:

 dμ(4) Extra open brace or missing close brace =∑¯nDnDmD¯mD(¯T¯nDδ¯nD∖{c}nD∖{c}TnD)δ¯ncmcδ¯mcnc(¯T¯mDδ¯mD∖{c}mD∖{c}TmD), (2)

where, for any set , we denote . The model is stable for . Each invariant can be represented as a -colored graph with two white and two black vertices and signifies a gluing of four -simplices, two positively oriented and two negatively oriented. The coupling counts the number of positively oriented simplices. When evaluating the partition function in perturbation theory one obtains Feynman graphs having colors. Each such graph is dual to a -dimensional triangulation.

The generating function of the moments of is:

The partition function is the generating function at zero external sources and is denoted by . The cumulants (connected moments) of can be computed as the derivatives of the logarithm of :

The cumulants are linear combinations of trace invariant operators [35]:

where the sum runs over all the -colored graphs [5] with white vertices. Somewhat abusively, we call the coefficient also a cumulant and we define the rescaled cumulant corresponding to the graph :

 K(B,μ(4))≡K(B,μ(4))ND−2k(B)(D−1)−C(B),

where denotes the number of connected components of .

3.1 The intermediate field representation

The intermediate field representation of the quartic melonic models has already been discussed in detail in [20] and [17]. Let us recall here the main steps in its derivation. Each quartic interaction in expressed in the intermediate field (Hubbard Stratonovich) representation as an integral over an auxiliary hermitian matrix:

 eND−1g22∑c∈DV(4)c(¯T,T)=∫[dHc]e−12ND−1Tr[HcHc]+gND−1∑nc¯ncHc¯ncnc∑nD∖{c}¯nD∖{c}(¯T¯nDδ¯nD∖{c}nD∖{c}TnD),

where the integral over is normalized to for . As there are quartic melonic interactions, one obtains intermediate matrix fields . Each is an matrix and the indices of have the color . The integral over and is now Gaussian and can be computed explicitly to obtain:

 Z(4)(J,¯J) =∫(D∏c=1[dHc])e−S(H)+1ND−1JR(H)¯J,\default@crS(H) =12ND−1D∑c=1Trc[HcHc]−TrD[lnR(H)],

where is the trace over the index of color , is the trace over all the indices, and the resolvent is:

 R(H)=11D−g∑Dc=1Hc⊗1D∖{c},

where is the identity matrix with indices of color and for any subset we denote .

In the rest of this paper will be considered real, positive, and smaller than . Observe that in this range of the coupling constant the resolvent can be singular. However one can still make sense of the integral defining by modifying the contour of integration for the field in the complex plane so as to avoid the singularities.

3.2 Equations of motion

Taking into account that:

 ∂∂Hc¯acbcTrD[lnR(H)]=g∑¯pDqDR(H)qD¯pD(δ¯pD∖{c}qD∖{c}δ¯pc¯acδqcbc),

the classical equations of motion in the intermediate field representation write:

 0=∂S(H)∂Hcbcac=ND−1Hcacbc−g∑pDqDR(H)qDpD(δpD∖{c}qD∖{c}δpcbcδqcac).

The configuration is not a vacuum of the theory as, for , is not a solution of the classical equations of motion. In order to find solutions to the classical equations of motion we express them in matrix form:

 Hc−g1ND−1TrD∖{c}[11D−gD∑c′=1Hc′⊗1D∖{c′}]=0, (3)

where denotes the trace over the indices with colors in . A simple inspection of these equations reveals that for small enough the stable vacuum of the theory is invariant under conjugation by the unitary group and invariant under color permutation, . The invariant solutions of eq. (3) are then obtained for satisfying the self consistency equation:

 a=g1−gDa.

This self consistency equation has two solutions:

• The melonic vacuum. A first solution, denoted and called the melonic vacuum, is the sum of a power series in :

 a0=1−√1−4Dg22gD=g∑n≥01(n+1)!(2n)n(Dg2)n.

We will see below that this solution is the stable vacuum of the theory for small enough. In the region of interest (), is real, positive, and bounded by:

 a0=1−√1−4Dg22Dg=2g1+√1−4Dg2≤1√D,

as is increasing with and attains its maximum for .

• The instanton. The second solution, denoted , is an instanton solution:

 ainst=1+√1−4Dg22gD.

As , it follows that in the region of interest (), is real, positive, and .

For the two solutions collapse: .

4 Translating the intermediate field

In order to study the effective theory around a solution of the classical equations of motion we must translate the field to . This translation is a translation (in the complex plane) by of the contour of integration of all the diagonal entries of . The effective theory is obtained by Taylor expanding in . We first discuss a translation of by some arbitrary constant . We denote and we define:

 b(a)≡g1−gDa.

Observe that if is real and is real, then is also real. The melonic vacuum and the instanton are the two solutions of the self consistency equation . We will use the shorthand notation instead of but the reader should keep in mind that is a function of .

In terms of the perturbation field , the resolvent and the trace of its logarithm write as:

 R(a1D+M)=1(1−gDa)1D−gM=(11−gDa)11D−bM,\default@cr TrD[lnR(a1D+M)]=−NDln(1−gDa)+TrD[ln11D−bM].

After translation by the action becomes:

and the generating function writes in terms of the perturbation field :

 Z(4)(J,¯J) =e−ND(Da22+ln(1−gDa)) ×∫(D∏c=1[dMc])e−ND−12∑Dc=1Trc[McMc]−aTrD[M]+TrD[ln11D−bM]+1ND−11(1−gDa)J(11D−bM)¯J.

The logarithm of the resolvent contains quadratic terms in , which must be reabsorbed in the quadratic part of the action. Let us define the subtracted vertex function:

 Q(M)=TrD[ln11D−bM]−TrD[aM]−12TrD[(bM)2],

which, by definition, has no quadratic terms in . The generating function is then:

 Z(4)(J,¯J) =e−ND(Da22+ln(1−gDa)) ×∫(D∏c=1[dMc])e−ND−12∑Dc=1Trc[McMc]+12TrD[(bM)2]+Q(M)+1ND−11(1−gDa)J(11D−bM)¯J. (4)

Integral form of the subtracted vertex function. One can give a closed integral formula for the subtracted vertex function . A Taylor expansion with integral rest yields:

 TrD[ln11D−bM]=\default@cr ={∂tTrD[ln11D−tbM]}t=0+12{∂2tTrD[ln11D−tbM]}t=0+12∫10dt(1−t)2∂3tTrD[ln11D−tbM],

and substituting the derivatives:

 ∂rtTrD[ln11D−tbM]=(r−1)!TrD[1(1D−tbM)r(bM)r],

we obtain:

 Q(M)=(b−a)TrD[M]+∫10dt(1−t)2TrD[1(1D−tbM)3(bM)3],

that is is the sum of a linear piece and a piece starting at order .

Linear term. At the price of modifying the covariance, the translation of by a generic introduces some univalent vertices in the theory. Choosing comes to choosing the weight of the univalent vertex as the counterterm of the melonic graphs [12, 13], and by translating to the melonic vacuum, one subtracts the contribution of the entire melonic family: as , the linear term which represents the univalent vertices vanishes. As expected, in this case is a solution of the classical equations of motion due to the fact that the remaining piece of starts at order .

No such interpretation exists for the translation to the instanton although the linear term cancels also in that case.

Free energy. We can compute the large free energy of the model in the perturbative regime using Schwinger Dyson equations. According to [13], the large covariance and the large free energy are:

 1−K2+Dg2K22=0 ⇒ K2=1−√1−4Dg22Dg2=a0g,\default@cr W∞=limN→∞(1NDlnZ(4)(0,0))=1+ln(K2)−K2+Dg22K22=−Dg22K22+ln(K2).

The large free energy writes in term of as:

 W∞=−D2a20−ln(1−gDa0),

that is precisely the explicit prefactor in Eq. (4) for . We will re-derive this result below.

It ensues that the translation to the melonic vacuum resums the contribution of the melonic sector, and the remaining integral over (which, as we will see below, only contributes at lower orders in ) is precisely the effective theory of a perturbation field around the melonic phase.

We note that the case of matrices, , is quite different. A careful inspection shows that for matrices, even after translating to , the integral over the perturbation contributes at leading order in . This is natural because for matrices the melonic graphs are only a subset of the planar graphs, and one cannot subtract the contribution of the planar sector by a single counterterm.

4.1 The effective theory for the fluctuation field

The space of Hermitian matrices is a dimensional real vector space with inner product . The quadratic part of the action is a quadratic form on the direct sum of such Hilbert spaces . Denoting a generic element in this direct sum space by:

 ^M=⎛⎜ ⎜⎝M1⋮MD⎞⎟ ⎟⎠,

the inner product in is:

 \Braket^P^M=D∑c=1Trc[PcMc].

The effective theory for the fluctuation field is somewhat involved due to the fact that, while the quadratic part of the action is a quadratic form on the direct sum space , the interaction is a trace in the tensor product space . This is the root of all the subtleties we will need to deal with below.

We now work out the effective covariance for the fluctuation field. The quadratic part of the action of the fluctuation field in Eq. (4) is:

 ND−1D∑c=1Trc[McMc]−TrD[(bM)2]=\default@cr =ND−1(1−b2)D∑c=1Trc[(Mc)2]+ND−2b2D∑c=1(Trc[Mc])2−ND−2b2(D∑c=1Trc[Mc])2.

We denote by the element of consisting in the identity matrix in and zero on the other components and by the element consisting in the identity matrix in all the s:

 ^Ic=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋮1⋮0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,^I=⎛⎜ ⎜⎝1⋮1⎞⎟ ⎟⎠=D∑c=1^Ic.

The orthogonal projectors on the one dimensional subspaces generated by and are:

 Pc^M =\Braket^Ic^M\Braket^Ic^Ic^Ic=Trc[Mc]N^Ic⇒\Braket^MPc^M=Trc[Mc]NTrc[Mc]\default@crP^M =\Braket^I^M\Braket^I^I^I=∑Dc=1Trc[Mc]DN^I⇒\Braket^MP^M=∑Dc=1Trc[Mc]DN(D∑c′=1Trc′[Mc′]).

Let us denote the identity operator on . The quadratic part writes then as where the operator is:

 O=ND−1(1−b2)I+ND−1b2D∑c=1Pc−ND−1Db2P.

The covariance of the effective theory is the operator . In order to compute it, it is most convenient to diagonalize .

Lemma 1.

The operator is diagonalized as:

 O=ND−1(1−b2)[I−D∑c=1Pc]+ND−1[D∑c=1Pc−P]+ND−1(1−Db2)P,

where , and are the projectors on the eigenspaces of . The eigenvalues of and their degeneracies are:

 Λ1=ND−1(1−b2) ,dim(Im[I−D∑c=1Pc])=DN2−D,\default@crΛ2=ND−1 ,dim(Im[D∑c=1Pc−P])=D−1,\default@crΛ3=ND−1(1−Db2) ,dim(Im[P])=1.
Proof.

It is enough to check that the operators , and are three (mutually orthogonal) orthogonal projections. The three operators are obviously Hermitian and:

 Pc′Pc^M=Trc[Mc]N\Braket^Ic′^IcN^Ic