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Phase transitions in spinor quantum gravity on a lattice

We construct a well-defined lattice-regularized quantum theory formulated in terms of fundamental fermion and gauge fields, the same type of degrees of freedom as in the Standard Model. The theory is explicitly invariant under local Lorentz transformations and, in the continuum limit, under diffeomorphisms. It is suitable for describing large nonperturbative and fast-varying fluctuations of metrics. Although the quantum curved space turns out to be on the average flat and smooth owing to the non-compressibility of the fundamental fermions, the low-energy Einstein limit is not automatic: one needs to ensure that composite metrics fluctuations propagate to long distances as compared to the lattice spacing. One way to guarantee this is to stay at a phase transition. We develop a lattice mean field method and find that the theory typically has several phases in the space of the dimensionless coupling constants, separated by the second order phase transition surface. For example, there is a phase with a spontaneous breaking of chiral symmetry. The effective low-energy Lagrangian for the ensuing Goldstone field is explicitly diffeomorphism-invariant. We expect that the Einstein gravitation is achieved at the phase transition. A bonus is that the cosmological constant is probably automatically zero.


I. INTRODUCTION
We live in a world with fermions, and they must be included into General Relativity.
The standard way one couples Dirac fermions to gravity is via the Fock-Weyl action [1,2]: Fermions interact with the frame field e A µ (also known as vierbein, repère or tetrad) and with the spin connection ω AB µ being the gauge field of the local Lorentz group. The frame and the spin connection are a priori independent field variables. The bosonic part of the action has to be written through e µ and ω µ accordingly. This is known for the last 90 years as Cartan's formulation of General Relativity [3]. Speaking generally, it is distinct from the classic Einstein-Hilbert formulation based on the Riemann geometry, since it allows for a nonzero torsion. We stress that the presence of fermions in Nature forces us to make a definite choice in favor of the Cartan, as contrasted to the Riemann geometry.
In practice, however, it is hardly possible to detect the difference. In the leading order in the gradient expansion of the gravitational action written down in terms of e µ and ω µ , the saddle-point equation for ω µ says that torsion is on the average zero. Therefore, Cartan's theory reduces to that of Einstein.
In the next order in p 2 /M 2 P where M P is the Planck mass and p is the characteristic momentum, a four-fermion contact interaction appears from integrating out torsion. Its strength is many orders of magnitude less than that of weak interactions [4] therefore this correction will hardly be detected any time soon in the laboratory. In principle, it modifies e.g. the Friedman cosmological evolution equation that follows from the purely Riemannian approach. However the correction remains tiny as long as fermions in the Universe have Fermi momentum or temperature that are much less than M P [5]. If they reach that scale such that the four-fermion correction becomes of the order of the leading stress-energy term, the theory itself fails since the gradient expansion [5,6] from where it has been derived, becomes inapplicable. There is no agreed upon idea how the theory looks like at the Planck scale; in particular, quantum gravity effects are supposed to set up there.
Being indistinguishable from Einstein's equation in the range where observations are performed, Cartan's theory, however, has a critical feature when one attempts to quantize it. The bosonic part of the action is written in terms of e µ and ω µ . To preserve the required general covariance or invariance under the change of coordinate system, called diffeomorphism, any action term is necessarily odd in the antisymmetric Levi-Civita tensor ǫ κλµν . That makes all possible diffeomorphism-invariant action terms not sign-definite [7].
The simplest example is the invariant volume itself or the cosmological term, d 4 x det(e).
If the frame field is allowed to fluctuate, as supposed in quantum gravity, the sign of det(e) can continuously change from positive to negative or vice versa. Of course, det(e) = 0 is a singularity where the curved space effectively looses one dimension but it is not possible to forbid such local happenings in the world with a fluctuating metric, see the illustration in  In the standard Riemannian formulation, one writes the invariant integration measure with the help of det(g) where g µν = e A µ e A ν is the metric tensor, hence det(g) = (det(e)) 2 is sign-definite. Its square root, however, should be understood as det(g) = det(e) and can have any sign. If it passes through zero it changes sign by continuity [8].
This fundamental pathology of any diffeomorphism-invariant quantum theory has not been stressed before, probably for two reasons. First, one commonly deals with the perturbative quantization about flat or e.g. de Sitter metric such that the main concern is the absence of runaway fluctuations from that point only. However, when quantizing gravity, one has to be concerned with large non-perturbative fluctuations as well. Second, usually Minkowski space-times are considered where the integration measure exp(i Action) is oscillating anyway independently of the action sign. However, a theory with a sign-indefinite action in Euclidian space where the weight is exp(−Action) is usually fundamentally sick also in Minkowski space. An illustration is provided by the scalar φ 3 theory, see Fig. 2.
Perturbation theory exists there in the usual sense near φ = 0. However, if in Euclid-ian space the theory does not exist, in Minkowski space one cannot define properly the non-perturbative Feynman propagator. There will be also other pathologies related to the possibility of tunneling to a bottomless state. In gravity theory, Euclidian formulation has its own right, for example in problems related to thermodynamics and to tunneling, like in the Hawking radiation problem where paradoxes are encountered just because we do not know how to quantize Euclidian gravity. If a theory is well defined for Euclidian signature, it is usually possible to Wick-rotate it to the Minkowski world. Therefore, for clearness we shall discuss here Euclidian gravity.
Any diffeomorphism-invariant action, with any number of derivatives, is not sign-definite in Euclidian space and hence cannot serve to define quantum gravity non-perturbatively.
At this time, we see only one way to overcome the sign problem, and that is to use in part fermionic variables in formulating quantum gravity microscopically, rather than only bosonic ones. Integrals over anticommuting Grassmann variables are well defined irrespectively of the overall sign in the exponent of a fermionic action. The reason is that in fermionic integrals introduced by Berezin [9] one actually picks up only certain finite order in the Taylor expansion of the exponent of the action, such that the overall sign does not matter.
More specifically, we suggest [7] (see also Ref. [16]) that at the fundamental, microscopic level gravity theory is defined as a theory of certain fundamental anticommuting spinor fields ψ † , ψ. We wish to preserve local gauge Lorentz symmetry exactly at all stages, and for that we need the explicit connection field ω µ . The frame field e µ and the metric tensor g µν will be composite fields making sense only at low energies. The basic independent variables will be ψ † , ψ and the gauge field ω µ , the same type of degrees of freedom as in the Standard Model. We believe that using the same type of variables as in the Standard Model will help to unify all interactions [7]. As far as only gravity is concerned, the fundamental spinor fields ψ † , ψ may or may not be related to the fundamental matter fields. We introduce the main building blocks of the theory in Section II.
A quantum field theory is well defined if it is regularized in the ultraviolet. We shall regularize spinor quantum gravity by introducing simplicial lattice (made of triangles in 2 dimensions, tetrahedra in 3 dimensions, 5-cells or pentachorons in 4 dimensions, etc.) covering an abstract space, such that the simplex vertices are characterized and counted by integers i. Only the topology of this abstract number space matters, e.g. the number of nearest neighbors, etc.
Each vertex i in the number space corresponds to the real world coordinate by a certain map x µ (i). Diffeomorphism-invariance means that the theory should not depend on the coordinates x µ (i) we ascribe to the vertices. Also, in the continuum limit (implying slowly varying fields) the action should be of the form d d x L(x) and invariant under diffeomorphisms, x µ → x ′µ (x). The integration measure over the fields in the path integral formulation should be also diffeomorphism-invariant. In addition, we require exact gauge invariance under local Lorentz transformations. We build fermionic actions satisfying these conditions in Section III, and regularize them by putting on a lattice in Section IV.
After constructing a completely well-defined lattice-regularized quantum theory, the next question to address is whether the continuum limit can be achieved and whether it reduces to the Einstein-Cartan theory in the low-energy limit. The continuum limit is obtained when and if field correlations spread over a large distance in lattice units. The trouble is that the quantum theory one deals with in this approach is a typical strong-coupling theory where most of the correlations die out over a few lattice cells. Contrary to the standard lattice gauge theory where long-range correlations are ensured by simply taking the weak-coupling limit β → ∞, in spinor quantum gravity there is no obvious handle to make the correlations long-ranged.
The main trick and the invention of this paper is to ensure long-range correlations by adjusting the bare dimensionless coupling constants to the point (or line, or surface) where the theory undergoes a phase transition of the 2 nd kind. At such point, all correlation functions become long-ranged, and the Einstein theory will be guaranteed in the low-momenta limit by the inherent diffeomorphism invariance.
Second-order phase transitions occur in theories where there is an order parameter, usually related to the spontaneous breaking of a continuous symmetry. Our primary goal in this project is to demonstrate that second-order phase transitions are typical in the kind of diffeomorphism-invariant theories we consider. We develop in Section V an original mean field method well suited for the search of the phase transitions, and check its accuracy in the Appendix where it is probed in an exactly solvable model, with very satisfactory results.
Using this method we unveil the phase diagram of a generic 2-dimensional lattice spinor gravity, in the space of the bare coupling constants, Section VI. The model has two continuous symmetries: the U(1) chiral symmetry and the U(1) symmetry related to the fermion number conservation; both can be in principle spontaneously broken.
It turns out that there is a range of bare couplings where the fermionic lattice system experiences spontaneous breaking of chiral symmetry. In the particular model we studied we did not observe spontaneous breaking of the fermion number, however it can happen in other models. This is an interesting finding per se but it may be also of use in the attempts to unify quantum gravity with the Standard Model. On this route, one expects one or several spontaneous breakups of continuous symmetries.
The 2-dimensional model we consider in some detail has certain nice features. First, the physical (invariant) volume V is extensive i.e. proportional to the number of lattice points taken. This is not altogether trivial since nonperturbative metric fluctuations allow, in principle, "crumpling" of the space, and that is what some researchers indeed typically observe in alternative nonperturbative approaches to gravity. In spinor gravity, it is a natural result following from the non-compressibility of fermions. Second, the quantum average of the curvature turns out to be zero such that the empty space without sources is effectively flat. This is also a welcome feature since the natural result in nonperturbative gravity is that the curvature is of the order of the cutoff, that is of the Planck mass, which is unacceptable.
Third, despite flatness the theory definitely describes a fluctuating quantum vacuum, as exemplified by the fact that the physical volume variance or susceptibility V 2 − V 2 is nonzero.
As a result of the spontaneous breaking of continuous symmetry (here: chiral symmetry), a Goldstone field appears. We check by an explicit calculation in Section VII that the low-momentum effective ("chiral") Lagrangian for the Goldstone field is diffeomorphisminvariant as expected. This invariance is rooted in the way we construct the original lattice action for spinors. The appearance of a Goldstone particle means that a definite bilinear combination of fermions is capable of propagating to large distances. However this is not enough: in order for the system to totally loose memory about the original lattice, all degrees of freedom have to propagate to long distances in lattice units. This happens only at a phase transition where we expect that the Einstein-Hilbert action emerges as a low-energy effective action for the classical metric, with the cosmological constant being automatically zero, see Section VIII. In Section IX we discuss the dimensions of various quantities and fields used throughout the paper. We summarize in Section X.

II. COMPOSITE FRAME FIELDS
Following Ref. [7] we introduce a composite frame field e A µ built as a bilinear fermion "current". In d dimensions the frame field transforms as a vector of the SO(d) Lorentz gauge group: Since A, B, . . . = 1, . . . , d are flat Euclidean indices we can equivalently write them either as subscripts or superscripts. The frame field transforms also as rank-one tensor (world vector) with respect to diffeomorphisms x µ → x ′µ (x): Our basic objects are fermion fields ψ(x), ψ † (x) assumed to be world scalars under diffeomorphisms, and transforming according to the spinor representation of the Lorentz group, The dimension of the spinor representation is d f = 2 [d/2] , see, e.g. [17].
We introduce the covariant derivative in the spinor representation, where ω AB µ is the spin connection in the adjoint representation of the SO(d) group, and Σ AB are its d f × d f generators, built from Dirac matrices γ A satisfying the Clifford algebra, In the adjoint (antisymmetric tensor) representation the corresponding covariant derivative Its commutator defines the curvature where One can built two distinct bilinear combinations of the fermion fields, transforming as the frame field (1,2): To check that e A µ and f A µ transform as a vector (1) one needs the relation between the matrix V rotating spinors (3) and the matrix O rotating vectors, Given that ψ, ψ † anticommute, the above bilinear operators are Hermitian.
We can define the bilinear fermion operator that plays the rôle of the torsion field, for example, and similarly for the other composite frame field f µ (11).

III. DIFFEOMORPHISM-INVARIANT ACTION TERMS
One can now construct a sequence of many-fermion actions that are invariant under local Lorentz transformations and also diffeomorphism-invariant, using either e A µ or f A µ (or both) as building blocks: where ǫ µ 1 µ 2 ...µ d is the totally antisymmetric (Levi-Civita) tensor. Notice that S 0 is the analog of the cosmological term but there are many of them since one can replace any number of e A µ 's by f A µ 's, S 1 is the analog of the Einstein-Hilbert-Cartan action linear in curvature, and the last action term S [d/2] for even d is a full derivative. Apart from full derivatives, there are 3 possible action terms in 2d, 6 terms in 3d, 8 terms in 4d, 12 terms in 5d, etc.
The use of ǫ µ 1 µ 2 ...µ d is obligatory to support diffeomorphism-invariance. In principle, one can construct Lorentz-invariant action terms by contracting the flat indices with Kronecker deltas instead of ǫ A 1 ...A d , however that will make the action term P -and T -odd. For example, there is a well known P, T -odd term in four dimensions, called sometimes the Holst action, [19,20], but we do not consider such terms here. One can add to the list of admissible action terms any of the actions (13) multiplied by any power of the world and Lorentz-group scalar (ψ † ψ); we shall consider such kind of terms later on in relation to the spontaneous breaking of chiral symmetry.
All action terms (13) are apparently invariant under two global U(1) rotations: • phase rotation related to the fermion number conservation, ψ → e iα ψ, ψ † → ψ † e −iα • chiral rotation for even dimensions d, ψ → e iβγ d+1 ψ, since both e A µ and f A µ are invariant under these transformations. The two corresponding Nöther currents are conserved. However, both symmetries can be spontaneously broken by interactions, and we shall see that this is what indeed typically happens.

IV. SPINOR GRAVITY ON THE LATTICE
In order to formulate quantum theory properly one has to regularize it at short distances.
The most clear-cut regularization is by (lattice) discretization, however diffeomorphisminvariance imposes severe restrictions on it, see recent discussion by Wetterich [13][14][15]. We impose two basic requirements: • explicit invariance under local gauge transformations of the Lorentz group, small or large (as in lattice gauge theory) • if the fields vary slowly in lattice units, i.e. in the continuum limit, the lattice action reduces to one of the diffeomorphism-invariant action terms (13) and the like.

A. Triangulation by simplices
To that end, we introduce an abstract discretized space where only the topology of vertices and edges connecting neighbor vertices is chosen beforehand and fixed. We find that the simplest hypercubic topology does not work. Only in two dimensions it is possible, for accidental reasons, to fulfill item 2 above by introducing a square lattice. In higher dimensions, the simplest but sufficient construction is to use a simplicial lattice. For uniformity, in two dimensions we also consider a triangle lattice made of three-vertex cells. In 3d simplices are tetrahedra or 4-cells, in 4d these are pentachorons or 5-cells, and so on.
It is always possible to cover the whole d-dimensional space by (d + 1)-cells or simplices, although the number of edges entering one vertex may not be the same for all vertices.
Alternatively, the number of edges coming from all vertices is the same but then the edges lengths may vary, if one attempts to force the lattice into flat space. Since only the topology of the nearest neighbors matters and the abstract "number" space does not need to be flat, this is also acceptable. The important thing is that the chosen set of cells should fill in the space without holes and without overlapping [18].
All vertices in a simplicial lattice can be characterized by a set of d integers. For brevity we label these d numbers by a single integer i. Each vertex has its unique integer label i, supplemented with a rule what labels are ascribed to the neighbor vertices forming elementary cells. We shall denote the d + 1 labels belonging to one cell by i = 0, 1, . . . , d. In this section, we write down the full lattice action as a sum over actions for individual simplicial cells, therefore we shall not be concerned with the precise geometric arrangement of the cells.
Each vertex in the abstract number space corresponds to the real world coordinate by a certain map x µ (i). The goal is to write possible action terms in such a way that if the fields vary slowly from one vertex (or link) to the topologically neighbor one, the action reduces to one of the possible diffeomorphism-and local Lorentz-invariant action term in Eq. (13).
We start by writing the volume of an elementary cell (simplex) in a given coordinate system in d dimensions. It can be presented as a determinant of a d × d matrix, where x µ 0 is the coordinate ascribed to one of the vertices, and x µ i , i = 1 . . . d are the coordinates ascribed to all the other vertices. We introduce the notion of a "positive order" of vertices i in the cell: it is such that for smooth functions x µ i the volume (14) is positive. An odd permutation of vertices in this set makes a "negative order".
It will be convenient to use the antisymmetric symbol if i k does not belong to a given cell, With the help of this symbol the cell volume (14) can be written as

B. Lattice action
The building blocks of our construction are anticommuting spinor fields ψ i , ψ † i that are world scalars and "live" on lattice vertices i, and the parallel transporter U ij . As in any lattice gauge theory, we replace the connection ω µ by a unitary matrix "living" on lattice links [7], In terms of these lattice variables the discretized versions of the composite frame fields (10,11) are:ẽ The difference betweenẽ andf is that the first has both fermions in the same vertex whereas in the second fermions are residing in the neighbor vertices.
Expanding all fields in Eqs. (18,19) around the center of a cell x = 1 d+1 d i=0 x i we obtain: where e A µ , f A µ are given by their continuum expressions (10,11), and the correction term is proportional to the derivatives of the fields and to the squares of the lengths of the cell edges. If the fields are slowly varying, meaning that the derivatives are small, the correction term can be neglected. This is what we mean by the continuum limit.
We also need the discretized version of the curvature tensor F AB µν : it is a plaquette. In our case the plaquettes are triangles, and we define the parallel transporter along a closed triangle spanning the i, j, k vertices: Expanding P ijk around the center of the cell x we obtain: and Using the above ingredients one can easily construct the lattice regularized version of the action terms (13). For example, the discretized cosmological term S 0 has the form: where any number ofẽ's can be replaced byf 's. In the continuum limit one uses Eqs. (20,21) and obtains The coordinate factors combine into the volume of the cell (16) and one gets where det(e) is composed from the continuum tetrad (10) and is attributed to the center of a cell. Eq. (25) proves that the lattice action (23) becomes the needed continuum action (13) if the fields involved are slowly varying from one lattice vertex to the neighbor ones.
Similarly, one finds the lattice version of all other action terms S k of Eq. (13): where the total number of plaquette factors P (22) is k, k = 0, 1 . . . [d/2]. In fact one can write a variety of such action terms replacing any number of composite frame fieldsẽ (18) by the composite frame fieldsf (19).

C. Lattice partition function
The lattice-regularized partition function for the spinor quantum gravity is quite similar to that of the common lattice gauge theory. One integrates with the Haar measure over link variables U ij living on lattice edges, and over anticommuting fermion variables ψ i , ψ † i living on lattice sites. The lattice, though, must be simplicial, otherwise the trick used e.g. in Eq. (24) to get the diffeomorphism-invariant action in the continuum limit, would not work.
Because of the requirement of diffeomorphism-invariance, the lattice action is quite different from those used in common lattice gauge theory. Typically one has many-fermion terms in the action. There are no action terms without fermions. One can write 3 action terms in 2d (all of them are 4-fermion), 6 terms in 3d (four are 6-fermion and two are 2-fermion), 8 terms in 4d (five are 8-fermion and three are 4-fermion), etc. We assume that spinor fields are dimensionless since we normalize the basic Berezin integrals as hence all quantities in Eq. (26) are dimensionless. Therefore, the "coupling constants" λ k one puts as arbitrary coefficients in front of the action termsS k (26) are all dimensionless.
The partition function is whereS (m) k are lattice actions of the type (26) with any number of composite frame fieldsẽ (18) replaced by the other composite frame fieldsf (19).

V. MEAN-FIELD APPROXIMATION
The partition function (28) defines a new type of a theory, and new methods -exact, numerical and approximate -have to be developed.
In principle, in order to compute the partition function (28)  In practice, however, the arising combinatorial problem is tremendous, and we did not manage yet to find a computational algorithm that would be faster than the exponent of the lattice volume. So far we have done a toy model in 1d exactly (see the Appendix) and succeeded in computing numerically correlation functions in 2d for limited volumes. There is a hope that the 2d model may be solved exactly but the method can hardly be extended to higher dimensions.
Therefore, for this pilot study, we have developed an approximate mean-field method to get the first glance on the dynamics of the new interesting theory at hand. Comparing the results with an exactly solvable model we see that the mean field accuracy is within a few percent. In the 2d model there are a few exact functional relations that are satisfied with the accuracy better than 15%, and this can be systematically improved. More important, the mean field approximation reveals a nontrivial phase structure of the theory in the space of the coupling constants λ k . This is the main finding of this study that may have important physical implications, see the Introduction.
The mean-field approximation we use is an extension of methods developed in condensed matter physics, that go under the name of "dynamical mean-field approach" or "local impurity self-consistent approximation" or "cavity method", see Ref. [21] for a review. Roughly speaking, the idea of the method is the following: One first picks up a simple element of the lattice (e.g. one simplex or a group of simplices with or without the boundary, let us call this fixed element "the cavity"), and calculates the effective action for the fields inside the cavity in the collective background of the fields outside it, replacing the background by the supposed mean field. At the second stage, one makes the method self-consistent, namely one calculates the mean field by integrating over the "live" variables inside the chosen cavity using the effective action found and expressed through the mean field at the first stage. As a result one gets a system of highly nonlinear self-consistent equations for a set of mean values of the field operators. Solving those equations one obtains the mean-field values as function of the coupling constants λ k . This gives the phase diagram of the theory in the space of the coupling constants.
The method has the advantage that it can be systematically improved by enlarging the chosen cavity. In the limit when the cavity covers the whole lattice it is an exact calculation.
Also, it is known that the accuracy of the mean field method is better the more nearest neighbors there are [21]. In simplicial lattices the number of neighbor cells is large, and the mean field method becomes exact in the limit d → ∞.
Let us formulate the method more mathematically. We choose the cavity, for example the elementary simplex with the boundary. We label the "live" fields belonging to the cavity by m, n, . . . and the fields outside the cavity (that will be replaced by mean fields) by i, j, . . ..
The full partition function can be written symbolically as where S mn is the part of the action that contains only fields from the cavity, S ij contains only fields from outside the cavity and S mi contains both. The link elements U mi are connecting vertices from the cavity with their nearest neighbors outside.
The last integral in Eq. (29) is the full partition function with the cavity cut out. When the lattice volume goes to infinity cutting out a finite cell does not change the averages of operators as compared to the averages computed on a full lattice; we denote them as The integration over the links U mi connecting the cavity with the outside neighborhood must be performed explicitly in Eq. (29). We expand e S mi in powers of the mixed action; since S mi is a fermion operator the power series is finite. Integrating over U mi splits all terms involved into a sum of products of operators composed of the cavity fields O ψ † m , ψ m , U mn and those living outside the cavity where the sum goes over various fermion operators labeled by p. Operators built from the cavity fields are left intact whereas the outside operators are replaced by the averages according to Eq. (30). We, thus, obtain the effective action for the fields inside the cavity: Finally, we make the calculation self-consistent by requesting that the operator averages O p computed from the cavity fields alone with the effective action (32) coincide with the full ones O p : Since S eff depends on the averages O p the self-consistency Eq. Of special interest are the cases where certain operator averages (the "condensates") violate the continuous symmetries of the original theory. It signals the spontaneous breaking of symmetry and leads to a nontrivial phase diagram for the theory. In the next section we illustrate it in a general 2d model.

VI. TWO-DIMENSIONAL SPINOR GRAVITY
The partition function is defined by Eq. (28) where the action has in general three terms with three arbitrary coupling constants λ 1,2,3 , The lattice-regularized version of it is, according to Eq. (23), where i, j, k = 0, 1, 2 label the vertices of a cell which in 2d is a triangle. Using integration by parts the first term in (34) can be rewritten as −F 12 12 (ψ † ψ) 2 . It gives an alternative discretization for the same continuum action: Although in the continuum limit the lattice actions (35)  ψ axial transformation : Therefore, both the continuum (34) and the lattice (35) actions possess these two global symmetries also; the corresponding Nöther currents are conserved. The vector symmetry means that the fermion number is conserved whereas the axial means that the difference between the numbers of "left-handed" and "right-handed" fermions (described by the upper and lower components of the spinors, respectively) is also conserved. It is also called the helicity conservation, or chiral symmetry.

A. Exact results
In the 2d partition function (28) Since there are two types of frame fields, e and f , we can define three types of "physical" or invariant volumes of the generally curved space, averaged over quantum fluctuations of the composite frame fields, The immediate conclusion from Eqs.(39-42) is that the average action is irrespectively of the coupling constants, where M is the number of triangle cells, which is twice the number of vertices N for large simplicial lattices.
Further on, one can introduce "physical volume susceptibility" or variance Therefore from Eq. (39) we know exactly the average physical volumes and volume susceptibilities at least at the edges of the parameter space Λ d = (λ 1 , λ 2 , λ 3 ): where M = 2N is the total number of simplicial cells in the lattice. The proportionality of these quantities to M is a very general property (valid not only at the edges of the parameter space but everywhere) following from Eq. (39). It shows that the physical volume is an extensive quantity, as it should be. This is not altogether trivial since nonperturbative metric fluctuations allow, in principle, "crumpling" of the space, or the formation of "branched polymers", and that is what some researchers observe in alternative nonperturbative approaches to gravity. In spinor gravity, it is a natural result following physically from the non-compressibility of fermions and mathematically expressed by Eq. (39).
The susceptibilities (48) There are theorems for mixed derivatives, valid in the whole parameter space Λ, that can be used to check the accuracy of approximate calculations, for example, no other result could be correct. It is illuminating, however, to see how "microscopically" the Euler theorem works for fluctuating spaces. In higher dimensions det(e)R is not a full derivative but it still may be possible to find its average in a similar way.
At the same time, owing to quantum fluctuations the average curvature squared is generally nonzero and extensive, implying that the volume-independent combination dies out in the thermodynamic limit, Eqs.(47,50,52) mean that although we apparently deal with a quantum fluctuating vacuum, the space is on the average large and flat in the absence of external sources. Therefore, one can say that the model describes a flat background metric G µν that is a unity matrix in a particular frame representing the flat space but transforms as a tensor under the change of coordinates. We shall use this notion in Section VII.

B. Mean-field approximation for one simplex cavity
In this Subsection we apply the mean-field method formulated in Section V to the lattice action (36) where we first put for simplicity λ 3 = 0. At the end of this Section we formulate the main results for λ 3 = 0.
In the first approximation to the mean-field method, we choose the elementary triangle cell (m, n, p) as the "cavity", see Fig. 3. The fields inside the triangle cavity are considered as real quantum fields, whereas the fields outside the cavity are combined into certain gauge-invariant operators that are frozen to their mean-field values. The triangle cavity is surrounded by three "black" triangles of the type (i, m, n) with a common edge, and by nine "white" triangles of the type (i, j, m) with a common vertex. The effective action for the fields inside the cavity gets contributions from both types of neighbors. Following the method of Section V, we expand the action exponent for every border cell, and integrate over the link variables U mi connecting the cavity with the outer lattice. As the result we obtain the product of operators built of the fields inside the cavity and those built of the outside fields. The latter operators are replaced by the averages to be found later from the self-consistency condition. We stress that after integrating over U im the operators on both sides can be only gauge-invariant.
For the single cell cavity we obtain operators of two types: single-site operators (they arise from the "white" cells), and double-site operators built from fermions at adjacent vertices (they arise from "black" cells).
Here is the list of operators that appear in this calculation. First of all, there are operators that are invariant under the U(1) V × U(1) A transformations described in Subsection A: O 1 is a single-site operator while O 2 and O 3 are double-site operators.
To be able to study the potential breaking of the U(1) V ×U(1) A symmetries we introduce operators that transform under those rotations. The chiral non-invariant operators that transform under U(1) A (38) are Fermion number violating operators transforming under U(1) V (37) are All operators are Hermitian.
The effective action for the fields inside the triangle cavity is computed as described in Section V. From the "black" neighbors we obtain the double-site effective action e S(m,n) = 1 + 8 9 where all double-site operators refer to the cavity vertices m and n. From the "white" neighbors we obtain the one-site effective action e S(m) = 1 + 8 9 Actually the operator averages in (56,57) imply averaging over the cyclic permutation of lattice sites in the cavity: e.g. This calculation is straightforward but the equations are rather lengthy. Therefore, we just comment here on its most important features.
First of all, we notice that S eff is quadratic in the symmetry breaking operators C 1,2 and W 1,2 , as it should be, therefore one gets a system of linear homogeneous self-consistency equations on the averages C 1,2 , W 1,2 that always have a zero solution, unless the deter- In the absence of symmetry breaking one puts C 1,2 = W 1,2 = 0 and solves the system of three nonlinear equations on the averages O 1,2,3 . There are in general several solutions but none are real in the whole Λ space. We pick up the solution that is real near the line λ 2 = 0. However it develops a cut and becomes complex at the lines |λ 2 | = 8.69|λ 1 | signalling that there can be a phase transition along these lines. A careful study in the next Subsection shows that indeed these are the border lines separating the phase with spontaneous chiral symmetry breaking, see Fig. 4. Outside this domain, i.e. at |λ 2 | < 8.69|λ 1 | chiral symmetry is not broken, the solution for the normal, symmetry-preserving operators is real, and one can approach the line λ 2 = 0 where we can check the accuracy of the mean field method by comparing the average physical volume V 1 ≈ cells det(ẽ) where the average is computed over one (cavity) cell with the effective action (58), with the exact result (47). We find numerically where M is the total number of lattice cells. We note that the functional dependence on We have also tested a more primitive mean-field approximation where the cavity is taken in the form of two neighbor vertices connected by a link. It is also capable of detecting the spontaneous breaking of chiral symmetry but the accuracy is, of course, worse: it is at level of 40%.

C. Spontaneous chiral symmetry breaking
An accurate way to study spontaneous breaking of a continuous symmetry is to introduce a small term in the action that violates the symmetry in question explicitly. Since we are interested in the spontaneous breaking of the U(1) A or chiral symmetry we introduce the simplest diffeomorphism-invariant "mass term" that is not invariant under chiral rotations (38). Its discretized lattice version is obvious, see the first term in Eq. (35).
Adding this term we repeat the same mean-field derivation of the effective action for the triangle cavity as in Eqs.(56,57) which now obtain an addition Let us note that the terms linear in the mass parameter m are also linear in the chirality-odd operators C 1,2 .
With this addition to the previous effective action (58) we now turn to solving the self-

D. No fermion number violation
The effective action (58) contain operators W 1,2 violating the U(1) V symmetry (37) related to the fermion number conservation. If W 1,2 = 0 it signals the spontaneous violation of this symmetry. Fermion number conservation is spontaneously broken e.g. in ordinary superconductors and "color" superconductors in QCD. However, in contrast to chiral symmetry that is also broken in QCD, spontaneous fermion condensation usually happens not in the vacuum but at nonzero chemical potential for fermions, since interactions are effectively amplified near the Fermi surface. In this Subsection we look for the spontaneous fermion number non-conservation in the same 2d model where we observe spontaneous breaking of the U(1) A symmetry in the mean-field approximation.
Following the same logic as in the previous Subsection we introduce a small action term that violates the U(1) V symmetry explicitly, This operator preserves the chiral U(1) A symmetry. The correction to the effective one-triangle action is: We solve again the self-consistency equations on the operator averages but now with this addition, and look for non-analytic dependence on the small parameter b. In contrast to the case of spontaneous chiral symmetry breaking we do not find such solutions in the whole We conclude that the fermion number conservation is not broken spontaneously in the model, except maybe along the line of the chiral phase transition. We did not study the inclusion of a chemical potential for fundamental fermions -that would explicitly violate Lorentz symmetry but presumably make the phase diagram of the model more rich.
There are no reasons why fermion number conservation would not break spontaneously, say, in 4d, and the mean field method suggested here is a simple way to detect it.

E. Full phase diagram
The full action compatible with the principles proclaimed has in 2d three terms and consequently three coupling constants. In the previous Subsections we have restricted our study to the case of λ 3 = 0.
Actually we repeat all the steps described above also for λ 3 = 0. The algebra becomes more cumbersome but still doable. We find that the chiral symmetry breaking phase I occupies the cone shown in Fig. 7; Fig. 4 is its section at λ 3 = 0. We remark that the accuracy of the meanfield approximation for some reason deteriorates as λ 3 grows. Still the exact relation (43) holds even at λ 3 → ∞ up to a factor of 1.6.

VII. LOW-ENERGY ACTION FOR PROPAGATING FIELDS
The theory defined by the partition function (28)  However there are ways to guarantee that long-range correlations appear; moreover that can be checked in the mean-field approximation. An example which we consider here is provided by the Goldstone theorem: If a global continuous symmetry is broken spontaneously the associated Goldstone bosons are exactly massless and hence propagate to large distances.
In the previous Section we have shown that the continuous U(1) A or chiral symmetry is spontaneously broken in a broad range of the space of the coupling constants. Supposing the coupling constants are chosen inside that range (inside the cone in Fig. 7), there is a massless Goldstone excitation α(x) being the phase of the U(1) A rotation (38).
Under this rotation the chirality-violating operators C 1,2 and C 1,2 transform as To derive the low-energy action for the Goldstone field we allow the phase α to vary slowly from cell to cell: We parameterize the operator averages C ± 1,2 in the same way and re-derive the effective action (58) for the fields inside the triangle cavity, taking now into account that the operator averages have slightly different phases in the cells surrounding the cavity. Then, integrating over the fields inside the cavity we find the effective one-cell partition function Z 1 modified by the varying nearest neighborhood. If α is the same for all neighboring cells it is the same expression as in Section V, let us call it Z 10 . However, there will be further terms depending on the gradients of α(x), we are now after.
The full partition function is, in the mean field approximation, a product of Z 1 's over all cells whose number is M. Therefore the action for the Goldstone field α(x) is where α i is the value of the phase attributed to one of the 12 neighbor cells i, and ∆α i is the difference between α i and α 0 attributed to the central cavity cell; the summation goes over all neighbor cells. It is important that the dependence of Z 1 on α i starts from quadratic terms, which is the consequence of chiral symmetry; hence ∂Z 1 /∂α i = 0, and we are left with second derivatives.
Ignoring the first α-independent term in Eq. (66) we find that the action is quadratic in the jumps ∆α from one cell to the neighbor ones, We now introduce a coordinate system by mapping the centers of the cells to coordinates x µ (i) (Section IV). If the changes of α from a cell to neighbor cells are small we can expand where ∆x µ i = x µ (i)−x µ (0) is the distance between the coordinate attributed to the cell i and that attributed to the cavity cell, in a given coordinate frame x µ (i). Putting this expansion into Eq. (67) we obtain The first factor M, the full number of cells on the lattice can be written as where V (cell) is the cell volume in a given frame, see Eq. (14). The combination  We thus arrive at a diffeomorphism-invariant low-energy action for the massless Goldstone field: This field can propagate infinitely far in lattice units since its masslessness is guaranteed by the Goldstone theorem.
To complete our study of the spontaneous chiral symmetry breaking we derive the analog of the Gell-Mann-Oakes-Renner relation for the pion mass in QCD, expressed through the quark masses. If chiral symmetry is broken explicitly by a small fermion mass term (60) the phase of the chiral condensate becomes a pseudo-Goldstone field with a mass proportional to the square root of the fermion mass m.
Indeed, the addition of the mass term (60) changes the effective one-cavity partition function: The Z m piece depends explicitly on the chiral condensate phase α introduced in Eq. (65), and from symmetry considerations it is clear that the expansion starts from the α 2 term; direct calculation confirms it.
Summing up the mean-field action over the whole lattice one uses the relation (70) where the r.h.s. behaves as ∼ d 2 x √ G according to the transformation properties under the change of the coordinate system x µ (i) attributed to the lattice. We obtain thus the action for the pseudo-Goldstone mode in the continuum limit where µ is proportional to the pseudo-Goldstone boson mass. We see that it is proportional to the square root of the mass parameter m that breaks chiral symmetry explicitly. In QCD, this is known as the Gell-Mann-Oakes-Renner relation for the pion mass. The coefficient in this relation depends on the coupling constants λ 1,2,3 . At the 2 nd order phase transition surface of the cone in Fig. 7 the pseudo-Goldstone mass goes to zero at fixed m.
There is a famous Mermin-Wagner theorem stating that a continuous symmetry cannot be spontaneously broken in 2d as the resulting Goldstone bosons would have an unacceptably large, actually divergent free energy. Since the mean field approximation misses the Goldstone physics, one can argue that the spontaneous chiral symmetry breaking we observe is an artifact of the approximation. If, however, the Goldstone field α(x) is Abelian as here, the actual phase is, most likely, that of Berezinsky-Kosterlitz-Thouless where the chiral condensate ρ e iα indeed vanishes owing to the violent fluctuations of α(x) defined on a circle (0, 2π), but the correlation functions of the type e iα(x) e −iα(y) have a power-like behavior, and there is a phase transition depending on the original couplings of the theory.
In any case our primary goal here is to learn how to deal with the lattice regularized spinor quantum gravity which is a new type of a theory. The mean-field approximation is one possible approach that is expected to work even better in higher dimensions where, as a matter of fact, the Mermin-Wagner theorem does not apply.

VIII. HOW TO OBTAIN EINSTEIN'S LIMIT?
The apparent diffeomorphism-invariance of Eq. (75) is built in by our construction of the lattice and lattice action in Section IV. As soon as there are degrees of freedom that can propagate to long distances, their low-energy effective action is diffeomorphism-invariant in the continuum limit.
In the previous Section the appearance of long-propagating mode has been guaranteed by the Goldstone theorem. However, it concerns only the specific Goldstone modes associated with the spontaneous breaking of continuous symmetry. Other degrees of freedom remain heavy: their correlation functions decay exponentially after a few lattice cells. If one attempts to write an effective low-energy action for the classical metric tensor g cl µν (see below its exact definition) it will have the diffeomorphism-invariant form, with the constants c 1,2 computable, in principle, from the original coupling constants of the lattice-regularized theory. However, if one does not take special measures, the ratio c 1 /c 2 , playing the rôle of the graviton mass, will be on the order of the inverse lattice spacing. In such a situation it is senseless to introduce the metrics in the first place. It makes sense only if c 1 /c 2 happens to be zero or very small, such that the graviton and the Newton force propagates to large distances.
To ensure it, it is sufficient to stay,e.g., at the phase transition of the second order, where all degrees of freedom become massless. The classical metric tensor g cl µν and the effective action functional Γ[g cl µν ] can be introduced by means of the Legendre transform (proposed in this context also by Wetterich [14]). One introduces first the generating functional for the stress-energy tensor Θ µν as an external source, where S the fermionic action andĝ µν is a 4-fermion operator built from the frame fields (10,11) or, after discretization, from their lattice versions (18,19). The classic metric field is by definition This equation can be solved to give the functional Θ µν [g cl ]. Using it one can construct the effective action as the Legendre transform: At the phase transition fluctuations are long-ranged. For long-range fluctuations it is legal to take the continuum limit of the lattice, which is diffeomorphism-invariant. The low-energy limit of diffeomorphism-invariant actions for a quantity transforming as a metric tensor is uniquely given by Eq. (76). Moreover, the cosmological term necessarily has zero coefficient, c 1 = 0, since otherwise the graviton would propagate to a finite distance c 2 /c 1 , which contradicts the masslessness of the fluctuations at the phase transition. This is how one can recover Einstein's gravity from the lattice-regularized spinor theory.
In principle, the effective Einstein-Hilbert action from spinor quantum gravity can be derived in the mean-field approximation similarly to our derivation of the low-energy effective chiral Lagrangian in Section VII. However, in 2d where we have so far succeeded in developing the mean-field method the Einstein-Hilbert action is a full derivative and there are no gravitons or the Newton force. Therefore, the derivation of Eq. (76) has to be postponed till higher dimensions are studied along the lines of the present paper.

IX. DIMENSIONS
In this paper, we use unconventional dimensions of the fields, which, however, we believe are natural and adequate for a microscopic theory of quantum gravity. The fermion fields are normalized by the Berezin integral (27) and are dimensionless, hence the composite frame field (10) has the dimension 1/length and the metric tensor has the dimension 1/length 2 , in contrast to the conventional dimensionless metric tensor. On the other hand all diffeomorphism-invariant quantities are dimensionless in our approach. In this section we explain why it is convenient, and what is the relation to the usual approach.
The historic tradition in General Relativity is that the space-time at infinity is flat, therefore one can safely choose the coordinate system such that g µν is a unity matrix there. This sets the traditional dimensions of the fields. In particular, the scalar curvature has the dimension 1/length 2 , the fermion fields have the dimension 1/length 3 2 , etc. However, in a diffeomorphism-invariant quantum theory where one can perform arbitrary change of coordinates x µ → x ′µ (x) not necessarily identical at infinity, for example, a dilatation x µ → x µ /ρ, and where g µν can a priori strongly fluctuate at infinity, this convention is not convenient.
The natural dimensions of the fields are those that are in accordance with their transformation properties: any contravariant vector transforms as x µ and has the dimension of length, a covariant vector, in particular the frame field e µ transforms as a derivative and has the dimension 1/length, g µν has the dimension 1/length 2 , etc. World scalars like the scalar curvature and the fermion fields are, naturally, dimensionless. In fact it is a tautology: a quantity invariant under diffeomorphisms is in particular invariant under dilatations and hence has to be dimensionless.
In this convention, any diffeomorphism-invariant action term is by construction dimensionless and is accompanied by a dimensionless coupling constant, as in Eq. (28).
Let us suppose that we have a microscopic quantum gravity theory at hand that successfully generates the first terms in the derivative expansion of the effective action, where c 1,2 are certain dimensionless constants expressed through the dimensionless couplings λ 1,2,... of the original microscopic theory. The ground state of the action (1) is the space with constant curvature R = 2c 1 /c 2 , represented e.g. by a conformal-flat metric where x 0 and ρ are arbitrary. At the vicinity of some observation point x 0 it can be made a unity matrix by rescaling the metric tensor, where the rescaling factor m has the dimension of mass, andḡ µν has the conventional zero dimension. At this point one can rescale other fields to conventional dimensions, in particular introduce the new fermion fieldψ of conventional dimension m 3/2 : The new composite dimensionless tetrad field compatible both with Eqs. (82) and (83) is One can now rewrite the action (80) together with the fermionic matter in terms of the new rescaled fields denoted by a bar, Underbraced are the cosmological constant and the Plank mass squared, respectively; numerically, λ = 2.39 · 10 −3 eV, M P = 1.72 · 10 18 GeV. The dimensionless ratio of these values, is the only meaningful quantity in pure gravity theory, independent of the arbitrary scale parameter m. If a fermion obtains an effective mass, e.g. as a result of the spontaneous chiral symmetry breaking, leading to an additional term in the effective low-energy action then the "theory of everything" has to predict also other dimensionless ratios. For example, taking the top quark mass m t = 172 GeV one has to be able to explain the ratio In other words, one can measure the Newton constant (or the Planck mass) or the cosmological constant in units of the quark or lepton masses or the Bohr radius. Only dimensionless ratios make sense and can be, as a matter of principle, calculated from a microscopic theory.
To that end it is convenient and legitimate to use natural dimensions when g µν has the dimension 1/length 2 whereas all world scalars are dimensionless, be it the scalar curvature R, the interval ds, the fermion field ψ or any diffeomorphism-invariant action term. There must be special physical reasons for massless excitations in the theory, for which the continuum limit makes sense and diffeomorphism-invariance becomes manifest. One such reason is spontaneous breaking of continuous symmetry where the existence of massless fields is guaranteed by the Goldstone theorem. To show that spontaneous breaking may be typical in such kind of theories, we have developed a new mean-field approximation. We have checked its accuracy in a 1d exactly solvable model, and in a full 2d theory where certain exact relations can be derived. The exact relations tell us nice things: the physical or invariant volume occupied by the system is extensive as due to the non-compressibility of fermions, the volume variance (or susceptibility) is also extensive showing that it is a true quantum vacuum, and the average curvature is, at least in 2d, zero meaning that the quantum space is on the average flat. They also tell us that our mean-field approximation is rather accurate, and the accuracy can be systematically improved.
We show, within the mean-field method, that the spontaneous breaking of chiral symmetry happens in a broad range of the coupling constants and that in this range the low-energy action for the Goldstone field (or pseudo-Goldstone if we add a term explicitly breaking symmetry) is diffeomorphism-invariant, as expected.
To obtain the low-energy Einstein limit one has to stay at the second-order phase transition surface in the space of the coupling constants. There the masslessness of excitations, and not only of the Goldstone ones, is guaranteed. Hence one can go to the continuum limit where the diffeomorphism-invariance is also guaranteed by construction. Therefore, we expect that the effective low-energy action for the classical metric tensor, derived through the Legendre transform, is just the Einstein-Hilbert action, with the zero cosmological term.
This can be probably seen already in the mean-field approach for dimensions higher than two. This work is in progress.
ψ α , ψ † α , α = 1, 2, that transform under the U(1) gauge transformation as ψ → V ψ, ψ † → ψ † V † , V = exp(iασ 3 ). The gauge field is represented by link variables U ij = exp(−iω ij σ 3 /4) that transform as U ij → V i U ij V † j . We construct the lattice version of the two "frame" fields, as in Eqs. (18,19), (which however do not have the meaning of frame fields in 1d), and form the action that is quite similar to the full 2d action (36): where λ 1,2 and µ are the coupling constants. The partition function is defined as a product of Berezin integrals on the 1d lattice with N points: We imply antiperiodic boundary conditions for fermion fields and periodic boundary conditions for link variables.
The partition function (A.3) is exactly computable by a kind of transfer-matrix method.
Diagonalizing the transfer matrix we obtain a nontrivial result: The fact that the partition function has the form of a sum of extensive exponents means that actually it describes simultaneously four independent phases or states of the system that do not compete and hence do not mix up in the thermodynamic limit N → ∞. The stable phase is the one with the lowest free energy, that is with the largest partition function.
In phase 0 fermions in the neighbor lattice sites form an ordered state of the type "pair-gappair-gap..." all over the lattice, where the "pair" means that there are four link matrices in the integration over link variables, and "gap" means zero matrices. It can be realized only on even-N lattices, hence it is a lattice artifact, and we do not consider it further. Phases 1 and 2 are states where two link matrices appear in all link integrations. We concentrate of phases 1 and 2 only in what follows.
We also calculate exactly average values of the following 4-fermion operators: We now turn to constructing the mean field approximation to the model, to check its accuracy against the exact calculation. We apply the general method of Section V, which is rather straightforward in this simple case. We first take the "cavity" in the form of two neighbor lattice sites connected by a link (1 st approximation), and then three adjacent sites connected by two links (2 nd approximation). Both mean-field approximations give satisfactory accuracy when compared to the exact results but the second is, of course, better.
In both cases the cavity boundary is just the neighbor sites connected to the cavity by link variables U mi . Expanding e S mi up to the second power (higher powers are zero because of too many fermion operators) and integrating over U mi we obtain several operator structures.
Splitting them into the product of operators composed of fields inside the chosen cavity, and the operator built of the outside fields, we replace the latter by the averages, to be found self-consistently. Most of the operators break the σ 3 -symmetry of the original action, and we ignore them. The only operator with proper symmetries left is O = (ψ † i ψ i ) 2 . We find the effective action for the two-site cavity The results for the averages of three operators in the first and second approximations as well as their exact values (A.5,A.6,A.7) are presented in Fig. 9. There is a "phase transition" between phases 1 and 2 at λ 1 = 0. We see that the second mean-field approximation corresponding to a three-site, two-segment cavity gives a very satisfactory accuracy.