Superconducting and Anti-Ferromagnetic Phases of Spacetime

A correspondence between the $SO(5)$ theory of High-T${}_C$ superconductivity and antiferromagnetism, put forward by Zhang and collaborators, and a theory of gravity arising from symmetry breaking of a $SO(5)$ gauge field is presented. A physical correspondence between the order parameters of the unified SC/AF theory and the generators of the gravitational gauge connection is conjectured. A preliminary identification of regions of geometry, in solutions of Einstein's equations describing charged-rotating black holes embedded in deSitter spacetime, with SC and AF phases is carried out.


INTRODUCTION
Two of the outstanding problems in theoretical physics today are those of high temperature superconductivity (HTC) on the one hand and quantum gravity (QG) on the other. In the case of HTC, it has been demonstrated that the anti-ferromagnetic (AF) and superconducting (SC) phases d-wave [5,6,22,23] and p-wave [11] superconductors can be given a unified explanation in terms of a non-linear sigma model for a field which behaves as a vector transformation under SO(5) rotations. In QG research it is known that general relativity with non-zero cosmological constant (Λ = 0)can be obtained from a socalled BF model (a topological field theory) for a gauge field, valued in either SO(3, 2) (for Λ < 0) or a SO(4, 1) (for Λ > 0), by a symmetry breaking mechanism [8,21]. This mechanism was first outlined in a seminal paper by MacDowell and Mansouri [9] in 1977 whose motivation was to construct a unified theory of gravity and supergravity.
In this work we demonstrate the equivalence between these two theoretical frameworks. The picture resulting from our line of reasoning is that of a spacetime with non-zero Λ, as described by classical general relativity, emerging via symmetry breaking of a topological quantum field theory (TQFT) defined on a four-dimensional manifold. The superconducting and anti-ferromagnetic phases can be identified with regions of spacetime where Λ > 0 and Λ < 0 respectively. The boundary between these phases can be viewed as the horizon of black holes. The interiors of these black holes have a description in terms of a anti-deSitter (Λ < 0) geometry.

BACKGROUND
The AdS/CFT conjecture, as formulated by Maldacena, states that there is an exact correspondence between supergravity (SUGRA) in the interior of a 5dimensional AdS spacetime and a maximally supersymmetric (N = 4) Yang-Mills theory living on the four dimensional boundary of this space. While this connection is remarkable in its beauty, there are some interpreta-tional issues. We clearly don't live in a 5-dimensional world. All physical phenomena we have understood so far can be described by the physics in a four or lowerdimensional spacetime. To be sure there remain many problems which don't have a clear solution, such as the energy excess in cosmic rays or that of gamma ray bursts (GRBs). The possibility remains that these problems might ultimately be best described as by physics in spacetimes with dimension d > 4. In the meantime we are left with an inadequate understanding of how the Maldacena Conjecture and the Holographic Principle can fit into a more complete description of the real four dimensional world we observe on scales stretching from the galactic to the subatomic.
The situation is similar in the primary theoretical alternative to string theory as a candidate theory of quantum gravity -Loop Quantum Gravity (LQG). LQG has had remarkable success so far in providing a microscopic description of the degrees of freedom whose excitations correspond to quanta of geometry, i.e. volume and area operators. The kinematical Hilbert space H kin consists of spin-network states which are abstract graphs whose edges are labeled by irreducible representations of su (2) and whose vertices are labeled by invariant tensors known as intertwiners. Here two-dimensional surfaces are used as "probes". When an edge of a spin-network intersects a given surface that surface is endowed with a quantum of area. A space-like 2-sphere immersed in such a space is given an amount of area determined by the number and types of resulting punctures on its surface. Many different configurations of punctures can add up to give the same value for the total area of a 2-sphere which can then be identified with a black hole horizon of given classical area A. This degeneracy is identified as the source of black hole entropy in LQG. These successes, however, have not yet led to a complete understanding as to how a superposition of spin-network states can yield a classical spatial three-dimensional manifold.
There is also an alternative viewpoint -that the correct theory for describing "our" 3+1 world is ultimately a variant of Yang-Mills, which itself lives on the boundary of a 4+1 dimensional spacetime, of which our observable Universe constitutes the boundary. This picture leads to the following problem. Let us say that YM fields in our 3+1 world correspond to some theory gravity in a 4+1 bulk. But then there is no reason that one should not consider Yang-Mills (and all other fields), living in the 5-dimensional bulk, consistent with the diffeomorphism and other local gauge symmetries! Then the holographic idea would suggest that these fields, residing in the 5 dimensional bulk, are themselves the boundary degrees of freedom of some 5+1 dimensional bulk gravity theory. In other words, following this hypothesis in the wrong direction, so to speak, leads to a "tower of turtles".
The gravitational theory we observe and which fits the vast majority of observations on all scales, describes a four-dimensional world. Correct, and minimal, application of the holographic idea then suggests that we try to retrieve the gravitational physics of a 3+1 world by considering a gauge theory living on 2+1 boundaries living in the bulk. This can be accomplished by following a different line of thought, in which 2+1 CFTs encode a description of the dynamics of 3+1 GR. Gravity in 2+1 dimensions is trivial. There are no non-trivial vacuum solutions of Einstein's equations in 2+1 dimensions. The only complication arises if we include matter. In 2+1 all this does is introduce conical singularities on the space-like surfaces whose deficit angle corresponds to their mass. The gravitational interaction between these defects is then determined solely by considering the statistics of such particles under exchanges. This statistical interaction can then be modeled by a gauge field which would have the precise interpretation as the CFT living on the boundary, i.e. 2+1 dimensional spacetime is special in that, on it, gravity and gauge theory are interchangeable.
In this paper we first present a construction for which we later supply a physical interpretation. We identify the bulk AdS 4 spacetime as a description of the interior (S − ) geometry of a black hole. The exterior (S + ) geometry is described an asymptotically deSitter 4D spacetime. We identify the 2+1 dimensional boundary (∂S) between these two spaces as the black hole horizon, or more simply the black horizon. This boundary plays the role of a phase boundary between two different phases of geometry. The phase described by the AdS/CFT correspondence is hidden from observers living in the exterior of the horizon. However an observer inside the horizon would reach the same conclusion about the boundary physics as an observer in the exterior. This is the condition required to be able to consistently glue these different geometries along a common boundary. The ratio of the vev of the condensate in the exterior and interior regions Λ − /Λ + can then be interpreted as the size of the AdS 4 region with respect to the asymptotically dS 4 universe within which it is contained.

SO(5) MODEL OF HIGH TEMPERATURE SUPERCONDUCTIVITY
One can ask why is it that the group SO(5) should have anything to do with the description of the SC or AF phases in condensed matter systems, and for that matter, why do we need a unified description of the two phases in the first place. There are three reasons [23] to believe that this should be the case: 1. In 1988, Chakravarty, Halperin and Nelson [4] demonstrated that the non-linear sigma model for a field with SO(3) symmetry gives a good description of the properties of a two-dimensional (2+1) Heisenberg anti-ferromagnet in the lowtemperature, long wavelength regime.
2. The behavior of the superconducting state is known to be well-described by a so-called "XY" model for a U (1) gauge field.
3. Both d-wave SC and AF can be described in terms of the behavior of singlet pairs in the Hubbard model at half-filling. These singlet pairs can describe either an AF phase, a SC phase or a so-called "spin-bag" phase where both the phases co-exist. Now, if both AF and SC arise in different regimes of a system with the same underlying physics -the Hubbard model at half-filling -and can co-exist under certain conditions, it follows, that one would be well-advised to seek out a low-temperature, long wavelength effective field theory which can describe both phases. Such a theory should contain a SO(3) × U (1) symmetry, which should arise after some symmetry breaking transition. The smallest gauge group that can accomodate such a symmetry among its subgroups is SO(5).
In [5,6,11,22,23] it was shown that a non-linear sigma model for a field with an SO(5) gauge symmetry, can describe the physics of both the AF and SC phases. Four of the elements of the generators of the group algebra can be identified with the AF and SC order parameters. The remaining six can be identified with operators which generate transformations between the AF and SC phases.
As argued in the beautiful paper by Zhang [23], the physical picture of the transition between the AF and SC states is the following. The overall system is described by some microscopic Hamiltonian describing the interaction between electrons on a lattice. Below some characteristic temperature T MF , electrons on neighboring sites tend to from singlet bound pairs or dimers. The AF and the SC phase are different states which this dimer collective can form. When the dimers are not free to move -due to the lack of vacancies on the lattice -the collective forms a dimer "solid" which corresponds to the AF phase. As a certain system parameter is varied the dimer solid begins to melt and forms a fluid which corresponds to the SC phase. At the transition between the two phase one will have regions where both the solid and liquid phases are present. This corresponds to the "spin-bag" phase where both AF and SC co-exist.
First let us introduce some notation. c † p,↑↓ (c p,↑↓ )is the operator which creates (resp. destroys) an electron with momentum p and given spin (↑ or ↓). σ i , {i = x, y, z} are the Pauli spin-matrices. With these in hand we can define the operators for spin (S), momenta (π) and total charge (Q) for electrons near the Fermi surface. These are as follows: Here g(k, p) ≡ g(k − p) is a function of the electron momenta in terms of which the operator for the superconducting gap ∆ can be written as [22,23]: Thus g(k) possesses the symmetries of the gap function ∆ k . For the case of d-wave HTSC, it has the form: These operators can be arranged in the form of a 5 × 5 matrix L IJ as in: It can be shown that, if |g(k)| 2 = 1, the elements of this matrix satisfy the commutation relations: which are the commutation rules satisfied by the generators of the Lie alebgra of the group SO(5). Furthermore the vector S = (S x , S y , S z ) can be identified as the order parameter of the ferro/anti-ferromagnetic (F/AF) phase, while Q is the order parameter of the superconducting (SC) phase. The operator π = (π x , π y , π z ) rotates the AF order parameter into the SC order parameter and vice-versa [23].
In his work [22,23] suggests that the behavior of high-T TC superconductors can be characterized by introducing a five-dimensional superspin vector n I , whose components can be identified with the various AF and SC order parameters as follows: where q = (π x , π y , π z ) is the AF order parameter; the operators ∆, π have been defined previously in (1) and (2).
The matrix L IJ and the vector n I satisfy the following commutation relations: and can be seen to be conjugate variables [5], just as the momentum and position p, q are in the ordinary harmonic oscillator. Thus in terms of these objects we can write down the Hamiltonian for SO(5) effective theory of AF and SC: where the various terms correspond to, respectively, the kinetic energy of SO(5) rotors (∼ L 2 ), the coupling between rotors on different sites (∼ n 2 , < .. > denotes sum over nearest neighbors), coupling between an external field and the momenta of the rotors (∼ BL) and a symmetry breaking term (V (n)) which breaks the SO(5) symmetry down to SO(3) × U (1). As we shall see in the following discussion the Hamiltonian for gravity in the Einstein-Cartan formalism, along with an associated "Higgs" field v I , resembles this condensed matter Hamiltonian.
In addition we can also include terms for the elastic energy of the SO(5) generators [11]: where v IJ = K (L IK ∇L JK − L JK ∇L IK ). Such a term is the measure of the spatial rigidity of the system and allows us to construct a unified description of p-wave SC and ferromagnetism [11].

CARTAN DECOMPOSITION
We now come to the gravity side of the picture. Our ingredients are a four-dimensional manifold M 4 on which we have a SO(4, 1) or SO(3, 2) connection A I µ , depending on whether Λ > 0 and Λ < 0 respectively. There is no metric structure on this manifold to begin with. This connection can then be decomposed into two parts [21]: where ω I µ is identified with an so(3, 1) connection and e I µ is a four-dimensional frame field. ǫ = +1 when Λ > 0 and ǫ = −1 when Λ < 0. The curvature F [A] of the connection can then be written as: where I, J, K = 0, 1, 2, 3, 4 label the elements of the so(4, 1) (resp. so(3, 2)) matrices, and the spacetime indices are suppressed. In terms of these indices, 10 can be written as: where a, b = 0, 1, 2, 3. This can be more clearly seen in the explicit matrix form: 1, 2, 3) are the generators of boosts, ω i j (i, j = 1, 2, 3; i = j) are the generators of spatial rotations (ω 0 1 = −ω 1 0 and ω 1 2 = −ω 2 1 ) and e a /l (a = 0, 1, 2, 3) are the generators of translations. The gauge curvature can then be expanded as follows. For the so(3, 1) part: Similary the R(3, 1) part of the curvature is given by: where D ω is the antisymmetrized covariant derivative operator w.r.t the connection ω. Finally we see that the various components of the gauge field strength can be written as: where R is the curvature of a so(3, 1) connection ω and e is a so(3, 1) valued one-form.

BF THEORY
The action for a topological theory on a manifold M with local gauge group G is given by: where B and F are a (n − 2)-form and a 2-form respectively on M and which takes values in the Lie-algebra g of G. F is the field strength for a connection A. The configuration variables are the gauge connection A and the two-form field B and the action is invariant under SO(5) transformations of the gauge field. Varying the action w.r.t. these variables we find the two equations of motion [8]: where F IJ = dA IJ + A I K ∧ A KJ is the curvature tensor and D A is the covariant derivative w.r.t. the gauge connection A. We have made use of the fact that: followed by a partial integration in order to obtain (18). Since the field strength is identically zero everywhere, in the present form, this action describes a system with no local degrees of freedom. The value of S BF when evaluated on a given manifold, for any choice of B and A, will only yield information about the topology of the manifold. Thus (17) is the action for a topological field theory or TFT and as such has no correspondence with classical general relativity. The situation changes, how-ever, when we add a term to the action quadratic in the B field which corresponds to the breaking of the SO(5) symmetry of the theory resulting in a theory with propagating local degrees of freedom. The modified action is as follows [8]: where v M is a fixed SO(5) vector pointing in a preferred direction. It is this choice of a preferred direction that breaks the SO(5) symmetry, in much the same way as the choice of a preferred direction for spins breaks the symmetry of the Ising model and allows the ferromagnetic phase to appear from an initially disordered phase where the spins point in arbitrary directions.
The equations of motion for the modified action are: Now we can always choose our co-ordinates in the SO(5) space such that v M has only one non-vanishing component, such that v M := (0, 0, 0, 0, α/2). Then the equation of motion for the B field in (21) becomes: where a, b ∈ {0, 1, 2, 3}; whereas the e.o.m for the gauge connection is unchanged. The second of these equations in combination with (16b) tells us that: i.e., the torsion of the gauge connection is zero. Contracting both sides of (22a) with ǫ ef ab we obtain: where ⋆ is the Hodge dual operator (contraction with ǫ abcd ). Substituting the solution for B (23) into the modified action and using the fact that F a4 = 0 (22b), we find: where in the third line we have utilized the identity (16a).
Finally we have: The last two terms give us the Palatini action 1 for general relativity with a cosmological constant, while the first term is a topological term whose variation vanishes due to the Bianchi identity.

PHYSICAL INTERPRETATION
It is straightforward to see the correspondence between the operators for charge, rotations and translations (acting on the electron wavefunction which) form the components of the SO(5) connection (4) and the operators defined in the spacetime connection given in (13). First let us write down the form of the 5 × 5 matrix generators of the Lie algebras of so(4, 1), iso(3, 1) and so (3,2) in the following suggestive form [21, p. 10]: (26) where J i are the generators of rotations, B i generate boosts and P a = (P 0 , P i ) generate translations. The value of the factor ǫ determines the type of the algebra. If ǫ is −1 , 0 or 1, the Lie-algebra the above matrix describes is so(4, 1), iso(3, 1) or so(3, 2) respectively. Table I illustrates this correspondence.

DISCUSSION: PHASES OF SPACETIME
The notion of obtaining Einstein gravity from the symmetry breaking of a gauge theory is not a new idea, having first been suggested more than two decades ago by MacDowell and Mansouri [9] in 1977 and by Stelle and West in 1980 [16]. More recent work on this topics are the papers by Freidel and Starodubstev [8] Randono [13,14], and Westman, Zlosnik and collaborators [10,20]. The notion that geometry should have various phases is also suggested by the numerical work in the field of Causal 1 The connection formalism and the first order Palatini action for General Relativity are reviewed in a forthcoming review article on LQG [19]. Dynamical Triangulations [1]. The connection between the spin-networks used in Loop Quantum Gravity and the Ising model has recently been discussed in [7]. What is new in the present work, to the best of our knowledge, is that it is the first to connect the symmetry breaking on the gravitational side with a well-established model on the condensed matter side. In any situation where a symmetry is spontaneously broken, it is crucial to not only be able to identify the underlying microscopic dynamics which causes the symmetry to break and also to be able to identify and classify the various phases that result from this process. Here we are able to take the first tentative steps towards achieving both these goals.
On the condensed matter side, it is understood that the SO(5) formalism for High-T C superconductivity and anti-ferromagnetism is only an approximation (or effective field theory) [2,3] that arises in the long-wavelength low-energy limit of the physics of some underlying fundamental dynamics. In [5,12] several examples of microscopic Hamiltonians are given who long-wavelength theory explicitly exhibit the SO(5) symmetry. A recurring example of an exact microscopic Hamiltonian in the case of High-T C SC/AF is that of the tight-binding Hubbard model. In [17,18] we pointed out that the behavior of black hole entropy in LQG suggests a connection between the physics of a black hole horizon and that of the quantum hall effect. There we suggested the Hubbard model as a candidate microscopic Hamiltonian for describing the physics of a black hole horizon. The present work provides support for this proposal. This addresses the question of the microscopic origin of the effective SO(5) theory in the gravitational context.
Knowledge of the detailed phase diagram of High-T C SC/AF also allows us to make concrete suggestions regarding the possible phases which spacetime geometry can manifest. The important aspect is the ability to identify the various phases -superconducting, antiferromagnetic, ferromagnetic, spin-bag, etc. -with the various solutions of Einstein's equations. To do so we can refer to the dictionary given in table I.
In the AF phase the order parameter is given by the Neel vector S = (S x , S y , S z ). The dictionary table I tells us that on the gravitational side this corresponds to the components of the so(5) connection which correspond to spatial rotations (−ω 3 2 , ω 3 1 , −ω 2 1 ) in the symmetry broken theory. Thus in order to associate a geometric configuration with an AF phase, we should look for a solution of Einstein's equations where rotations in the spatial planes are determined. An example is spacetime of a Kerr-deSitter 2 black hole, which describes a rotating black hole. Observers outside a Kerr-deSitter black hole will experience a spacetime with broken rotational invariance -with the rotation axis of the black hole defining a preferred direction in space -and far from the horizon, the generators of spatial rotations (−ω 3 2 , ω 3 1 , −ω 2 1 ) will reach a constant, non-zero value. Thus, the geometry experienced by observers far from the horizon of a rotating black hole can be identified with the anti-ferromagnetic phase.
For the SC phase it is, at present, not clear to us as to what geometric configuration should be identified with with it. A guess would be that the geometry near or inside the hozion of a charged -Reissner-Nordstormblack hole can be identified with a SC phase. If we consider the case of black hole which is both rotating and charged -Kerr-Newman 3 -then it would appear that the AF phase can be identified with the bulk geometry far from the horizon and the SC phase with the bulk geometry in the interior of the black hole. Of course, this identification is, as yet, speculative and requires detailed analytical investigation before it can be taken seriously. However, this tells us the general direction one must follow for identifying phases of geometry with the phases encountered in condensed matter.
Further implications, if any, of the correspondence conjectured here for cosmological solutions of Einstein's equations and for the general program of quantum grav- 2 The theory we are considering has Λ = 0, thus one has to work with the deSitter/anti-deSitter generalization of the Kerr spacetime. 3 once again with the caveat that the black hole is embedded in a bulk deSitter spacetime For the reader's convenience let us clarify some aspects of the notation used in this paper. For the most part, spacetime indices µ, ν, . . . are suppressed. One-forms correspond to objects with one spacetime index: V µ . Two-forms are objects with two spacetime indices F µν , which are antisymmetric in those indices, i.e., F {µν} = 0, where {..} denotes symmetrization over the enclosed indices.

Quantity
The tetrad e µ I can be thought of as a spacetime field (labeled by µ) which, at each point of our spacetime manifold, gives us a vector (labeled by I) -or more precisely an element of the five-dimensional Clifford algebra, which rotates under the respective gauge transformations.
When we say a one-form is "lie-algebra valued" or "takes values in the Lie-algebra of ...", we are talking about spacetime fields which have one or more internal gauge degrees of freedom. Generators T IJ of the liealgebra of SO(5) are labeled pairs of indices IJ, where I, J, K, . . . ∈ {0, 1, 2, 3, 4}. For e.g., the gauge connection A µ ≡ A IJ µ T IJ and the field strength F µ ≡ F IJ µ . The "wedge" product between one-forms and two-forms is defined as the completely antisymmetric outer product between two given objects. For instance, given a oneform e µ and a two-form F µν , the wedge product between the two would give a three-index object completely antisymmetric in all the indices: where on the right we have written the wedge product with spacetime indices suppressed and on the right with spacetime indices shown explicitly. The action for BF theory written with spacetime indices shown explicitly is: The operators d denotes the antisymmetric exterior derivative. For instance the action of d on a one-form e µ is given by: D denotes the covariant derivative w.r.t. the gauge field. Its action on fields which are scalars under the gauge group coincides with that of the partial derivative: D µ e ν ≡ ∂ µ e ν . For fields valued in the Clifford algebra its action is given by: D µ e ν I ≡ ∂ µ e ν I + A µ I J e µ J . D denotes the exterior (completely anti-symmetrized) covariant derivative: De ≡ 1 2 (D µ e ν I − D ν e µ I ).

The deSitter Hamiltonian as a Spin System
As an aside we would like to point the formal similarity between the Hamiltonian for GR with positive Λ and condensed matter systems, in particular the spin-ice model. Recently it was proposed by Dittrich and Hynbida that intertwiners, invariant tensors that live on the vertices of spin-networks, possess a structure similar to that of the Ising model. This observation supports our own ideas regading the relationship between spin-networks and Isinglike models in what follows. Following Smolin [15], we have for the Hamiltonian constraint for GR with Λ > 0: where our degrees of freedom are spins S i placed at the vertices of a hexagonal lattice ⋆ L. The dual L of this lattice is a triangular lattice. The spins can also be seen as being located on the faces of L. This makes sense from the quantum geometry framework where the area operator of a surface is the Casimir J 2 of a system of spins j i labeling each point on the surface p i which is pierced by a loop carrying a flux of the gravitational connection. The deSitter Hamiltonian H deS can then be interpreted as being the sum of the terms corresponding to the kinetic energy and the nearest neighbor, two and three body interaction energies of spins E ai placed at the vertices of the hexagonal lattice ( ⋆ L). The two and three body interaction energies are: (28) where i, j, k label vertices in ⋆ L and a, b, c label the possible states of each spin variable. From the form of the above equations it is clear that the "spins" in this case have to live in a three-dimensional hilbert space H 3 . The two-body interaction term contains the kinetic energy term which is given by: The remaining components of the two-body term can be interpreted as exchange energies: where the sign in the last term determines the statistics the particles E i a obey under exchange. When restricted to a 2D space, in addition to fermionic and bosonic statistics we can have anyonic statistics, i.e. exchanging two identical objects can lead to a phase change of e ıθ . The exchange term should then be written as: where the anyon phase factor is included. This completes our discussion of the formal description of the deSitter Hamiltonian as a spin-system. Much further analytical and numerical work is required in order to provide a more concrete basis for this correspondence.