A model of a 2d non-Fermi liquid with SO(5) symmetry, AF order, and a d-wave SC gap

# A model of a 2d non-Fermi liquid with So(5) symmetry, AF order, and a d-wave SC gap

## Abstract

Demanding a consistent quantum field theory description of spin particles near a circular Fermi surface in leads to a unique fermionic theory with relevant quartic interactions which has an emergent Lorentz symmetry and automatically has an internal symmetry. The interacting theory has a low-energy interacting fixed point and is thus a non-Landau/Fermi liquid. Anti-ferromagnetic (AF) and superconducting (SC) order parameters are bilinears in the fields and form the -dimensional vector representation of . An AF phase occurs at low doping which terminates in a first order transition. We incorporate momentum dependent scattering of Cooper pairs near the Fermi surface to 1-loop and derive a new kind of SC gap equation beyond mean field with a d-wave gap solution. Taking into account the renormalization group (RG) scaling properties near the low energy fixed point, we calculate the complete phase diagram as a function of doping, which shows some universal geometric features. The d-wave SC dome terminates on the over-doped side at the fixed point of the RG, which is a quantum critical point. Optimal doping is estimated to occur just below . The critical temperature for SC at optimal doping is set mainly by the universal nodal Fermi velocity and lattice spacing, and is estimated to average around for LSCO. The pseudogap energy scale is identified with the RG scale of the coupling.

## I Introduction

In the renormalization group (RG) framework, Landau’s theory of Fermi liquids is characterized by the irrelevance of the interactions of particles near the Fermi surface, in other words the low energy fixed point is simply a free theory of fermions. The underlying reasons for the wide success of Landau/Fermi liquid theory are well-understood(1); (2); (3); (4); (5), and consequently the known models of non-Landau/Fermi liquids are relatively rare. (Henceforth referred to simply as non-Fermi liquids.) An important exception is the Luttinger liquid and other related models consisting of quartic interactions of Dirac fields in spatial dimension. Here the non-Fermi liquid behavior can be attributed to the fact that in , quartic interactions of Dirac fields are marginal operators in the RG sense. In higher dimensions quartic interactions of Dirac fermions are irrelevant and this is one of the reasons why candidate non-Fermi liquid models were not found in the previous works. Whereas more exotic non-Fermi liquid models have been proposed which typically involve gauge fields, the lack of non-Fermi liquid models in appears paradoxical when one considers even the simplest models of itinerant electrons with quartic interactions, such as the Hubbard or t-J model, which are believed to be at strong coupling. Since such models have been proposed as good starting points for thinking about high superconductivity in the cuprates(6); (7); (9), it is certainly worthwhile to continue to try and construct relatively non-exotic models of continuum fermions with quartic interactions that have some resemblance to the Hubbard model and have non-Fermi liquid behavior in the normal state.

Though the search for a novel kind of non-Fermi liquid in provided one of the main initial motivations for the formulation of the model that will be presented and analyzed in this work, the model turns out to have many unexpected bonus features, almost all of which are intrinsic to . We list the most prominent:

The 4-fermion interaction is unique for spin electrons and automatically has symmetry. In the interactions are relevant and the model has a low energy interacting fixed point with non-classical exponents which can be computed perturbatively.

The model generalizes to flavors, where corresponds to spin electrons, and has symmetry. Since this provides a underlying framework based on a microscopic theory for exploring the ideas of Zhang based on (10); (11). In particular one can derive the effective Ginzburg-Landau theory.

Because of the symmetry the model naturally has both anti-ferromagnetic (AF) and superconducting (SC) order parameters that form the 5-dimensional vector representation of . For repulsive interactions the model has AF order and no SC order in mean field approximation.

When one incorporates momentum dependent scattering to 1-loop to go beyond mean field, an attractive d-wave channel opens up and the momentum dependence of the gap can be calculated. This d-wave SC phase terminates on the over-doped side at the RG fixed point, which is a quantum critical point. Due to mathematical properties of the d-wave gap equation, it also terminates on the under-doped side yielding a “dome”. Due to the properties of the RG flow, this attractive d-wave instability exists for arbitrarily strong repulsive interactions at short distances.

Although the model may be at arbitrarily strong coupling at short distances, the low energy fixed point is at a relatively small coupling , and this renders the model perturbatively calculable. We are thus able to calculate the main features of the complete phase diagram as a function of a doping variable, including the phase boundary of the d-wave superconducting dome and estimate the optimal doping fraction, which is near . This phase diagram depends on a single parameter which encodes the ratio of the strength of the coupling at short versus long distances. In the figure below, we summarize the results of our calculations for , which corresponds to infinite coupling at short distances. The overall scale of temperature is set by the universal nodal Fermi velocity and the lattice spacing.

Many of the above properties were highlighted on a list of the most important features of high superconductivity compiled early on in the subject(6); (9). Because of the importance of the planes, high is believed to be essentially a phenomenon. This, along with the detailed properties of the solution of our model, in particular the phase diagram, led us to propose it as a model of high superconductivity(12). If our theory turns out to be the correct description, it reveals that the phenomenon of high superconductivity is remarkably universal, with a single energy scale, and its main features follow from the existence of the low energy fixed point in . It is truly a beautiful phenomenon that has managed to realize some subtle theoretical loopholes in the usual requirements of unitarity, the spin-statistics theorem, and the Mermin-Wagner theorem, which are only possible in . Our theory represents a significant departure from the models considered thus far in connection with high , which are reviewed in (13); (14); (16); (15), along with reviews of experimental results. On the other hand, we believe it represents a particular scaling limit of the Hubbard model at and just below half-filling, and is in this sense conservative in comparison with other more exotic ideas, and is thus in line with the early ideas concerning the rôle of AF order and the Heisenberg and Hubbard models(6); (7); (8). However our model isn’t simply a direct scaling limit of the Hubbard model with no attention paid to the Fermi surface, since the latter only has an symmetry, whereas our theory has the symmetry. In our theory the “fermion sign” problem is solved by doing analytic, perturbative calculations in a fermionic theory from the beginning, and relatively simple 1-loop calculations already reveal the main features.

Irregardless of whether our model has been exactly realized in the laboratory, it can serve as a useful tool for exploring many of the paradigms in the area of strongly correlated electrons and also for developing new methods. For instance, we develop new gap equations that take into account higher order scattering of Cooper pairs near the Fermi surface. Our analysis shows clearly how in one can obtain a momentum dependent gap with a d-wave structure from a rotationally invariant continuum field theory, i.e. without an explicit lattice that breaks the rotational symmetry. This is interesting especially since the precise origin of the d-wave symmetry of the SC gap has been unclear. We also show how to introduce doping in terms of the coupling and RG scale, and a small non-zero temperature as a relativistic mass coupling.

For the remainder of this introduction we outline the organization of the paper and summarize our main results. In section II we motivate the model by showing how it can approximately describe particles and holes near a circular Fermi surface. The manner in which we expand around the Fermi surface is in the same spirit as in (1); (2); (3); (4); (5) but differs in some important ways. For a single spin-less fermion one thereby obtains a free hamiltonian of particles and holes with a massless, i.e. relativistic dispersion relation. In section III we insist on a local quantum field description of the effective theory near the Fermi surface with a consistent quantization. Since the particles are massless, the only known candidate field theories are either Dirac or “symplectic” fermions, which differ primarily by being first versus second order in space-time derivatives respectively. For the remainder of the paper we focus on symplectic fermions since unlike the Dirac fermions, the interactions are relevant. The model was first proposed in this context by one of us(17), where the groundwork was done on the low energy non-Fermi liquid fixed point and in part the AF properties; at the time the SC properties were unknown. As explained in the present paper, the central idea of this previous work, that the AF order parameter is bilinear in symplectic fermion fields and that the low energy RG fixed point describes a quantum critical point, appears to be correct; however as we will see, the quantum critical point terminates the SC rather than AF phase. Quantum critical points in the context of high were emphasized earlier by Vojta and Sachdev(19). The issue of the unitarity of our theory was mostly resolved in (18) by noting that the hamiltonian is pseudo-hermitian and this is sufficient for a unitary time evolution. In this paper, the expansion around the Fermi surface provides a new view on the pseudo-hermiticity and it is explained how it is related to the kinematics of particles versus holes. The critical exponents were computed to 2-loops in (18), which corrected some errors in(17). In this paper we analyze many more properties, in particular the AF and d-wave SC ordering properties for the first time.

Since the consistency of the quantization of a fermionic theory with a lagrangian that is second order in time derivatives is at the heart of the unitarity issue, in section IV we work out in detail the dimensional quantum mechanical case where all the subtle consistency issues are present. In this section we also construct the conserved charges for the symmetry. In section V the field theory version in spatial dimensions is defined and spin and charge are identified for the case of . In this section we also define the order parameters for AF and SC order.

In section VI we sketch an argument that the resistivity is linear in temperature in the limit of no interactions. In the next section we consider small thermal perturbations near . By comparing with the specific heat of a degenerate electron gas, we argue that a small non-zero temperature can be incorporated as a coupling in the lagrangian corresponding to relativistic mass and we estimate the constant .

In section VIII the mean field analysis is carried out with potential competition between AF and SC order. As we explain, these two phases actually do not compete in our model in this approximation. As a check of the formalism, we reproduce some of the basic features of the BCS theory for an s-wave gap in the case of an attractive coupling in section IX. For repulsive interactions we find only AF order is possible in mean field approximation and this is studied in section X. There we first argue that this phase must be anti-ferromagnetic by comparing our model with the low energy non-linear sigma model description of the Heisenberg anti-ferromagnet. This gives another motivation for our model at half-filling away from the circular Fermi surface, and explains how the same model can interpolate between a SC phase and an AF one. We argue that the AF phase terminates in a first-order phase transition. The AF gap is then the solution of a transcendental equation that is analyzed in various limits.

In section XI, orbital symmetries of momentum dependent gaps are studied in a model independent way and we explain how a d-wave gap can arise. This analysis is based on a gap equation which is derived in Appendix A. In section XII we compute the 1-loop contributions to the scattering of Cooper pairs and show that at low energies the d-wave channel is attractive if the number of components . Since the theory is free for , this means that only the physically relevant case has d-wave SC. This also means that the d-wave pairing cannot be studied with large methods.

Section XIII is devoted to describing our RG prescription which is specific to . This is necessary for a proper understanding of the phase diagram. In section XIV we present global features of the phase diagram, which is characterized by some universal geometric relations, and bears a striking resemblance to the cuprates. The SC phase terminates at a second order phase transition precisely at the low energy RG fixed point, and is thus a quantum critical point. We also estimate optimal hole doping. In section XV we present detailed numerical solutions to the AF and SC d-wave gap equations at non-zero temperature. For reasonable values of the lattice spacing and universal nodal Fermi velocity, we estimate on average for SC in LSCO. In section XVI we describe the interpretation of the pseudogap within our model.

Although we do not give a complete and rigorous derivation of our model from lattice fermion models, in order to motivate and point out relations, we have collected some known results about the latter in Appendix B.

## Ii Expansion around the Fermi surface

### ii.1 Kinematics

Let us first ignore spin and consider a single species of fermion described by the free hamiltonian in momentum space

 H=∫(ddk)(ε(k)−μ)c†kck (1)

where is the chemical potential, we have defined , and

 {c†k,ck′}=(2π)dδ(d)(k−k′) (2)

At finite density and zero temperature, all states with are filled, where the Fermi energy depends on the density, and at zero temperature . The Fermi surface is the manifold of points satisfying .

We wish to consider a band of energies near as shown in Figure 2 for . Let be any wave-vector in such a band, and let denote a ray from the origin to infinity along the direction of . We further assume that the Fermi surface is sufficiently smooth, such that intersects only once. The latter implies that can be uniquely expressed as

 k=kF(k)+p(k) (3)

where is the vector from the origin to the intersection of with . Whereas the two vectors and are by construction parallel, the vector is either parallel or anti-parallel to . Let us fix to be a small vector parallel to , i.e. pointing radially outward, as shown in Figure 2. Since now is uniquely determined by , we may write . Furthermore, since the particles below the Fermi surface correspond to , the energies near the Fermi surface are approximately given by

 ε(k)=εF±p⋅vF(k) (4)

where corresponds to above or below the Fermi surface, and

 vF(k)=→∇ε(k)|kF (5)

is the Fermi velocity normal to .

Let us now assume that the Fermi surface is rotationally invariant, i.e. depends only on . In the Fermi surface is thus a circle. This leads to the simplification that and are independent of . For any in the band,

 k=(kF±p)ˆp (6)

where , and . Furthermore, since is normal to , the energies are linear in :

 ε(k)=εF±vF|p| (7)

For non-relativistic particles with , .

Since the map from to is one-to-one, we can define the following operators:

 ap=ckF+p,     bp=c†kF−p (8)

where it is implicit that depends on . The above is a canonical transformation, since

 {a†p,ap′}={b†p,bp′}=(2πd)δ(d)(p−p′) (9)

After normal ordering, the hamiltonian for the particles in the band is defined to be

 H=∫|p|<Λc(ddp)[(vF|p|−ˆμ)a†pap+(vF|p|+ˆμ)b†pbp] (10)

where is zero at zero temperature. For the remainder of this paper we mostly set . Since we are only interested in a band of energies near the Fermi surface, we have introduced a cut-off . The vacuum is defined to satisfy . This corresponds to , which correctly implies that all states with are filled. The and thus correspond to particles and holes respectively.

There is an approximation made in obtaining the above hamiltonian having to do with the density of states, and this is crucial to understanding how our expansion differs from previous works. In the rotationally invariant case, for particles above the Fermi surface:

 ∫ddk=∫dΩ∫dp(kF+p)d−1 (11)

where are angular integrals. Note that due to eq. (6), the angular integrals for and are identical. At least two approximations to the above are meaningful. The first favors low energies where one approximates . This is the approximation that is commonly made in the literature(1); (2). One the other hand, expanding out the , at high energies the leading term is , and is the most sensitive to the short distance physics and spatial dimensionality. A possible short-coming of the first, low-energy approximation is that the high energy physics is discarded from the beginning. It cannot be recovered by the RG flow to low energies since the latter is irreversible. In the physical problem we are considering, the short-distance physics of the strong Coulomb repulsion is known to be important for understanding the AF phase, so it makes sense to adopt an approximation that favors high energies from the beginning and to then incorporate their effects by an RG flow to lower energies. We thus keep the most important term at short distances and set , i.e. . This is in line with the usual RG idea that it is important to fix the high-energy physics as accurately as possible, and then flow down to lower energies. Finally our choice is necessary for the effective field theory description in the next section. However one should not conclude that every theory with a circular Fermi surface can be described by a relativistic field theory. One signature of a relativistic description is a density of states that is linear in energy: .

The above expansion around the Fermi surface is thus not identical to the expansion in (1); (2); (3); (4); (5), where the integration over is taken to be normal to times the angular integrations, and it is assumed that . This leads to the choice , and the constant is absorbed into the definitions of the operators. Thus in the approach followed in (1); (2); (3); (4); (5), although the angular integrals obviously depend on , the scaling analysis of the dependence leads to marginal 4-fermion interactions for any , and the resulting theory is effectively -dimensional, or a collection of such theories, one for each angular direction. Notably, it was not possible to obtain a non-Fermi liquid based on 4-fermion interactions in this approach(2).

In contrast, in the approach developed in this paper there is a strong dependence on , as in other critical phenomena, and this will turn out to be very important. In particular, it leads to a non-Fermi liquid in . There are other important justifications for this choice. In particular, near half-filling where the interacting lattice model can be mapped to the Heisenberg anti-ferromagnet, there is known to be a relativistic description of the low energy, long wavelength limit in terms of the non-linear sigma model. It will be shown in section X that our choice of field theory near the circular Fermi surface can be extrapolated to half-filling in that an independent derivation of it can be provided exactly at half-filling.

For general processes, physical momentum conservation of the ’s is not equivalent to conservation. However, consider a zero-momentum process proportional to . If the ’s are all exactly on the Fermi surface, then this implies . For even numbers of particles, since all vectors on the Fermi surface have the same length, this is satisfied by pairs of particles with opposite . Allowing now small deviations from the Fermi surface, one has

 δ(∑iki)=δ(∑particlespi−∑holespi) (12)

Because of the particle/hole transformation for the ’s in eq. (8), this is equivalent to overall conservation. Note that by construction, it is not possible for the momentum of a particle and a hole to add up to zero, so in the above -function, holes are paired with other holes, and particles with other particles. Therefore, spatial translational invariance of our local field theory will ensure physical momentum conservation of the ’s for this class of processes.

The important allowed processes are shown in Figure 3. Examples of un-allowed process are shown in Figure 4. The distinction between allowed and un-allowed processes can be made explicit by introducing an operator that distinguishes particles and holes:

 CapC=ap,       CbpC=−bp (13)

where is a unitary operator satisfying so that . An eigenstate with pairs of particles and/or pairs of holes is then required to have . We will return to this in connection with the pseudo-hermiticity of symplectic fermions in the sequel.

### ii.2 Lattice fermions

In the sequel, our field theory model will be related to lattice models of intinerant electrons such as the Hubbard model, although we do not claim a precise equivalence. The known square lattice model results we will need to make the comparison are all contained in Appendix B. In this section we consider only the free, hopping term. In momentum space the 1-particle energy is (190)

 εk=−2t(coskxa+coskya) (14)

where is the lattice spacing. Equal energy contours in the first Brillouin zone are shown in Figure 5. The Fermi surface at half-filling is the square diamond with corners on the axes. Note that one does not have to be very far below half-filling for the contours to be approximately circular. The free local field theory model in the next section can thus be viewed as an approximate effective theory for free particles on the lattice below half-filling.

An important point is that our model is not simply a direct continuum limit of the lattice model since, without additional care, the latter does not take into account the Fermi surface at finite density. For instance, whereas the Hubbard model has at most an symmetry(36), our continuum model has the larger symmetry. Furthermore, as will be explained in section X, the success of our model can be attributed to the fact that an alternative justification of it can be given right at half-filling so that it can actually interpolate between half-filling and below.

## Iii Requirements on the free local field theory

The main requirements we impose for a local field theory description of the last section are:

(i) The theory has a lagrangian description with a consistent quantization.

(ii) In momentum space the hamiltonian reduces to equation (10) for particles and holes of energy . The latter is a relativistic dispersion relation for massless particles.

In order to motivate our arguments, let us start from non-relativistic particles with . The second-quantized description consists of a single field with lagrangian

 L=∫ddx(iΨ†∂tΨ−12m∗→∇Ψ†⋅→∇Ψ) (15)

The field has the momentum space expansion

 Ψ(x,t)=∫(ddk)cke−iεk+ik⋅x (16)

Expanding around the Fermi surface one finds

 Ψ(x,t)=e−iεFt∫(ddp)˜Ψp(x,t)eikF(p)⋅x (17)

where

 ˜Ψp(x,t)=ape−ivF|p|t+ip⋅x+b†peivF|p|t−ip⋅x (18)

We wish to find an effective theory for , which satisfies the relativistic wave equation:

 (∂2t−v2F→∇2)˜Ψp(x,t)=0 (19)

Thus, due to the kinematics of the expansion around the Fermi surface we identify an emergent Lorentz symmetry. In this Lorentz symmetry is . The case of is special in that the Fermi surface consists of only two disconnected points , and the decomposition (17) naturally separates into left and right movers. For higher dimensions there is no such separation since all points on the Fermi surface are related by spatial rotations and continuously connected. There are many additional reasons why is the exceptional case, and these will be pointed out where appropriate in the sequel since some of the literature attempts to draw analogies between and .

There are only two known candidate field theories which differ in whether the lagrangian is first or second order in derivatives. First consider the case of first-order. One then needs to factor the operator into two first-order multiples. The only way to accomplish this is to promote to a multi-component field and introduce a matrix representation , , of the Clifford algebra:

 {γμ,γν}=2ημν (20)

where . One then has

 ∑μ,νγμγν∂μ∂ν=∑μ∂μ∂μ=∂2t−→∇2 (21)

(We have adopted the relativistic notation and , and henceforth, repeated indices are implicitly summed over.) The lagrangian is then the standard first-order Dirac lagrangian:

 L=∫ddx i¯¯¯¯ψγμ∂μψ (22)

where . The smallest representation of the Clifford algebra is two dimensional: , and where are the standard Pauli matrices. In this simplest case, although there is a doubling of components, they are constrained by the Dirac equation of motion and the spectrum still consists of one species of particles and holes with hamiltonian eq. (10).

Interactions are relevant to the low energy physics if the operator characterizing them has scaling dimension less than . The classical scaling dimension of the Dirac field is in spatial dimensions, thus a quartic interaction has dimension and is thus only relevant for . It is not even perturbatively renormalizable for . Thus, the Dirac theory should lead to ordinary Landau-Fermi liquid behavior in . It is noteworthy that once again is special, and this helps to explain how for instance the Hubbard model can be mapped onto interacting Dirac fermions, and the low energy fixed point found using special bosonization techniques and spin-charge separation(35). Simply based on the fact that the interactions in the Hubbard model are strong in , one can rule out a description in terms of Dirac fields with quartic interactions since the latter are irrelevant. Furthermore, it is already understood that one normally needs additional special properties in order to obtain the first-order Dirac theory. For example, it is known to arise when one expands around special nodes (Dirac points) on the Fermi surface for a hexagonal lattice(21), as in graphene, and the multiple components of the Dirac field correspond to different sub-lattices. There is no reason to expect this here for a square lattice, as in the cuprates.

The other candidate field theory is second-order in derivatives, with kinetic term

 S=∫dtddx ∂μχ−∂μχ+ (23)

For fermionic (Grassman) fields, this is a very unconventional theory, since it potentially has problems with the spin-statistics theorem and unitarity; in high energy elementary particle theory it usually corresponds to ghost fields. These issues will be discussed in detail and resolved completely in the next two sections. Here, let us give the main arguments for why this should be the right starting point:

(i) As shown in the next two sections, the free theory in momentum space corresponds precisely to the hamiltonian (10) for particles and holes near a circular Fermi surface. This is of course a perfectly hermitian theory with no negative norm states.

(ii) The fundamental field has scaling dimension and thus quartic interactions have dimension which is actually relevant for . Thus it can have non-Fermi liquid behavior.

(iii) Although we are led to consider this model for the nearly circular Fermi surface below half-filling, a simple argument leads to the same model at half-filling. It is well-known that a low-energy description of excitations above the staggered AF state is described by the non-linear sigma model for a field constrained to have constant length with the action

 S=∫dtddx ∂μ→ϕ⋅∂μ→ϕ (24)

(See Appendix B.) In our model the anti-ferromagnetic order parameter is bilinear in the fields . The non-linear constraint on the fields follow from imposing a similar constraint on the fields: . This was pointed out in(17). Inserting this into the above action one finds that one obtains the second order action (23) for the fields up to irrelevant operators (eq. (104) below). Thus the symplectic fermion model with interactions can in principle describe AF order, and in the sequel we will show that this is indeed the case. This is explained in greater detail in section X.

Since only serves to convert dimensions of time and space, it plays the rôle of the speed of light; we can set it to unity since it can always be restored by dimensional analysis. The important point here is that is fixed and universal in our model, in particular it does not depend on the coupling. In the sequel the doping will be related to the coupling, so does not depend on doping either. For high materials, we believe the “speed of light” has actually already been measured(20). The Fermi surface at half-filling is closest to the nearly circular surface below it in the nodal to direction, so our should correspond to this nodal Fermi velocity. Remarkably, the latter was measured to be universal at low energies, i.e. independent of doping in (20). Whereas this universality of has not been explained theoretically up to now, it is a necessary aspect of our theory. Taking the slope of the curves in(20) we estimate for LSCO. As we will show in section XV, this gives very reasonable estimates of . Another signature that the system may be in a relativistic regime is a density of states that is linear in energy, as explained in the last section.

## Iv Symplectic Fermion Quantum Mechanics

As stated above, symplectic fermions are primarily characterized by a lagrangian that is second-order in space and time derivatives. Since this is unfamiliar to most readers and there are some delicate issues in the quantization of such theories, let us first start with the simplest case of quantum mechanics.

### iv.1 Canonical quantization

In order to draw comparisons, let us first consider a first order lagrangian as in the Dirac theory:

 L=N∑α=1(ic†α∂tcα−ωc†αcα) (25)

It is well-understood that this lagrangian has two consistent quantizations, i.e. one can impose either canonical commutation relations or canonical anti-commutation relations . In both cases the hamiltonian is . It’s clear that both options are possible since eq. (15) is a proper second quantized description of either bosons or fermions. For future reference we note that the model has a manifest symmetry.

The second order bosonic version of the above is just the ordinary harmonic oscillator with . Since the first-order lagrangian can be consistently quantized as a fermion or boson, one expects that the second-order case should also be quantizable as a fermion, and as we now describe, this is indeed the case. In order to have a fermionic version, we need at least 2 degrees of freedom since fermionic variables square to zero. Let us therefore consider the lagrangian

 L=˙χ−˙χ+−ω2χ−χ+ (26)

where are Grassman variables:

 {χi,χj}=0, (27)

which implies , and we have defined . The canonical momenta are and , which leads to the canonical anti-commutation relations

 {χ−,˙χ+}=−{χ+,˙χ−}=i (28)

The canonical hamiltonian is simply

 H=˙χ−˙χ++ω2χ−χ+ (29)

The equation of motion is . Because this is second-order, the mode expansion involves both positive and negative frequencies:

 χ−(t) = 1√2ω(a†e−iωt+beiωt) χ+(t) = 1√2ω(−b†e−iωt+aeiωt) (30)

The canonical anti-commutation relations (28) then require

 {a,a†}={b,b†}=1 (31)

with all other anti-commutators equal to zero. The hamiltonian is

 H=ω(a†a+b†b−1) (32)

### iv.2 Pseudo-hermiticity

The only subtle aspect of the above quantization is the extra minus sign in the expansion of in eq. (30), which was necessary in order to have the canonical relations (31). This minus sign implies that is not the hermitian conjugate of . One can understand this feature more clearly, and also keep track of it, with the operator that distinguishes particles and holes in eq. (13):

 χ+=C(χ−)†C. (33)

In terms of the original variables, the hamiltonian is pseudo-hermitian, . However after using the equations of motion and expressing it in terms of ’s, since it is quadratic in ’s, the hamiltonian (32) is actually hermitian. This issue will be revisited when interactions are introduced in the next section.

### iv.3 Symmetries

We now study the symmetries of the -copy theory. Introduce variables , , and define the lagrangian

 L=12∑i,j,αϵij(˙χiα˙χjα−ω2χiαχjα) (34)

where is the anti-symmetric matrix . The hamiltonian is

 H=12∑i,j,αϵij(˙χiα˙χjα+ω2χiαχjα) (35)

Arrange into a -component vector and consider the transformation , where is a dimensional matrix. Then the lagrangian is invariant if where , and is the transpose. This implies that is an element of the group of dimension . Interestingly, of the classical Lie groups, is the only one that doesn’t play any known rôle in elementary particle physics(25). Note that the bosonic version of the theory with real components has the symmetry with dimension , thus the fermionic version always has a larger symmetry. The N-component symplectic fermion also has a larger symmetry than the N-component first order fermionic action which has a symmetry or an symmetry if the complex fermions are rewritten in terms of real fields.

The conserved charges that generate the dimensional Lie algebra of are easily constructed. Define

 Q0αβ = −i(χ−α˙χ+β+χ+β˙χ−α) Q−αβ = −i(χ−α˙χ−β+χ−β˙χ−α) (36) Q+αβ = i(χ+α˙χ+β+χ+β˙χ+α)

One can easily verify that the hamiltonian commutes with all the charges using

 {χ−α,˙χ+β} = −{χ+α,˙χ−β}=iδαβ {χiα,χjβ} = {˙χiα,˙χjβ}=0 (37)

In “momentum” space, the charges are

 Q0αβ = a†αaβ−b†βbα Q−αβ = a†αbβ+a†βbα (38) Q+αβ = b†αaβ+b†βaα

and they satisfy the hermiticity properties

 (Q0αβ)†=Q0βα,     (Q−αβ)†=Q+αβ (39)

## V Field theory version with spin and interactions

### v.1 Lagrangian and Hamiltonian

The field theory version in spatial dimensions follows straightforwardly from the above case with the addition of spatial or momentum integrals. Introducing fields , the action is

 S=12∫dtddx∑i,j,αϵij(∂μχiα∂μχjα−m2χiαχjα) (40)

and the equations of motion are

 (∂μ∂μ+m2)χ=0. (41)

The momentum space expansion is

 χ−(x,t) = ∫(ddp)√2ωp(a†pe−ip⋅x+bpeip⋅x) χ+(x,t) = ∫(ddp)√2ωp(−b†pe−ip⋅x+apeip⋅x) (42)

where and . (We do not display the indices since they just correspond to identical copies.) The canonical anti-commutation relations are

 {χ−(x,t),˙χ+(x′,t)}=−{χ+(x,t),˙χ−(x′,t)}=iδ(d)(x−x′) (43)

which in momentum space leads to

 {ap,a†p′}={bp,b†p′}=(2π)dδ(d)(p−p′) (44)

(All other anti-commutators are zero.)

The hamiltonian is

 H=∫(ddp)N∑α=1ωp(a†p,αap,α+b†p,αbp,α) (45)

In the limit , we obtain the effective hamiltonian near the Fermi surface in eq. (10), as desired. The mass in this section is an infra-red regulator and is unrelated to the non-relativistic mass above. In section VII we will show that it can be viewed as proportional to the temperature.

The field theory has the same symmetry as the quantum mechanical version. The expressions for the conserved charges are identical to the eqs. (36,38) with additional integrals over or .

There is a unique 4-fermion interaction that preserves the symmetry:

 Sint=−π2g∫dtddx(χ−ϵNχ+)2=−4π2g∫dtddx (∑αχ−αχ+α)2 (46)

(We have included an overall so that the RG equations below for have no ’s; our convention is the same as in (18).) For , even without imposing the symmetry, there is a unique interaction due to fermionic statistics since there are only independent fields, which implies that higher order terms beyond quartic interactions are zero by Fermi statistics. This interaction is automatically invariant. Positive corresponds to repulsive interactions. Since the field has classical scaling dimension , the interaction is a dimension operator which is relevant for as previously mentioned.

### v.2 Pseudo-hermiticity

The symplectic fermion action (40) has a Lorentz invariance if is understood to be a Lorentz scalar. Since is fermionic, to a particle physicist this model would appear to violate the spin-statistics theorem. There are two separate aspects of this issue. First of all, in the condensed-matter context, rotational spin is an internal flavor symmetry (spin fermions corresponds to ) which is not viewed as embedded in the Lorentz group. This implies that spin particles are not forced to be described by the first-order Dirac theory. There is no violation of the spin-statistics connection in our theory since we are quantizing spin particles with fermionic fields. The potential problem rather has to do with unitarity, as explained in the quantum mechanical case studied in the last section, and is manifested in the pseudo-hermiticity property (33). This explains how the proof of the spin-statistics connection is circumvented: the proof assumes that the hamiltonian is built out of fields and their hermitian conjugates and thus doesn’t allow for different fields being related by pseudo-hermitian conjugation(5). Furthermore, in the end the free hamiltonian in momentum space is a perfectly hermitian theory with no negative norm states.

Whereas the free theory is hermitian in momentum space, for the interacting theory, it follows from (33) that the hamiltonian is pseudo-hermitian:

 H†=CHC (47)

where the unitary operator and . This sort of generalization of hermiticity was understood to give a consistent quantum mechanics long ago by Pauli(22), and more recently in connection with symmetric quantum mechanics(24); (23). Let us summarize the main properties enjoyed by pseudo-hermitian hamiltonians:

(i) Define a C-hermitian conjugation as follows:

 A†c=CA†C (48)

Then the usual rules are satisfied:

 (AB)†c=B†cA†c,     (aA+bB)†c=a∗A†c+b∗B†c (49)

(ii) Define a -conjugate inner product:

 ⟨ψ′|ψ⟩c≡⟨ψ′|C|ψ⟩ (50)

Then time-evolution is unitary:

 ⟨ψ′(t)|ψ(t)⟩c=⟨ψ′|eiH†tCe−iHt|ψ⟩=⟨ψ′(0)|ψ(0)⟩c (51)

(iii) The eigenvalues of are real:

 (E−E∗)⟨ψE|ψE⟩c=⟨ψE|(CH−H†C)|ψE⟩=0 (52)

(iv) Diagonal matrix elements of -pseudo-hermitian operators are real. This will be important in the sequel since it guarantees the reality of vacuum expectation values of pseudo-hermitian order parameters. For our model

 H†c=H,       (χ−)†c=χ+,     (χ−χ+)†c=χ−χ+ (53)

In the present work there is a new aspect of the pseudo-hermiticity that relates to the kinematics of the expansion around the Fermi surface. As explained in section II, for the 4-particle processes near the Fermi surface, conservation of relative to the Fermi surface is equivalent to conservation of the physical momentum if particles are paired with particles and holes with holes. Since for particles versus holes, on physical grounds we should restrict to eigenstates with even numbers of holes and even numbers of particles with . Let denote an eigenstate of which is also an eigenstate of . Then . Thus for the eigenstates of interest, .

### v.3 Charge and Spin

In order to describe spin electrons, as usual we treat the spin as a flavor and thus consider the theory. The symmetry of the free theory is . A subgroup of this large (10-dimensional) symmetry can be identified with rotational spin and charge.

It was pointed out in (18) that there are potentially two ways to identify electronic spin, and the focus in that work was the subalgebra that exists for all and acts on the indices of . It turns out that the other identification is more natural in the present context since, as explained in section II, the two components corresponding to the indices are already necessary for the expansion of a single spinless fermion near the Fermi surface. There is actually an analog of the identification of spin and charge for arbitrary . Let be an element of the group in the defining dimensional representation. Using the relation

 MtϵNM=ϵN, (54)

the elements of the Lie algebra satisfy

 mtϵN=−ϵNm, (55)

and a basis is the following: , where are the Pauli matrices, is an dimensional anti-symmetric matrix, and are symmetric matrices(25). Clearly has an sub-algebra generated by where with the in the -th entry. For this corresponds to an symmetry. However since this sub-algebra does not mix the flavors, it is not the right sub-group for identification with spin as a flavor. The correct identification is the sub-algebra generated by and where now is also traceless. There is also a which commutes with the corresponding to . The is generated by the charges , which form a closed algebra:

 [Q0αβ,Q0α′β′]=i(δαβ′Q0α′β−δβα′Q0αβ′) (56)

The we identify with electric charge is generated by .

Let us now return to the physically interesting case of and label the two components as , corresponding to up and down spins. The spin symmetry is generated by

 Qz = 12(Q0↑↑−Q0↓↓)=−i2∫ddx(χ−↑˙χ+↑+χ+↑˙χ−↑−χ−↓˙χ+↓−χ+↓˙χ−↓) (57) = ∫(ddp)(12(a†p↑ap↑−a†p↓ap↓)−12(b†p↑bp↑−b†p↓bp↓)) Q+ = 1√2Q0↑↓=−i√2∫ddx(χ−↑˙χ+↓+χ+↓˙χ−↑)=1√2∫(ddp)(a†p↑ap↓−b†p↓bp↑) Q− = 1√2Q0↓↑=−i√2∫ddx(χ−↓˙χ+↑+χ+↑˙χ−↓)=1√2∫(ddp)(a†p↓ap↑−b†p↑bp↓)

satisfying the Lie algebra:

 [Qz,Q±]=±Q±,     [Q+,Q−]=Qz (58)

(As before .) The charge is generated by

 Qe=Q0↑↑+Q0↓↓ = −i∫ddx(χ−↑˙χ+↑+χ+↑˙χ−↑+χ−↓˙χ+↓+χ+↓˙χ−↓) (59) = ∫(ddp) ((a†p↑ap↑+a†p↓ap↓)−(b†p↑bp↑+b†p↓bp↓))

The conserved electric current corresponding to the above charge is

 Jeμ=−i∑α=↑,↓(χ−α∂μχ+α+χ+α∂μχ−α) (60)

The fields have electric charge . Commutations of the fields with the generators shows that form a doublet whereas is the conjugate:

 [Qz,χ±↑]=∓12χ±↑,