1 Introduction

Abstract

We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D -- model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D -- model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D -- model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.

April 8, 2010

Partially gapped fermions in 2D

[2mm] Jonas de Woul and Edwin Langmann

[2mm] Theoretical Physics, KTH

SE-10691 Stockholm, Sweden

jodw02@kth.se and langmann@kth.se

[5mm]


1 Introduction

This is the second in a series of papers that aim to develop a method to do reliable computations in a spinless 2D lattice fermion model of Hubbard type; see [1] for a concise summary. In the first paper in this series [2], an effective model for the low-energy physics of the lattice system was derived. It was shown that parts of the fermion degrees of freedom in the model can be treated exactly using bosonization. In this paper, we apply mean field methods to the remaining degrees of freedom. We present analytical and numerical results showing that our method is useful for obtaining quantitative physical information about the lattice fermions.

1.1 Motivation

The difficulty to do reliable computations in 2D lattice fermion models of Hubbard-type has remained an outstanding challenge in theoretical physics for many years. One can hope that a solution to this problem will be a key step towards a satisfactory theory of high-temperature superconductors [3]. One simple example is the so-called 2D -- model describing spinless fermions on a square lattice with local hopping and density-density interactions; see Section 2.1 for a precise definition. One of us (EL) proposed a particular partial continuum limit of this lattice system [1, 2]. This leads to an interacting model of so-called nodal fermions, which have linear band relations, coupled to so-called antinodal fermions with hyperbolic band relations. It was found that this is a natural 2D analogue of the Luttinger model, not only in that it arises as a continuum limit of the 2D -- model, but also since the nodal fermions can be bosonized and thus treated exactly. In particular, it is possible to integrate out the bosonized nodal fermions and thus obtain an effective model for the antinodal fermions; see Section 2.2.

In this paper we use mean field theory to address the question if and when the antinodal fermions in this 2D Luttinger model have a gap. These are key questions since if the antinodal fermions are gapped they do not contribute to the low-energy physics. We then obtain an effective Hamiltonian of nodal fermions that is exactly solvable. We find a significant parameter regime away from half-filling where this is indeed the case. As a motivation for our work, we also present mean fields results for the 2D -- model that show that there is an interesting region away from half filling where a direct application of mean field theory fails. We find that this regime becomes accessible to mean field theory by using the 2D Luttinger model.

We recall the key parameters of these two models. The 2D -- model is characterized by the nearest-neighbor (nn) hopping constant , the next-nn (nnn) hopping constant , the nn density-density coupling strength , and the filling factor (see Section 2.1 for more details). The 2D Luttinger model depends on two additional parameters, and , which have the following significance. To derive the model, it is assumed that there is an underlying Fermi surface that, when the system is near half-filling, is a line segment in each nodal region. These are the so called nodal Fermi surface arcs, and the parameter determines the size of these arcs. Furthermore, the parameter fixes the point about which the nodal band relations are linearized; the details are given in Section 2.2. One important question addressed in this paper is how to fix the parameter .

1.2 Mean field phase diagrams

Two-dimensional lattice fermion systems with repulsive interactions are often insulators at half-filling, but away from half-filling there are competing tendencies that lead to rich Hartree-Fock (HF) phase diagrams. A well-known example is the 2D Hubbard model which, at half-filling, has an insulating antiferromagnetic (AF) HF ground state [4]. However, away from half-filling, unrestricted HF theory yields intricate solutions that include domain walls, vortices, polarons etc. imposed on an AF background; see e.g. [5] and references therein. These solutions suggest that the pure AF state is only stable at half-filling and that additional holes or particles tend to distribute so as to perturb the AF state as little as possible. Furthermore, away from half-filling, these solutions break translational invariance in a complicated manner and are highly degenerate. One thus expects that the low-energy properties of the lattice model away from half-filling should be dominated by fluctuations between these degenerate solutions. Unfortunately, it is difficult to formulate a useful low-energy effective model for this situation.

Unrestricted HF theory is computationally demanding and thus applicable only for moderate lattice sizes. However, it is possible to find “mixed” regions in the phase diagram, i.e. regions with intricate and highly degenerate HF solutions, already in mean field theory. By the latter we mean HF theory restricted to states invariant under translations by two sites [6, 7]. This requires little computational effort and thus allows to explore the full phase diagram for arbitrarily large lattice sizes. In this manner one can identify the regions in the phase diagram in which an exotic physical behavior can be expected. It is remarkable how rich the resulting mean field diagrams for the 2D Hubbard model are [7].

In this paper we present mean field phase diagrams for the 2D -- model, which is a spinless variant of the Hubbard model and therefore somewhat simpler. We find a stable charge-density-wave (CDW) mean field ground state at half-filling , a translation-invariant normal (N) state far away from half-filling, and extended doping regions with mixed phases in-between. We indicate the mixed phases in our phase diagrams by horizontal lines to emphasize that mean field theory fails here, i.e. it does not allow any specific conclusions to be drawn; see Figure 1(a). Put differently, the mixed regions are ”unknown territories” (analogous to white spots on ancient maps) for mean field theory, but the knowledge of their existence is nevertheless important physical information.

Figure 1: Comparison between the coupling () vs. filling () mean field phase diagrams of the 2D -- model (a) and the 2D Luttinger model (b) for and . Shown are the charge-density-wave (CDW) and normal (N) phases as a function of coupling and filling at zero temperature. The regimes marked by horizontal lines are mixed, i.e. neither the CDW nor the N phase is thermodynamically stable. The CDW phase in (a) exists only at half-filling . The CDW phase in (b) corresponds to a partially gapped phase.

Our main result is to show that the 2D Luttinger model provides a tool to explore these ”unknown territories” using mean field theory for the antinodal fermions. To be more specific, we compute mean field phase diagrams for the effective antinodal Hamiltonian defined in Section 2.2. We find that the antinodal fermions indeed have a CDW gap in a significant region of the parameter space, as conjectured in [1, 2]. We refer to this also as a CDW phase, but we emphasize that, in general, it corresponds to a partially gapped phase of the 2D Luttinger model. There are also the nodal fermions that are gapless, and these fermions can dope the system even if the antinodal fermions remain gapped and half-filled. In this way a large part of the mixed regions of the phase diagram for the 2D -- model is filled in; see Figure 1(b). We also study how sensitive the occurrence of a partially gapped phase is to variations in the model parameters, especially and .

1.3 Related work

The derivation of the 2D Luttinger model was inspired by important work of Mattis [8], Schulz [9], Luther [10], and Furukawa et al. [11]; see [2] for discussion and further references.

The phase diagram of the 2D -- model, at and away from half-filling, has been studied using various techniques. Recent work close to ours is [12] in which the possibility of phase separation is investigated; see also [13, 14]. In particular, Figure 1(a) in [12] is the same as our Figure 1(a). Note though, unlike [12], we do not necessarily interpret the horizontally lined region in our Figure 1(a) as a phase-separated state. Instead, we only conclude that the considered mean field theory fails to give a stable homogeneous phase there.

One motivation for our work are experimental results on high temperature superconductors. It is known from angle-resolved photoemission spectroscopy [15] that these materials can have an electronic phase in which parts of the underlying Fermi surface do not have gapless excitations. In particular, for hole-doped materials, one finds near half-filling that the degrees of freedom in the antinodal regions are gapped, while in each nodal region there is an ungapped Fermi surface arc; see e.g. Figure 5 in [16] or Figure 1 in [17].1 We believe that this suppression of accessible electronic states at the Fermi level seen in experiments provide strong support for our approach. However, we stress that the emphasis of the present work is mathematical, and a confrontation of our results with experiments is postponed to future work.

1.4 Notation and conventions

We use the symbol “” to emphasize that an equation is a definition. We denote by and the real- and imaginary parts of a complex number , and is its complex conjugate. We write “” short for “ and ” etc. We use bold symbols for matrices, e.g.  is the identify matrix. By “” we mean a numerical result “”.

The fermion models considered in this paper are defined by Hamiltonians of the following generic type

(1)

with fermion operators labeled by a finite number of one-particle quantum numbers . Note that equals the number of one-particle degrees of freedom that are included in the model. Our normalization is such that

(2)

The model parameters and correspond to the matrix elements of the kinetic energy and two-body interaction potential, respectively, and is the chemical potential. The model is defined on a fermion Fock space with a vacuum annihilated by all . Expectation values with respect to a given state of the model (both zero and non-zero temperatures) are denoted by . The state can be either the exact thermal equilibrium state or an approximate Hartree-Fock state; it will always be clear from the context which is meant. The inverse temperature is denoted by .

1.5 Plan of paper

The two models we consider are defined in Section 2. Some results presented here are (minor) generalizations of the corresponding ones in [2], as further elaborated in Appendix A. Section 3 explains the method we use to compute mean field phase diagrams, with some technical details deferred to appendices B and C. Our results are given and discussed in Section 4. Section 5 gives some closing remarks.

2 Models

In this section, we define the two studied models and discuss the relation between them. We note in passing that the notation used here is slightly different from that in [2]; see Appendix A.1 for details.

2.1 2d -- model

The 2D -- model describes spinless fermions on a square lattice with sites and lattice constant . The fermions hop with amplitudes and between nn and nnn sites, and fermions on nn sites interact with a density-density interaction of strength . The Hamiltonian is given in Fourier space by

(3)

with the tight-binding band relation ()

(4)

and the interaction vertex2

(5)

with

(6)

The fermion operators are labeled by momenta in the Brillouin zone

(7)

Other conventions used are explained in Section 1.4 (with the there corresponding to our here). Note that the number of different one-particle quantum numbers equals the system volume, . The chemical potential is to be determined such that the fermion density

(8)

has a fixed specified value. We refer to as filling factor or filling, and is called doping. The filling lies in the range , and corresponds to half-filling. We always assume , , and that is an integer. To simplify notation we set in all figures, i.e. energies are measured in units of . Figure 2 shows the Brillouin zone of this model and a typical example of a non-interacting Fermi surface defined by .

Note that we use anti-periodic boundary conditions and a large-distance cutoff different to that used when deriving the 2D Luttinger model in [2]. This is legitimate since we are interested in the thermodynamic limit , and finite size effects are negligible for the system sizes we use in our numerical computations.

Figure 2: Fermi surface for the tight-binding band relation in 4 with , and (bent dashed curves). Also indicated are the eight regions labeled by and discussed in the main text (large squares for , , rectangles for , , and small squares for , ). Note that the Brillouin zone is equal to the union of these regions. The two antinodal points and four nodal points are depicted by dots and rings, respectively. The parameters used in the plot are and , and corresponds to .

The invariance of the 2D -- model under particle-hole transformations provides an important guide for us. The interested reader can find more details in Appendix A.1.

2.2 2D Luttinger model

A detailed derivation of the 2D Luttinger model and its partial solution by bosonization were given in [2]. Here we first describe this model and then summarize the results from [2] that we actually need.

The 2D Luttinger model involves six fermion flavors labeled by a pair of indices with and . These flavors correspond to different regions in the Brillouin zone of the 2D -- model, as shown in Figure 2 (two large squares for and four large rectangles for ). The sizes of these regions are determined by a parameter in the range (we used in Figure 2). There are two more fermion flavors with and (two small squares in Figure 2), but in this paper we assume that the parameters are such that the fermions are far away in energy from the Fermi level and thus their dynamics can be ignored [2] (we discuss this point in Section 5, Remark 2).

The momenta in the regions with can be written as with

(9)

and in

(10)

Close to the points , , the band relation in 4 can be well approximated by the lowest-order non-trivial terms in a Taylor series expansion with

(11)

i.e.  correspond to saddle points of if . This also explains why we impose these bounds on . Similarly, the band relation for fermion degrees of the freedom corresponding to can be well approximated by bands linear in close to the points with another free parameter ( or ). We use standard terminology [15] and refer to the fermions with as antinodal and the fermions with as nodal, respectively. In Figure 2 the antinodal points are indicated by dots and the nodal points by circles.

The 2D Luttinger model is defined by a Hamiltonian of the form where and include terms depending only on the nodal and antinodal fermions, respectively, and are interaction terms with both kinds of fermions. It is obtained from the 2D -- model by certain approximations that amount to a particular partial continuum limit [2]. A key assumption is that there is an underlying Fermi surface in the nodal regions consisting of line segments (“Fermi surface arcs”) and containing the nodal points for some determined by , and ; see 49. In [2] we fixed by the condition , but in the present paper we work in the grand canonical ensemble and thus allow (i.e. ) to be arbitrary in intermediate steps of our computations. No assumption is made on the Fermi surface in the antinodal regions.

As explained in [2], Sections 2 and 6.2, we need to restrict ourselves to parameters such that

(12)

with the same bound for as for . Moreover, we require and to be different from , since otherwise one has additional back-scattering terms in the 2D Luttinger model which spoil a simple treatment of the nodal fermions using bosonization; see [2]. These conditions define the parameter regime of interest to us.

The nodal fermions in the 2D Luttinger model can be bosonized and integrated out exactly. This yields an effective model for the antinodal fermions that, in the local-time approximation [2], is given by a Hamiltonian of the form

(13)

This Hamiltonian is also of the generic type discussed in Section 1.4 (with the one-particle quantum numbers used there to be identified with here). Note that there are different momenta , and that the number of one-particle degrees of freedom in the model is . The filling factor of the antinodal fermions is therefore

(14)

while the total filling factor of the 2D Luttinger model is (including the nodal fermions etc.)

(15)

see [2]. It is in 15 that is to be identified with the filling factor of the 2D -- model.

A main result in [2] are explicit formulas for the parameters and in terms of the other model parameters:

(16)

and

(17)

with

(18)

The parameter can be computed from the following identity

(19)

with

(20)

and where we have introduced a convenient short-hand notation

(21)

The constant corresponds to a renormalization of the bare antinodal interaction and arises from integrating out the bosonized nodal fermions.

The 2D -- Hamiltonian is equivalent (in a low-energy approximation) to the effective antinodal Hamiltonian in 13 only if one takes into account the additive constant (see Appendix A.2)

(22)

with

(23)
(24)

and

(25)

The constant is derived in [2], but in the present paper we only need that it is independent of the chemical potential:

(26)

As we show later, the parameter can be fixed by the self-consistency condition . Thus the effective antinodal model contains one more free parameter as compared to the 2D -- model, namely .

As already mentioned, the results in [2] are restricted to the special case in which is explicitly fixed by the condition , and they are written in a slightly different form. The interested reader can find details about how to obtain the results given here from the ones in [2] in Appendix A.2.

It is not essential to work with the Taylor expansion of the antinodal band relations, and one can equally well use the full band relations

(27)

instead of 11. As we will discuss, the results for the band relations in 11 and 27 agree quite well for smaller values of , but for close to one there are some quantitative differences. Furthermore, in the derivation of the 2D Luttinger model, certain approximations were done on the interaction vertex of the 2D -- model; see Section 5 in [2]. It was argued that this had no important consequences for low-energy scattering processes. We note here that these approximations were only necessary for processes that include nodal fermions, and it would be possible to use the full interaction vertex for processes involving just antinodal fermions. With this, and using the full band relation 27, one could derive a refined 2D Luttinger model for which the 2D -- model is recovered by setting .

We finally mention that the local-time approximation in [2] was only done for simplicity, and it is possible to generalize our treatment here to take into account the full time dependent interaction. We hope to come back to this in the future.

3 Method

In this section, we discuss the mean field Hamiltonians used for deriving phase diagrams and the procedure that allows us to identify the mixed regions. More explicit details can be found in appendices B and C.

The fermion models considered in the previous section are defined by Hamiltonians of the type 1. Conventional HF theory at zero temperature and fixed particle number for such models amounts to considering the set of all variational states of the form with and orthonormal one-particle states.3 Using standard terminology, we refer to the states as Slater determinants. The are to be chosen so as to minimize the HF energy

(28)

It is straightforward to compute an explicit formula of in terms of the one-particle density matrix ; see Appendix B.

In the present paper, we work at (mainly) small but non-zero temperatures (unless otherwise indicated ). This means that we minimize the full grand canonical potential (including the entropy) with respect to all HF Gibbs states; see Appendix B for a full discussion. We will however set the temperature to zero in the current section to not burden the presentation.

3.1 2d -- model

It is convenient to choose the Slater determinant as ground state of a reference Hamiltonian

(29)

with the non-interacting Hamiltonian obtained from the one in 3 by setting , and the matrix elements of the HF potential . This allows to parametrize HF theory by the one-particle Hamiltonian defined in 29. As explained below, it is important to use the grand canonical ensemble, i.e. to fix the particle number by adjusting the chemical potential [7]. One thus obtains

(30)

with and the eigenvalues and corresponding orthonormal eigenvectors of and the Heaviside function. Note that the chemical potential is included in .

Unrestricted HF theory amounts to determining the HF potential that minimizes under the filling constraint

(31)

This method is computationally demanding and thus restricted to small system sizes.

By mean field theory we mean the restriction of HF theory to states that are invariant under translations by two sites. As explained in Appendix C.1, this corresponds to considering the following restricted set of HF potentials:

(32)

with and three real variational parameters , and . For this corresponds to a normal (N) state that is translation invariant, and for one has a charge-density-wave (CDW) state for which translation invariance is broken down to translations by two lattice sites. The computational problem is now very easy: there are only three variational parameters, and can be computed analytically by Fourier transformation; the interested reader can find details in Appendix C.1.

Figure 3: Hartree-Fock energies of the charge-density-wave (CDW) and normal (N) states vs. chemical potential for the 2D -- model at , , . The energy of the thermodynamically stable phase is given by the minimum of these curves. The constant slope of the CDW curve in implies that the CDW phase can exist only at half-filling, and the kinks at , where the system changes from the CDW to the N phase, imply mixed regions where neither the CDW nor the N phase is thermodynamically stable. (A convenient overall constant has been added to .)

It is important to note that, by working with the grand canonical ensemble, we not only can detect variational ground states given by Slater determinants (with ) or (with ), but it is also possible that the state with lowest energy is mixed and of the form

(33)

for some .4 In general, the two Slater determinants correspond to different filling factors and , and the state in 33 corresponds to total filling

(34)

The possibility of obtaining 33 can be seen by computing the HF energies for the CDW and N states, for and , as functions of the chemical potential ; see e.g. Figure 3 for , and . One finds two different regimes: there is an interval where , i.e. the system clearly has a CDW mean field ground state ; for and we find .5 The second kind of regime is for and where one finds , i.e. a N ground state . However, not all doping values can be realized in this way: according to 31 we can obtain the doping from the slope of

but this is not continuous at , , where the CDW state changes to the N state. The mean field phases can be determined by the following four doping values,

(35)

as follows. The system is in the CDW phase in the doping regime and in the N phase for and . However, in the regions and the mixed state in 33, with determined by 34, has lower energy than either of the pure Slater states. We refer to the latter as a mixed phase. We emphasize that mixed phases occur in large parts of the phase diagram; e.g. for , and we find and .

The results described above have the following physical interpretation (we only discuss the regions close to since the other one is similar). In the CDW phase one has the effective band relations (see Appendix C.1)

(36)

This shows that the CDW phase has a band gap , and as long as is in this gap, changing it cannot affect doping. Thus the HF energy is a linear function of with slope (half-filling) in this region. The N phase is not gapped and doping can be monotonically increased by increasing , and therefore the HF energy is a strictly concave function of . Thus, as we try to increase doping by increasing in the CDW phase, the HF energy of the N phase decreases faster than the HF energy of the CDW phase, and when both energies become equal at the doping in the N phase is significantly larger than the doping in the CDW phase.

A possible interpretation of the mixed state in 33 is a phase-separated state in which parts of the system are in the CDW phase and parts in the N phase [7, 12]. We can therefore conclude that, for , a phase-separated state has lower variational energy than any simple mean field state. However, we emphasize that the occurrence of a mixed phase does not necessarily imply phase separation, but it nevertheless proves that a true HF ground state is very different from any state that can be described by a simple mean field ansatz 32 (a true HF ground state can in principle be found by unrestricted HF theory). Thus mean field theory allows to determine those regions in phase space where non-conventional physics (not describable by mean field theory) is to be expected.

It is interesting to note that, for non-zero , it is possible to have a CDW phase also away from half-filling. This can be seen by computing a plot similar to Figure 3 but with , for example. One still finds that the CDW energy as a function of is a straight line in most of the interval , but as approaches from the left, this curve starts to bend so that . There is no bending of close to , however, and . Thus at , , it is possible to dope the CDW state on the particle side (i.e. for ) but not on the hole side (). This shows that the parameter affects the mean field phase diagrams both quantitatively and qualitatively.

Our results for the 2D -- model were obtained with MATLAB using the system size (i.e. lattice sites). We checked that this is large enough so that finite size effects are essentially negligible. However, we note that some phase boundaries are slightly affected by finite size effects even at this system size, as discussed in more detail below.

3.2 2D Luttinger model

We use HF theory as explained for the 2D -- model in the previous section. The reference Hamiltonian can now be written as (using a convenient matrix notation)

(37)

with variational parameters , and .

The grand canonical potential corresponding to this reference Hamiltonian, evaluated for the Hamiltonian in 13, is denoted by ; see Appendix C.2.3 for explicit formulas. It is important to note that this is only the antinodal contribution and that the total grand canonical potential of the 2D Luttinger model is

(38)

with the energy constant in 2226 taking into account the contributions from the other fermion flavors , and . Note that is a function of (rather than ), and that the filling constraint following from our general discussion of HF theory in Appendix B is

(39)

On the other hand, the filling constraint in 31 should still hold true but with replaced by in 38. This implies the following consistency condition

(40)

which must be fulfilled for the parameters and given in Section 2.2. This identity is non-trivial and provides an important check of our computations. We therefore include details of its verification in Appendix C.3.

The calculation of the grand canonical potential of the antinodal CDW phase is done as follows (the normal phase is treated identically). For fixed , we make an ansatz for the antinodal filling . Solving 19 for then gives us using 16. We proceed by minimizing the grand canonical potential 38 with respect to the variational parameters , and . This in turn gives us a specific value for the antinodal filling to be compared with our initial guess. We repeat the above procedure until a self-consistent solution is obtained for . An example of a resulting curve for vs. is given in Figure 4.

Figure 4: Hartree-Fock energies vs.  as in Figure 3 but now for the 2D Luttinger model at and (, , ). Shown are the energies of the antinodal charge-density-wave (CDW) and normal (N) phases. Again there is an interval for which the CDW energy is lower, but unlike Figure 3, this energy now deviates from a straight line (dashed in the figure). This is due to the nodal fermions that change the total filling even though the antinodal fermions are half-filled and gapped. This kind of plot proves the existence of a partially gapped mean field phase in the 2D Luttinger model.

For the numerical calculation of , we fix the number of momenta in the antinodal Fourier space regions 10 to . Then , which is large enough so that finite size effects are smaller than the symbol size in our figures. Furthermore, at this system size we can safely replace the Riemann sums in Appendix A.2.3 by integrals.

4 Phase diagrams

In this section, we present and discuss mean field phase diagrams for the 2D -- model and the 2D Luttinger model.

4.1 2d -- model

Figures 1(a), 5 and 6 give various phase diagrams of the 2D -- model. Shown are the phase boundaries of the CDW phase, the N phase, and the mixed phase (indicated by horizontal lines) and how they depend on filling , coupling and nnn hopping .

As seen in Figure 1(a), for the CDW phase is only stable at half-filling , and there is a significant mixed region away from half-filling. The size of this region grows with increasing . The invariance of the phase diagram under is a consequence of particle-hole symmetry and . We found that all phase boundaries in this figure are insensitive to finite size effects.

The dependence of the phase boundaries on and for is shown in Figure 5. Due to particle-hole symmetry this phase diagram is invariant under , and we therefore only discuss . For , the effect of is small and, in particular, the CDW phase exists only at half-filling. However, for it is possible to dope the CDW phase on the particle (but not on the hole) side, as discussed in Section 3.1. For even larger values of , the CDW phase can be doped both on the particle and the hole side, and the mixed region becomes smaller with increasing and eventually vanishes. Note that the phase boundaries between the N and the mixed phases do not change much with , for example, and for and , respectively. Moreover, the phase boundary between the CDW and the mixed phases at the hole side for is, to a good approximation, a straight line. The small wiggles of the phase boundary between the CDW and the mixed phases on the particle side for are due to (minor) finite size effects, but all other phase boundaries are quite insensitive; the same is true for the phase diagrams in Figure 6.

Figure 5: Mean field phase diagram of the 2D -- model at zero temperature: next-nearest-neighbor hopping vs. filling at and . Shown are the charge-density-wave (CDW), normal (N), and mixed (horizontal lines) phases.

The dependence of the phase boundaries on can also be seen in Figures 6(a) and (b) showing the vs.  phase diagrams for and , respectively. Note that, for non-zero , there is a critical coupling value below which no CDW phase exists (e.g.  for ), and that there is a phase boundary between the CDW and N phases. Moreover, for strong coupling, the CDW phase broadens out and increasingly dominates the phase diagram (only visible in (b)). The phase boundary between the CDW and N phases is quite sensitive to finite size effect, and it is difficult to determine from our numerical data if it is a first- or second order phase transition. Finally, to see how the phase diagram evolves with it is instructive to compare Figures 1(a) with Figures 6(a) and (b).

Figure 6: Mean field phase diagrams of the 2D -- model at zero temperature: coupling vs. filling at in (a) and in (b) (). Shown are the charge-density-wave (CDW), normal (N), and mixed (horizontal lines) phases.

4.2 2D Luttinger model

Apart from Figure 1(b), all our phase diagrams for the 2D Luttinger model have been computed for , and or . These parameter choices are partly motivated by our results on the 2D -- model. To be specific, for , the phase diagram displays several, qualitatively different, features near half-filling, while is still of the same order of magnitude as ; see Figure 6(a). Furthermore, there is an interesting transition near and at which it becomes possible to both particle- and hole-dope the CDW phase. Other than this, there is nothing special about these parameter values.

As mentioned in the introduction, a main result of this paper is that the 2D Luttinger model indeed has a phase in which the antinodal fermions are gapped and half filled, as conjectured in [1, 2]. Figure 1(b) shows one example with fixed values of and (as explained later, we can in fact eliminate the parameter in this figure by the requirement ). Similar to the 2D -- model, we again find a CDW phase, a N phase, and a mixed phase in-between. However, for , the mixed phase is typically much smaller than for the 2D -- model; cf. Figure 1(a).

Figure 7: Mean field phase diagrams of the 2D Luttinger model at zero temperature, , , , and for different values of the antinodal region size and nodal linearization point . Shown are the charge-density-wave (CDW), normal (N), and mixed (horizontal lines) phase boundaries of the antinodal fermions. (a) vs. total filling at (results for other values of are similar). (b) vs. nodal Fermi surface location at the four phase boundaries in Figure (a). (c) vs.  at . (d) vs.  at . Note that we have to restrict to as indicated by the dashed line. Due to particle-hole symmetry, the corresponding result for can be obtained as .

An important question is how sensitively the results depend on and . We find that the qualitative features of the phase diagrams are robust and the quantitative dependence on is weak. However, the quantitative dependence on is more pronounced. To be more specific, Figures 7(a)–(d) give representative examples for : (a) shows how the phase boundaries depend on if is fixed, while (c) and (d) show how they change with for fixed . We note that the almost vertical phase boundaries found in Figure 7(a) for near is not special to the current choice of and . Particle-hole symmetry and imply that the phase boundaries are invariant under . This and explain the symmetry of Figure 7(c). Recall that is restricted by 12 to lie in the range .

Figure 8: Mean field phase diagrams of the 2D Luttinger model at zero temperature, , and (left) respectively (right): antinodal region size vs. filling . Shown are the charge-density-wave (CDW), normal (N), and mixed (horizontal lines) phases of the antinodal fermions. The parameter is determined such that the nodal points are located on the nodal Fermi surface arcs at each phase boundary, i.e. .

Figure 7(b) shows as a function of the nodal Fermi surface location, parameterized by , at the four phase boundaries in 7(a). These figures suggest that one can fix by the requirement using the following iterative procedure: given one can compute by solving the mean field equations and then set . We generally find that the sequence converges quickly independent of the starting value for . One can thus eliminate the parameter and obtain phase diagrams depending only on .

Figure 8 shows two examples of such phase diagrams, the left for and the right for . Comparing the left diagram with Figure 7(c), one finds that the filling values at the four phase boundaries agree up to an error . Since varies over an extended interval in Figure 8, while it is fixed at in 7(c), this further demonstrates the insensitivity of the phase boundaries to changes in . The same feature holds true for the right diagram. Moreover, Figure 8 shows again the qualitative changes of the phase diagram induced by non-zero . For , the hole side of the phase diagram is similar to the one for , but on the particle side, one no longer finds a mixed region. Instead there is a continuous transition between the CDW and the N phase.

Figure 9: Comparison between the temperature () vs. filling () mean field phase diagrams of the 2D -- model (a) and the 2D Luttinger model (b) for , and . In (b), and is fixed by the condition at the phase boundaries. Shown are the charge-density-wave (CDW), normal (N), and mixed (horizontal lines) phases. Note the different temperature scales in the two plots.

Figure 9 compares the effect of varying the temperature on the phase diagrams of the 2D -- model (a) and the 2D Luttinger model for and (b). For the 2D -- model at finite temperature, it becomes possible to dope the CDW phase away from half-filling. Moreover, the mixed phase decreases in size as temperature is raised from zero, and it completely disappears at . For larger values of , there is a continuous transition between the CDW and N phase. The qualitative features of the 2D Luttinger model at non-zero temperature are quite similar to the 2D -- model, except that the CDW phase is only partially gapped and the overall temperature scale is reduced. For example, the mixed phase now disappears at . We note that in computing the filling contribution from the nodal fermions, we have assumed for simplicity that there are only bosonic excitations from the ground state, i.e. the total number of nodal fermions is independent of temperature. We leave it to future work to investigate whether or not this is a justified assumption.

Figure 10: Comparison between the results in Figure 8 obtained with the Taylor series approximated band relations 11 (full circles), and the ones obtained using the full band relations 27 (open circles). Note that the results agree well for , but there are some differences for and .

Finally, Figure 10 compares phase diagrams of the 2D Luttinger model obtained using the exact antinodal band relation in 27 (open circles) to the ones obtained using the Taylor series approximated ones in 11 (full circles). Note that, for (left), there are only small quantitative differences. For (right), however, there are larger deviations, in particular for and : in the former case we find a mixed region between the CDW- and N phases, but in the latter case this mixed region is absent.

4.3 Discussion

In this section, we discuss some general features of the mean field results for the 2D Luttinger- and 2D -- models that we observed in our numerical computations. These observations are also based on phase diagrams not included in the present paper.

Whenever the CDW and the N phase share a phase boundary in a diagram (i.e. there is no mixed region in between), the CDW order parameter goes continuously to zero at the boundary. Such phase boundaries are more difficult to determine numerically than transitions to a mixed phase.

For we only found a CDW phase in the 2D Luttinger model with , i.e. the antinodal fermions are half-filled. However, as for the 2D -- model, for non-zero it is possible to have a CDW phase with . In fact, as a general rule of thumb, if the parameters and are such that the CDW phase in the 2D -- model can be doped, then the 2D Luttinger model has a partially gapped phase with . The converse is not always true. We expect that the physical properties of the 2D Luttinger model in a gapped phase with is qualitatively different to one with .

Furthermore, as exemplified in Figure 10, when the phase diagram of the 2D Luttinger model is hardly changed if one replaces the Taylor series approximated band relation in 11 by the exact one in 27. For and close to 1, the results are more sensitive to this replacement.

We note that it is not obvious that the phase boundaries can be fixed unambiguously by the requirement that since the -value for which this occurs is, in general, different for the N and the CDW phase. However, we found that using the -value from the N and the CDW phase leads to results that are very similar: for the discrepancy is typically smaller than the symbol size in our figures, and this is also true for , apart from the case when is close to one and the full band relation in 27 is used. In Figures 8-10 we determined the phase diagrams using from the CDW phase. When using the -values from the N phase, the CDW phase increases slightly in size, while the N phase decreases slightly in size.

We now discuss how the size and location of the nodal arcs evolve in the left diagram of Figure 8 as and are varied. This serves as a representative example for the general case. Consider first fixed and ( is analogous). When the system is in the half-filled and partially gapped CDW phase, one finds .6 As the nodal fermions are hole doped, decreases until eventually the CDW phase becomes unstable at some filling . For even smaller values of , the system is first in a mixed phase, followed by a N phase. When the system goes from the CDW-mixed phase boundary to the N-mixed boundary, the value will first increase towards . As an example, when , is on the former boundary and on the latter. Furthermore, when the size of the nodal arcs is increased (i.e. decreasing the value of ), the value of on the phase boundaries approaches asymptotically .

Finally, in mean field theory, the antinodal fermions in the 2D Luttinger model behave very much as the fermions in the original 2D -- model scaled by a factor . To give an example, when the antinodal fermions are half-filled (), the size of the CDW gap is, to a good approximation, proportional to . Likewise, the qualitative features of the temperature vs. filling phase diagrams for the 2D -- - and 2D Luttinger model are almost identical if the temperature scale of the latter is reduced by a factor .

5 Final remarks

1. Mean field theory is a variational method and therefore not necessarily restricted to weakly coupled systems. For example, there exist interacting fermion Hamiltonians of the type 1 for which mean field theory is exact (examples include Hartree- and BCS-like models; see e.g. [18] for details). Moreover, for many models describing electrons in conventional 3D metals, it is known that mean-field type approximations can give accurate results despite of the presence of strong Coulomb interactions. Nevertheless, for lattice fermion systems of Hubbard-type, standard mean field theory fails in a large part of the parameter regime [7]. In this paper, we demonstrated that it is possible to circumvent this problem by treating parts of the fermion degrees of freedom exactly using bosonization, as proposed in [1, 2]. It is straightforward to extend this approach to the 2D Hubbard model [19].

2. The results in [2] and the present paper suggest that the 2D -- model has a qualitatively different behavior in different filling regimes that can be described by different effective Hamiltonians. For example, at half filling there is, on the mean field level, a fully gapped CDW phase that is adequately described by the antinodal Hamiltonian in 13 with . Upon doping the system, both the nodal- and the antinodal degrees of freedom become relevant, and the low-energy physics is governed by the 2D Luttinger model with . For the special case when the antinodal fermions are gapped, it is possible to describe the system by a pure nodal fermion model that can be solved exactly by bosonization [2]. For large filling ( close to one), the low-energy physics is expected to be dominated by the out-fermions with and . In this regime, the appropriate effective model describes non-interacting fermions with a band relation [2]. Corresponding statements hold for the in-fermions at small filling ( close to zero).

3. A main result of this paper is that the 2D Luttinger model indeed has a partially gapped phase with gapless nodal fermions and gapped antinodal fermions. The possibility to obtain such a phase is insensitive to changes of parameter values and model details like the band relation.

4. For all degrees of freedom in the 2D Luttinger model are treated in mean field theory (there are no nodal fermions then). One should therefore expect that the phase diagram of the 2D Luttinger model for should be qualitatively similar to the one of the 2D -- model. Our results show that this is indeed the case, but there are some quantitative differences; compare Figures 5 and 8. Most of these differences are explained by the Taylor series approximation of the antinodal bands 11; cf. Figure 10. The remaining discrepancy is due to the approximation of the interaction vertex mentioned at the end of Section 2.2. As discussed there, it is possible to improve on this approximation [2] and derive a refined 2D Luttinger model that gives back the 2D -- model when setting .

5. There exist parameter values such that the renormalized antinodal coupling constant in 17 is negative: we found, for example, for , , , , and . However, this and all other such cases we found are barely within the domain of validity of our method; recall 12. We therefore restricted our discussion in this paper to .

6. It is worth noting that our results do not violate the Luttinger theorem [20]: the proof of this theorem assumes a standard connected Fermi surface, and it therefore requires modifications if there is a (partial) gap.

7. In this paper, we only presented mean field results for the simplest consistent charge-density-wave ansatz (i.e. only , and non-zero, see Appendix C). Another interesting possibility is the so-called d-wave charge-density-wave (DDW) phase that has been suggested in the context of high-temperature superconductors [22]. It corresponds to an ansatz with , and non-zero (this is a special case of the general ansatz in 73). One can find mean field solutions for which gives lower energy than . However, we never obtained a DDW solution that has lower energy than the CDW solution discussed in the main text. The same result hold for the other mean field order parameters given in Appendix C.

Note added: Recently, numerical simulations of the 2D -- model using fermionic PEPS [21] produced a phase diagram that is remarkably similar to our Figure 1(a); cf. Figure 22 in [21].


Acknowledgments

This work was supported by the Göran Gustafsson Foundation and the Swedish Research Council.

Appendix A Model details

The results in the present paper are a slight generalization of the ones derived in [2]. In this appendix we explain how these generalizations are obtained. We also point out the (minor) differences in notation as compared to [2]. In the following, we write “(I.33)” short for “Equation (33) in [2]” etc.

In [2], the lattice constant was an important parameter when taking the partial continuum limit in the nodal region. However, in the present paper we can without loss of generality set it to . The relation between the fermion operators used here and the ones in [2] is as follows,

(41)

a.1 2d -- model

The Hamiltonian in 3 is obtained from the one in (I.19) and (I.25)–(I.28) by inserting and 41, and using

i.e. normal-ordering of the interaction with respect to the trivial vacuum does not generate any additional terms.

We recall that the 2D -- model is invariant under the following transformation of parameters:

(42)

To see this, we make the particle-hole transformation