A Conventions

ITP–UU–12/27

SPIN–12/25

CERN-PH-TH/2012-240


Holographic models for undoped Weyl semimetals

[8mm] Umut Gürsoy, Vivian Jacobs, Erik Plauschinn, Henk Stoof, Stefan Vandoren

[5mm] Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland

[3mm] * Institute for Theoretical Physics and Spinoza Institute, Utrecht University

3508 TD Utrecht, The Netherlands

[5mm] Umut.Gursoy@cern.ch; V.P.J.Jacobs, E.Plauschinn, H.T.C.Stoof, S.J.G.Vandoren@uu.nl

[3mm]

 

Abstract

We continue our recently proposed holographic description of single-particle correlation functions for four-dimensional chiral fermions with Lifshitz scaling at zero chemical potential, paying particular attention to the dynamical exponent . We present new results for the spectral densities and dispersion relations at non-zero momenta and temperature. In contrast to the relativistic case with , we find the existence of a quantum phase transition from a non-Fermi liquid into a Fermi liquid in which two Fermi surfaces spontaneously form, even at zero chemical potential. Our findings show that the boundary system behaves like an undoped Weyl semimetal. 1

 

1 Introduction

Over the past years, the AdS/CFT correspondence has become a more and more popular and widespread tool which offers the opportunity to apply ideas from string theory to realistic materials studied in condensed-matter physics, see e.g. [1, 2, 3, 4, 5] and references therein. More specifically, it is potentially very useful when considering strongly coupled or critical condensed matter, which is generically not possible to describe within perturbation theory. However, in the context of the AdS/CFT correspondence these systems can be understood and investigated more conveniently using a dual theory in a higher-dimensional curved spacetime, i.e., in the framework of general relativity. Most commonly, the duality exists between a weakly coupled, classical gravity theory in a curved anti-de-Sitter (AdS) spacetime on the one hand, and a strongly coupled, conformal field theory (CFT) living on the flat boundary of the AdS spacetime on the other hand. Such a conformal field theory describes for example a quantum critical point of the condensed-matter system under consideration, and the observables that are most readily available from the AdS/CFT correspondence are usually correlation functions of the composite operators that classify the conformal field theory. For our purposes, most relevant are correlators of fermionic operators, since they may show Fermi or non-Fermi liquid-like behavior most easily [6, 7, 8, 9, 10].

However, in condensed-matter physics it is more natural to think in terms of fermionic single-particle operators, i.e., creation and annihilation operators in a Fock space, that satisfy the following (equal-time) anti-commutation relations

(1.1)

where labels the spin of the electron. Furthermore, in experiments on condensed-matter systems, the quantity that is measured is essentially always related to single-particle or two-particle correlation functions and not to the above-mentioned composite operators. For example, the retarded single-particle correlation or Green’s function is measurable in electronic systems using angle-resolved photoemission spectroscopy (ARPES). For this reason, we are interested in finding of a strongly interacting condensed-matter system using holographic methods. Inspired by previous work on this topic, such as [11] and [12, 13], we have shown recently how the usual holographic prescription can be modified in such a manner that it allows for the construction of the retarded single-particle Green’s function [14]. In a sense, our method can be seen as a bulk derivation of the semi-holographic description advocated in [12].

A crucial consequence of the anti-commutation relations in equation (1.1) is the existence of a sum rule for the corresponding single-particle two-point correlation function, given by

(1.2)

This sum rule is essential in determining whether the quantity under consideration is the correlator of a single-particle or a composite operator, and therefore plays a central role in our construction.

The quantum critical points described by the AdS/CFT correspondence are characterized by their dynamical scaling exponent . In the usual correspondence, an anti-de-Sitter background leads to a quantum critical point with relativistic scaling, i.e., . But from a condensed-matter perspective we are also interested in quantum critical points that have a dynamical scaling exponent different from one. To achieve this, the usual anti-de-Sitter background is generalized into a so-called Lifshitz background, which leads to a non-relativistic, i.e., scaling on the boundary [15]. Non-zero temperature effects can then be studied by placing a black brane in the Lifshitz spacetime [16], see also [17] for a more recent discussion. The pure Lifshitz geometry without a black brane develops singularities due to diverging tidal forces [18]; as a result the zero-temperature limit might become ill-defined2. While this could be of possible concern for our analysis, our findings show that the fermionic Green’s function is well defined at zero temperature and the limit is smooth, so at least the spectral-weight function for fermions does not suffer from any singularities.

In [14] we have described how to construct the retarded Green’s function of a strongly interacting, but particle-hole symmetric system of chiral fermions with an arbitrary dynamical exponent using a Lifshitz black-brane background. The aim of the present paper is to analyze the physics that follows from our prescription. In particular, we calculate the retarded single-particle Green’s function in various cases, at zero and non-zero temperatures, and present the corresponding spectral-weight function and dispersion relations. Although we give some results for , we mostly focus on the case and four boundary spacetime dimensions. With this number of dimensions, the boundary fermions are Weyl fermions and the boundary system behaves like an interacting Weyl semimetal. A semimetal is a gapless semiconductor. In addition, a Weyl semimetal [21, 22] is a semimetal with touching valence and conduction bands based on chiral two-component fermions that in the non-interacting limit and at low energies, satisfy the Weyl equation . Here, the denotes the chirality of the fermion. Because the total system has to be chirality-invariant, the most simple realizations of such a Weyl semimetal considered in the literature usually contain two of such linear-dispersion cones with opposite chirality, separated in momentum-space [21]. The single-particle propagator presented here represents the physics of one of these chiral cones. Since the holographic boundary theory is that of Weyl fermions, and one cannot write down mass terms for Weyl fermions, the system will automatically be gapless at zero chemical potential. Due to the Weyl character of the holographic boundary fermions, the topology of the band structure is therefore protected, just like in Weyl semimetals.

We would like to stress that holography is employed in a bottom-up manner in this paper. That is, the holographic model is an effective low-energy theory, satisfying the above definition of a Weyl semimetal, whereas the microscopic structure of the actual condensed-matter system remains hidden. Of course, all holographic AdS/CMT models have this latter feature in common. We just assume that the Weyl semimetal under consideration leads to a low-energy theory that is strongly coupled, chiral and scale-invariant. The advantage of a bottom-up approach is that we can explore a range of possible situations.

In this paper, we are studying the properties of the ground state of such a holographic model for a semimetal at zero doping, so there is a zero total charge density. But as will be discussed, this ground state has in fact very interesting properties. In particular, zero chemical potential corresponds to particle-hole symmetric systems, and there are many-body correlations due to the possibility of creating particle-hole pairs. This indicates that we are not describing a single fermionic excitation, but a fermionic many-body problem. By computing the momentum distribution of the particles and holes, we find for that our holographic model for a Weyl semimetal contains a quantum phase transition between a non-Fermi-liquid phase and a Fermi-liquid phase with two Fermi surfaces, one for the particles and one for the holes, even at zero doping. In this phase, there is a non-zero density of both particles and holes, and conduction can take place.

The scale set by the Fermi surface is determined by an appropriate combination of dimensionless parameters and that enter in our prescription for the Green’s function on the holographic boundary [14], and the quantum critical point is at . One can think of as a spin-orbit coupling constant and of as an effective interaction parameter coupling the elementary fermion to the conformal field theory. These parameters are not present in the standard formulation of holography, but enter our holographic prescription for the fermionic single-particle Green’s function. We discuss the nature of these parameters in more detail in the next section.

The phase transition that takes place is schematically illustrated in figure 1, where we show the non-interacting band structure and how it is populated in the ground state of each phase. In particular, for the conduction band is empty and the valence band is completely filled, whereas for the conduction band contains a Fermi sea of particles and the valence band a Fermi sea of holes. Note that we present here an idealized picture that neglects the renormalization of the Fermi surfaces involved, which is discussed at length in this paper.

(a)
(b)
Figure 1: An idealized picture of the ground state that we find for and . Grey areas refer to filled energy levels. In this picture, we only show the effect of the interactions on the occupation numbers and not on the dispersion relations.

The layout of the paper is as follows. In section 2, we briefly summarize some basic formulas from [14] that are needed here. In section 3, we present analytic and numerical results for the single-particle spectral function for the cases and . In section 4 we consider the single-particle momentum distribution, and describe properties of the quantum phase transition and the Fermi liquid involved, that does allow for well-defined but strongly renormalized quasi-particles and quasi-holes near the two Fermi surfaces. The main part of the paper ends with conclusions in section 5. Finally, there are a number of appendices which contain our conventions and give more details about the calculations.

2 Single-particle Green’s function for Lifshitz fermions

In this section, we determine solutions to the Dirac equation in a Lifshitz spacetime, from which we construct the retarded fermionic single-particle Green’s function. To keep the paper self-contained, we start by defining our notation and conventions. Readers who are only interested in the result of the calculation may skip directly to equation (2.11). Further details on the derivation can be found in the appendices, in particular, for remarks on how to switch from natural to SI units we would like to refer to appendix A.1. Some of the results in this section have already appeared in more detail in our previous work [14], other related work on fermions in Lifshitz backgrounds can be found in [23, 24, 25].


We begin by specifying the gravitational background metric of a Lifshitz black brane [16], i.e.,

(2.1)

where denotes the number of spatial dimensions. The extra spatial coordinate runs from the horizon at to the boundary at , and the temperature of the black brane is obtained by demanding the absence of a conical singularity at , leading to [16]

(2.2)

The metric (2.1) enjoys the Lifshitz isometry, which will be inherited by the boundary theory as we will show shortly, and which reads

(2.3)

Next, we introduce a Dirac fermion into the above gravitational background. Following [11, 14], we include a boundary action on an ultra-violet (UV) cut-off surface at . While many of the results can be phrased for arbitrary dimensions, for later purposes we specify to spatial dimensions already here. In the conventions of [14], the total action for the bulk fermion that we consider is

(2.4)

where the first term on the right-hand side is the Dirac action in the bulk and the second term is the boundary action. Our notation is such that with being anti-hermitian, and a natural choice for the Dirac matrices employed in this paper is given in appendix A.2. Furthermore, we have decomposed into chiral components according to its eigenvalue under [8, 9] in the following way

(2.5)

The normalization constants and appearing in (2.4) are left unspecified for the moment. The coupling of the fermions to the gravitational field is through the vielbeins and the spin connection , which are determined from the geometry and whose explicit form can be found in appendix A.2. The symbol appearing in the boundary action is the usual Dirac operator for a fermion in a Lifshitz background with arbitrary dynamical exponent , and denotes the determinant of the induced metric on the boundary.

Following [26], we also define Fourier-transformed spinors on each constant slice as

(2.6)

where and denote the frequency and momenta of the plane wave. The Dirac equation resulting from (2.4) (see [14] for more details) and in-falling boundary conditions that we shall employ at the horizon imply a relation between and . It can be expressed as

(2.7)

and can be used to integrate out from the action. Together with (2.4) and (2.6) this results in a holographic effective action for the field on the cut-off surface as

(2.8)

The Green’s function of that follows from (2.8) for our geometry (2.1) reads

(2.9)

where we have rescaled so that it acquires a canonically normalized kinetic term. The final step is now to take a double-scaling limit

(2.10)

As we take this limit, we make sure that the resulting effective action has a kinetic term expected from a theory with dynamical scaling . In the case , this can be achieved by renormalizing away a relativistic kinetic term and adding appropriate counter-terms to the action [14]. The result for then is

(2.11)

where is an arbitrary real number in our approach and where in the following we denote . Before we discuss the self-energy , we would like to spend some words on the nature and interpretation of the parameter , both from a holographic point of view, and from a condensed matter point of view of the boundary.

The parameter appeared first in [14], where it was denoted (see the discussion between (3.18) and (3.19) in that reference). It is a coefficient that is determined by the holographic renormalization procedure for Lifshitz geometries in the presence of bulk fermions that has not been not carried out. In our set-up, it is the parameter that multiplies the counterterm for the fermions on the UV brane involving spatial derivatives, after sending the UV cut-off to infinity. Given a particular bulk model, is a fixed number. For instance, for a dynamical exponent , and relativistic symmetry, (or in natural units, being the speed of light). For other values of the dynamical exponent, we do not know the magnitude of , neither do we know its sign. It is fixed and not tunable for a given bulk action. Nevertheless, we treat it here as a variable, and allow it to vary in the phase diagram of the boundary system, in much the same way as the bulk fermion mass . In this manner we can explore all the possible physical properties of the system, even though at present we do not precisely know its value for the background that we are using. Moreover, as explained in Section 4.5, the same phase diagram can be obtained from a fixed value of , but by varying the bulk mass over a broader range, see also (2.14). From this point of view, treating as a variable is perfectly viable, as long as one has enough bulk models (or string compactifications) with sufficiently many different choices for . This is completely consistent within our approach, as we have not specified the bulk action. We only used the metric that couples to the bulk fermions.

Forgetting about holography, and looking at the boundary system from a condensed-matter point of view, the parameter is the strength of the spin-orbit coupling . What the value of the spin-orbit coupling is depends on the underlying microscopic model for the fermions, and we cannot compute it without specifying the model. In fact, it may depend on the properties of the material. Hence we treat it as a parameter that encodes this ignorance, and we allow to vary it. In other words, we study how the physics changes as a function of , and the result of this analysis is the main content of our paper. In particular, we found that the sign of appears to be crucial in determining whether the system displays Fermi-liquid behavior or not. In a certain sense, we are doing model building, and thus we checked that for all values of , important physical consistency conditions such as sum rules and Kramer-Kronig relations are satisfied.

After having discussed the significance of the parameter , we return to the self-energy appearing in the Green’s function (2.11). In our set-up, the holographic self-energy is by construction an effective description of the interactions between the elementary chiral fermions. This interaction term arises from coupling the elementary field to the conformal field theory encoded in the Lifshitz background. As can be seen by comparing with (2.9), it is related to the quantity introduced in (2.7) by

(2.12)

where is the coupling constant which stays finite in the limit (2.10) and reads

(2.13)

The definition of the self-energy in (2.12) is valid for all values of the momentum , however, for the allowed range of is extended to . This can be derived from the asymptotic behavior of near the boundary, and for more details we refer the reader to appendix B. Furthermore, it is possible to extend the relation between and even to for non-zero momentum, which requires introducing certain counter-terms on the cut-off surface at before taking the limit . Since we will also be interested in the range , using the results in appendix B, we can show that the relation (2.12) in this case should be modified to

(2.14)

Note that the second term in this expression, which is divergent for , removes the divergence in and yields a finite result for .

The transfer matrix defined in equation (2.7) is a complex two-by-two matrix. In the case we are interested in, that is, , it can be diagonalized by choosing the Weyl basis of gamma matrices (A.15). A first-order differential equation for the eigenvalues of can be derived, which was achieved in the anti-de-Sitter case () in [26] and generalized to arbitrary in [14]. The resulting differential equations read

(2.15)

where we used rotational invariance to set . Imposing in-falling boundary conditions at the black-brane horizon, which corresponds to considering the retarded Green’s function, leads to the boundary condition

(2.16)

The functions as well as the resulting self-energy have various symmetries, which are discussed in more detail in appendix C.

From the imaginary part of the retarded Green’s function , the total spectral weight, or spectral function, can be obtained, which is of great importance in condensed-matter physics and is also directly observable in experiments. It is defined as the trace of the two-by-two matrix , i.e.,

(2.17)

It was shown in [14] that the spectral-weight function corresponding to (2.11) satisfies the sum rule

(2.18)

in the physically allowed range for the scale dimensions of the CFT operator coupled to . In [14], this range was found to be , where the lower bound is due to unitarity in the field theory, and the upper bound is a consequence of the requirement that the coupling of the elementary fermion to the CFT is irrelevant in the ultra-violet. Within this allowed physical range for , we have checked numerically that the Kramers-Kronig relations are satisfied and the sum rule (2.18) is obeyed. The latter is a consequence of the fact that is the Green’s function for an elementary field and its hermitian conjugate , that satisfy anti-commutation relations. In conclusion, the prescription (2.11) thus allows us to holographically compute single-particle Green’s functions, which are important from a condensed-matter perspective.

Figure 2: Illustration of our holographic construction of the self-energy.

To close this section, we summarize and illustrate our holographic construction in figure 2. In particular, equation describes chiral single-particle propagators on the boundary of the spacetime. These single-fermion excitations are modeled as dynamical sources coupled to the fermionic composite operators in the conformal field theory. These chiral single-fermions interact with each other, which is described via the coupling with the conformal field theory due to a fermionic excitation of opposite chirality that travels into the bulk spacetime, feels the bulk gravitational effects classically, and comes back to the boundary, forming the self-energy of the single fermions. In the ultra-violet (UV), the points where the chiral dynamical source emits and reabsorbs the fermion of opposite chirality are very close together and the latter fermion cannot travel far into the bulk. Hence, it only notices the flat background on the cut-off surface in the bulk spacetime, which leads to a free fermionic propagator in the UV. For the infra-red (IR) case, however, the fermion of opposite chirality can indeed travel far into the curved part of the bulk spacetime, and as a result the IR dynamics will be dominated by interactions and will generally not be free. This corresponds physically to the usual “energy-scale” interpretation of the extra spatial direction. For clarity, we recite the three different couplings that one should distinguish here.

  1. The first coupling is the analog of the ’t Hooft coupling that describes the effective coupling in the conformal field theory. This is assumed to be large in our model, hence the corresponding string states in the bulk are assumed to decouple.

  2. The second coupling is the inverse of Newton’s constant, , that is proportional to , where is the degree of the gauge group governing the conformal field theory. This is assumed to be large, allowing us to treat the fermion bulk action as a perturbation on the action of general relativity. In particular, we may ignore the back-reaction of the bulk fermions on the spacetime in the large- limit. This corresponds to the fact that the connected parts of four-point functions of composite operators in the conformal field theory vanish.

  3. Finally there is what we call , the coupling between the dynamical source and the dominant channel in the conformal field theory.

The first two coupling constants are implicit in our model, as the gravitational action is not specified. A strongly interacting conformal field theory is characterized by non-trivial scaling dimensions, different than the engineering dimensions. This is indeed the case modeled here. In order to produce a non-trivial self-energy for the single fermion, the first and last coupling constants should be large, at least of order 1. As a consequence, there is always a momentum-space region in the IR where the self-energy is dominant over the kinetic term in the single-fermion propagator . This is what we mean when saying that the single fermions are ”strongly interacting”. In the context of the Weyl semimetal, the single-particle propagator from models the excitations of one of the chiral cones.

3 Single-particle spectra

In the previous section, we have outlined the construction of the retarded single-particle Green’s function. In the present section, we now study this Green’s function for both the relativistic and non-relativistic case in more detail, and work out its physical properties. The zero-temperature results were first obtained in [14], and here we consider non-zero temperatures as well. However, let us stress that the systems we describe are always at zero chemical potential.

3.1 Relativistic case

In the case that the elementary fermion interacts with a relativistic CFT, the background is given by an AdS black brane and is described by the metric shown in equation (2.1) for . The mass range for the bulk Dirac fermion is then restricted to lie within the interval .

Zero temperature

For vanishing temperature and , the self-energy can be computed analytically for arbitrary frequencies and momenta . The result for the full retarded Green’s function has been worked out in [14] and reads

(3.1)

where we defined as well as the constant (for arbitrary values of )

(3.2)

Note that (3.1) can be extended into the complex plane by allowing for complex momenta , which is important when determining the pole structure of the Green’s function. Because of the branch point at , we need to introduce a branch cut which is taken to run from to for later convenience. Furthermore, we find that in order for (3.1) to be free of singularities in the upper half plane, i.e., to satisfy the Kramers-Kronig relations, we have to demand

(3.3)

which is derived in detail in appendix D.1. It is then straightforward to show that the Green’s function (3.1) also satisfies the sum rule (2.18), as it was done in [14] and summarized in appendices D.2 and D.3. Furthermore, as mentioned above, (3.1) is valid on the complex plane, with the prescription that on the real line the self-energy is found by using

(3.4)

This directly follows from (3.1) by noting that the region for real frequencies is obtained from the region by , that is and , while the region is obtained via . Note also that the Green’s function for has no imaginary part.

Non-zero temperature

We have studied the effects of non-zero temperature on the structure and form of the Green’s function by numerical integration of the Dirac equation. To illustrate our results, we have included figures 3 and 4 which show the total spectral-weight function (2.17) for and , respectively. Furthermore, we made a distinction between positive and negative masses , and we employed the rotational symmetry to set . Let us discuss these figures in some more detail:

(a)
(b)
Figure 3: Total spectral function for , and .
(a)
(b)
Figure 4: Total spectral function for , and .
  • First, we note that a non-vanishing temperature results in a smearing out of the features of the Green’s function, as can be seen by comparing figures 3 and 4.

  • For we observe a large spectral weight approximately at , which is due to the “light cone” of relativistic physics. In the zero-temperature case (3.4), the imaginary part of the Green’s function is strictly zero outside of the light cone in the region . For non-zero temperatures, there is a small contribution in this region, which increases with temperature.

    Note that the peak in the spectral weight is not caused by a pole in the Green’s function, but by the aforementioned branch cut. Therefore, it has no interpretation as a well-defined quasi-particle excitation, but it is still possible to approximately determine the dispersion relation, as it is shown in the next subsection.

  • For negative masses , the spectrum has a similar form. Indeed, there is again a large peak at . However, for low , the imaginary part of the Green’s function goes to zero at as . It has a maximum in between, which leads to a broad maximum in the spectrum. Its approximate location is determined analytically in section 3.1.3.

    We have also investigated the ratio of the location of the maximum and its width as a function of . This ratio is approximately constant, which means that its shape becomes broader as we increase its position, and vice versa. The spectral function at looks very similar to what is plotted in figure 7 for the non-relativistic case with . It is not possible to tune the parameters such that the peak is located sufficiently far away from the origin at while its maximum remains sharp. Therefore, we cannot interpret this feature in the spectral function as a massive particle. Again, we give some further analytic arguments of this statement at zero temperature in section 3.1.3.

Dispersion relation

In the last subsection 3.1.2, we have illustrated that the spectral weight is peaked at particular functions . In this section, we now investigate the corresponding dispersion relation of the theory. For vanishing temperature this can be done analytically via the single-particle Green’s function by solving

(3.5)

By definition, the solutions to this equation determine the dispersion relation of the would-be (quasi-)particle. The interpretation as a particle only becomes justified if the width is small compared to the energy of the particle. Indeed, a large peak in the spectral density function is only obtained when both Re() and are minimal, as follows from the identity

(3.6)

Let us first solve (3.5). Using the explicit expression (3.1), equation (3.5) generically gives two possible dispersion relations, namely

(3.7)

The first solution describes a free, massless relativistic excitation which is always present. The second equation in (3.7) should be studied separately in the two regions and . In the first case, using (3.4) we obtain the solution

(3.8)

Since and the left-hand side of (3.8) should be real, we conclude that there is no solution for . In the second case, , the solution reads

(3.9)

For , we do not find a solution since . On the other hand, for , we obtain

(3.10)

which is indeed close to the locations of the maxima in the spectral-weight function that were found for non-zero temperature. For zero temperature, equation (3.10) yields the exact result.

Now we can look at the width, by computing the imaginary part of the self-energy at the values (3.10). A straightforward calculation shows that, at zero temperature, we have

(3.11)

with given by (3.10). However, for momenta small compared to the gap (in the restframe of the would-be particle), we can approximate

(3.12)

The width is then comparable to the gap, and therefore the peak in the spectral density does not have a quasi-particle interpretation. One might try to make the width smaller by taking the value of close to the unitarity bound, . However, notice that then the gap also narrows down. We conclude therefore that no true quasi-particles exist.

3.2 Non-relativistic case

In principle, every value of the dynamical exponent can be considered using the prescription shown in (2.11). However, for our purposes we are mostly interested in where the elementary fermion interacts with a CFT exhibiting Lifshitz scale invariance.

Zero temperature

Unlike the relativistic case studied above, for an arbitrary dynamical exponent we were not able to obtain analytically for the both and non-vanishing. But it is possible to determine the zero-temperature result for and separately, and then restrict the general expression to a great extent.

Let us therefore start with the case and . From [14] we recall the expression for the Green’s function as

(3.13)

where we employed the definition (3.2). In the case with and we instead find [14]

(3.14)

where again denotes . The generic case at vanishing temperature when both and are non-zero can then be restricted as follows. Using the scaling and rotational symmetries of the self-energy (see appendix C.1), and defining for notational simplicity, we have

(3.15)

where and are complex functions of and only. Furthermore, we can derive conditions on the functions using the symmetry of under the change , as discussed in appendix C.4. The asymptotic behavior of the functions and in the limits and is fixed by the expressions (3.13) and (3.14) given above. We therefore have

(3.16)

Unfortunately, these conditions do not seem to be sufficient to determine the analytic form of (3.15). Therefore we have to study the Green’s function numerically. Yet, we can determine the qualitative form of the dispersion relation by analytic arguments as we describe below in section 3.2.3.

Non-zero temperature

We have studied the retarded Green’s function and the corresponding spectral-weight function for numerically as a function of , , and . Using the symmetries summarized in appendix C, that is chirality and particle-hole symmetry, we observe that the components of the spectral-weight function obey the following relations

(3.17)

Consequently, it suffices to consider separate components instead of the trace over chiral components.

Next, we recall that for the Green’s function (2.11) contains the parameter , which we have not determined analytically. Treating as a free parameter, we observe that under a change of sign of , the far UV behavior of one component of the spectral weight function asymptotes the UV behavior of the other component with the original sign, that is,

(3.18)

Since can be interpreted as a spin-orbit coupling constant, we use the following convention for plotting components of the spectral-weight function: when is positive (negative), the plus(minus) component of the spectral-weight function is shown. In this manner we make sure that we always compare spectra with equal group velocity in the UV, also when changes sign. We make this choice because qualitatively the UV physics is then, apart from the topology of the band structure, independent of the sign of . This allows us to compare more clearly the physics for positive and negative values of as we will see shortly.

In figures 5 and 6 we show numerical results for separate components of the corresponding spectral-weight function. The parameters are chosen as , and , for a number of different values for . We discuss some of the features in turn:

  • In figure 5, showing the results for positive , we see that the spectral-weight function is very sharply peaked in the UV and behaves as a free chiral fermion with a quadratic dispersion. In the IR, the self-energy becomes dominant which changes and smears out the form of the free-fermion dispersion due to strong interactions. When switches sign, the spectral-weight function changes in the IR, with the convention such that the far UV behavior stays the same. For , the band structure is similar to that of a Weyl semimetal, as can be seen by combining it with the density plot of for (not shown) and comparing it with figure 1. Indeed, the observed dependence and the absence of a gap are defining properties of the Weyl semimetal. For , the system is gapless too, and similar to a Weyl semimetal in the presence of Fermi surfaces.

  • In figure 6, where is negative, we observe a phenomenon similar to the relativistic case with . In particular, the spectral weight has to vanish for as , so there is a maximum in the spectral-weight function at non-zero . However, the ratio of the location and the width of the peak, , again remains constant as we change , so there is no gap generation with a quasi-particle interpretation. This is further illustrated in figure 7.

(a) Density plot of at
(b) Density plot of at
(c) Density plot of at
(d) Density plot of at
(e) Density plot of at
(f) Density plot of at
Figure 5: Density plot of for , , , .
(a) Density plot of at
(b) Density plot of at
(c) Density plot of at
(d) Density plot of at
(e) Density plot of at
(f) Density plot of at
Figure 6: Density plot of for , , , .

, , , blue linered linegreen line
Figure 7: Spectral-weight function for vanishing momentum determined from (3.13). Since the ratio of the location and the width of the peak remains constant ( in the figure), there is no quasi-particle interpretation.

To close our discussion about non-zero temperature effects in the case of , let us consider the strict IR (or hydrodynamic) limit , . As explained in appendix B.3, in this limit the contribution from the free propagator vanishes and the Green’s function reduces to the inverse of the self-energy . An analytical result for all values of the dynamical exponent and dimension and for non-zero temperature has been obtained for this case in equation (B.21), which we recall here for convenience

(3.19)

Dispersion relation

We now consider the dispersion relation for at zero temperature, which can be derived from the general form of the Green’s function given in (3.15). In particular, the dispersion relation is obtained by solving (3.5), which in the present case reads

(3.20)

In the following we consider the upper-spin component for definiteness. The latter means that is replaced by , where we again employed the rotational symmetry to align the momenta in the -direction. For positive (negative) momentum the dispersion relation then becomes

(3.21)

In order to determine the qualitative form of the dispersion relation, we study various limits of the equation above.

  • We first consider the UV limit , for which there are the three distinct possibilities: , and The first two do not allow for a solution of (3.21), but the third possibility leads to

    (3.22)

    in the allowed mass range . This result confirms our general picture that the interaction of the elementary fermion with the CFT is irrelevant in the UV, hence the dispersion becomes that of a free fermion.

  • Next, we consider the limit , . In the mass range we determine the dispersion curve from (3.13) as

    (3.23)

    Noting that and , we find that the only solution to this equation is for . On the other hand, for we obtain the only solution as

    (3.24)

    These solutions correspond to the points where the dispersion curve touches the axis at .

    Studying the dispersion relation for the case of and , we find another solution at , consistent with the particle-hole symmetry. This is the non-relativistic analog of the result we obtained in equation (3.10). As mentioned above, and similar to the case with , the corresponding peaks in the spectral-weight function cannot be interpreted as a true gapped quasi-particle excitation. This is further illustrated by a plot of the analytic result in figure 7.

  • The most interesting case is when and . As one can see from figure 5, this situation is quite different from and . The dispersion curve for crosses the axis three times, instead of only once at . It therefore seems that two Fermi surfaces appear at the momenta for which . This signals a phase transition as one crosses from positive to negative . The remainder of this paper is devoted to a detailed study of this phase transition.

4 Quantum phase transition

In figures 5 and 6 we have already seen that for the parameter controls features of the spectral density significantly. In particular, when switching the sign of the number of zeros of the dispersion relation changes, as becomes clear when comparing for instance figures 5(a) and 5(f). In the present section, we study this phenomenon in more detail. For simplicity we focus on the case since the quantum phase transition discussed here is also present for . However, we also briefly consider the case in section 4.5.

4.1 Momentum distribution

Let us start by defining the momentum distribution function. We choose again a convention such that the group velocity of a particle with one of the spin components always has the same sign at large momenta, irrespectively of the sign of . More concretely, with the Fermi distribution we write

(4.1)

The behavior of when changes sign is illustrated in figure 8 for non-zero temperatures, where we again employed the rotational symmetry to set . In particular, for the momentum distribution indicates a Fermi surface with a certain width, on which we comment later. When switches sign, two extrema appear that develop into sharp discontinuities as increases. This is a clear signature of a Fermi surface. The locations of the jumps are determined analytically in the next section.

(a)
(b)
(c) ,
(d) ,
(e) ,
(f) ,
Figure 8: Momentum distribution of the minus component for , , . For it has a smooth kink-like behavior whereas for two Fermi surfaces develop. The red dots give the analytic value (4.3) of . For small and negative , the Fermi surfaces are smeared out because of the non-zero value of the temperature.

4.2 Fermi momentum

We now study analytically how the number of putative Fermi surfaces changes as we vary the parameter . To do so, we recall that Fermi surfaces are determined by the poles of the Green’s function at vanishing frequencies . The latter can be computed by setting to zero the denominator in (2.11). Note also that the self-energy is given by (2.12) which satisfies the particle-hole symmetry derived from (C.16). Employing the general form of the self-energy implied by (3.15), we see that in the limit the imaginary part of vanishes and the condition for the presence of a Fermi surface coincides with the presence of zeros of the dispersion relation (3.5). Thus, the loci of the Fermi surfaces are given precisely by the points where the dispersion curve crosses the axis.

We therefore consider equation (3.21) in the limit . Using (3.16) and the expression (3.2) for , we obtain the following formula for the loci of the Fermi surfaces

(4.2)

Then, as we have mentioned in equation (3.3), to avoid violation of causality we have to require . Equation (4.2) can thus have non-trivial solutions in the range only when , that is

(4.3)

Next, let us consider again the numerical results for the imaginary part of the Green’s function shown in figure 5. For , there are zero-energy modes at non-vanishing momentum, suggesting that the system indeed has two Fermi surfaces at zero temperature. However, due to the small but non-zero temperature which we have to employ in our numerics, the spectrum shown in the plots has a finite width and the locations of the zero modes are approximately at the Fermi momentum (4.3). To investigate this point further and to confirm that we are indeed dealing with a genuine Fermi surface, in the following we compute the quasi-particle weight as a function of and , as well as the effective mass and the lifetime at the Fermi surface as a function of , , and .

Quasi-particle residue

In order to scrutinize the Fermi surfaces, in this subsection we determine the quasi-particle residue by linearizing the dispersion relation around the Fermi surface at , . For convenience, we choose the lower component of the Green’s function which results in the spectral-weight function shown in figures 5(d)-(f). Furthermore, we employ the rotational symmetry to set with , and thus in the following. We then compute up to first order in derivatives

(4.4)

Using (4.4) in the retarded Green’s function, close to the Fermi surface we obtain the expression

(4.5)

where the wavefunction renormalization factor, or quasi-particle residue , is given by

(4.6)

The residue is equal for both spin components because the derivatives of are equal at . Furthermore, note that depends on the coupling explicitly due to the factor in , but also implicitly via the dependence of on shown in (4.3). The dependence on is only through . To be able to determine the quasi-particle residue, the real part of the self-energy has to be linear in , which is indeed the case, as shown in figure 9. The first derivative of the real part of the self-energy is shown for in figure 10(a).

Let us illustrate the calculation of for a particular value of and , namely and . For , the first derivative of the real part of the self-energy has the non-zero value at . This leads to a finite quasi-particle residue of about , which is precisely the height of the step in the momentum distribution at , as shown in figure 11. Repeating this calculation for different values of , we have determined , which is shown in figure 12. Note that as is required. Furthermore, the large deviation from 1 of the quasi-particle residue for small and negative , demonstrates that we are indeed describing a strongly interacting system.

(a)
(b)
Figure 9: Imaginary and real part of the self-energy evaluated at the Fermi momentum for , , and . The imaginary part is zero around and the real part is linear around , which are both defining properties of a Fermi liquid.
(a)
(b)
Figure 10: This figure shows the first derivative of and the second derivative of , both at , , and , evaluated at the Fermi momentum.
Figure 11: The momentum distribution for , and . The quasi-particle residue is given by the distance between the dotted lines: the difference in height is the numerical value of , which is 0.56 for . The red dots give the analytic value of .
red curveblue curve
Figure 12: Wavefunction renormalization factor as a function of for , and for different temperatures. is only defined for negative . The quasi-particle weight lies between one and zero, as expected. It approaches unity for very large and negative values of , i.e., very far away from the phase transition. For it vanishes only strictly at . Due to non-zero temperature effects, the quasi-particle residue has a non-zero minimum at , which