On the renormalization of Coulomb interactions in two-dimensional tilted Dirac fermions
We investigate the effects of long-ranged Coulomb interactions in a tilted Dirac semimetal in two dimensions by using the perturbative renormalization-group method. Depending on the magnitude of the tilting parameter, the undoped system can have either Fermi points (type-I) or Fermi lines (type-II). Previous studies usually performed the renormalization-group transformations by integrating out the modes with large momenta. This is problematic when the Fermi surface is open, like type-II Dirac fermions. In this work, we study the effects of Coulomb interactions, following the spirit of ShankarShankar (), by introducing a cutoff in the energy scale around the Fermi surface and integrating out the high-energy modes. For type-I Dirac fermions, our result is consistent with that of the previous work. On the other hand, we find that for type-II Dirac fermions, the magnitude of the tilting parameter increases monotonically with lowering energies. This implies the stability of type-II Dirac fermions in the presence of Coulomb interactions, in contrast with previous results. Furthermore, for type-II Dirac fermions, the velocities in different directions acquire different renormalization even if they have the same bare values. By taking into account the renormalization of the tilting parameter and the velocities due to the Coulomb interactions, we show that while the presence of a charged impurity leads only to charge redistribution around the impurity for type-I Dirac fermions, for type-II Dirac fermions, the impurity charge is completely screened, albeit with a very long screening length. The latter indicates that the temperature dependence of physical observables are essentially determined by the RG equations we derived. We illustrate this by calculating the temperature dependence of the compressibility and specific heat of the interacting tilted Dirac fermions.
The Weyl fermions in solid state materials have attracted intense theoretical and experimental interests in condensed matter community in recent years. These materials are topological since the Weyl nodes, at which the conduction and the valence bands touch in the momentum space, act as the magnetic monopoles in the momentum spaceXWan (). Therefore, these Weyl nodes can be characterized by the “magnetic charges” they carry. On the other hand, many of the electromagnetic responses or the transport properties of the Weyl semimetal are deeply rooted in nontrivial phenomena in the quantum field theory, such as the chiral anomalyZyuzin (); ZWang (); CXLiu (); PGoswami (); Hosur (). Very recently, Weyl fermions have been detected experimentally in the non-centrosymmetric but time-reversal preserving materials such as TaAs, NbAs, TaP, and NbPCShekhar (); BQLv (); SYXu (); BLv (); LYang (); SYXu2 (); NXu ().
Due to the lack of a fundamental Lorentz symmetry, the spectra of Dirac/Weyl semimetals realized in solid state materials do not have to be isotropic. In particular, they can be tiltedAASoluyanov (). When the tilting angle is large enough, the electron and hole Fermi surfaces can coexist with the band-touching Dirac/Weyl nodes. This leads to a new kind of materials, which are commonly referred to as type-II Dirac/Weyl semimetalsAASoluyanov (). In three dimensions (D), the tilted Weyl cones were proposed to be realized in a material WTeAASoluyanov (), while in two dimensions (D), the tilted Dirac cones were proposed to be realized in a mechanically deformed graphene and the organic compound -(BEDT-TTF)IMOGoerbig (). Recently, type-II Dirac fermions are experimentally discovered in two materials: PdTeHJNoh (); FCFei () and PtTeMZYan ().
In contrast with type-I Dirac/Weyl semimetals whose Fermi surface is point-like, type-II Dirac/Weyl semimetals have an extended Fermi surface. In D, it consists of two straight lines crossing at the Dirac node, while it is hyperboloids touched at the Weyl nodes in D. Due to the nonvanishing density of states (DOS) at the Fermi energy, the physical properties of type-II Dirac/Weyl semimetals are expected to be distinct from those of type-I Dirac/Weyl semimetals. On the other hand, the open Fermi surface in the type-II materials will result in behaviors different from the usual Fermi liquid (FL) which has a closed Fermi surface. Even for type-I materials, the nonzero tilting parameter may lead to observable effects. Some theoretical studies along these directions mainly for non-interacting fermions have been performed, including the conductance and noise for tilted Dirac (D) or Weyl fermions (D)MTrescher (), longitudinal magnetoconductivity for tilted Dirac fermionsPros (), thermodynamic and optical responses for tilted Weyl fermions in the presence of magnetic fieldsTEOBrien (); ZMYu (); MUdagawa (); STchoumakov (), anomalous Hall effect for tilted Weyl fermionsAAZyuzin (); JFSteiner (), and anamolous Nernst and thermal Hall effect for tilted Weyl fermionsFerreiros (); Saha ().
In the present work, we study the effects of long-range Coulomb interactions on the tilted Dirac fermions in terms of the renormalization group (RG). Due to the vanishing DOS at the Fermi level, it is well-known that the Coulomb interaction for untilted Dirac fermions in D is marginally irrelevant in the sense of RGJGonzalez (); JGonzalez2 (); DTSon (). Moreover, the value of Fermi velocity will increase at low enegies. This leads to logarithmic corrections to various thermodynamic response functions, which can be computed in terms of the RG equations. The results fit the experimental data for graphene quite wellDESheehy ().
What will happen for Dirac fermions in D when the tilting parameter is not zero? Previous RG analysisHIsobe (); ZMHuang () shows that the values of velocities also increase logarithmically toward infinity, as what happens in graphene. By taking into account the fluctuations of transverse gauge fields, the velocities of fermions are renormalized up to the speed of light at low energiesHIsobe2 (). Moreover, according to the analysis in Refs. HIsobe2, and ZMHuang, , the magnitude of the tilting parameter flows to zero at low energies for both types of Dirac fermions. For type-II Dirac fermions, this indicates its instability in the presence of Coulomb interactions. Within such a scenario, interacting type-II Dirac fermions is stable only when the screening length is short enough so that the above RG flow stops and turns its direction before the instability occurs.
All the above mentioned RG studies on tilted Dirac fermions employ some types of cutoff functions in the momenta, such as a direct momentum cutoff or the dimensional regularization. This is certainly fine for type-I Dirac fermions since the Fermi surface is a point. However, for systems with open Fermi surface like type-II Dirac fermions, any momentum cutoff function, which is a curve in the momentum space, will intersect with the Fermi surface. By integrating out the modes with momenta larger than the mometum curoff to implement the RG transformation, one integrates out not only the high-energy modes but also the low-energy modes around the Fermi surface. Hence, the RG transformation is not to scale to the Fermi surface, and the conclusions about type-II Dirac fermions inferring from such a RG analysis is dubious.
As emphasized by ShankarShankar (), the proper way of doing RG for FL should be to scale toward the Fermi surface. The usual FL has a closed Fermi surface. Therefore, there exists a characteristic momentum, the Fermi momentum, to separate the high-energy modes from the low-energy modes. For type-II Dirac fermios, its Fermi surface is open so that such a characteristic momentum simply does not exist. To implement the RG transformation properly, we label the excitations directly by their energies and other ”angle” variables. By integrating out the high-energy modes, we obtainn the RG functions for velocities of fermios and the tilting parameter. Our main findings are as follows.
(i) For type-I Dirac fermions, our results are identical to the previous ones. That is, the values of velocities and the magnitude of the tilting parameter monotonically increase and decrease at low energies, respectively. This is not surprising since the Fermi surface is point-like in this case.
(ii) For type-II Dirac fermions, the values of velocities are also increase monotonically at low energies. In contrast with the previous studies, the RG functions for the two velocities of fermions are different so that they have different RG flow even if the bare values are the same. Moreover, the magnitude of the tilting parameter increases as lowering the energy, indicating the stability of type-II Dirac fermions. In other words, the long-range Coulomb interaction helps to stabilize type-II Dirac fermions, in contrast with previous studies.
(iii) In terms of the RG equations, we calculate various thermodynamic response functions for both types of Dirac fermions: the temperature dependence of specific heat at low temperatures, and the density and temperature dependence of the isothermal compressibility.
(iv) Since there is a nonzero DOS at the Fermi level for type-II Dirac fermions, it is argued that the Coulomb interaction will be screened at long distances.HIsobe2 (); ZMHuang () In order to address this issue, we calculate the vacuum polarization at zero frequency. We find that the polarization function is singular at zero momentum for both types of Dirac fermions. This singularity can be removed if we sum the diagrams with leading divergences in terms of RG, following the method employed in Ref. RRBiswas, . From this RG-improved polarization function, we find that there is indeed complete screening for type-II Dirac fermions, albeit with a very long screening length. This indicates that the scaling of physical observables with temperatures in a large temperature range is indeed described by the RG equations we derived in this paper.
The organization of the rest of the paper is as follows. The model is defined and discussed in Sec. II. We present the RG equations and its implications in Sec. III. The calcutations of thermodynamic response functions for both types of Dirac fermions are shown in Sec. IV. Sec. V is about the screening of the Coulomb potential and the Coulomb impurity problem. The last section is devoted to conclusive discussions. The details of calculations of the RG equations and the vacuum polarization are put in appendix A and B, respectively.
Ii The model
We start with the minimal model of non-interacting tilted Dirac fermions described by the HamiltonianMOGoerbig ()
where denote the valley degeneracy, account for the spin degeneracy, and are the standard Pauli matrices. The fields and , which obey the canonical anticommutation relations, describe Dirac fermions around the Dirac nodes at the points and in the first Brillouin zone (BZ). and are the “speeds” of Dirac fermions along the and directions, respectively. Without loss of generality, we take . In Eq. (1), we have set the energy of the Dirac nodes to be zero. The dimensionless constant is the tilting parameter.
The spectrum of is given by
where with . It is clear that the Dirac cones are tilted along the -axis when , as shown in Fig. 1. When , the Dirac cones are slightly tilted, corresponding to type-I Dirac fermions, while the Dirac cones are tipped over when , corresponding to type-II Dirac fermions. are the Lifshitz transition points which separate the two types of Dirac fermions. When the chemical potential , the Fermi surface consists of two straight lines:
for type-II Dirac fermions, where .
The Coulomb interaction between electrons is described by the Hamiltonian
where is the normal-ordered electron density operator with the annihilation operator and creation operator for electrons with spin-, and
is the Coulomb potential. In Eq. (5), is the dielectric constant and is the charge carried by the electron. In terms of the low-energy degrees of freedom around the Dirac nodes, the electron operator can be written as
where the fermion field and is the area of the system. With the help of Eq. (6), the density operator can be written as
and contains those terms with the factors or . Due to the fast oscillating nature, these terms will generate short-ranged repulsive four-fermion interactions at low energies, and we shall neglect them.
Our working Hamiltonian is . When , is invariant against the “particle-hole” (PH) transformation
This PH symmetry forbids terms like , , or since they are odd under the PH transformation. We shall see later that this PH symmetry together with gauge invariance guarantee the renormalizablity of this theory.
Iii The RG equations
To derive the RG equation, we perform a Hubbard-Stratonovich transformation so that the action in the imaginary-time formulation can be written as
where , , , , , and denotes the Fourier transform of . Since the auxillary field is real, . We have extended the number of fermion fields to pairs such that . Physically, due to the spin degeneracy. still preserves the PH symmetry as long as we require that transforms as
under the PH transformation.
As we have discussed in the introduction, the proper way to implement the RG transformation for a system with an open Fermi surface is to integrate out an energy shell each time, instead of a momentum shell. To achieve this gola, we have to parametrize the euqla-energy curves first. These curves are given by the equations . For type-I Dirac fermions (), these equal-energy curves are ellipses and can be parametrized as
for given , where . On the other hand, for type-II Dirac fermions (), these equal-energy curves are hyperbolas and can be parametrized as
for given , where . In Eq. (13), the and signs correspond to the right and the left branches of the hyperbola, respectively.
for type-I Dirac fermions, and
for type-II Dirac fermions, where is the UV cutoff in energies and . In Eq. (15), the first and the second integrals for given correspond to the integrations over the right and the left branches of the hyperbola, respectively. In fact, it suffices to consider the integrals over or since the involved two bands have been taken into account by the Pauli matrices. However, this regularization breaks the PH symmetry at . Hence, we define the momentum integral by Eqs. (14) or (15), and include a prefactor .
Before plunging into the calculation of the RG equations, we discuss some constraints on the renormalization of the various terms in the action . First of all, is invariant against the gauge transformation:
By integrating out the fast modes which have energies within the range with , becomes
where denotes the terms with higher scaling dimensions. Here we do not explicitly write down the renormalized Hamiltonian since it is irrelevant in the derivation of the Ward identity. The gauge invariance leads to the Ward identityJGonzalez (); GYCho ()
Next, the non-analytic dependence of the term on prohibits its renormalization under integrating out the fast modesJGonzalez (); GYCho (). Thus, it is not necessary to introduce the wavefunction renormalization of the field. This fact together with Eq. (17) result in the non-renormalization of .
We now integrate out the fast modes to the one-loop order to get the RG equations by assuming the weak-coupling limit . From the above discussions, it suffices to compute the self-energy of fermions. After integrating out the fast modes and rescaling the energy, angle variable, frequency, and fields by , , , , and , we find the following facts: (i) to the one-loop order, which leads to for both types of Dirac fermions. (ii) and acquire nontrivial renormalization. (iii) is not renormalized to the one-loop order for type-I Dirac fermions, while it acquires nontrivial renormalization for type-II Dirac fermions. The details of the claculations are left to appendix A.
The resulting one-loop RG equations for type-I Dirac fermions are
where the quantities with the subscript indicate the renormalized parameters, while those without the subscript correspond to the bare ones. Here we have taken . Equation (18) is identical to the one in Ref. HIsobe, . On the other hand, the one-loop RG equations for type-II Dirac fermions are
Since , Eqs. (19) and (20) indicate that the values of and become large at low energies. This implies that the dimensionless coupling is irrelevant around the Gaussian (non-interacting) fixed point, and thus justifies our perturbative calculations.
The RG flows for the ratio and the tilting parameter are shown in Figs. 2 and 3, respectively. We see that although both and increase at low energies, the rate of change of is faster than that of so that the ratio decreases at low energies. Therefore, even if we take , at . This is different from type-I Dirac fermions where if . On the other hand, the value of always increases at low energies. This implies the stability of type-II Dirac fermions against weak Coulomb repulsions. Our result is distinct from the conclusion in Ref. ZMHuang, .
Iv Thermodynamics at low temperaures or small densities
Now we are in a position to extract the temperature or density dependence of various thermodynamic response functions at low temperatures or small densities with the help of the RG equations, following the method proposed in Ref. DESheehy, .
Before doing it, we have to determine the RG flows of the temperature and the chemical potential . At finite temperature, we find that
according to the rescaling of . Hence, we get or
The solution of Eq. (22) is .
To extract the scaling equation for , we add the term
to the action. By integrating out the fast modes, this term becomes
to the one-loop order. Performing the rescaling of , , , and , we find that
That is, or
The solution of Eq. (23) is .
The first physical quantity we would like to study is the isothermal compressibility , which is defined as where is the average density of electrons. Since is a physical quantity, upon renormalization, we find that
where . We may regard as the average density of the renormalized system where the effective coupling is small and the effective temperature is high. With an appropriated choice for the renormalization scale , we can put the renormalized theory into a regime in which the calculation becomes simple.
From Eq. (24) and the solutions of Eqs. (22) and (23), we find that . We first consider the case with , and run the RG flow to the scale such that where is the bandwidth which is the UV cutoff in energies for the model we use to describe the Dirac fermions. From the solution of Eq. (22), we have . In terms of this expression, we get where denotes the renormalized variable at scale . Since the effective coupling is irrelevant, we may replace by the result of non-interacting fermions.
For type-I Dirac fermions, we have
As a result, we get
Equation (25) reduces to the one for graphene when DESheehy (). It indicates that the compressibility at given temperature is enhanced due to a nonzero tilting parameter. On the other hand, for type-II Dirac fermions, we have
Figure 4 exhibits temperature dependence of in units of for both types of Dirac fermions, i.e.,
for type-I Dirac fermions, and
for type-II Dirac fermions, where
is the compressibility at . We see that for both types of Dirac fermions, the compressibility at low temperatures is suppressed by the Coulomb interaction through the enhancement of the velocities. In particular, the compressibility is a constant for non-interacting type-II Dirac fermions. Hence, the behavior of at low temperatures deviates from the one of non-interacting fermions indicates the effect of Coulomb interactions.
Next, we consider the case with , and run the RG flow to the scale such that , leading to . For type-I Dirac fermions, we have
which leads to
and . From Eq. (27), we find that
where , and thus we obtain
On the other hand, for type-II Dirac fermions, we have
Hence, we get
From Eq. (29), we find that . Substituting this expression into , we obtain
where and .
Now we determine the temperature dependence of the specific heat at constant volume, which can be extracted from the the free energy density through the relation . Upon renormalization, we find that
Hence, we get where .
For type-I Dirac fermions, we have
which leads to
which results in
We have to emphasize that the specific heat for type-II Dirac fermions at low temperatures is not linear in because , , and are functions of . This deviation from the linear behavior suggests the effect of Coulomb interactions.
V Screening of the Coulomb potential
One may wonder whether or not the above RG flows we obtained are cut at low energies due to the screening of the long range Coulomb interaction. To answer this question, we compute the vacuum polarization to the one-loop order, where the vacuum polarization is defined by the Dyson equation
In Eq. (34), is the the full propagator of the field and is the bare propagator as well as the Fourier transform of the bare Coulomb potential . The Fourier transform of the renormalized Coulomb potential is then given by .
v.1 The screened Coulomb potential
As we have shown, our RG equations for type-I Dirac fermions are identical to those by regularizing the theory with a momentum cutoff. Hence, we may compute in terms of dimensional regularizationDTSon (), yielding
When , Eq. (35) reduces to the one for grapheneDTSon (). On the other hand, for type-II Dirac fermions, we have to employ our parametrization for momenta [Eq. (13)]. An exact evaluation of is difficult. Fortunately, to answer the question of screening, it suffices to determine , and we find that for
where is the band width, , is the DOS at the Fermi level, and is a nonuniversal constant. The details of the calculations are left to appndix B.
for type-II Dirac fermions and type-I Dirac fermions with . This implies that is a singular point of as well as .
For type-II Dirac fermions, we remove this singularity by summing the leading divergent diagrams, following the method employed in Ref. RRBiswas, . Since this theory is renormalizable, this can be achieved by replacing and in Eq. (36) by the energy dependent functions and and scaling and to the energy scale . Thus, the polarization function becomes
As we have shown, is an increasing function of , while is a decreasing function of . Moreover, the product increases with increasing . Hence, at low energies (corresponding to small momenta), we may take such that Eq. (37) can be approximated as . In terms of this expression, becomes
which holds for , where is the Thomas-Fermi wavenumber. When , the screened Coulomb potential behaves like
instead of the bare one . This indicates that the RG flow we have obtained holds only when the energy scale is much larger than , where we have taken . On account of the renormalization of the Coulomb interaction, is smaller than the value for non-interacting fermions, and thus we expect that our results for the compressibility and specific heat hold for a large temperature range. Notice that the anisotropy of the screened Coulomb potential due to tilting is removed at long distances.
v.2 Coulomb impurity
In terms of the above polarization function, we can also study the Coulomb impurity probelm for tilted Dirac fermions. Consider an impurity of charge located at the origin, where the charge carried by an electron is . The charge density induced by this impruity is given by
For type-II Dirac fermions, we employ Eq. (38) and get
which holds only for . The total induced charge is then given by
which implies the complete screening of the impurity charge. The presence of a nonvanishing screening length and the complete screening of the impurity charge rely on a nonzero DOS at the Fermi energy. In contrast with the usual FL with a finite Fermi momentum, for type-II Dirac fermions, we have to sum the leading divergent diagrams beyond the RPA approximation.
For type-I Dirac fermions, we obtain