Soft Modes and Nonanalyticities in a Clean Dirac Metal

TL;DR

This paper investigates how electron-electron interactions in a clean Dirac metal lead to unique soft excitations and nonanalytic behaviors in physical observables, distinguishing Dirac Fermi liquids from conventional Fermi liquids.

Contribution

It introduces the concept of a Dirac Fermi liquid and analyzes the nonanalytic effects of soft modes on physical properties, supported by general arguments and explicit calculations.

Findings

Identification of soft two-particle excitations in Dirac metals.

Nonanalytic dependencies of observables like density of states and susceptibility.

Comparison of nonanalyticities between Dirac and Landau Fermi liquids.

Abstract

We consider the effects of the electron-electron interaction in a clean Dirac metal, i.e., a Dirac system with the chemical potential not tuned to the Dirac point. We introduce the notion of a Dirac Fermi liquid (DFL) and discuss the soft two-particle excitations in such systems, which are qualitatively different from the corresponding excitations in a conventional, or Landau, Fermi liquid (LFL). These soft excitations lead to nonanalytic dependencies of observables, including the density of states, the spin susceptibility, and the specific heat, on the temperature, magnetic field, and wave number, which first appear at second order in the interaction. We determine and discuss the nature of these nonanalyticities in a DFL and compare them with the corresponding effects in a LFL. Our results are based on very general arguments regarding the existence and nature of the soft modes, and corroborated by explicit perturbative calculations. We also discuss their consequences for magnetic quantum phase transitions in Dirac metals.

Figures

Equations

$$H_v=v({\pi }_3\otimes \mathbf{\sigma })\cdot \mathbf{k}.$$

$$H_{\Delta }=\Delta ({\pi }_1\otimes {\sigma }_0).$$

$$H_{ϵ}=({ϵ}_{\mathbf{k}}-\mu )({\pi }_0\otimes {\sigma }_0).$$

$$H_0$$

$$-h({\pi }_0\otimes {\sigma }_3).$$

$$E_{\mathbf{k}}^{1\pm }$$

$$E_{\mathbf{k}}^{2\pm }$$

$$k_{\text{F}}^2=2m^2\left(\mu /m+v^2\pm \sqrt{(\mu /m+v^2{)}^2-{\mu }^2/m^2}\right).$$

$$k_{\text{F}}^{\pm }=\sqrt{2m\mu }\pm v/2m+O(v^2)(v^2\ll \mu /m).$$

$$k_{\text{F}}=\mu /v(v^2\gg \mu /m).$$

$$G_n(\mathbf{k})={\left[i{\omega }_n({\pi }_0\otimes {\sigma }_0)-H_0\right]}^{-1},$$

$$G_n^{\Delta =0}(\mathbf{k})={\pi }_{+}\otimes {\mathfrak{g}}_n(\mathbf{k},\mathbf{h})+{\pi }_{-}\otimes {\mathfrak{g}}_n(-\mathbf{k},\mathbf{h})$$

$${\mathfrak{g}}_n(\mathbf{k},\mathbf{h})=\frac{(i{\omega }_n-{\xi }_{\mathbf{k}}){\sigma }_0+(v\mathbf{k}-\mathbf{h})\cdot \mathbf{\sigma }}{(i{\omega }_n-{\xi }_{\mathbf{k}}{)}^2-(v\mathbf{k}-\mathbf{h}{)}^2}$$

$$\left(G_{n}^{\Delta=0}({\bm{k}})\right)^{\alpha\alpha^{\prime}}=\delta_{\alpha\alpha^{\prime}}\,\frac{(i\omega_{n}-\xi_{\bm{k}})\sigma_{0}+(\alpha v{\bm{k}}-{\bm{h}})\cdot{\bm{\sigma}}}{(i\omega_{n}-\xi_{\bm{k}})^{2}-(\alpha v{\bm{k}}-{\bm{h}})^{2}}\$$

$$G_{n}^{h=0}({\bm{k}})=\frac{1}{(i\omega_{n}-\xi_{\bm{k}})^{2}-v^{2}{\bm{k}}^{2}-\Delta^{2}}\ \bigl{[}(\pi_{0}\otimes\sigma_{0})(i\omega_{n}-\xi_{\bm{k}})+(\pi_{3}\otimes{\bm{\sigma}})\cdot v\,{\bm{k}}+(\pi_{1}\otimes\sigma_{0})\Delta\bigr{]}$$

$$G_{n}({\bm{k}})\approx\frac{\left[(i\omega_{n}-\xi_{\bm{k}})^{2}-(v^{2}{\bm{k}}^{2}+\Delta^{2})\right]\bigl{[}(\pi_{0}\otimes\sigma_{0})(i\omega_{n}-\xi_{\bm{k}})+(\pi_{3}\otimes{\bm{\sigma}})\cdot v\,{\bm{k}}+(\pi_{1}\otimes\sigma_{0})\Delta\bigr{]}}{\left[(i\omega_{n}-\xi_{\bm{k}})^{2}-(v^{2}{\bm{k}}^{2}+\Delta^{2}+h^{2}-2h\sqrt{v^{2}k_{z}^{2}+\Delta^{2}})\right]\left[(i\omega_{n}-\xi_{\bm{k}})^{2}-(v^{2}{\bm{k}}^{2}+\Delta^{2}+h^{2}+2h\sqrt{v^{2}k_{z}^{2}+\Delta^{2}})\right]}$$

$${\left(G_n^{\Delta =0}\right)}^{\alpha {\alpha }^{\prime }}(\mathbf{k})\approx {\delta }_{\alpha {\alpha }^{\prime }}{G}_n^{\alpha }(\mathbf{k})$$

$$G_n^{\alpha }(\mathbf{k})=\frac{1}{2}\left(F_n^{\alpha -}(\mathbf{k})M(-\alpha \hat{\mathbf{k}})+F_n^{\alpha +}(\mathbf{k})M(\alpha \hat{\mathbf{k}})\right)$$

$$F_n^{\alpha \pm }(\mathbf{k})=\frac{1}{i{\omega }_n-{\xi }_{\mathbf{k}}\pm |\alpha v\mathbf{k}-\mathbf{h}|}$$

$$M(\hat{\mathbf{k}})={\sigma }_0-\mathbf{\sigma }\cdot \hat{\mathbf{k}}.$$

$${\left(G_n^{v=\Delta =0}\right)}^{\alpha {\alpha }^{\prime }}(\mathbf{k})={\delta }_{\alpha {\alpha }^{\prime }}{G}_n(\mathbf{k})$$

$$G_n(\mathbf{k})=\frac{1}{2}\left(G_{n\downarrow }(\mathbf{k})M(\hat{\mathbf{h}})+G_{n\uparrow }(\mathbf{k})M(-\hat{\mathbf{h}})\right).$$

$$G_{n\uparrow\downarrow}({\bm{k}})=F_{n}^{\alpha\pm}({\bm{k}})\Bigl{|}_{v=0}=\frac{1}{i\omega_{n}-\xi_{\bm{k}}\pm|{\bm{h}}|}\ .$$

$$M(\hat{\mathbf{k}})M(\hat{\mathbf{k}})$$

$$M(\hat{\mathbf{k}})\mathbf{\sigma }M(\hat{\mathbf{k}})$$

$$M(\hat{\mathbf{k}})M(-\hat{\mathbf{k}})$$

$$M(\hat{\mathbf{k}})\mathbf{\sigma }M(-\hat{\mathbf{k}})$$

$$\text{tr}\left(M(\hat{\mathbf{k}})M(\hat{\mathbf{p}})\right)$$

$$S_0=\sum _k\sum _{\sigma {\sigma }^{\prime }}\sum _{\alpha {\alpha }^{\prime }}{\bar{\psi }}_{\sigma }^{\alpha }(k)\left[i{\omega }_n{\delta }_{\sigma {\sigma }^{\prime }}{\delta }_{\alpha {\alpha }^{\prime }}-(H_0{)}_{\sigma {\sigma }^{\prime }}^{\alpha {\alpha }^{\prime }}\right]{\psi }_{{\sigma }^{\prime }}^{{\alpha }^{\prime }}(k).$$

$$S_{0}=\sum_{k}\sum_{\alpha}\Bigl{(}\bar{\psi}^{\alpha}(k),\bigl{(}(i\omega_{n}-\xi_{\bm{k}})\sigma_{0}+h\sigma_{3}-\alpha\,v\,{\bm{\sigma}}\cdot{\bm{k}}\bigr{)}\psi^{\alpha}(k)\Bigr{)}\ \\$$

$$+\sum _k\sum _{\alpha \ne {\alpha }^{\prime }}\left({\bar{\psi }}^{\alpha }(k),\Delta {\sigma }_0{\psi }^{{\alpha }^{\prime }}(k)\right).$$

$$S_{\text{int}}$$

$$S_{\text{DL}}=S_0+S_{\text{int}}.$$

$$G_{n_1{\sigma }_1}(\mathbf{k}+\mathbf{q}/2)G_{n_2{\sigma }_2}(\mathbf{k}-\mathbf{q}/2)$$

$$G_{\pm }(\mathbf{k},\omega \pm i0)=\frac{1}{\omega -{\xi }_{\mathbf{k}}\pm |h|}\mp i\pi \delta (\omega -{\xi }_{\mathbf{k}}\pm |\mathbf{h}|)$$

$$F_{n_1}^{{\alpha }_1{\beta }_1}(\mathbf{k}+\mathbf{q}/2)F_{n_2}^{{\alpha }_2{\beta }_2}(\mathbf{k}-\mathbf{q}/2)=-\frac{F_{n_1}^{{\alpha }_1{\beta }_1}(\mathbf{k}+\mathbf{q}/2)-F_{n_2}^{{\alpha }_2{\beta }_2}(\mathbf{k}-\mathbf{q}/2)}{i{\Omega }_{n_1-n_2}-\mathbf{k}\cdot \mathbf{q}/m+{\beta }_1|{\alpha }_{1}v(\mathbf{k}+\mathbf{q}/2)-\mathbf{h}|-{\beta }_2|{\alpha }_{2}v(\mathbf{k}-\mathbf{q}/2)-\mathbf{h}|}$$

$${\phi }_{{\beta }_1{\beta }_2}^{{\alpha }_1{\alpha }_2}(\mathbf{q},i{\Omega }_n;i{\omega }_m)=\frac{1}{V}\sum _{\mathbf{k}}{F}_k^{{\alpha }_1{\beta }_1}{F}_{k+q}^{{\alpha }_2{\beta }_2}$$

Full text

††preprint: arXiv:1812.04592

I Introduction

The electronic spectrum of solids in the vicinity of special points in the Brillouin zone where two bands touch has been the subject of investigations since the early days of solid-state physics.Herring (1937); Abrikosov and Beneslavskii (1970) If one of the touching bands is a conduction band and the other a valence band, and if the chemical potential is tuned to the touching point (sometimes called a diabolical point), one has the situation of a semimetal, a material that in a certain sense is in between a metal and an insulator.Lifshitz and Pitaevskii (1991) Even if the chemical potential lies within the conduction band, the physics that underlies the existence of the touching point can lead to unusual properties. Much interest in recent years has focused on topological properties of such materials, which manifest themselves on the surface, and their relation to topological insulators.Wan et al. (2011); Yang et al. (2011); Burkov and Balents (2011) We will focus on a different aspect, namely, bulk properties such as the density of states and the magnetic susceptibility for which topological considerations are not important, but which nonetheless are strongly influenced by the spin-orbit coupling that also leads to the existence of a touching point.

A special case that is of conceptual interest is a system where the spin-orbit interaction leads to a linear crossing of doubly degenerate bands, such that the effective Hamiltonian in the vicinity of the crossing point is reminiscent of a massless Dirac Hamiltonian,Zhang et al. (2009) although the effects we will discuss persist if the crossing point is gapped out (i.e., if the Dirac system is massive rather than massless). We will consider a situation where the chemical potential lies within the conduction band, so that he material is a true metal with an extended Fermi surface (as opposed to a semimetal with a point-like Fermi surface) and well-defined quasi-particles. In this sense, we will consider a Fermi liquid. However, the spin-orbit interaction that leads to the band crossing, whether or not the latter is gapped out, modifies the properties of the Fermi liquid in ways that make it qualitatively different from an ordinary Landau Fermi liquid (LFL) that is realized in the absence of a spin-orbit interaction. This leads us to the notion of a Dirac Fermi liquid (DFL), which differs from a LFL in the single-particle spectrum, but more importantly in two-particle correlations that are important for the behavior of various thermodynamic properties, as well as the density of states and transport coefficients. As we will see, the spectrum of soft two-particle correlations is qualitatively different in a DFL than in a LFL. Interestingly, this is true even if the spin-orbit interaction is weak and qualitatively affects the single-particle spectrum only asymptotically close to the Dirac point (in gapless systems), or not at all (in gapped ones). By ‘Dirac Metal’ we mean a metal whose conduction electrons form a DFL.

It has long been realized that generic soft modes in condensed-matter systems lead to generic scale invariance and nonanalyticities in the thermodynamic and transport properties as functions of the wave number, frequency, temperature, or external fields.Belitz et al. (2005) Prominent examples include long-time-tail effects in classical fluids,Pomeau and Resibois (1975); Dorfman (1975); Dorfman et al. (1994) the long-range nature of the longitudinal susceptibility in the ordered phase of a classical Heisenberg ferromagnet,Vaks et al. (1967); Brézin and Wallace (1973) and weak-localization effects in disordered electron systems.Lee and Ramakrishnan (1985); Altshuler and Aronov (1984) More recently, these effects have been shown to have dramatic consequences for quantum phase transitions. A prime example is the quantum ferromagnetic transition in (Landau) metals, which is discontinuous as a result of soft modes in a LFL at zero temperature, rather then continuous as the corresponding transition in classical systems.Brando et al. (2016)

We will focus on effects in a Dirac Metal that are universal in the sense that they are determined by the properties of the electron system in the limit of long wavelengths and small frequencies (the “hydrodynamic regime”) and hence independent of detailed microscopic properties of the material other than the underlying Dirac crossing point. Of particular importance are excitations that are soft or massless in this regime, as they can lead to nonanalytic behavior of observables as a function of wave number, frequency, or an external field. We will discuss the existence and nature of such soft modes in a DFL in some detail, and compare and contrast them with the analogous soft modes in a LFL. We will then explore the consequences for the density of states, the spin susceptibility, and the specific heat in a Dirac Metal.

II Model

II.1 Single-particle Hamiltonian, and symmetries

A linear crossing of two bands can be described by a term $\pm v{\bm{k}}\cdot{\bm{\sigma}}$ in the single-particle Hamiltonian in momentum space,Abrikosov and Beneslavskii (1970) where ${\bm{\sigma}}\equiv(\sigma_{1},\sigma_{2},\sigma_{3})$ are the spin Pauli matrices, and $v$ is a characteristic velocity whose value reflects the strength of the spin-orbit interaction. The sign of this term reflects the chirality of the electrons: plus and minus describe left-handed and right-handed electrons, respectively. This term is invariant under time reversal (TR), which changes the sign of both the momentum ${\bm{k}}$ and the angular momentum ${\bm{\sigma}}$. Spatial inversion (SI) changes the sign of ${\bm{k}}$ as well as the chirality, but not the sign of ${\bm{\sigma}}$, so in a system with spatial inversion symmetry the relevant part of the Hamiltonian must include both chiralities.chi If we represent the chirality degree of freedom by a second set of Pauli matrices ${\bm{\pi}}=(\pi_{1},\pi_{2},\pi_{3})$, the part of the Hamiltonian that represents the spin-orbit interaction can be written

[TABLE]

The most general single-particle Hamiltonian that is invariant under both TR and SI also includes a term that mixes the chiralities in a symmetric way,

[TABLE]

with $\sigma_{0}$ the $2\times 2$ unit matrix in spin space, and the usual band Hamiltonian

[TABLE]

Here $\pi_{0}$ is another copy of the $2\times 2$ unit matrix, and $\epsilon_{\bm{k}}$ is the single-particle energy which for small ${\bm{k}}$ is bilinear in the components of ${\bm{k}}$. For simplicity we will assume $\epsilon_{\bm{k}}$ to be isotropic, $\epsilon_{\bm{k}}={\bm{k}}^{2}/2m$, with $m$ an effective mass. $\mu$ is the chemical potential. $H_{\epsilon}+H_{v}+H_{\Delta}$ is the most general single-particle Hamiltonian for a system that is invariant under both TR and SI and whose spin-orbit interaction is described by $H_{v}$; it is equivalent to the effective Hamiltonian derived in Refs. Zhang et al., 2009; C-X. Liu et al., 2010 for the Bi2Se3 family of topological insulators. In our simple isotropic model we also have invariance under simultaneous rotations in spin space and real space, which leaves ${\bm{\sigma}}\cdot{\bm{k}}$ invariant. Coupling the fermions to an external magnetic field ${\bm{h}}=(0,0,h)$ via a Zeeman term (which breaks TR, of course), we arrive at a single-particle Hamiltonian

[TABLE]

II.2 Single-particle spectrum, Green function, and action for a Dirac gas

II.2.1 Single-particle spectrum

The single-particle spectrum is readily obtained by finding the eigenvalues of the $4\times 4$ Hamiltonian $H$. The single-particle energy $E_{\bm{k}}$ has four branches:

[TABLE]

In order to discuss the various possibilities for different ranges of parameter values, let us introduce an atomic-scale momentum $p_{0}$ (on the order of an inverse lattice spacing), velocity $v_{0}=p_{0}/2m$, and energy $E_{0}=p_{0}^{2}/2m$. We then measure $E_{\bm{k}}$, $\Delta$, and $h$ in units of $E_{0}$, $v$ in units of $v_{0}$, and ${\bm{k}}$ in units of $k_{0}$.

Let us start with the case of a small spin-orbit coupling. For $v=0$ the spectrum consists of four degenerate parabolic bands, two for each spin projection and two for each chirality. A small spin-orbit coupling $v$ splits the bands, for $\Delta=h=0$ the Fermi surface for a nonzero chemical potential consists of two two-fold degenerate sheets, and the split bands display a linear crossing at $k=0$, see Fig. 1(a). $\Delta\neq 0$ gaps out that band crossing, Fig. 1(b), whereas $h\neq 0$ lifts the two-fold degeneracy and splits the crossing band pairs vertically, Fig. 1(c). If both $\Delta$ and $h$ are nonzero the band crossings are shifted to nonzero values of $k$ if $h>\Delta$, and they disappear if $\Delta>h$, see Figs. 1(d,e). At the Fermi surface (for a sizable chemical potential) all of these effects appear as a small splitting of the four-fold degenerate parabolic band in the absence of a spin-orbit coupling. However, as we will discuss, the properties of the resulting Fermi liquid are qualitatively different from the parabolic-band case and the systems can be characterized, in a well-defined sense, as a Dirac metal (Fig. 1(a,b)) or Weyl metal (Fig. 1(d-e)).

For large values of $v$, when the spin-orbit term $H_{v}$ dominates over $\epsilon_{\bm{k}}$ in $H_{0}$, the spectrum can be described as follows: For $\Delta=0$ and $h=0$ it consists of two degenerate Dirac cones. The lower cone eventually bends over, but for large $v$ this happens only for wave vectors large compared to the inverse lattice spacing, so this branch will not contribute to the Fermi surface, see Fig. 2(a). For a chemical potential tuned to the crossing point, $\mu=0$, the system is a Dirac semimetal, and for $\mu>0$ it is natural to call it a Dirac metal. $\Delta>0$ gaps out the crossing point, and the system is a Dirac insulator for $\mu<\Delta$, and a Dirac metal for $\mu>\Delta$, Fig. 2(b). $h>0$ separates the Dirac cones in wave-number space, leading to a Weyl semimtal if $\mu=0$ and a Weyl metal if $\mu>0$, Fig. 2(c). This structure persists for $0<\Delta<h$, Fig. 2(d), whereas for $0<h<\Delta$ the cones are gapped out and one has a Weyl insulator for $\mu=0$ and a Weyl metal for $\mu\gg\Delta$, Fig. 2(e).

Even though the single-particle spectra in Figs. 1, 2 are related in obvious ways, they appear very different at wave numbers that are not asymptotically small; this is true in particular at the Fermi surface for generic values of the chemical potential. However, as we will show in Sec. III, the soft-mode spectrum is the same for both small and large values of the spin-orbit coupling $v$,lar and the systems depicted in Fig. 1 therefore constitute Dirac and Weyl metals in the same sense as those in Fig. 2 for a sufficiently large value of $\mu$. For our purposes the distinction between the Dirac and Weyl cases is not important, and we will refer to all of the metallic systems depicted in Figs. 1, 2 as ‘Diral metals’. We also note that in systems that are not invariant under SI, or if TR is broken by effects other than a magnetic field, there are additional possibilities, in addition to the Dirac and Weyl cases, that we will not discuss, see, e.g., Ref. Mitchell and Fritz, 2015.

For $\mu\gg\Delta$ and $h=0$ the various Fermi surfaces are characterized by Fermi wave numbers that are solutions of the quadratic equations

[TABLE]

Our explicit calculations in Sec. IV will be performed in the limit of a small spin-orbit interaction, $v^{2}\ll\mu/m$. $v$ then splits the spherical Fermi surface into two two-fold degenerate sheets, see Figs. 1(a,b), and we have

[TABLE]

The second solution of the quadratic equation, $k_{\text{F}}=2mv$, does not correspond to a physical Fermi surface if it is larger than a reciprocal lattice vector.

II.2.2 Single-particle Green function

The single-particle Green function is defined as

[TABLE]

with $\omega_{n}$ a fermionic Matsubara frequency. It is a $4\times 4$ matrix in chirality-spin space with matrix elements $\left(G_{n}({\bm{k}})\right)_{\sigma\sigma^{\prime}}^{\alpha\alpha^{\prime}}$. The complete explicit expression is quite complicated, but we will not need it for our purposes. Rather, we list various special cases and approximations that suffice for our objectives.

For $\Delta=0$ the Green function is diagonal in the chirality index and one finds

[TABLE]

Here we have defined $\xi_{\bm{k}}=\epsilon_{\bm{k}}-\mu$. A similar simplification occurs in zero field, where one finds

[TABLE]

A nonzero field appears in both the numerator and the denominator of the Green function, as is illustrated by the special case $\Delta=0$ in Eqs. (7). For reasons that will become clear in Sec. IV (see the remark before Eq. (36)) we do not need to retain the occurrences of $h$ in the numerator for our purposes. In an approximation that suffices for determining the leading nonanalytic behavior of the density of states and the spin susceptibility we thus keep $h$ only in the denominator and find

[TABLE]

This reduces to Eqs. (7) in the same approximation if the gap $\Delta$ is zero, or if it is is negligible compared to the additive terms in Eq. (9). Anticipating that the wave vector ${\bm{k}}$ will be close to the Fermi surface, we see that the relevant criterion is $\Delta\ll vk_{\text{F}}$, with $k_{\text{F}}$ the Fermi wave number for the Fermi surface under consideration. This is the limit we will investigate in Sec. IV. As we will discuss in Sec. V.1 the presence of a small $\Delta$ does not qualitatively change the results.

For $\Delta=0$ the Green function is diagonal in the chirality index, see Eq. (7c). In the same approximation as in Eq. (9), i.e., ignoring the ${\bm{h}}$ in the numerator, it can be written

[TABLE]

Here $\hat{\bm{k}}={\bm{k}}/|{\bm{k}}|$ is the unit vector in ${\bm{k}}$-direction. In writing the Green function in the form of Eq. (10b) we have performed a partial fraction decomposition of the quadratic denominator. The partial Green function $F_{n}^{\alpha\pm}$ therefore has two indices: $\alpha$ is the chirality index, and the additional $\pm$ represents the two branches of the quadratic denominator. Since spin is not a good quantum number in the presence of a spin-orbit interaction, these two branches cannot be interpreted as representing two different spin projections.

Equations (10) no longer allow for the Landau limit of zero spin-orbit coupling to be taken, because of the approximation involved. However, performing a partial-fraction decomposition on Eqs. (7), which are still exact, shows that the exact Landau Green function can be written in a form analogous to Eqs. (10), viz.,

[TABLE]

are the Green functions for up-spin ($\uparrow\equiv+$) and down-spin ($\downarrow\equiv-$) electrons, respectively.

For later reference we list five properties of the matrix $M$, which can easily be checked by direct calculation:

[TABLE]

II.2.3 Action for a Dirac Fermi gas

The action for a noninteracting Fermi system governed by the Hamiltonian $H_{0}$, or a Dirac Fermi gas, is given in terms of the inverse Green function,

[TABLE]

Here $\bar{\psi}$ and $\psi$ are fermionic fields with spin index $\sigma=(+,-)\equiv(\uparrow,\downarrow)$ and chirality indes $\alpha=(+,-)$, and $k=({\bm{k}},\omega_{n})$ is a 4-vector that comprises the wave vector ${\bm{k}}$ and the Matsubara frequency $\omega_{n}$. Introducing spinors $\psi^{\alpha}=(\psi_{\uparrow}^{\alpha},\psi_{\downarrow}^{\alpha})$ and a scalar product $(\bar{\psi},\psi)=\sum_{\sigma}\bar{\psi}_{\sigma}\psi_{\sigma}$, we can write $S_{0}$ more explicitly in the form

[TABLE]

[TABLE]

To summarize the symmetry properties, this action is invariant under SI, TR (for $h=0$), and simultaneous rotations in spin space and real space.

II.3 Electron-electron interaction, and action for a Dirac Fermi liquid

In order to study correlation effects, we need to add a four-fermion interaction $S_{\text{int}}$ to the noninteracting electron system described by $S_{0}$. Underlying all correlations is the Coulomb interaction, but the effective interaction in a fully renormalized theory will contain all contributions that are allowed by symmetry. In our case, the noninteracting action is invariant under simultaneous rotations in spin space and momentum space,spi as well as under SI; this also implies invariance under TR. A four-fermion interaction that is consistent with all of these requirements can be written

[TABLE]

Here the prime on the sum over $q$ indicates a restriction $|{\bm{q}}|<\Lambda$, with $\Lambda\ll k_{\text{F}}$ a momentum cutoff. This is necessary in order to avoid overcounting, as explained in Appendix A. The leading nonanalyticities we are interested in will not depend on the cutoff.

The six terms in Eq. (LABEL:eq:2.15) are separately invariant under the required symmetries, and come with six interaction amplitudes. They are graphically represented in Fig. 3. As we will see in Sec. III, interactions that mix chiralities ($\Gamma_{\text{s},2},\Gamma_{\text{s},3},\Gamma_{\text{t},2},\Gamma_{\text{t},3}$) play a qualitatively different role that those that do not ($\Gamma_{\text{s},1},\Gamma_{\text{t},1}$). A sketch of how this effective interaction is generated from a Coulomb interaction between density fluctuations is provided in Appendix A.

The full action for our model of interacting Dirac fermions, i.e., a Dirac Fermi liquid, is given by

[TABLE]

It still contains the Landau Fermi liquid as a limiting case if one puts the parameters $v$ and $\Delta$ in $H_{0}$ equal to zero.

III Soft Modes in a Dirac Metal

The nonanalytic behavior of various observables we are interested in is the result of four-fermion correlation functions that are soft or massless at zero temperature and the dependence of these soft modes on the fields conjugate to the respective observables. Prior to performing explicit calculations it is helpful to discuss the existence, nature, and origin of these soft modes. We start by discussing the soft modes in an ordinary LFL, and then generalize to the case of a DFL.

III.1 A Ward identity for a Landau Fermi liquid

Consider the LFL Green function as given by Eqs. (11). It is easy to check that the following identity holds:

[TABLE]

where $\Omega_{n_{1}-n_{2}}=\omega_{n_{1}}-\omega_{n_{2}}$ is a bosonic Matsubara frequency, and $h=|{\bm{h}}|$. This is a generalization of Velicky’s Ward identity Velicky (1969) for noninteracting electrons to a nonzero magnetic field. Notice that the denominator on the right-had side has the structure of the differential operator of a Boltzmann equation in the absence of the collision operator (i.e., a time derivative plus a streaming term). It also holds, with the ballistic dynamics replaced by diffusive ones, for electron systems with quenched disorder if one takes $G$ to be the unaveraged Green function and takes a disorder average on either side of Eq. (17).Velicky (1969); Wegner (1979); dif It further still holds, with a slightly more complicated frequency structure and $m$ replaced by an appropriately renormalized effective mass, for interacting electrons at zero temperature, with or without quenched disorder, if the left-hand side is replaced by the appropriate four-fermion correlation function that factorizes into a product of two Green functions in the noninteracting limit.Belitz and Kirkpatrick (1997, 2012) This is a consequence of the adiabatic correspondence between single-quasiparticle states in the noninteracting and interacting Fermi systems, respectively, that is inherent in Landau Fermi-liquid theory. In an interacting system at nonzero temperature a dephasing rate appears in the denominator of the right-hand side of the Ward identity and its generalization discussed below. A nonzero temperature thus gives a mass to all of the soft modes we will discuss. However, for our purposes there is a stronger effect of a nonzero temperature that is most easily seen by considering convolutions of Green functions, see Eq. (20c) below.

In order to see the significance of the Ward identity, analytically continue the Green function in Eq. (11c) to complex frequencies $z$ and consider its analytic structure. The function $G_{\pm}({\bm{k}},z)=1/(z-\xi_{\bm{k}}\pm|{\bm{h}}|)$ has a cut on the real axis, and close to the real axis it is given by

[TABLE]

We see that the behavior of the right-hand side of Eq. (17) is drastically different depending on whether the two fermionic Matsubara frequencies $\omega_{n_{1}}$ and $\omega_{n_{2}}$ have the same sign, or opposite signs. In latter case, the numerator on the right-hand side is nonzero in the limit $\Omega_{n_{1}-n_{2}}\to 0$, ${\bm{q}}\to 0$ for all frequencies or energies where the spectrum of the Green function is nonzero. The denominator, on the other hand, is small of $O(\epsilon)$, where $\epsilon$ stands for $\Omega$, or $q=|{\bm{q}}|$, or $h$, all of which scale the same way. In particular, it vanishes in the limit of zero frequency $\Omega$ and wave number $q$ if the two spin projections are the same ($\sigma_{1}=\sigma_{2}$). The four-fermion correlation on the left-hand-side is thus a soft mode of $O(1/\epsilon)$. Whereas, if $n_{1}$ and $n_{2}$ have the same sign, then both the numerator and the denominator are small of $O(\epsilon)$, and the four-fermion correlation is of $O(1)$. Similarly, products of $N$ Green functions $G_{n_{1}}G_{n_{2}}\ldots G_{n_{N}}$ are of $O(1/\epsilon^{N-1})$ if the $N$ frequency indices $n_{i}$ do not all have the same sign, but of $O(1)$ if they do.

It is illustrative to derive the same result as a Ward identity that results from a unitary transformation of the fermionic fields that represents a rotation in frequency space,Wegner (1979) see Appendix B. This shows that the soft four-fermion excitations can be interpreted as the Goldstone modes of a spontaneously broken continuous symmetry, viz., the symmetry between retarded (positive Matsubara frequency) and advanced (negative Matsubara frequency) degrees of freedom. The relevant order parameter is the spectrum of the Green function.

III.2 A Ward identity for a Dirac Fermi liquid

We now consider the analogous identity involving the Green function $F_{n}^{\alpha\beta}({\bm{k}})$ given in Eq. (10c). We find

[TABLE]

For the numerator, the same discussion as in the LFL case applies, but the denominator now allows for more possibilities. Anticipating that in perturbation theory with respect to the electron-electron interaction we will encounter convolutions of Green functions we define a correlation function

[TABLE]

Depending on the relative values of $\alpha_{1,2}$ and $\beta_{1,2}$ there may or may not be a mass in the denominator of Eq. (20b). Before we discuss the various cases, we consider the dependence on the Matsubara frequency $\Omega_{n}$ and its consequences for the soft-mode structure at nonzero temperature. To this end it is useful to perform the sum over the fermionic Matsubara frequency. We define

[TABLE]

The factor of $i\Omega_{n}$ in the numerator is a consequence of the frequency-mixing requirement, which in Eq. (20b) was reflected by the $\Theta$ function: Only products of retarded and advanced Green functions can produce convolutions that are potentially soft, which makes the frequency summation trivial. In any frequency sum over products of convolutions the term with $n=0$ therefore comes with zero weight, and the smallest frequency index that contributes is $n=1$. The $i\Omega_{n}$ in the denominator at nonzero temperature thus effectively acts as a mass that is linear in $T$. This is a stronger effect than the dephasing rate mentioned after Eq. (17), which vanishes faster than $T$ for $T\to 0$, and it is responsible for the temperature scaling as the wave number and the magnetic field.

III.3 Soft modes in Landau and Dirac Fermi liquids

We now discuss two limiting cases.

First case: $h\gg vk_{\text{F}}$. To zeroth approximation we can put $v=0$ and recover the LFL case. $\varphi$ is then independent of the chirality index, and $\beta=\pm\equiv\uparrow\downarrow$ labels the spin projection. Consistent with Eq. (17), we have two types of singular behavior. The first type is realized by the $\uparrow\downarrow$ and $\downarrow\uparrow$ combinations:

[TABLE]

which we recognize as the hydrodynamic part of the Lindhard function. Here the hydrodynamic singularity is not cut off by a magnetic field. We will refer to this type as “soft modes of the second kind".

Second case: $h\ll vk_{\text{F}}$. In this case we can ignore $h$ compared to $vk_{\text{F}}$ in Eq. (20b). The correlations with $\beta_{1}=-\beta_{2}$ then become independent of the chirality index, and we have

[TABLE]

They are soft of the first kind for correlations that mix the chiralities ($\alpha_{1}\neq\alpha_{2}$), and soft of the second kind for correlations within a given chirality ($\alpha_{1}=\alpha_{2}$). We note that the soft modes of the first kind have a chirality structure that is different from the one necessary for forming a density of a spin density, which is diagonal in $\alpha$:

[TABLE]

These are nevertheless the modes that produce a nonanalyticity in the spin susceptibility, as we will discuss next.

We also note that models of the graphene-type mentioned after Eq. (1a), while allowing for a formally very similar discussion, lead to a drastically different physical interpretation. If the Pauli matrix in the $\bm{\sigma}\cdot{\bm{k}}$ term is a pseudo-spin, then the coupling constant $v$ does not give a mass to the modes that were soft of the first kind in a LFL. This has important consequences for the effects of the soft modes on the spin susceptibility, as we will discuss in Sec. V.

III.4 Physical consequences of soft modes: Nonanalytic behavior of observables

In order to determine the effects of the soft modes on observables, we need to generalize the notion of soft modes of the first and second kinds from the last subsection. Consider an observable $\cal O$ and its conjugate field $\mathfrak{h}$. Each soft mode can then be classified as being of the the first or second kind with respect to $\mathfrak{h}$, depending on whether or not $\mathfrak{h}$ cuts off the singularity. In Sec. III.3, $\mathfrak{h}$ was the magnetic field $h$, and $\cal O$ was the spin density. As illustrated by Eq. (22b), the correlations we denoted by $\varphi_{\beta\beta}^{\alpha\alpha^{\prime}}$ are of the first kind with respect to $h$ if $\alpha\neq\alpha^{\prime}$, and of the second kind if $\alpha=\alpha^{\prime}$. However, if we take $\mathfrak{h}$ to be the chemical potential $\mu$, which makes $\cal O$ the number density, then all of the soft modes are of the second kind with respect to $\mu$. Now consider the partition function $Z$ as a generating functional $Z[\mathfrak{h}]$ that depends on $\mathfrak{h}$. Then the free energy is given by $f[\mathfrak{h}]=-(T/V)\ln Z[\mathfrak{h}]$, the observable by ${\cal O}=\partial f/\partial{\mathfrak{h}}$, and the corresponding susceptibility by $\chi_{\cal O}=\partial^{2}f/\partial{\mathfrak{h}}^{2}$. Now $f$, and all of its derivatives, will be nonanalytic functions of $\mathfrak{h}$ if some mode is soft of the first kind with respect to $\mathfrak{h}$. Generically, the field will scale linearly with the frequency or temperature, and with the external wave number, and scaling implies that $\chi_{\cal O}$ will also be a nonanalytic function of those. Necessary conditions for these nonanalyticities to be realized are of course the mixing of positive and negative frequencies, and a coupling to the relevant soft modes of the first kind. The first condition requires an interacting fermion system, and the second one implies that only interaction amplitudes in certain channels will lead to nonanalyticities.

We will now apply these considerations, which were first discussed in Ref. Belitz et al., 2002, to various specific observables. We will first show that they correctly reproduce the known nonanalyticities in a LFL, and then use them to predict the corresponding results in a DFL.

III.4.1 Nonanalyticities in a Landau Fermi liquid revisited

Consider the static spin susceptibility $\chi_{s}=\partial^{2}f/\partial h^{2}$ in a clean LFL. In the absence of an interaction there is no frequency mixing, and hence $\chi_{s}$ in a Fermi gas, i.e., the Lindhard function, is analytic at zero wave number and zero field. It is easy to see in perturbation theory that there still is no frequency mixing to first order in the interaction amplitudes, but starting at second order an electron-electron interaction mixes retarded and advanced degrees of freedom, and the relevant soft modes of the first kind are given by Eq. (21a). They involve a mixing of the spin projections, and therefore at least one spin-triplet interaction amplitude $\Gamma_{t}$ (more precisely, one of the two components of the spin triplet that are transverse with respect to the field) is required in order to produce a nonanalytic behavior of $\chi_{s}$. This can be seen from Eq. (LABEL:eq:2.15): Only the terms that involve the Pauli matrices $\sigma^{1}$ and $\sigma^{2}$ have the requisite $\uparrow\downarrow$ spin structure. We thus expect $\chi_{s}$ in a LFL to be a nonanalytic function of the wave number $Q$, the magnetic field $h$, and also the temperature (which scales the same as the frequency) with a prefactor of the nonanalyticity that for weak interactions is quadratic in the $\Gamma$ and contains at least one $\Gamma_{t}$. Simple scaling argumentsBelitz and Kirkpatrick (2014) show that the leading nonanalyticity takes the form $\delta\chi_{s}=\chi_{s}(Q)-\chi_{s}(Q=0)\propto Q^{d-1}$ in generic spatial dimensions $d$, and $\delta\chi_{s}\propto Q^{2}\ln Q$ in $d=3$. Using only the soft-mode structure of the LFL we thus conclude that the spin susceptibility as a function of either $Q$, or $h$, of $T$, with the other two variables equal to zero, has a nonanalytic behavior

[TABLE]

in generic dimensions, and $\delta\chi_{s}\propto Q^{2}\ln Q$ etc. in $d=3$.

This is indeed the case, as is well known from explicit calculations,Belitz et al. (1997); Betouras et al. (2005) with one exception: In 3-d systems the leading temperature nonanalyticity at $O(\Gamma_{t}^{2})$ has a zero prefactor.Carneiro and Pethick (1977) This zero is accidental, however, and has no bearing on the general argument. Analogous considerations and results hold for LFLs in the presence of quenched disorder,Altshuler and Aronov (1984); Belitz and Kirkpatrick (1994) only the soft modes are diffusive rather than ballistic. As a result, the singularity is stronger, $\delta\chi_{s}\propto Q^{d-2}$ etc, and the prefactors are linear in $\Gamma_{t}$ for weak interactions.

It is also illustrative to note that none of the soft modes are cut off by the chemical potential, which is the field conjugate to the electron number density. The density susceptibility $\chi_{n}=\partial n/\partial\mu=\partial^{2}f/\partial\mu^{2}$ therefore can not be a nonanalytic function of the wave number or the temperature. The same is true for the susceptibilities of the density current, as well as the related higher-rank tensor correlation functions. For instance, a source term $({\bm{J}}\cdot{\bm{k}})\,\bar{\psi}(k)\psi(k)$ in the action generates the number-current density. Such a term can be absorbed into the single-particle energy $\epsilon_{\bm{k}}$ by completing the square, and hence does not cut off any of the soft modes. This is also confirmed by explicit calculations,Belitz et al. (1997) and we will discuss the consequences in Sec. V.

The extreme opposite case is represented by a nonzero temperature, which cuts off all of the soft modes, see the discussion after Eq. (20b) above. Focusing on $d=3$ from here on, we thus expect the specific-heat coefficient $\gamma_{V}=\partial^{2}f/\partial T^{2}$ in a clean 3-d LFL to have a $T^{2}\ln T$ nonanalyticity to which both spin-singlet and spin-triplet soft modes contribute. This is indeed the case,Chubukov et al. (2006) and an analogous result holds for disordered electrons.Belitz and Kirkpatrick (1994) In a magnetic field two of the three spin-triplet modes become massive, and the corresponding $T^{2}\ln T$ contributions turn into $h^{2}\ln h$ terms.

We finally consider the density of states, which is given in terms of the spectrum of the Green function. It is not a thermodynamic quantity but can be generated if we add to the action a source term that has the structure of a frequency-dependent chemical potential:

[TABLE]

with $\delta\mu(\omega_{n})=-\delta\mu(-\omega_{n})$ a source field that is an odd function of the Matsubara frequency and discontinuous at $\omega_{n}=0$, $\delta\mu(0+)=-\delta\mu(0-)\neq 0$. It is not a physically realizable field, but generates the density of states $N$ at the Fermi level via

[TABLE]

This source field gives all of the soft modes a mass, just as a nonzero temperature does: In the denominator on the right-hand side of the Ward identity, Eq. (17) or (19), a term $\delta\mu(\omega_{n_{1}})-\delta\mu(\omega_{n_{2}})$ appears, which for $n_{1}\to 0+$, $n_{2}\to 0-$ becomes $2\delta\mu(0+)$.

We thus expect the density of states, as a function of the frequency or energy $\omega$ at zero temperature, to have a leading nonanalytic contribution that is proportional to $\omega^{2}\ln\omega$, or to $T^{2}\ln T$ for the density of states on the Fermi surface as a function of the temperature, and with both the spin-singlet and the spin-triplet modes contributing. As in the case of the specific-heat coefficient, a magnetic field will give two of the three spin-triplet modes a mass and turn the corresponding $T^{2}\ln T$ or $\omega^{2}\ln\omega$ into $h^{2}\ln h$. In generic dimensions $1<d<3$, all of these nonanalyticities turn into power laws with an exponent $d-1$.

III.4.2 Nonanalyticities in a Dirac Fermi liquid

In the preceding subsection we saw that one can deduce the known results for nonanalyticities in a LFL from the structure of the soft modes alone. We will now employ the same line of reasoning to deduce the nature of the nonanalyticities in a DFL. For simplicity, we will restrict ourselves to $d=3$. It is obvious from power counting what the corresponding results are for $1<d<3$. In Sec. IV we will demonstrate that explicit calculations of the spin and density susceptibilities, as well as the density of states, confirm these results.

For the density susceptibility we obtain the same null result as in the LFL case: The chemical potential $\mu$, which is the field conjugate to the particle number, gives none of the soft modes a mass. The free energy therefore is an analytic function of $\mu$, and the density susceptibility $\partial n/\partial\mu$ has no nonanalyticities.

The fields conjugate to the density of states and the specific-heat coefficient, viz., the frequency-dependent chemical potential and the temperature, respectively, give all of the soft modes a mass. The density of states will thus have a nonanalytic contribution

[TABLE]

Contributions that do not mix chiralities will remain unaffected by the field. By the same argument, we obtain for the specific-heat coefficient

[TABLE]

with all interaction amplitudes contributing to the former result, but only the $\Gamma_{\mu,(2,3)}$ to the latter.

For the spin susceptibility the relevant soft modes are the ones that are made massive by a nonzero magnetic field. From Eq. (22b) we see that the only soft modes that contribute are those that mix chiralities. We thus expect

[TABLE]

In all three cases the prefactors will depend only on the inter-chirality interaction amplitudes. Note that the modes that caused the nonanalyticities in $\chi_{s}$ in a LFL are massive in a DFL, see Eq. (22a). As a result, an interaction that couples different chiralities is necessary in order to obtain any nonanalytic behavior of $\chi_{s}$ in a DFL.

IV Nonanalytic Behavior of Observables: Explicit Calculations

We now demonstrate that the results predicted by the general arguments in Sec. III.4.2 are confirmed by explicit perturbative calculations. We perform the calculations for the case of a spin-orbit interaction that is weak in the sense $v\ll\sqrt{\mu/m}$. In particular, we ignore the differences between the various values of $k_{\text{F}}$, $v_{\text{F}}$, and $N_{\text{F}}$. We will ignore all contributions from modes that are massive due to $v>0$. This implies that our results are observable for wave numbers $q/k_{\text{F}}\lesssim v/v_{\text{F}}$, see Eq. (22a). In particular, this implies that we will not be able to recover the LFL limit by taking $v\to 0$. We will further work for $\Delta=0$; as we will show in Sec. V.1 the results remain qualitatively valid for $\Delta\neq 0$ as long as $\Delta\ll vk_{\text{F}}$.Zer We start with the spin susceptibility of a DFL.

IV.1 Spin susceptibility

The spin susceptibility is given by the two-point correlation function of the spin density, Eq. (23). We consider the static longitudinal susceptibility

[TABLE]

with $Q=({\bm{Q}},i0)$ the external wave vector/frequency. We are interested in the dependence of $\chi_{\text{s}}^{\text{L}}$ on the external wave number $|{\bm{Q}}|$, the magnetic field $h$, and the temperature $T$. Any nonanalytic dependence on these parameters requires that one goes to at least second order in perturbation theory with respect to the electron-electron interaction, and we denote this contribution to the spin susceptibility by $\delta\chi_{\text{s}}^{\text{L}}$. With the interaction given by Eq. (LABEL:eq:2.15), and to lowest order in a loop expansion, there are four structurally different diagrams that contribute to $\delta\chi_{\text{s}}^{\text{L}}$, which are shown in Fig. 4.two

As mentioned above, we keep only contributions that are not made massive by the spin-orbit coupling $v$ via Eq. (22a). That is, we keep only Green function convolutions where all of the Green functions $F^{\alpha\beta}$ share the same branch index $\beta$.

IV.1.1 Wave-number dependence

We first consider the dependence on the external wave number $Q$ for $h=T=0$. Diagrams (i) and (ii) in Fig. 4 lead to products of convolutions of three Green functions each. They can be realized with the interaction amplitudes $\Gamma_{\text{s},1}$ and $\Gamma_{\text{t},1}$, or with $\Gamma_{\text{s},2}$ and $\Gamma_{\text{t},2}$ (the $\Gamma_{\mu,1}$ and $\Gamma_{\mu,2}$ do not mix). From the discussion in Sec. III.4 we do not expect these processes to contribute, since the corresponding soft modes are of the first kind with respect to $h$. Indeed, the contributions from the individual diagrams cancel due to the symmetry properties of convolutions of Green functions listed in Appendix C: Diagrams (i) and (ii), if realized with $\Gamma_{\mu,1}$ and $\Gamma_{\mu,2}$, cancel due to Eq. (56b), and diagrams (iii) and (iv) cancel due to Eq. (58c).

This leaves $\Gamma_{\mu,3}$ as the only candidates for producing singular behavior, as expected from the general arguments in Sec. III.4. An analysis of the spin structure of the diagrams alone shows that any realization that contains a $\Gamma_{\text{s},3}$ cannot contribute to the leading singularities since the leading terms contain spin-matrix products of the type shown in Eq. (12c), which vanish. Therefore, the only terms that require a full calculation are ones proportional to $(\Gamma_{\text{t},3})^{2}$.

Realizing the diagrams in Fig. 4 by means of $\Gamma_{\text{t},3}$ leads to the following expression for $\delta\chi_{\text{s}}^{\text{L}}$ in terms of integrals:

[TABLE]

Power counting shows that all of the integrals in Eq. (31b) scale as a constant plus $|{\bm{Q}}|^{d-1}$ for spatial dimensions $d<3$, and as ${\bm{Q}}^{2}\ln|{\bm{Q}}|$ for $d=3$, which is the expected nonanalytic behavior. Determining the prefactor for $d<3$ would be hard. However, for $d=3$, and to leading logarithmic accuracy, we can expand the integrands in powers of ${\bm{Q}}$ to $O({\bm{Q}}^{2})$, and replace the resulting logarithmic divergence with $\ln|{\bm{Q}}|$. After lengthy calculations we obtain, for $h=T=0$ and $v\ll v_{\text{F}}$,

[TABLE]

where we have omitted the constant contribution.

While this result has the same functional form as the corresponding one in a LFL,Belitz et al. (1997) we stress that the underlying soft-mode structure is very different: The LFL soft modes that were responsible for the nonanalytic behavior in the absence of a spin-orbit interaction have become massive, see Eq. (22a), and the soft modes that are responsible for the present result mix chiralities, which is why only the interaction amplitude $\Gamma_{\text{t},3}$ contributes. We also note that our result is independent of $v$ and nonzero in the limit $v\ll v_{\text{F}}$. This does not imply that it remains unchanged in the true $v\to 0$ limit: We have neglected terms that have become massive due to $v\neq 0$, so we can no longer take the $v\to 0$ limit. If one keeps the terms that are soft for $v=0$, but massive for $v>0$, then one finds that in the limit $v\to 0$ they cancel the contribution shown in Eq. (32), leaving behind the known result for a LFL.Belitz et al. (1997) We will come back to these points in Sec. V.

IV.1.2 Temperature dependence

We now turn to the temperature dependence of $\delta\chi_{\text{s}}^{\text{L}}$ at ${\bm{Q}}=h=0$. For ${\bm{Q}}=0$ it is possible to identically rewrite all of the terms in Eq. (31b) as products of convolutions of three Green functions. This can be achieved by means of partial integrations with respect to the $z$-component of the hydrodynamic wave vector ${\bm{q}}$. We find

[TABLE]

We now use a spectral representation for the $\cal F$ to perform the summation over the hydrodynamic frequency. This yields

[TABLE]

Here the ${\cal F}{}^{\,"}$ are the spectra of the $\cal F$, which contain a continuous part as well as delta-function contributions, and $n(x)=1/(e^{x}-1)$ is the Bose distribution function. The low-$T$ behavior can now be determined by standard asymptotic analysis. Power counting shows that the leading singularity in $d<3$ spatial dimensions is $T^{d-1}$, and $T^{2}\ln T$ in $d=3$, as expected. Focusing on $d=3$ again, structural considerations show that ${\cal F}^{(2)}$ does not contribute to the singular behavior, while ${\cal F}^{(1)}$ and ${\cal F}^{(3)}$ do. The contributions from ${\cal F}^{(1)}$ cancel internally, and those from ${\cal F}^{(3)}$ yield the final result

[TABLE]

As expected from the general arguments in Sec. III, there is a nonanalytic temperature dependence whose functional form and sign are the same as for the ${\bm{Q}}$-dependence at $T=0$. While the result is formally independent of $v$, the comments made after Eq. (32) apply here as well: We have neglected modes that are massive for $v>0$. If kept, they will cancel the term shown in the limit $v\to 0$. In contrast to the case of the $Q$-dependence, here the cancellation is exact, and the prefactor of the $T^{2}\ln T$ term in a LFL is zero, at least to second order in the interaction.Carneiro and Pethick (1977); Belitz et al. (1997) The nonzero result in a DFL underscores the point that the absence of a $T^{2}\ln T$ term in a in a LFL is not of structural significance, but due to a prefactor that is accidentally zero in $d=3$, see the discussion in Sec. V.

IV.1.3 Magnetic-field dependence

Finally, we consider the dependence on the magnetic field $h$ at ${\bm{Q}}=T=0$. Since ${\bm{Q}}=0$, we can write $\delta\chi_{\text{s}}^{\text{L}}$ in the form of Eqs. (33). In order to extract the leading nonanalyticity in $d=3$ we expand the integrand to $O(h^{2})$ and realize that the resulting logarithmic divergence is cut off by $h$. The leading singular behavior is obtained from terms that maximize the number of Green functions in the integrands. This justifies the approximation we made in Sec. II.2.2, where we neglected all occurrences of $h$ in the numerator of the Green function. To leading logarithmic accuracy we obtain

[TABLE]

Again, this is consistent with the prediction based on the general arguments in Sec. III. As explained there, this is, in a well defined sense, the most fundamental nonanalyticity, since $h$ is the field conjugate to the observable whose susceptibility we are calculating, viz., the magnetization. The nonanalytic dependences on ${\bm{Q}}$ and $T$ follow from the one on $h$ due to scaling.

IV.2 Density susceptibility, and related correlation functions

We next ascertain that the density susceptibility $\chi_{\text{n}}$ has no nonanalytic behavior, as predicted by the general arguments of Sec. III. To second order in the interaction the diagrams are the same as in Fig. 4, but the external vertices now carry a unit matrix in spin space rather than a $\sigma_{3}$, which changes the result of performing the spin traces. The realizations of the diagrams in terms of $\Gamma_{\mu,1}$ and $\Gamma_{\mu,2}$ vanish for the same symmetry reasons as in the case of $\chi_{\text{s}}^{\text{L}}$. For the realizations in terms of $\Gamma_{\mu,3}$ we find

[TABLE]

As in the case of $\delta\chi_{\text{s}}^{\text{L}}$, the expression in terms of integrals, Eq. (37a), is valid only for the leading nonanalytic contributions to $\delta\chi_{\text{n}}$. We see that Eq. (37a) is very similar to the corresponding result for $\delta\chi_{\text{s}}^{\text{L}}$, Eq. (31a). However, the symmetry relations from Eqs. (57b) and (59c) now lead to terms being equal and opposite that were equal in the case of $\chi_{\text{s}}^{\text{L}}$. As a result, the contributions proportional to $\Gamma_{\mu,3}^{2}$ cancel as well, as they must according to the general arguments. We conclude that the density susceptibility $\chi_{\text{n}}$ is an analytic function of the wave number, the temperature, and the magnetic field. Consistent with the very general reasons underlying this result, the cancellation occurs at a structural level, and it is not necessary to evaluate any of the integrals in order to see it.

The symmetry relations that lead to this null result are independent of the angular factors encoded in the function $f(\hat{\bm{k}})$ in Eqs. (56) - (59). As a result, the absence of nonanalyticities also applies to the density current susceptibility and to all high-rank tensor susceptibilities related to the density. This is important for the stability of Fermi liquids, as we explain in Sec. V.

IV.3 Density of states

The density of states (DOS) $N(\omega)$, with $\omega$ the difference in energy space from the Fermi surface, is given by the trace of the Green function via

[TABLE]

For noninteracting electrons, $N(\omega\to 0)=4N_{\text{F}}$. We are interested in corrections $\delta N(\omega)$ that are nonanalytic functions of the frequency $\omega$ due to the effects of the soft modes. To second order in the interaction there is only one diagram that is of one-loop type in the sense of Ref. two, , which is shown in Fig. 5. Realizing this diagram in all possible ways with the interaction amplitudes defined in Eq. (LABEL:eq:2.15), and keeping only terms with hydrodynamic content, we find seven different contributions. In terms of convolutions of the Green function $F$ defined in Eq. (10c) these are

[TABLE]

Power counting shows that these integrals scale as a constant plus $|\omega|^{d-1}$ for spatial dimensions $1<d<3$, and as $\omega^{2}\ln|\omega|$ in $d=3$, which is the behavior expected from the general arguments in Sec. III.4.1. An explicit calculation shows that the nonanalyticity is absent for $d=3$ (but not for $d<3$) in $\delta Q_{\text{tt},1}$, $\delta Q_{\text{tt},2}$, and the third contribution to $\delta Q_{\text{tt},3}$.hig We obtain

[TABLE]

On the Fermi surface, $\omega=0$, we find an analogous result for $\delta N(T)$ as a function of the temperature: The result has the same form as Eq. (40a) with $\omega^{2}\ln|\omega|$ replaced by $T^{2}\ln T$, only the numerical prefactor is slightly different.

A nonzero magnetic field affects only $\delta Q_{\text{tt},3}$, which is the only contribution containing convolutions that mix chiralities. Accordingly, the term proportional to $(\gamma_{\text{t},3})^{2}\omega^{2}\ln|\omega|$ gets replaced, for $h>\omega,T$, by $(\gamma_{\text{t},3})^{2}h^{2}\ln h$, again with a different numerical prefactor. The other contributions remain unaffected.

V DIscussion

We conclude by discussing various aspects of our results, and by elaborating on some points we have mentioned in passing only.

V.1 Different types of Fermi liquids

V.1.1 Landau Fermi liquids versus Dirac Fermi liquids

A Fermi liquid in general is characterized by the existence of well-defined quasiparticles that are continuously related to the excitations in a Fermi gas. As we have mentioned in the Introduction, the action given by Eq. (16), with or without the spin-orbit coupling $v$, does indeed describe a Fermi liquid. This can be seen explicitly by considering the electron self energy, which is given by the diagram in Fig. 5 without the two external Green functions. If we average over the Fermi surface, we obtain the expressions in Eqs. (39) with the factor $F_{k}^{\alpha\beta}F_{k}^{\alpha\beta}$ common to all of them replaced by a delta function $\delta(\xi_{\bm{k}})$ that pins ${\bm{k}}$ to the Fermi surface. Given that the DOS scales as $\omega^{d-1}$, it is obvious that the self energy scales as $\omega^{d}$, i.e., well-defined quasiparticles exist for all dimensions $d>1$.

However, the nature of the Fermi liquid is drastically different depending on whether or not the spin-orbit coupling $v$ is present. As illustrated in Eqs. (21, 22), the modes that are soft of the first kind with respect to the magnetic field $h$ for $v=0$ are no longer soft for $v>0$. This is related to the fact that the spin-orbit coupling destroys spin conservation. The modes that are soft instead are excitations that mix chiralities, see Eq. (22b), a concept that disappears in the limit $v\to 0$ when the two chirality channels become degenerate. For our purposes, this is the crucial difference between a Landau Fermi liquid (LFL) at $v=0$ and a Dirac Fermi liquid (DFL) at $v>0$. As noted in connection with Eq. (23), the soft excitations in a DFL have a structure that is qualitatively different from that of density fluctuations. In this sense they are as different from the soft modes in a LFL as particle-particle or Cooper-channel excitations are from particle-hole excitations.

V.1.2 Effects of the gap $\Delta$, and the limit of large $v$

We still need to show that our results are not qualitatively altered if one considers the general model with $\Delta\neq 0$. As can be seen from Eq. (9), the gap affects the Green function in two distinct ways. A trivial effect is that the expression $(\alpha v{\bm{k}}-{\bm{h}})^{2}$ in the denominator of the Green function gets modified according to

[TABLE]

which modifies $F_{k}^{\alpha\beta}$, Eq. (10c). A less trivial effect is that the Green function is no longer diagonal in the chirality index due the term $(\pi_{1}\otimes\sigma_{0})\Delta$ in the numerator of Eq. (9). This allows for realizations of the diagrams in Fig. 4 that are not possible if $\Delta=0$. However, in order for convolutions of the $F$ to be soft of the first kind with respect to the magnetic field they still need to mix chiralities. This can now be achieved by means of the Green functions rather than the interaction amplitudes, but the soft-mode structure remains unchanged and the integrals are still the same as in Eqs. (31) (except for the modified $F$). The only effect of the gap is therefore to produce small corrections of $O((\Delta/vk_{\text{F}})^{2})$ to the results we obtained in Sec. IV.

Our explicit results have been obtained in the limit $v\ll v_{0}$, with $v_{0}$ the atomic-scale velocity. An obvious question is what happens in the limit $v/v_{0}\lesssim 1$. Consider the spin susceptibility $\chi_{\text{s}}$ as an example. The general considerations in Sec. III show that the soft modes of the first kind with respect to the magnetic field are given by convolutions with all branch indices $\beta$ the same, while the chirality indices must not be all the same. Furthermore, in order for the magnetic field dependence to not disappear upon a simple shift of the wave number, the two soft modes in any one-loop contribution to $\chi_{\text{s}}$ must carry different branch indices. This is consistent with the structure of the branch and chirality indices in, e.g., Eq. (33a). As a result, the interaction couples a branch $E^{1\beta}$ of the single-particle specrum, Eqs. (3), with a branch $E^{2\beta}$. However, one of these two branches will not contribute to the Fermi surface in the limit of large $v$, as can be seen from Fig. 2. This suggests that in this limit the spin susceptibility does not have a nonanalyticity in systems with $\Delta=0$. $\Delta\neq 0$, on the other hand, will eliminate the condition that the two soft modes must carry different branch indices and will likely restore the nonanalyticity, albeit with a prefactor that is proportional to $\Delta$. These remarks apply to the spin susceptibility, other observables will behave differently. For instance, for the density of states all of the soft modes are of the first kind. The restrictions that are in place for $\chi_{\text{s}}$ therefore do not apply, and we expect the behavior for large $v$ to be qualitatively the same as for small $v$. These qualitative considerations suggest that the behavior in the limit of large $v$ is quite complex, and different in various aspects from that for small $v$. This limit requires a separate detailed investigation.

V.1.3 Different types of Dirac Fermi liquids

A linear spectrum at small momenta is not sufficient to uniquely define a Dirac Fermi liquid. This can be seen by comparing the model for a Dirac metal we have discussed, where the Pauli matrices in the $v\bm{\sigma}\cdot{\bm{k}}$ term represents the physical spin degree of freedom, with the graphene-type models mentioned after Eq. (1a), where they represent a pseudo-spin. In the former case, the discussion in Sec. V.1.1 applies. In a graphene-type model, on the other hand, the situation is different. Since the Pauli matrices in the $v\bm{\sigma}\cdot{\bm{k}}$ term and in the Zeeman term represent different degrees of freedom, the soft modes of the first kind with respect to the magnetic field are unaffected by the coupling constant $v$, and the physics behind the nonanalytic spin susceptibility the same as in a LFL. We note, however, that these remarks do not pertain to, e.g., the density of states. Since the nonanalyticity of the latter relies on modes that are soft with respect to the frequency-dependent chemical potential introduced in Eq. (25), rather than the magnetic field, the physical difference between the two models does not apply.

We also note that other types of spin-orbit interactions exist, which leads to different types of single-particle spectra. For instance, the touching point may be quadratic rather than linear.Abrikosov and Beneslavskii (1970) For some of these systems the behavior in the metallic regime will likely be the same as for the Dirac metals discussed here. For a two-dimensional electron system with a Rashba spin-orbit coupling the analog of Landau Fermi-liquid theory has been developed in Ref. Ashrafi et al., 2013, which also calculated the effective mass and various thermodynamic quantities in terms of Fermi-liquid parameters. A similar study for the two-dimensional helical Fermi liquid on the surface of certain bulk topological insulators was carried out in Ref. Lundgren and Maciejko, 2015. The spin susceptibility for a two-dimensional electron system with a spin-orbit interaction of Rashba type was considered in Ref. Zak et al., 2010a, and the consequences for nuclear spin polarization were discussed in Ref. Zak et al., 2010b. Our result that the chirality-independent modes, Eq. (22a), become massive as a result of the spin-orbit interaction, are consistent with the results of these papers. However, we find that as a result of the chirality degrees of freedom there are different soft modes that remain soft for $v>0$, see Eq. (22b). It is the existence of the latter that leads to a DFL having nonanalyticities very similar to those in a LFL, although as a result of a different set of soft modes.

V.2 Nature of the soft modes

As we have discussed in Sec. III, the soft modes that underly all of the nonanalyticities we have derived are phase-space four-fermion correlation functions that are local in frequency space and of the general form

[TABLE]

For noninteracting electrons, the correlation factorizes into a product of two Green functions and we obtain, in the LFL case, Eq. (17), but at $T=0$ it remains soft even in the case of interactions.Belitz and Kirkpatrick (2012) Their physical interpretation is in terms of the Goldstone modes of a spontaneously broken rotational symmetry in frequency space, see Appendix B. In calculations that are perturbative with respect to the interaction they enter via convolutions of noninteracting Green functions, the most basic one for a DFL is given in Eqs. (20). We note that these Goldstone modes do not describe density fluctuations: The latter are local in time space, whereas the Goldstone modes are local in frequency space. This distinction gets blurred in the case of noninteracting electrons (e.g., Eq. (21b) is just the hydrodynamic part of the Lindhard function, which does describe density fluctuations), but it is physically important in general. For instance, the physical density susceptibility in an interacting system has no dephasing rate, and remains soft, as a result of particle number conservation. This is not true for the Goldstone modes, which do acquire a mass at nonzero temperature. In a DFL, the chirality structure invalidates the interpretation as density fluctuations of even the noninteracting reference system, see the discussion after Eq. (22b)

As the discussion in Sec. III.4 has shown, the properties of the soft modes in the presence of the external field conjugate to the observable under consideration are crucial. Specifically, only modes that are soft of the first kind with respect to the conjugate field, i.e., acquire a small mass upon the application of a small field, can lead to nonanalyticities of the corresponding variable. This observation also underscores the fact that the fundamental nonlinear behavior of any susceptibility is the one with respect to the relevant conjugate field. The nonanalytic dependences on anything else (e.g., wave number, frequency, or any fields other than the conjugate one) are derivative and follow from how these parameters scale with the conjugate field. As an example, the nonanalytic dependence of the spin susceptibility on the magnetic field $h$ expressed in Eq. (36) is the fundamental one for this observable. The nonanalyticities as a function of the temperature, Eq. (35), and the wave number, Eq. (32), are consequences of the fact that both the temperature and the wave number scale linearly with $h$.

Conversely, without a mode that is soft of the first kind with respect to the respective conjugate field there can be no nonanalytic dependence on any parameter. An example is the absence of a nonanalyticity in the density susceptibility and related correlation functions that is predicted by the arguments in Sec. III.4 and confirmed by the calculation in Sec. IV.2. This null result is of great physical importance: Since the diamagnetic susceptibility is related to the coefficient of the ${\bm{Q}}^{2}$ term in the wave-number dependent number-density current suceptibility,Pines and Nozières (1989) a nonanalyticity in the density channel analogous to the one in the spin-density channel would imply ideal diamagnetism.

Another consequence of the structure of the soft modes, and their dependence on various parameters, is the fact that the results for the nonanalyticity in the spin susceptibility, Eqs. (32, 35, 36) are nonzero in the limit $v\to 0$. This does not contradict the fact that the spin-orbit coupling is necessary in order to obtain these terms in the first place. The point is that other modes have acquired a mass due to the spin-orbit interaction and therefore do not contribute to any nonanalytic behavior. If one kept all contributions to the susceptibility, from massive modes as well as from massless ones, then some of the terms that are made massive by the spin-orbit interaction would, in the limit $v\to 0$, cancel the terms we have calculated, leaving behind the result for a LFL. As written, our results for the spin susceptibility are valid for $\Delta,h\ll vk_{\text{F}}\ll\epsilon_{\text{F}}$. Corrections of order $\Delta/vk_{\text{F}}$ and $v/v_{\text{F}}$ can be calculated if desired, they just make the prefactors of the nonanalyticities more complicated. In our calculation of the density of states, Sec. IV.3, we have relaxed the condition $vk_{\text{F}}\ll\epsilon_{\text{F}}$ since in this case some contributions vanish as $v\to 0$.

V.3 Importance of the soft-mode structure

An important conceptual point is that the general considerations in Sec. III suffice for determining the universal nonanalytic behavior of any observable. No explicit calculations are necessary unless one wants to determine the prefactor of the nonanalyticity, which is model dependent and hence non-universal in any case. The calculations in Sec. IV merely demonstrate that the results of explicit calculations are consistent with the general arguments, as they must be. It is also worth noting that the explicit calculations for the DFL are very involved, and guidance from the general structural arguments is very helpful for the asymptotic analysis needed to extract the leading nonanalytic behavior.

The structural arguments can be taken to a higher level by employing field-theoretic and renormalization-group (RG) techniques. In fact, some elements of such an analysis we already employed in Sec. IV by restricting the calculation to diagrams that correspond to one-loop terms in a field-theoretic description. More generally, a LFL is associated with a stable renormalization-group (RG) fixed point that characterizes the entire LFL phase (as opposed to a critical RG fixed point that characterizes a phase transition, see Ref. Ma, 1976 for a discussion of this important distinction). Such a fixed point has been identified in both a fermionic field theoryShankar (1994) and an effective bosonic one.Belitz and Kirkpatrick (2012) Our soft-mode analysis in Sec. III implies that the spin-orbit interaction is a relevant operator with respect to this fixed point and hence makes it unstable. In addition, there must be another unstable fixed point, one describing a Dirac semimetal, with respect to which the chemical potential is a relevant operator. In the presence of both the spin-orbit interaction and a nonzero chemical potential the RG flow must lead from either unstable fixed point to a stable one that describes a DFL. In such a description, the leading nonanalyticities we have derived will be associated with the least irrelevant operators at the stable fixed point. Such a description will do much more than just confirm the perturbative results. Since the nature the nonanalyticities is tied to the scale dimensions of the least irrelevant operators, it can provide a proof of the hypothesis that the perturbative results are exact as far as the exponents of the nonanalyticities are concerned. Higher orders in perturbation theory and/or a loop expansion will change the prefactor, but cannot change the exponents. This is impossible to ascertain within perturbation theory. It has been shown to be true for a LFL,Belitz and Kirkpatrick (2014) and it is important to establish the corresponding result for a DFL. Such an analysis will also underscore the point that the DFL is qualitatively different from a LFL, the similarity of the resulting nonanalyticities notwithstanding: The similarities just reflects the fact that the least irrelevant operators have the same scale dimensions in both cases, but they obscure the fact that they belong to different fixed points.

In $d=3$ spatial dimensions the least irrelevant operators with respect to the DLF fixed point must have a scale dimension of 2, which leads to the logarithmic behavior we have found. In this sense $d=3$ represents an upper critical dimension that marks the boundary between dimensions $d<3$ where the least irrelevant operators represent the leading corrections to the fixed-point behavior, and dimensions $d>3$ where the leading corrections are analytic. As we have seen in Sec. IV, the prefactors of some of the $T^{2}\ln T$ and analogous terms vanish in perturbation theory. There is no structural reason for this that we are aware of. These zeros are likely accidental and an artifact of the low order in the perturbation theory and/or the loop expansion. One example of such an accidental zero is the absence of a $T^{2}\ln T$ contribution to the spin susceptibility of a LFL to second order in perturbation theory in $d=3$.Carneiro and Pethick (1977); Belitz et al. (1997) By contrast, the specific-heat coefficient in $d=3$ does have a $T^{2}\ln T$ contribution,Chubukov et al. (2006) and an analysis of the spin susceptibility for generic dimensions yields a $T^{d-1}$ nonanalyticity with a $d$-dependent prefactor that changes sign and is zero for $d=3$.us_

V.4 Consequences for quantum phase transitions in Dirac metals

We finally mention that our results for the spin susceptibility of a DFL have important implications for various magnetic quantum phase transitions in Dirac metals. It is well known that a nonanalytic dependence of the spin susceptibility on the magnetic field $h$ leads to a corresponding nonanalyticity in the magnetic equation of state.Belitz et al. (1997, 1999); Brando et al. (2016) Equation (36) implies, at $T=0$ and in a mean-field approximation for the dimensionless magnetization $m$, an equation of state of the form

[TABLE]

Here $t$ and $u>0$ are Landau parameters, and $w>0$ due to the sign of the prefactor in Eq. (36).pre This implies that any quantum phase transition from a paramagnetic Dirac metal to a homogeneous ferromagnetic one is necessarily first order. (This applies only at zero temperature, and in the absence of quenched disorder.) There are, however, other possibilities, such as a transition to a state with modulated magnetic order, with the wave-number scale of the modulation given by the maximum in the wave-number dependent spin susceptibility. Analogous conclusions apply to spin-nematic quantum transitions in Dirac metals. Also of interest are transitions to helimagnetic states with a long pitch wavelength, but our the SI-invariant model studied in the present paper does not directly apply to those. These implications will be pursued in a separate project.

Acknowledgements.

This work was supported by the NSF under grant numbers DMR-1401449 and DMR-1401410. Part of this work was performed at the Aspen Center for Physics, supported by the NSF under Grant No. PHYS-1066293, and at the Telluride Science Research Center (TSRC). We thank George de Coster for discussions.

Appendix A Electron-electron interaction in a Dirac metal

Consider a Coulomb interaction between fluctuations of the particle number density

[TABLE]

Here and in what follows we use an obvious spinor notation: $\psi^{\alpha}(k)=(\psi_{\uparrow}^{\alpha},\psi_{\downarrow}^{\alpha})$, and $\psi(k)=(\psi_{\uparrow}^{+},\psi_{\downarrow}^{+},\psi_{\uparrow}^{-},\psi_{\downarrow}^{-})$, and the same four-vector notation $k=({\bm{k}},i\omega_{n})$, $q=({\bm{q}},i\Omega_{n})$ as in Sec. III. With $v({\bm{q}})$ the interaction potential, the interaction part of the action can be written

[TABLE]

These are three identical ways to write the interaction. The first two represent the small-angle and large-angle scattering processes in the particle-hole channel, respectively, from Ref. Belitz et al., 1997. The third one represents scattering in the particle-particle channel, i.e., scattering across the Fermi surface. The latter processes do not contribute to the leading singularities we are interested in, and we neglect them. Since we are interested in metals (as defined in Sec. I) we can consider the interaction potential $v({\bm{q}})$ to be screened and hence short ranged.

The singular behavior we are interested in manifests itself at small values of the scattering wave vector ${\bm{q}}$ (the ‘hydrodynamic regime’). If we restrict the sum over this hydrodynamic wave vector to values $|{\bm{q}}|<\Lambda$, with $\Lambda$ a UV cutoff,cut we can add the first two identical expressions for $S_{\text{int}}$ in Eq. (45) to write an effective interaction

[TABLE]

Here the prime on the sum over $q$ indicates the restriction $|{\bm{q}}|<\Lambda$.

The two terms in Eq. (46) can be decomposed into a spin singlet and a spin triplet by using the completeness relation for the Pauli matrices,

[TABLE]

and the chirality dependence can be written analogously. We obtain

[TABLE]

In the calculation of any observable, the wave vectors ${\bm{k}}$ and ${\bm{p}}$ will be pinned to the Fermi surface by the poles of the Green functions. One can thus expand $v({\bm{k}}-{\bm{p}})$ in a complete set of functions on the Fermi surface. For our isotropic model this results in an expansion in spherical harmonics, as in Landau Fermi-liquid theory.Abrikosov et al. (1963) For our purposes the $\ell=0$ (s-wave) term in this expansion suffices, which amounts to replacing $v({\bm{k}}-{\bm{p}})$ by its average over the Fermi surface, and for studying leading the hydrodynamic singularities we can replace $v({\bm{q}})$ by $v({\bm{q}}=0)$. Furthermore, the four terms in Eq. (48) are individually invariant under rotations in spin space and chirality space, and their respective coupling constants will behave differently under renormalization. We thus obtain Eq. (48) with four independent number-valued interaction amplitudes for the four different terms:

[TABLE]

Equation (49) represents the most general interaction in the s-wave particle-hole channel that is invariant under rotations in both spin and chirality space. In particular, it is the most general interaction in that channel for the model defined in Sec. II in the LFL limit, i.e., for a vanishing spin-orbit interaction, $v=0$. $v\neq 0$ breaks the rotational invariance in chirality space, and $S_{\text{int}}$ as written in Eq. (49) therefore still has a higher symmetry than the single-particle action. If we require invariance only under spatial inversion, rather than under rotation in chirality space, we see that the four terms in Eq. (49) give rise to six different structures that are all individually invariant under spatial inversion and simultaneous rotations in spin space and real space and come with six independent interaction amplitudes.ind These are given in Eq. (LABEL:eq:2.15).

Appendix B Ward identities for a Landau Fermi liquid from rotations in frequency space

Consider noninteracting electrons described by the action $S_{0}$ in Eq. (13) with $v=0$. The partition function is given by

[TABLE]

We augment the action by a source term for products $\bar{\psi}\psi$,

[TABLE]

Now consider transformations of the fermionic fields

[TABLE]

This defines four different transformations for $i=0,1,2,3$. The $T^{(i)}$ are defined as

[TABLE]

$n_{1}$ and $n_{2}\neq n_{1}$ are two fixed Matsubara frequency labels, and $\phi({\bm{x}},{\bm{y}})$ is a nonlocal mixing angle. Note that $\psi$ and $\bar{\psi}$ transform via $T^{(i)}$ and its complex conjugate $T^{(i)*}$, respectively.

All four of the transformations are unitary with respect to the scalar product of $\bar{\psi}$ and $\psi$ that is defined by Eq. (51a) with $J$ replaced by the unit matrix. The functional integration measure is therefore invariant under the transformations, and we have

[TABLE]

Here $\delta S_{0}$ and $\delta S_{J}$ are the variations of the action under any of the transformations. If we differentiate Eq. (54b) with respect to $J$ and put $J=0$ we obtain

[TABLE]

where $\langle\ldots\rangle$ is an average with respect to the action $S_{0}$.

Equation (LABEL:eq:B.6) represents four Ward identities, one for each of the four transformations defined above. Calculating the variations, and factorizing the four-fermion correlation on the left-hand side of Eq. (LABEL:eq:B.6) we find that the four identities provide two independent pieces of information: The transformations with $i=0$ or $i=3$ relate products $G_{n_{1}\sigma}G_{n_{2}\sigma}$ to the difference of the same Green functions, and those with $i=1$ or $i=2$ do the same for $G_{n_{1}\sigma}G_{n_{2},-\sigma}$. The results are equivalent to Eq (17).

This derivation emphasizes that the soft modes are the Goldstone modes of a broken rotational symmetry in frequency space that reflects the difference between retarded and advanced degrees of freedom. The order parameter associated with this broken symmetry is the spectrum of the Green function, which is nonzero everywhere inside the band. This interpretation of the soft modes as Goldstone modes was first discussed by Wegner in the context of disordered electron systems.Wegner (1979)

Appendix C Symmetry properties of correlation functions

Here we list symmetry properties of convolutions of Green functions that are useful for simplifying the integrals one encounters in perturbation theory. They can be checked by explicit calculations and are valid only for the leading singular behavior of the convolutions.

Define a convolution

[TABLE]

Similarly the convolutions

[TABLE]

Useful properties of convolutions of four Green functions involve

[TABLE]

Finally, the convolutions

[TABLE]

These symmetry properties are generalizations of analogous properties in the LFL case.Belitz et al. (1997)