Quantum Ferromagnetic Transition in Clean Dirac Metals

TL;DR

This paper investigates the nature of ferromagnetic quantum phase transitions in clean Dirac metals, revealing that chirality-induced soft modes prevent the transition from becoming continuous, maintaining its first-order character despite strong spin-orbit interactions.

Contribution

It demonstrates that in Dirac metals, chirality introduces new soft modes that keep the ferromagnetic transition first order, challenging previous expectations about quantum critical points.

Findings

Chirality leads to new soft modes in Dirac metals.

The ferromagnetic transition remains first order despite spin-orbit interactions.

Contradicts the expectation of a quantum critical point in such systems.

Abstract

The ferromagnetic quantum phase transition in clean metals with a negligible spin-orbit interaction is known to be first order due to a coupling of the magnetization to soft fermionic particle-hole excitations. A spin-orbit interaction gives these excitations a mass, suggesting the existence of a ferromagnetic quantum critical point in metals with a strong spin-orbit interaction. We show that this expectation is not borne out in a large class of materials with a Dirac spectrum, since the chirality degree of freedom leads to new soft modes that again render the transition first order.

Equations

$$G_k=\frac{1}{2}\sum _{\sigma =\pm }{G}_k^{\sigma }M(\sigma \hat{\mathbf{h}}).$$

$$G_k^{\sigma }$$

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

$${\phi }_{\sigma ,{\sigma }^{\prime }}(\mathbf{q},i{\Omega }_n;i{\omega }_m)$$

$${\chi }_{\text{s}}(h\to 0)={{\chi }_{\text{s}}}^{(0)}+{{\chi }_{\text{s}}}^{(2)}{h}^2\ln (1/h)+O(h^2).$$

$$S_{\text{Z}}=c\int d\mathbf{x}d\tau \mathbf{m}(\mathbf{x},\tau )\cdot {\mathbf{n}}_{\text{s}}(\mathbf{x},\tau )$$

$$f=t{m}^2+\tilde{u}{m}^2\ln (1/m)+u{m}^4.$$

$$H_0$$

$$S_{\text{int}}=\frac{T}{2V}\sum _q\sum _{\alpha \ne {\alpha }^{\prime }}\left({\Gamma }_{\text{t},3}{\mathbf{\Psi }}_q^{\alpha {\alpha }^{\prime }}\cdot {\mathbf{\Psi }}_{-q}^{{\alpha }^{\prime }\alpha }+{\Gamma }_{\text{t},4}{\mathbf{\Psi }}_q^{\alpha {\alpha }^{\prime }}\cdot {\mathbf{\Psi }}_{-q}^{\alpha {\alpha }^{\prime }}\right)$$

$$G_k=\frac{1}{2}\sum _{\alpha ,\beta =\pm }{F}_k^{\alpha \beta }{M}_{\alpha \beta }(\hat{\mathbf{k}})$$

$$\Delta =0$$

$$\Delta \gg v{k}_{\text{F}}$$

$${\phi }_{{\beta }_1,{\beta }_2}^{{\alpha }_1{\alpha }_2}(\mathbf{q},i{\Omega }_n;i{\omega }_m)=\frac{1}{V}\sum _{\mathbf{k}}{F}_k^{{\alpha }_1{\beta }_1}{F}_{k+q}^{{\alpha }_2{\beta }_2}=\int \frac{d{\Omega }_{\mathbf{k}}}{4\pi }\frac{-2\pi i{N}_{\text{F}}\text{sgn}({\omega }_m)\Theta (-{\omega }_m({\omega }_m+{\Omega }_n))}{N(\mathbf{q},i{\Omega }_n;i{\omega }_m)}$$

$$N(\mathbf{q},i{\Omega }_n;i{\omega }_m)=\{\begin{array}{l}i{\Omega }_n-v_{\text{F}}\hat{\mathbf{k}}\cdot \mathbf{q}+{\beta }_2|v(k_{\text{F}}\hat{\mathbf{k}}+\mathbf{q})-{\alpha }_2\mathbf{h}|-{\beta }_1|v{k}_{\text{F}}\hat{\mathbf{k}}-{\alpha }_1\mathbf{h}|\text{for}\Delta =0\\ i{\Omega }_n-v_{\text{F}}\hat{\mathbf{k}}\cdot \mathbf{q}+{\beta }_2|\Delta -{\alpha }_{2}h|-{\beta }_1|\Delta -{\alpha }_{1}h|\text{for}\Delta \gg v{k}_{\text{F}}\end{array}$$

Full text

Solids in which a strong spin-orbit coupling leads to the linear crossing of doubly degenerate bands Herring (1937); Abrikosov and Beneslavskii (1970) have attracted much attention in recent years, most of which has focused on the topological properties of such materials Armitage et al. (2018). If the chemical potential $\mu$ is tuned to the crossing point one has a semimetal, if the crossing point is gapped out and $\mu$ lies in the gap one has a topological insulator. If $\mu$ lies within the conduction band one has a true metal, but the crossing point, whether gapped out or not, still leads to unusual properties. In all of these cases the effective Hamiltonian in the vicinity of the crossing point is reminiscent of a massless (for a gapless system) or massive (for a gapped one) Dirac Hamiltonian. We will be interested in the case of a true metal, which we will refer to as a Dirac metal (DM). Specifically, we will investigate the nature of the quantum phase transition (QPT) from a paramagnetic DM to a ferromagnetic one. This question is of particular interest since some of the recently found Dirac materials are magnetic Liu et al. (2017).

In clean metals with a negligibly weak spin-orbit interaction the ferromagnetic QPT is discontinuous, or first order, in all spatial dimensions $d>1$ Belitz et al. (1997, 1999); Kirkpatrick and Belitz (2012); Brando et al. (2016). At nonzero temperature ($T$) there is a tricritical point on the phase boundary separating a line of second-order transitions at high $T$ from a line of first-order transitions at low $T$. In an external magnetic field there are tricritical wings that end in critical points at the wing tips Belitz et al. (2005). Numerous experiments have confirmed these predictions Brando et al. (2016).

A spin-orbit interaction splits the Fermi surface, which is expected to suppress the soft modes that cause the first-order transition in ordinary metals. A fundamental question then is whether a strong spin-orbit interaction restores a ferromagnetic quantum critical point in a DM, or whether there is another universal mechanism driving the transition first order that is operative even in a DM. We will show that, generically, the answer is the latter.

The first-order mechanism hinges on the nature of electronic soft modes that couple to the magnetization. It is useful to first discuss these soft modes in an ordinary metal. To make the salient point it suffices to consider the Green function for a Fermi gas, which can be written

[TABLE]

Here $k=(\omega_{n},{\bm{k}})$ comprises a fermionic Matsubara frequency $\omega_{n}$ and a wave vector ${\bm{k}}$, and $\xi_{\bm{k}}=\epsilon_{\bm{k}}-\mu$ with $\epsilon_{\bm{k}}$ the single-particle energy. $\sigma_{0}$ is the $2\times 2$ unit matrix, and $\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli matrices. ${\bm{h}}$ is a magnetic field with amplitude $h=|{\bm{h}}|$, and $\hat{\bm{h}}={\bm{h}}/h$. Let $q=(\Omega_{n},{\bm{q}})$ with $\Omega_{n}$ a bosonic Matsubara frequency, and consider wave vector convolutions of the form

[TABLE]

Here $\Omega_{\bm{k}}$ is the solid angle with respect to the wave vector ${\bm{k}}$, and the radial integration has been done in the well-known approximation that captures the part of the integral that is singular in the limit ${\bm{q}},\Omega_{n}\to 0$ Abrikosov et al. (1963).

Important features of this result are: (1) The correlations described by these convolutions are soft modes that scale as $1/q$. They are ballistic in nature, i.e., the frequency scales as the wave number. (2) The modes are soft only if the frequencies of the two Green functions have opposite signs. (3) The modes with $\sigma=\sigma^{\prime}$ (the spin singlet and the longitudinal part of the spin triplet) are soft independent of ${\bm{h}}$, whereas those with $\sigma\neq\sigma^{\prime}$ (the transverse parts of the spin triplet) acquire a mass if $h>0$. (4) Convolutions of $n>2$ Green functions scale as $1/q^{n-1}$ provided the $n$ frequencies do not all have the same sign. (5) Spin conservation is important for this structure. In particular, a spin-orbit interaction is expected to render massive the modes with $\sigma\neq\sigma^{\prime}$.

In a Fermi liquid, i.e., in the presence of an electron-electron interaction, these conclusions remain valid provided the product of $n$ Green functions is replaced by an appropriate $2n$-point correlation function that factorizes into the product of Green functions in the noninteracting limit. The simplest argument for this relies on Landau Fermi-liquid theory which states that the quasi-particle states of the interacting system are adiabatically related to those of the noninteracting one Abrikosov et al. (1963). A different argument is that the soft modes are the Goldstone modes related to a symmetry between retarded and advanced degrees of freedom that is spontaneously broken with or without interactions as long as there is a nonvanishing density of states Wegner (1979); Belitz and Kirkpatrick (2012); Kirkpatrick and Belitz (2019).

These soft modes are responsible for various nonanalyticities of observables in Fermi liquids that have been known for a long time. In particular, the spin susceptibility $\chi_{s}$ is a nonanalytic function of the wave number, temperature, or magnetic field Carneiro and Pethick (1977); Belitz et al. (1997); Betouras et al. (2005). For our purposes we focus on the magnetic field dependence, which in a 3-d system at $T=0$ has the form

[TABLE]

The coefficient $\chi_{\text{s}}^{(2)}$ is positive pos and for weak electron-electron interactions it is proportional to ${\Gamma_{\text{t}}}^{2}$, with $\Gamma_{\text{t}}$ a spin-triplet interaction amplitude int . The logarithmic $h$-dependence reflects the fact that the only soft modes that couple to $\chi_{\text{s}}$ are the two transverse spin-triplet channels ($\sigma^{\prime}\neq\sigma$ in Eq. (2)) that are cut off by $h\neq 0$.

As mentioned above, a spin-orbit coupling is expected to cut off this singularity and make $\chi_{\text{s}}$ an analytic function of $h$. We will confirm this expectation below.

The nonanalyticity shown in Eq. (3) has profound consequences for the ferromagnetic QPT in a clean metal. Since the soft modes couple to the spin density ${\bm{n}}_{\text{s}}$, they also couple to the magnetization ${\bm{m}}$ as the latter couples to the former via a Zeeman contribution to the action,

[TABLE]

where $c$ is a coupling constant and $\bm{x}$ and $\tau$ denote real-space position and imaginary time, respectively. Any order-parameter theory in terms of the magnetization therefore will contain correlation functions of ${\bm{n}}_{\text{s}}$ in its vertices, and in particular incorporate the nonanalytic behavior of $\chi_{\text{s}}$. Within a renormalized mean-field theory this leads to a free energy density Belitz et al. (1997, 1999); Kirkpatrick and Belitz (2012); Brando et al. (2016)

[TABLE]

$t$, $\tilde{u}$, and $u$ are Landau parameters. Importantly, $\chi_{\text{s}}^{(2)}>0$ implies $\tilde{u}>0$. As a result, the QPT described by Eq. (5) is necessarily first order. The fermionic fluctuations that lead to the nonanalytic term in Eq. (5) thus invalidate Hertz’s conclusion Hertz (1976) that the ferromagnetic QPT is second order with mean-field critical behavior. The first-order nature of the transition described by Eq. (5) is in excellent agreement with experiment Brando et al. (2016).

The central question of the current paper is: What will a spin-orbit interaction do to these effects? As mentioned above, it is expected to make massive the underlying soft modes, which suggests that a sufficiently strong spin-orbit interaction will restore a ferromagnetic quantum critical point. We will show that this conclusion is not correct for DMs as defined above. While the spin-triplet modes that are soft in a Landau Fermi liquid do indeed become massive, the chiral degree of freedom in a DM leads to a new class of soft modes that restore the nonanalyticity of $\chi_{\text{s}}$. We thus predict that, surprisingly, the ferromagnetic QPT in a DM is first order, just as it is in ordinary metals. A possible exception are DMs where a symmetry enforces a gapless spectrum, see below. These predictions are relevant for a large class of Dirac materials, some of which are magnetic Liu et al. (2017).

We consider systems in which the spin-orbit interaction causes a linear crossing of two bands via a term $\pm{\bm{k}}\cdot{\bm{\sigma}}$ in the single-particle Hamiltonian Abrikosov and Beneslavskii (1970). Within a simple isotropic model, the most general Hamiltonian that respects both time reversal (in the absence of a magnetic field) and spatial inversion symmetry reads Zhang et al. (2009)

[TABLE]

The Pauli matrices $\bm{\sigma}$ represent the physical spin degree of freedom. A related class of models where $\bm{\sigma}$ represents a pseudo-spin degree of freedom, as is the case in graphene Kotov et al. (2012), behave very differently with respect to the properties we will discuss. The $\pi_{i}$ in Eq. (6) are a second set of Pauli matrices that represent the chirality degree of freedom necessary to insure inversion symmetry. $v$ is a characteristic velocity that measures the strength of the spin-orbit interaction, and $\Delta$ produces a gap in the single-particle spectrum. The magnetic field ${\bm{h}}$ has been chosen to point in the 3-direction.

The single-particle spectrum $E_{\bm{k}}$ is obtained by finding the eigenvalues of $H_{0}$. We introduce an atomic-scale momentum $p_{0}$, velocity $v_{0}=p_{0}/2m$, and energy $E_{0}=p_{0}^{2}/2m$, with $m$ the effective electron mass, and measure $E_{\bm{k}}$, $\Delta$, and $h$ in units of $E_{0}$, $v$ in units of $v_{0}$, and ${\bm{k}}$ in units of $p_{0}$. Figure 1 shows the spectrum for $k_{x}=k_{y}=0$, $h=0$, and two values of $v$. We are interested in true metals, where $\mu>\Delta\geq 0$. For a given value of $v$ the spectrum near the Fermi surface is then qualitatively independent of whether $\Delta=0$ or $\Delta>0$. However, the nature of the ferromagnetic QPT may depend on whether or not the spectrum is gapped, see below.

The Fermi liquid we are interested in is governed by $H_{0}$ plus all interaction amplitudes that are compatible with the symmetry of $H_{0}$. Since we are dealing with a true metal, screening works and the interactions can be localized in both space and time. Only amplitudes in the spin-triplet channel contribute to the nonanalytic behavior of the spin susceptibility. Furthermore, for reasons that will become clear later, only scattering processes that mix chiralities are relevant for our purposes. There are two interaction processes that are consistent with these criteria, viz.

[TABLE]

Here ${\bm{\Psi}}^{\alpha\alpha^{\prime}}_{q}=\sum_{k}{\bar{\psi}}^{\alpha}(k){\bm{\sigma}}\psi^{\alpha^{\prime}}(k-q)$ with $\bar{\psi}$ and $\psi$ fermionic spinor fields. $\Gamma_{\text{t,4}}$ was not considered in Ref. Kirkpatrick and Belitz, 2019. It breaks the conservation of the number of particles with a given chirality. The gap $\Delta$ in $H_{0}$ breaks the same symmetry, while the other terms in $H_{0}$ respect it. $\Gamma_{\text{t,4}}$ is therefore not allowed in systems where $\Delta=0$ due to a crystal symmetry, as is the case in some materials Wang et al. (2013); Neupane et al. (2014).

We now discuss the soft modes in the chiral Fermi liquid described by Eqs. (6, 7) that are analogous to Eq. (2). The Green function for the Hamiltonian $H_{0}$ can be written as a generalization of Eq. (1a):

[TABLE]

$F$ and $M$ for arbitrary parameter values are complicated, but simplify in the limits $\Delta=0$ and $\Delta\gg vk_{\text{F}}$, respectively, where $k_{\text{F}}$ is the Fermi wave number. We find

[TABLE]

with $M(\hat{\bm{k}})$ from Eq. (1c). The generalization of the convolution $\varphi$ in Eq. (2) thus reads

[TABLE]

For $v=\Delta=0$ the cone index $\beta$ reverts to the spin projection index $\sigma$ and we recover Eq. (2). For a chiral Fermi gas the modes with $\beta_{1}\neq\beta_{2}$ acquire a mass due to $v$, or $\Delta$, or both. That is, the modes that led to the nonanalyticity of $\chi_{\text{s}}$ in an ordinary Fermi gas, Eq. (3), cannot do so in a chiral one. However, for $\beta_{1}=\beta_{2}$ the modes in Eq. (11a) are soft, and for $\alpha_{1}\neq\alpha_{2}$ these soft modes are cut off by $h$ cut . These conclusions remain valid in a chiral Fermi liquid for the same reasons an in an ordinary Fermi liquid. The chirality thus restores the possibility of a nonanalytic $\chi_{\text{s}}$ that has the form shown in Eq. (3), and provide a new mechanism for the ferromagnetic QPT to be first order. The details depend on the parameter values in $H_{0}$, as we will now discuss.

We first consider the generic case of a gapped DM. The qualitative features are captured by the limit $\Delta\gg vk_{\text{F}}$, which greatly simplifies the calculations. For $\beta_{1}=\beta_{2}$ the modes are soft, and for $\alpha_{1}\neq\alpha_{2}$ the singularity is cut off by a magnetic field. The problem then maps onto the ordinary Fermi liquid, with the chirality degree of freedom in the chiral case playing the role of the spin projection in the non-chiral one. $\chi_{\text{s}}$ is given by Eq. (3) with both $\Gamma_{{\text{t}},3}$ and $\Gamma_{{\text{t}},4}$ contributing to the coefficient $\chi_{\text{s}}^{(2)}$. The arguments that lead to Eq. (5) then carry through and the ferromagnetic QPT is first order.

The case of a gapless DM is more complicated. Let us assume that $\Delta=0$ due to a lattice symmetry, which implies $\Gamma_{{\text{t}},4}=0$ as well. This case was considered in Ref. Kirkpatrick and Belitz (2019). To second order in $\Gamma_{{\text{t}},3}$, $\chi_{\text{s}}$ is given by integrals over products of two convolutions that are $3$-$F$ generalizations of Eqs. (11). Within each convolution the cone indices $\beta$ must all be the same, as explained above, but the overall integral is nonzero only if the two convolutions correspond to different cones. This is because for intra-cone scattering the magnetic field ${\bm{h}}$ can be eliminated by a shift of the hydrodynamic wave vector ${\bm{q}}$, and hence these processes cannot contribute to a nonanalytic $h$-dependence of $\chi_{\text{s}}$. This has important consequences: If $v$ is sufficiently large, then only the upper cone contributes to the Fermi surface, see Fig. 1. This suggests that if $v$ is large enough, $\chi_{\text{s}}$ is an analytic function of $h$ and the ferromagnetic QPT is second order since the soft modes cannot couple to the order parameter. In this case Hertz theory Hertz (1976) is expected to apply, and the critical behavior will be mean-field-like with the dynamics affected by a dangerous irrelevant variable Millis (1993). For smaller values of $v$ both cones contribute to the Fermi surface, see Fig. 1, $\chi_{\text{s}}$ is nonanalytic, and the transition is first order.

Finally, if $\Delta=0$ due to fine tuning (e.g., by doping a gapped DM), then there is no reason for $\Gamma_{{\text{t}},4}$ to be zero, and the situation is different again: The integral that contributes to the nonanalyticity of $\chi_{\text{s}}$ now is nonzero even if the cone indices of the two convolutions are the same. In this respect $\Gamma_{{\text{t}},4}\neq 0$ has the same effect as $\Delta\neq 0$. This was to be expected since the two terms break the same gauge symmetry, see the remark after Eq. (7). As a result, one cone contributing to the Fermi surface suffices for producing a nonanalyticity in $\chi_{\text{s}}$, and the ferromagnetic QPT is first order even for $v$ so large that only the up-cone contributes to the Fermi surface.

We conclude with several remarks. (1) While we have used many-body diagrammatic techniques for the calculations, the selection of the diagrams is informed by an underlying effective field theory as explained in Ref. Kirkpatrick and Belitz (2019). We have performed the calculation to one-loop order and in addition have restricted ourselves to second order in the electron-electron interaction. Renormalization-group arguments within the field theory show that $\chi_{\text{s}}$ must scale with $h$ as shown in Eq. (3) in $d=3$, and as $\chi_{\text{s}}\propto\text{const.}+h^{d-1}$ in generic dimensions $1<d<3$. This result is therefore exact as far as the functional form of the nonanalyticity is concerned, and the perturbative calculation merely confirms that the prefactor is nonzero.

(2) The preceding considerations imply that our results will not qualitatively change if one goes to higher order in the interaction expansion. For generic (i.e. gapped) DMs they cannot change by going to higher order in the loop expansion either. However, one open question is whether effects at higher loop order will restore the missing coupling in a DM that is gapless by symmetry and has a large spin-orbit coupling $v$, and thus render the QPT in such systems first order as well.

(3) For the velocity $v$, which measures the strength of the spin-orbit interaction, there are three relevant regimes: (i) $vk_{\text{F}}$ is small compared to the discontinuity of the magnetization at the first-order QPT for $v=0$, both measured in atomic units. While $v$ changes the soft-mode structure, the effect is too small to override the first-order mechanism that is operative for $v=0$. $v$ then is negligible and we have the case of an ordinary metal as discussed before Belitz et al. (1999); Brando et al. (2016). (ii) $v$ is not negligible, but still small enough for both cone branches to contribute to the Fermi surface, see Fig. 1. The soft-mode structure is now qualitatively different from an ordinary metal, the system is a DM, and the chiral soft modes are crucial for the conclusion that the QPT is still first order. This holds irrespective of whether the DM is gapped or not. (iii) $v$ is so large that only the up-cone branch contributes to the Fermi surface, Fig. 1. It now becomes important to distinguish between: (a) $\Delta\gg vk_{\text{F}}$. Then $vk_{\text{F}}=v\sqrt{2m(\mu-\Delta)}$, so this case can be realized if $\mu\gtrsim\Delta$, even if $v\lesssim v_{0}$. In this limit the analysis of the QPT maps onto the ordinary-metal case, even though the soft-mode structure is physically very different, and the QPT is again first order. (b) $\Delta\ll vk_{\text{F}}$. Then $vk_{\text{F}}=\sqrt{2mv^{2}\mu}$ if $mv^{2}\ll\mu$, or $vk_{\text{F}}=\mu$ if $mv^{2}\gg\mu$. In either case $\Delta<\mu$, as required for a metal. In this limit the QPT is second order in our one-loop calculation if $\Gamma_{{\text{t}},4}=0$, but first order if $\Gamma_{{\text{t}},4}\neq 0$.

(4) The graphene-type models that are given by Eq. (6) with $\bm{\sigma}$ representing a pseudo-spin rather than the physical spin behave very differently. In such systems, the soft-mode spectrum is the same as in ordinary metals and the ferromagnetic QPT is always first order.

(5) An alternative consequence of the nonanalytic $\chi_{\text{s}}$ is a QPT to an inhomogeneous magnetic state, which may compete with the first-order transition to a homogeneous ferromagnet. For ordinary metals this possibility was suggested in Ref. Belitz et al. (1997) and studied in detail in Refs. Conduit et al. (2009); Karahasanovic et al. (2012). An analogous investigation is needed for DMs.

This work was initiated at the Telluride Science Research Center (TSRC). We thank George de Coster for discussions.