Time reversal symmetry breaking and $d$-wave superconductivity of triple-point fermions

The Ginzburg-Landau free energy describing a finite-temperature second-order phase transition towards a $d$-wave order parameter is given by Mermin (1974); Boettcher and Herbut (2018a)

with the terms quadratic and quartic in the uniform order parameter as

The superconducting phase ensues when the quadratic coefficient $r<0$; the coefficient $q_{1}$ also needs to be positive for $F$ to be bounded from below. Note that the quartic term multiplied by the coefficient $q_{1}$ has the same value for all normalized states, so the value of the coefficient $q_{1}$ does not play a role in the selection of the superconducting order parameter. The signs and magnitudes of the two remaining coefficients $q_{2}$ and $q_{3}$ generally decide the broken symmetry state of the system. Let us first understand the role of the coefficient $q_{2}$, and assume $q_{3}$ to be sufficiently small. The value of the product $|\Delta_{a}\Delta_{a}|^{2}$ in our representation quantifies the overlap of the macroscopic superconducting state with its time-reversed counterpart Link et al. (2020). Hence, if $q_{2}<0$, the real order parameters which describe TR-preserving states maximize this term, and are therefore favored. The matrix $\phi$ can then be rotated into the diagonal form:

with the relative values of $\Delta_{1}$ and $\Delta_{2}$ that need to be determined from higher order terms, or by considering Gaussian fluctuations around the mean-field solution. If, on the other hand, $q_{2}>0$, the complex states with $\Delta_{a}\Delta_{a}=0$ and with maximally broken TR symmetry are preferred. There are also infinitely many order parameters which break TR symmetry maximally, so $q_{2}$ alone does not uniquely select the superconducting ground state. This brings us to the role of the remaining coefficient $q_{3}$. The sign of $q_{3}$ decides which of the complex superconducting states that break TR maximally is the superconducting ground state. The third term in the right-hand side of Eq.(18) with the coefficient $q_{3}$ may be shown to be related to the average magnetization of the state,Link et al. (2020) so that when $q_{2}>0$ and $q_{3}<0$, the state with maximal average magnetization and maximally broken TR symmetry is favored. This is the “ferromagnetic” state with

However, if the coefficients $q_{2}$ and $q_{3}$ both are positive, then the magnetization of the ground states needs to be minimized while keeping the TR symmetry maximally broken. These two requirements are not mutually exclusive, and are in fact fulfilled by the “cyclic” state, which is defined as

The details of arriving at Eqs. (19), (20) and (21) are presented in Appendix (B). The potential ground states configurations can also be systematically identified from a symmetry-based classificationKawaguchi and Ueda (2011); Herbut et al. (2019) of the $d$-wave order parameter. These states always show some residual symmetry, breaking the rotational $SO(3)$ invariance of the normal state. For instance, the ferromagnetic state and the cyclic state, which will be of our main interest eventually, have the remaining $SO(2)$ and tetragonal symmetry, respectively.

The complete phase diagram for rotationally invariant $d$-wave superconductors was first obtained by MerminMermin (1974) and has also been rederived and discussed at length in ref. Link et al., 2020. The one-loop values of the quartic coefficients have been computed in several examples so far. Most importantly, for the spherically symmetric Luttinger semimetals and standard BCS $d$-wave superconductors, $q_{3}$ has been shown to be exactly zeroMermin (1974); Boettcher and Herbut (2018a); Sim et al. (2019a); Herbut et al. (2019), whereas for the RSW semimetals $q_{3}$ turns out to be small and positive Link et al. (2020). In the case of three-band crossing under consideration here we show that a qualitatively novel situation arises with $q_{3}$ having either sign, depending on curvatures of the energy bands.

In the next section we describe the results of the computation of the quartic coefficients for triple-point fermions, and use it to determine the $d$-wave superconducting ground state of the system.

After performing the standard but lengthy one-loop integration over triple-point fermions (details are given in App. C), we find that for a finite and positive chemical potential the coefficients $q_{1}$ and $q_{2}$ are always positive irrespective of the values of the parameters $b$ and $c$ in the Hamiltonian, and have the standard temperature dependence of $\sim 1/T^{2}$ Stintzing and Zwerger (1997); Herbut (2000). In contrast, the coefficient $q_{3}$ changes its sign depending on the values of $b$ and $c$, as shown in Fig. 2, and it depends only logarithmically on temperature.

To understand the different temperature dependence of the quartic terms’ coefficients, and in particular why the sign of the coefficient $q_{3}$ is influenced by $b$ and $c$, let us consider $\mu>0$ and set $b=0$ for simplicity so that only the parameter $c$ determines the curvature of the energy bands. In this case, we find that $q_{1}$ and $q_{2}$ are positive for all physically relevant values $-v^{2}/(4\mu)<c$, whereas the coefficient $q_{3}$ is only positive if $c\leq 0$ and it is negative for $c>0$. The reason for this change in sign of $q_{3}$ lies in the analytical structure of the integrands of $q_{1,2,3}$, and, in particular, in the way that this structure depends on the curvature of the energy bands. To this end, we carry out the finite-temperature Matsubara sum of the one-loop integral over fermions that define the coefficients and Taylor expand around the Fermi momenta of the normal state. The coefficients $q_{1,2,3}$ then reduce to the following form:

and

where $\{a^{(1)}_{k_{\rm F},i},a^{(1)}_{t,i},b^{(1)}_{k_{\rm F},i},b^{(1)}_{t,i}\}$, $\{a^{(2)}_{k_{\rm F},i},a^{(2)}_{t,i},b^{(2)}_{k_{\rm F},i},b^{(2)}_{t,i}\}$, and $\{{\tilde{b}}_{k_{\rm F},i},{\tilde{b}}_{t,i}\}$ are numerical constants associated with quartic coefficients $q_{1}$, $q_{2}$, and $q_{3}$, respectively. Here, the sum is over all Fermi surfaces indexed by $i$, and $k_{{\rm F},i}$ and $v_{{\rm F},i}$ represent the Fermi momentum and Fermi velocity of the corresponding Fermi surface.

Finally, performing the momentum integration in Eq. (22), we find that $q_{1,2}$ acquire the usual $\sim 1/T^{2}$ temperature dependence to the leading order in inverse temperature, with the next order correction proportional to $\log(T)$. In the coefficient $q_{3}$, however, the leading order term in the integrand that would be of the form of $({\tilde{a}}_{k_{\rm F},i}v^{3}_{{\rm F},i}{|\textbf{k}-\textbf{k}_{{\rm F},i}|}^{3}+{\tilde{a}}_{t,i}T^{3})^{-1}$ is absent, leaving the first non-zero term in the Taylor expansion proportional to $({\tilde{b}}_{k_{\rm F},i}v_{{{\rm F},i}}|\textbf{k}-\textbf{k}_{{\rm F},i}|+{\tilde{b}}_{t,i}T)^{-1}$. The absence of the leading order term causes the different temperature dependence of $q_{1,2}$ and $q_{3}$, and it was also found in the example of $d$-wave pairing of the RSW fermions in ref. Link et al., 2020. We find that the coefficients near the critical temperature ($T_{c}$) are given by the following expressions to the leading order in $\mu/T_{c}\ll 1$:

and

where ${\eta}^{(1)}_{i}$, ${\eta}^{(2)}_{i}$, and $\tilde{\eta}_{i}$ are numerical coefficients.

What still remains to be explained is the dependence of the coefficient $q_{3}$ on parameters $b$ and $c$. As we already noted, parameters $b$ and $c$ determine which energy bands cross the Fermi level and thus control the number of Fermi surfaces in the normal state. Hence, the parameter $c$ sets the number of terms that contribute to the coefficients when we set $b=0$. For example, if $c<0$, there are two Fermi surfaces, and to obtain the quartic coefficients one needs to sum over $i=+1_{+},+1_{-}$. In other words, the coefficients then receive two contributions with the same Fermi velocity $v_{{\rm F},+1_{+}}=v_{{\rm F},+1_{-}}$ but with different Fermi momenta $k_{{\rm F},+1_{+}}$ and $k_{{\rm F},+1_{-}}$, as defined in Eqs. (8)-(10). If $c=0$, the energy band $E_{+1}$ crosses the Fermi level only once, and there is simply one Fermi momentum and the accompanying Fermi velocity that contributes to the quartic coefficients. The situation changes drastically for $c>0$ where all three energy bands cross the Fermi level; we then find three different Fermi surfaces and the sum is over $i=-1_{+},0,+1_{+}$ in Eqs. (24)-(25), with three different Fermi momenta $k_{{\rm F},0}$, $k_{{\rm F},-1_{+}}$, $k_{{\rm F},+1_{+}}$ and two different Fermi velocities $v_{{\rm F},0}$ and $v_{{\rm F},\pm 1_{+}}$. The leading order terms in the weak-coupling ($\mu/T_{c}\ll 1$) expressions for the quartic coefficients with $b=0$ are given as

$q_{2}(T_{c},c)=q_{1}(T_{c},c)/2$, and

with $C_{1}=1/(2^{1/3}3^{1/6}15120\pi)$, and $C_{3}=3/(35\pi^{2})$ . The coefficients $q_{1}$ and $q_{2}$ are positive for all values of $c$ since all terms in the sum $\sum_{i}{\eta}^{(1,2)}_{i}\frac{k_{{\rm F},i}^{2}}{v_{{\rm F},i}}$ arising from different Fermi surfaces are positive. The situation is, however, different for the $q_{3}$ coefficient; for $c\leq 0$, there are two Fermi surfaces that bring in two positive contributions to $q_{3}$ and yield a positive sign for $q_{3}$. On the other hand, when $c>0$, an additional Fermi surface appears and there is a competition between the contributions arising from different Fermi surfaces, i.e., a competition between $\tilde{\eta}_{0}/v_{{\rm F},0}$ and $\tilde{\eta}_{\pm 1_{+}}/v_{{\rm F},\pm 1_{+}}$. In other words, if we only consider contributions arising due to the expansions around $k_{{\rm F},\pm 1_{+}}$, which corresponds to $\tilde{\eta}_{{\pm 1}_{+}}/v_{{\rm F},{\pm 1}_{+}}\propto 3/\sqrt{v^{2}+4c\mu}$, the sign of $q_{3}$ would be positive. However, when we include the contribution from the expansion around $k_{{\rm F},0}$ which corresponds to $\tilde{\eta}_{0}/v_{{\rm F},0}\propto-2/\sqrt{c\mu}$, it always dominates and ultimately leads to a negative sign for $q_{3}$. As a result, for $b=0$, the cyclic state is favored if $c\leq 0$, and the ferromagnetic state is preferred when $c>0$.

As already mentioned earlier, for $\mu>0$ and $c<0$, we get positive contributions from two Fermi surfaces that originate from the same band $E_{+1}$. A similar situation also arises when one considers $\mu<0$ and $c>0$. In such a case, the $E_{-1}$ band intersects the Fermi level twice instead, forms two Fermi surfaces at two different momenta, and eventually leads to two positive contributions to $q_{3}$. One can easily check that the expressions for quartic coefficients stay invariant when one simultaneously transforms $\{\mu\rightarrow-\mu,b\rightarrow-b$, and $c\rightarrow-c\}$. This invariance suggests that the phase diagram for a positive chemical potential and the phase diagram for a negative chemical potential are related by two joint reflections, one around the $b$ axis and the other around the $c$ axis.

Finally, let us see how a finite parameter $b$ influences the superconducting state of the system. As shown in Fig. 2, in the case of the positive chemical potential, the cyclic state is the preferred superconducting state for $c\leq 0$ and finite $b$. However, for positive $c$ we find that the superconducting state could be either the cyclic or the ferromagnetic state depending on the value of $b$. To understand the phase diagram qualitatively, let us assume again a finite, positive value of $c$ and $b=0$. As we already noted, for $b=0$ and $c>0$, the negative contribution arising due to the expansion around $k_{{\rm F},0}$ always dominates the positive $k_{{\rm F},\pm 1_{+}}$ contribution and $q_{3}$ is consequently negative. However, coefficient $\tilde{\eta}_{0}/v_{{\rm F},0}$ gets progressively weaker with increasing $|b|$, and it even changes sign at a finite value of $b$. On the other hand, the two coefficients $\tilde{\eta}_{\pm 1_{+}}/v_{{\rm F},\pm 1_{+}}$ for $b+c>0$ ($\tilde{\eta}_{+1_{\pm}}/v_{{\rm F},+1_{\pm}}$ for $b+c<0$), always stay positive. As a consequence, one finds a phase boundary where the positive contribution from the Fermi surfaces at $k_{{\rm F},\pm 1_{\pm}}$ exceeds the negative contribution from near the Fermi momentum $k_{{\rm F},0}$, and any further increase of the absolute value of the parameter $b$ only yields the cyclic state as the superconducting ground state.

The curvature of the energy bands is thus found to be the “knob” that can tune between the cyclic and the ferromagnetic superconducting state.

In this section we analyze the energy spectrum of the Bogoliubov-de Gennes (BdG) quasiparticles which are described by the BdG-Hamiltonian $\sum_{\textbf{p}}\Psi^{\dagger}H_{\rm BdG}\Psi$ with

where the Nambu spinor is given by ${\Psi(\omega_{n},\textbf{p})=\big{(}\psi(\omega_{n},\textbf{p}),\mathcal{U}\psi^{*}(-\omega_{n},-\textbf{p})\big{)}}$. We focus on the quasiparticle energy spectrum for the cyclic and the ferromagnetic state, and find that both of these TR-symmetry-breaking states exhibit extended regions in the momentum space where the energy vanishes. These regions are known as Bogoliubov-Fermi (BF) surfacesTimm et al. (2017); Agterberg et al. (2017); Bzdušek and Sigrist (2017); Brydon et al. (2018); Menke et al. (2019); Lapp et al. (2020); Setty et al. (2020); Link and Herbut (2020); Herbut and Link (2021) and can be determined by identifying the momentum points that satisfy $\det[H_{\rm BdG}(\mathbf{p})]=0$. The emergence of these BF surfaces in the present case is elaborated on in Appendix D.

The mini BF surfaces that emerge for the cyclic and the ferromagnetic superconducting states are shown in Fig. 3. Note that the number of BF surfaces is tied to the number of Fermi surfaces present in the normal state. If the superconducting ground state is the cyclic state, for example, then it displays 8, 16, or 24 BF surfaces depending on whether it has one, two, or three Fermi surfaces in the normal state. The BF surfaces in this case generally appear along the diagonals of a cube centered at the origin of the momentum space. This picture changes in the ferromagnetic state; this state emerges in a narrow region of the phase diagram where the normal state generally has three Fermi surfaces. As a result, it typically exhibits six BF surfaces centered around the $z$-axis. We observe that despite of the triple-point fermion Hamiltonian’s lack of inversion symmetry, the BF surfaces appear nevertheless, in accord with the general arguments presented in ref. Link and Herbut, 2020.