Erratum: Anomalous Hall effects in chiral superconductors
- 1Center for Applied Physics and Superconducting Technologies, Department of Physics, Northwestern University, Darlington, IL, United States
- 2Fermi National Accelerator Laboratory, Batavia, IL, United States
- 3Initiative for the Theoretical Sciences, The Graduate Center, City University of New York, New York, NY, United States
- 4School of Physics, University of Sydney, Sydney, NSW, Australia
- 5Hearne Institute of Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, United States
We report theoretical results for the electronic contribution to thermal and electrical transport for chiral superconductors belonging to even or odd-parity E1 and E2 representations of the tetragonal and hexagonal point groups. Chiral superconductors exhibit novel properties that depend on the topology of the order parameter and Fermi surface, and—as we highlight—the structure of the impurity potential. An anomalous thermal Hall effect is predicted and shown to be sensitive to the winding number, ν, of the chiral order parameter via Andreev scattering that transfers angular momentum from the chiral condensate to excitations that scatter off the random potential. For heat transport in a chiral superconductor with isotropic impurity scattering, i.e., point-like impurities, a transverse heat current is obtained for ν = ±1, but vanishes for |ν| > 1. This is not a universal result. For finite-size impurities with radii of order or greater than the Fermi wavelength, R ≥ ℏ/pf, the thermal Hall conductivity is finite for chiral order with |ν|≥ 2, and determined by a specific Fermi-surface average of the differential cross-section for electron-impurity scattering. Our results also provide quantitative formulae for analyzing and interpreting thermal transport measurements for superconductors predicted to exhibit broken time-reversal and mirror symmetries.
1 Introduction
The remarkable properties of the spin-triplet, p-wave phases of superfluid 3He have stimulated research efforts to discover and identify electronic superconductors with novel broken symmetries and non-trivial ground-state topology [1–6], driven in part by predictions of novel transport properties. Chiral superfluids and superconductors are topological phases with gapless Fermionic excitations that reflect the momentum-space topology of the condensate of Cooper pairs. The A-phase of superfluid 3He was definitively identified as a chiral p-wave superfluid by the observation of anomalous Hall transport of electrons moving through a quasiparticle fluid of chiral Fermions [7, 8]. A chiral d-wave state was proposed for doped graphene [9, 10], while a chiral p-wave state is proposed for MoS [11]. There is evidence from μSR of broken time-reversal symmetry onsetting at the superconducting transition for the pnictide SrPtAs [12], and a chiral d-wave state has been proposed as the ground state [13]. Recent μSR experiments also provide evidence for chiral d-wave superconductivity in the pnictide LaPt3P [14]. The perovskite, Sr2RuO4, has been studied extensively and was proposed as a promising candidate for chiral p-wave superconductivity (Eu pairing with
The first superconductor reported to show experimental evidence of broken time-reversal symmetry was the heavy fermion superconductor, UPt3, based on μSR linewidth measurements [31]. This experiment followed theoretical predictions of broken time-reversal symmetry in the B-phase of UPt3, i.e., the lower temperature superconducting phase [32]. Another notable signature of broken time-reversal symmetry is the onset of Kerr rotation as UPt3 enters its low-temperature B-phase [33]. More recently, a neutron scattering experiment, using vortices as a probe for the superconducting state in the bulk, offers yet another piece of evidence for broken time-reversal symmetry in UPt3 [6]. These results support the identification of a chiral superconducting phase of UPt3, and they also support the basic theoretical model of a multi-component order parameter belonging to a two-dimensional representation of the hexagonal point group, D6h, in which a weak symmetry breaking field lifts the degeneracy of the two-component order stabilizing two distinct superconducting phases in zero magnetic field [32, 34]. In this theory the predicted A phase of UPt3 is time-reversal symmetric with pronounced anisotropic pairing correlations in the hexagonal plane [35, 36], is preferentially selected by the symmetry breaking field, and nucleates at
There are four two-dimensional representations of D6h: two even-parity representations, E1g and E2g, and two odd-parity representations, E1u and E2u, all of which allow for chiral ground states [37, 38]. The chiral ground states belonging to the E1 and E2 representations are defined by zeroes of the Cooper pair amplitude at points
2 Anomalous Hall transport
The winding number of the order parameter for a chiral superconductor reflects the topology of the superconducting ground state. For a fully gapped chiral superconductor ν is related to the Chern number defined in terms of the Bogoliubov-Nambu Hamiltonian in 2D momentum space, or for chiral superconductors defined on a 3D Fermi surface the effective two-dimensional spectrum at fixed pz ≠ 0,
For 2D chiral phases there is a spectrum of massless chiral Fermions confined on the boundary (edge states) with the zero-energy state enforced by the bulk topology. However, for a chiral order parameter defined on a closed 3D Fermi surface there is also a bulk spectrum of gapless Weyl-Majorana Fermions with momenta near the nodal points pz = ±pf, in addition to a spectrum of massless chiral Fermions confined on surfaces normal to the [1,0,0] and [0,1,0] planes [45].
2.1 Anomalous edge transport
For a fully gapped chiral p-wave ground state in two dimensions the spectrum of chiral edge Fermions is predicted to give rise to quantized heat and mass transport in chiral superfluids and superconductors [45–49]. In particular, an anomalous thermal Hall conductance is predicted to be quantized,
For a chiral superconductor defined on a 3D Fermi surface an anamolous thermal Hall current is predicted, but is not quantized in units of a fundamental quantum of conductance. Based on the linear response theory of Qin et al. [50] Goswami and Nevidomsky obtained a result for the anomalous thermal Hall conductivity of the B-phase of UPt3 for
The anomalous thermal Hall conductivity reflects the number of branches of chiral Fermions confined on the [1,0,0] or [0,1,0] surface, i.e., |ν| = 2 for the E 2u chiral ground state. The non-universality of the thermal Hall conductivity is reflected by the term Δp, which is the “distance” between the two topologically protected ν = 2 Weyl points at
Thus, heat transport experiments could decisively identify the broken symmetries and topology of superconductors predicted to exhibit chiral order. The thermal conductivity depends on both the topology of the order parameter and the Fermi surface. The anomalous thermal Hall effect, in which a temperature gradient generates heat currents perpendicular to it, results from broken time-reversal and mirror symmetries—a direct signature of chiral pairing3. A zero-field thermal Hall experiment can also be used as a signature of chiral edge states. However, zero-field thermal Hall transport has remained elusive thus far.
2.2 Impurity-induced anomalous transport
Here we consider zero-field Hall transport resulting from electron-impurity interactions in the bulk of the superconductor, which we show are easily several orders of magnitude larger than the edge contribution [52]. There are earlier theoretical predictions for impurity-induced anomalous thermal Hall effects in chiral superconductors based on point-like impurities by several authors [53–55]. As we show, the point-like impurity model, which includes only s-wave quasiparticle-impurity scattering, predicts zero Hall response except for Chern number ν = ±1 [52], i.e., only for chiral p-wave superconductors [53].
In the following we present a self-consistent theory incorporating the effects of finite-size impurities and show that such effects are essential for a quantitative description of Hall transport in chiral superconductors. Experimental observation of an impurity-induced anomalous thermal Hall effect would provide a definitive signature of chiral superconductivity. The bulk effect can easily dominate the edge state contribution to the anomalous Hall current, except in ultra-pure fully gapped chiral superconductors.
3 Transport theory
We start from the Keldysh extension [56] of the transport-like equations originally developed by Eilenberger, Larkin and Ovchinnikov for equilibrium states of superconductors [57, 58], and extended by Larkin and Ovchinnikov to describe superconductors out of equilibrium [59]. This formalism is referred to as “quasiclassical theory”. For reviews see Refs. [61–63]. The quasiclassical theory is formulated in terms of 4 × 4 matrix propagators for Fermionic quasiparticles and Cooper pairs that describe the space-time evolution of the their non-equilibrium distribution functions, as well as the dynamical response of the low-energy spectral functions and the superconducting order parameter. Here we are interested in the response to static, or low-frequency, thermal gradients and external forces that couple to energy, mass and charge currents. We follow as much as possible the notation and conventions of theory developed for thermal transport in unconventional superconductors by Graf et al. [64].
3.1 Keldysh-Eilenberger equations
The quasiclassical transport equations are matrix equations in particle-hole (Nambu) space which describe the dynamics of quasiparticle excitations and Cooper pairs. Physical properties, such as the spectral density, currents or response functions are expressed in terms of components of the Keldysh matrix propagator,
where
The nonequilibrium dynamics is described by a transport equation for the Keldysh propagator,
as well as transport equations for the retarded and advanced propagators,
where
is defined in terms of the excitation energy, ɛ, the coupling to external fields,
Note that ɛ is the excitation energy and t is the external time variable. The operator expansion for the convolution product is particularly useful if the external timescale, t ∼ ω−1 is slow compared to the typical internal dynamical timescales, ℏ/Δ and τ, i.e., ω ≪|ɛ|∼Δ and ω ≪ 1/τ. In this limit we can expand Eq. 6,
The quasiclassical transport equations are supplemented by the normalization conditions [57, 58],
3.2 Quasiclassical propagators
The quasiclassical propagators are 4 × 4-matrices whose structure describes the internal quantum-mechanical degrees of freedom of quasiparticles and quasiholes. In addition to spin, the particle-hole degree of freedom is of fundamental importance to our understanding of superconductivity. In the normal state of a metal or Fermi liquid there is no quantum-mechanical coherence between particle and hole excitations. By contrast, the distinguishing feature of the superconducting state is the existence of quantum mechanical coherence between normal-state particle and hole excitations. Particle-hole coherence is the origin of persistent currents, Josephson effects, Andreev scattering, flux quantization, and all other nonclassical superconducting effects. The quasiclassical propagators are directly related to density matrices which describe the quantum-statistical state of the internal degrees of freedom. Nonvanishing off-diagonal elements in the particle-hole density matrix are indicative of superconductivity, indeed the onset of non-vanishing off-diagonal elements is the signature of the superconducting transition.
The Nambu matrix structure of the propagators and self energies is
The 16 matrix elements of
where Nf is the normal-state density of states at the Fermi energy. The integration is over the Fermi surface weighted by the angle-resolved normal density of states at the Fermi surface, n(p), normalized to
The current densities are determined from Fermi-surface averages over the elementary currents, [evp], mass, [mvp], and energy, [ɛvp], weighted by the scalar components of the diagonal Keldysh propagator. In particular, the charge and heat current densities are given by
The off-diagonal components, fR,A,K and fR,A,K, are the anomalous propagators that characterize the pairing correlations of the superconducting state. Spin-singlet pairing correlations are encoded in fK, while fK is the measure of spin-triplet pairing correlations. Pair correlations develop spontaneously at temperatures below the superconducting transition temperature Tc. The anomalous propagators are not directly measurable, but the correlations they describe are observable via their coupling to the “diagonal” propagators, gR,A,K and gR,A,K, through the transport equations.
3.3 Coupling to external and internal forces
The couplings of low-energy excitations to electromagnetic fields are defined in terms of the scalar and vector potentials,
Note that
3.3.1 Mean-field self-energies
Superconductors driven out of equilibrium are also subject to internal forces on quasiparticles and Cooper pairs, originating from electron-electron, electron-phonon and electron-impurity interactions. These interactions enter the quasiclassical theory as self-energy terms,
The leading order contributions to the self-energy from quasiparticle-quasiparticle interactions correspond the mean-field self-energies,
Note that
In the Cooper channel the mean-field self energy from quasiparticle interactions is given by Eq. 17. The interaction vertex separates in terms of an even-parity, spin-singlet interaction, λs(p, p′), and an odd-parity, spin-triplet interaction, λt(p, p′), the latter resulting from exchange symmetry in the non-relativistic limit.5 In a rotationally invariant Fermi liquid like liquid 3He, the interactions in the Cooper channel further separates according to the irreducible representations of the rotation group in three dimensions, SO(3)L,
which are labeled by the orbital angular momentum quantum number l ∈ {0, 1, 2, … }, with the basis functions given by the spherical harmonics
3.3.2 Impurity self-energy
The effects of impurity disorder originate from the quasiparticle-impurity interaction,
where
where
Before proceeding to non-equilibrium quasiparticle transport we need to discuss the equilibrium state, including the effects of impurity scattering, on the equilibrium states of chiral superconductors and superfluids.
4 Equilibrium
For homogeneous systems in equilibrium the transport equations for the retarded and advanced propagators reduce to
where
A chiral superconducting ground state is defined by spontaneous breaking of time-reversal and mirror symmetries by the orbital state of the Cooper pairs. We restrict our analysis to unitary superconductors in which the 4 × 4 Nambu matrix order parameter obeys the condition,
Unitary states preserve time-reversal symmetry with respect to the spin-correlations of the pairing state. In the clean limit |Δ(p)| is the energy gap for quasiparticles with momentum p near the Fermi surface, i.e., the Bogoliubov quasiparticle excitation energy is doubly degenerate with respect to spin and given by
We consider four classes of chiral ground states corresponding to the even-parity, spin-singlet, E1g and E2g, and odd-parity, spin-triplet, E1u and E2u representations of the hexagonal point group, D6h. These representations all allow for chiral ground states with principle winding numbers, ν = ±1 (ν = ±2) for the E1 (E2) representations.7 Table 1 provides representative basis functions for these two-dimensional representations.
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Table 1. Representative orbital basis functions, expressed in the chiral basis, for the point groups D4h and D6h.
For even-parity, spin-singlet pairing the Nambu-matrix order parameter has the form,
For chiral E2g pairing the basis functions can be defined as
For odd-parity, ESP triplet states the Nambu-matrix order parameter takes the form,
4.1 2D chiral superconductors
Here we consider the fully gapped E1u and E2g chiral ground states defined on a 2D cylindrical Fermi surface. These two cases illustrate nearly all of the key physical phenomena responsible for anomalous thermal and electrical transport mediated by non-magnetic impurity scattering in chiral superconductors. At low temperatures, thermally excited quasiparticles and phonons are dilute, therefore quasiparticles interact predominantly with quenched defects. For randomly distributed impurities, the self-energy is given by
where Nf is the normal-state density of states per spin at the Fermi surface and
with δm the scattering phase shift in the mth cylindrical harmonic.9 Here and in the following, the directions
The mean field order parameter for unitary chiral states can be expressed in the following form,
in which case
for both S = 0 and S = 1. Thus, in the absence of external magnetic fields, magnetic impurities or spin-dependent perturbations, the spin structure of the order parameter can be transformed away by a unitary transformation, and as previously noted the quasiparticle excitation spectrum is doubly degenerate with respect to the quasiparticle spin.
This representation of the mean-field order parameter extends to the off-diagonal components of the impurity self energy. In Eq. 29 we chose Δ to be real. In this gauge the off-diagonal impurity self-energies reduce to
with the gauge condition,
The term proportional to the unit Nambu matrix,
The diagonal term proportional to
and similarly the impurity renormalized off-diagonal self energy is given by
Thus, for any of the chiral, unitary states described by Eq. 30, the equilibrium propagators that satisfy the transport equation and normalization condition, Eqs. 23, 24, are given by
Note that the functions gR,A and fR,A satisfy the symmetry relations:
4.1.1 Gap equation: mean-field order parameter
The pairing interaction combined with the off-diagonal component of the Keldysh propagator determines the mean-field pairing self-energy for any of the unitary chiral states is given by the “gap equation,”
where the pairing interaction in any of the two-dimensional E-reps defined on a cylindrical Fermi surface has the form
Thus, projecting out the amplitude of the chiral mean-field order parameter we obtain the gap equation,
In practice the pairing interaction strength λ|ν| is eliminated in favor of the critical temperature.
The equilibrium retarded and advanced propagators are given by
where
Upon solving Eq. 26, we obtain
The diagonal terms tm and
4.2 3D chiral superconductors
Here we consider chiral superconductors in 3D belonging to the two-dimensional E-representations of the tetragonal (D4h) and hexagonal (D6h) point groups, both even- and odd-parity E1 and E2 representations. These groups describe the discrete point symmetries of Sr2RuO4 and UPt3, respectively. See Table 1.
4.2.1 Symmetries of the order parameter
The mean-field pairing self-energy, after factoring the spin-structure using the unitary transformation in Eq. 28, has the structure,
The weak-coupling mean-field order parameter, “gap function”, is independent of energy and related to the equilibrium Keldysh pair propagator by the BCS gap equation,
where
where the interaction amplitude λΓ determines the critical temperature and
where Δ is the maximum value of the order parameter, and the normalized spherical harmonics are related to the standard spherical harmonics
4.2.2 Impurity self-energy
In the low temperature limit quasiparticle scattering from thermally excited quasiparticles and phonons is negligible compared to scattering off the random impurity potential. For a homogeneous uncorrelated random distribution of impurities the corresponding self-energy is a product of the mean impurity density nimp and the forward scattering limit of the single impurity t-matrix,
where the t-matrices are obtained as a solution of the integral equation,
The Keldysh component of the t-matrix then given by
Note that
where δl is the phase shift in the relative angular momentum channel, l. The normalization of the spherical harmonics is given by
An important feature of scattering theory by central force potentials, in this case the quasiparticle-impurity potential, is that the characteristic range R of the potential leads to phase shifts δl that decay rapidly to zero for l ≳ kfR, effectively truncating the summations over m and l.
4.2.3 Equilibrium properties
Below we present the framework for determining the self-consistent equilibrium propagators. To highlight the effects of chiral phase winding we consider systems that are gauge-rotation invariant, i.e., invariant under a rotation around the chiral axis combined with a specific element of the U(1). As a result the diagonal equilibrium propagator depends on
where the quasiparticle and pair propagators read
with
The equilibrium Keldysh propagator is determined by
Note that the retarded and advanced propagators are related by the symmetry relation,
where
The function D(ɛ) encodes the asymmetry in scattering rates for particles and holes, which has implications for the thermoelectric response of chiral superconductors [70].
Since the scattering potential is rotationally invariant we can expand the t-matrix Equation 48 into a set of decoupled equations for each cylindrical harmonic channel. Thus, we parametrize the t-matrix as follows
where the diagonal part of the t-matrix is given by
and the off-diagonal part is given by
Thus, by factoring out the dependence on the azimuth angle as shown in Eq. 48, we obtain integral equations for the cylindrical harmonics of the t-matrix,
where
5 Linear response theory
For small departures from equilibrium driven by a small temperature bias between different edges of the superconductor the heat current is proportional to the temperature gradient,
where
Here we consider the linear response functions for a static and homogeneous thermal gradient. For convenience we separate the Keldysh response into a spectral and anomalous part. The anomalous response encodes information about the non-equilibrium distribution function and is defined by,
where
where ∇Φ = ∇ tanh[ɛ/2T(r)] is the gradient of the local equilibrium distribution function. We added the subscript “eq” to denote the equilibrium propagators. We also adopt the shorthand notation,
with C and D defined in Eqs. 54, 57, respectively. It is also efficient to express the response functions as column vectors whose elements correspond to those of their corresponding matrices in particle-hole space,
The expression for the anomalous propagator (Eq. 65) can then be recast as
where the static thermal gradient leads to the perturbation,
The linear-response matrix
where |g|2 = gRgA, |f|2 = fRfA and
5.1 Self-energy—vertex corrections
The r.h.s. of Eq. 69 consists of two terms. The first is the contribution that is explicitly proportional to the external field, vp ⋅∇T. This term contributes only to the longitudinal thermal conductivity. Indeed the anomalous Hall conductivity arises solely from the non-equilibrium self-energy term. The self-energy corrections are the vertex corrections in the field-theoretical formulation based on Kubo response theory. These terms describe the response to perturbations by long-wavelength collective excitations of the interacting Fermi system [63]. In the context of the linear response theory developed for disordered chiral superconductors, the vertex corrections resulting from interactions of Bogoliubov quasiparticles with static impurities are obtained from the linear response corrections to the equilibrium t-matrix Eqs. 48, 49 obtained from the first-order non-equilibrium corrections to the full t-matrix Eqs. 21, 22. For the anomalous self-energy expressed in Nambu matrix form,
can be recast in column vector form as defined by Eq. 68,
where the impurity vertex-correction operator is given by
The retarded [advanced] t-matrix elements are evaluated at
In general the mean-field pairing self-energy also contributes a vertex correction (i.e.,
For point-like impurities, the vertex correction, and thus the anomalous Hall current, vanishes in all but chiral p-wave states. This can be shown by noting that for isotropic impurity scattering the vertex correction from Eq. 73,
5.2 Cylindrical harmonic decomposition for 2D chiral superconductors
For chiral superconductors with cylindrically symmetric Fermi surfaces and pairing interactions we can parametrize the non-equilibrium corrections to the propagators and self-energies in terms of cylindrical harmonics,
The response in different cylindrical harmonic channels can decoupled such that Eq. 69 reduces to
where
with
where the vertex-correction operators are given by
with
5.3 Spherical harmonic decomposition for 3D chiral superconductors
To exploit the axial symmetry of the Fermi surface and chiral symmetry of the order parameter, we write the anomalous propagator (δx → δg) and self-energy (δx → δΣ) as a sum of spherical harmonic components
with
The spherical harmonic components are then given by
The anomalous response in Eq. 69 can now be expressed in terms of solutions for each cylindrical harmonic component,
where the perturbation is
and the linear response matrix is given by
Similarly the vertex correction, Eq. 74, is recast as
where
Finally we use Eq. 88 to eliminate the self-energy term from Eq. 85, yielding
This equation is solved by matrix inversion.
6 Results for 2D chiral superconductors
To quantify the effects of finite-size impurities, we consider hard-disc scattering for which the scattering phase shifts are given by tan δm = J|m|(kfR)/N|m|(kfR) [71], where R is the hard-disc radius and, Jm(z) and Nm(z) are Bessel functions of the first and second kind, respectively. Results presented in this section were reported in Ref. [52]. They are included here to highlight the effects of disorder on fully gapped topological chiral superconductors and to compare with new results for 3D nodal chiral superconductors. We start with the effects of impurities on the equilibrium properties and the sub-gap excitation spectrum.
6.1 Suppression of Tc and pair-breaking
For temperatures approaching the critical temperature, Tc, from below temperature the order parameter approaches zero continuously at the second order transition. The resulting linearized gap equation yields the transition temperature in terms of the pairing interaction, λ, bandwidth of attraction (“cutoff”), ɛc, and the pair-breaking effect of quasiparticle-impurity scattering. The pairing interaction and cutoff can be eliminated in favor of the clean-limit transition temperature,
where Ψ(x) is the digamma function,
for a chiral order parameter with winding number ν. For s-wave pairing (ν = 0), σpb = 0 and consequently
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Figure 1. Impurity cross sections and critical temperature versus hard-disc radius for chiral states: ν =1 (solid) and ν =2 (dashed). For hard-disc radius kfR ≈3.05, pair-breaking cross sections and critical temperature of the two states coincide (filled circles). (Upper) Total cross section (black), transport cross section (solid purple) and pair-breaking cross section (purple). (Lower) Critical temperature for various impurity densities:
6.2 Density of states
The quasiparticle spectrum, N(ɛ) = Nf Im gR(ɛ), also depends sensitively on the winding number, ν, as shown in Figure 2. Note the existence of multiple sub-gap impurity bound states, which are broadened into bands with increasing impurity density. These states are generated by the combination of potential scattering by impurities and multiple Andreev scattering by the chiral order parameter. As a result, the number of bound states and their sub-gap energies are determined by not only the impurity potential, e.g., the ionic radius, but also the winding number ν. The impurity-induced sub-gap spectrum has important implications for all quasiparticle transport processes. In the low-temperature limit, T ≪|Δ|, the thermal conductivity is dominated by excitations at energies well below the clean limit gap edge. Diffusion within the lowest energy band of sub-gap states near the Fermi level determines the low temperature heat current as we discuss below.
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Figure 2. Density of states for chiral order ν =1 (left) and ν =2 (right), various impurity densities normalized by
6.3 Thermal conductivity and the anomalous Thermal Hall effect
In normal metals the thermal conductivity is limited by the transport mean free path for quasiparticles scattering off the random distribution of impurities, κN = (π2/3)NfvfLNT, where LN = 1/(σtr nimp) is determined by the transport cross-section.14
In the superconducting state the thermal conductivity depends on both the mean impurity density as well as the impurity cross-section via,
where we define the thermal transport lengths for the longitudinal and transverse currents by
Figure 3 shows the temperature dependence of longitudinal thermal conductivity for fully gapped chiral superconductors with ν = 1, 2. Note that the presence of impurities generally enhances the low-temperature thermal conductivity through the formation of sub-gap states, but the enhancement depends on winding number of the chiral order. For impurities with kfR = 1 note that a band of Andreev bound states with a finite density of states at ɛ = 0 develops for a chiral order parameter with ν = 2, but not for ν = 1 as shown in Figure 2. This is because the state with ν = 2 has more phase space for scattering on the Fermi surface with a nearly perfect sign change that leads to maximal pairbreaking (i.e., scattering with δϑ ≈ ± π/2) compared to the state with ν = 1 (scattering with δϑ ≈ π). Thus, for ν = 2 a gapless, diffusive, “metallic” band results in a low-temperature thermal conductivity which is linear in temperature as T → 0, i.e., κxx(T → 0) ∝ T. We also note that for ν = 1, such behavior only occurs for sufficiently large impurity densities where the impurity bands broaden to close the gap at ɛ = 0.
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Figure 3. Longitudinal thermal conductivity versus temperature for chiral superconductors with ν =1 (left) and ν =2 (right), impurity radius kfR =1, and normal-state transport lengths listed in the legend. The normal-state thermal conductivity is shown in black. Figure reproduced from Ref. [52] with permission of the APS and the authors.
Figure 4 illustrates perhaps the most pronounced effects of finite-size impurities on transport properties. Although the longitudinal conductivity is relatively insensitive to the impurity size or the winding number, the Hall conductivity depends strongly on both R and ν. For point-like impurities with radii smaller than the Fermi wavelength, kfR ≲ 1, the thermal Hall conductivity is finite for ν = 1, but is dramatically suppressed for chiral states with |ν| > 1, as is clear in the comparison between ν = 1 and ν = 2 for kfR = 0.2 shown in the lower two panels of Figure 4. This supports our previous argument that Hall currents vanish for point-like impurities, i.e., kfR ≪ 1, for all chiral winding numbers except |ν| = 1. Also note that as we increase the radius of the impurities such that kfR ≳ 1, the Hall conductivity for ν = 2 increases dramatically and can be substantially larger than that for ν = 1. Furthermore, for a fixed normal-state transport mean free path, the Hall conductivity exhibits a non-monotonic dependence on impurity size, reaching maximum at an intermediate radius. Thus, the details of the impurity potential, and thus the sub-gap spectrum, are of crucial importance for a quantitative understanding of anomalous Hall effects in chiral superconductors.
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Figure 4. Longitudinal (top) and transverse (bottom) thermal conductivity vs. temperature for chiral order ν =1 (left) and ν =2 (right), normal-state transport length LN/ξ0=7.5, and various impurity radii (see legend). Normal-state thermal conductivity shown in black. Figure reproduced from Ref. [52] with permission of the APS and the authors.
It is also instructive to compare the low-temperature limit of thermal Hall transport originating from the bulk topology in the form of chiral edge states with the bulk thermal Hall conductance from the random distribution of impurities embedded in the bulk of the superconductor. For chiral p-wave pairing the edge-state contribution to the thermal Hall conductance
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Figure 5. Thermal Hall transport length at ɛ =0 for chiral order ν =1 (left) and ν =2 (right), and varying impurity radii (see legend). Low-temperature transport requires quasiparticles states at ɛ =0, formed only with adequate impurity density. But high impurity density destroys superconductivity and thus rules out any anomalous Hall effects.
To compare the edge and bulk contributions to Kxy when both are present we consider typical values of the coherence length to Fermi wavelength, kfξ0, and the relative impurity size, kfR. For example, taking kfξ0 = 100, ν = 1 and kfR = 0.5 we find
7 Results for chiral superconductors in 3D
We have extended the analysis for chiral states in 2D to chiral states defined on closed 3D Fermi surfaces which often include symmetry enforced line and point nodes of the excitation gap. The results reported here include anomalous thermal Hall effects in candidates for 3D chiral superconductors belonging to tetragonal and hexagonal crystalline point groups, particularly the perovskite Sr2RuO4 and the heavy-fermion superconductor UPt3. To investigate the effects of ionic radius and the dependence on the ionic cross-section, we use the hard-sphere impurity potential for which the scattering phase shifts are analytically given in terms of the hard-sphere radius, R, and the Fermi wavevector [77],
where jl(z) and nl(z) are spherical Bessel functions of the first and second kind, respectively [78].
7.1 Critical temperature
For 3D chiral superconductors we obtain a result of the same form as Eq. 91 for the suppression of Tc by disorder, but with a pair-breaking cross-section appropriate for scattering of a 3D Fermi surface with finite-size impurities in 3D,
where the pair-breaking cross section is given by,
with
which determine the quasiparticle scattering lifetime and transport mean-free path, respectively. For point-like impurities all of the above cross sections coincide except for pairing in the s-wave channel, in which case σtot = σtr, but σpb = 0.
In the limit kfR ≪ 1 the total cross-section and pair-breaking cross-section both approach σtot = σpb = 4πR2, i.e., four times the geometric cross section of the hard sphere impurity. However, for kfR ≳ 1 the pair-breaking cross section is typically smaller than the total cross section as shown in Figure 6. In the limit kfR ≫ 1 σtr and
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Figure 6. (Upper) Total cross section (black) and pair-breaking cross section (purple) versus hard-sphere radius for pairing with total angular momentum J =1 (solid) J =2 (dashed) and J =3 (dotted). Note that the transport cross section is
7.2 Quasiparticle spectrum
Central to the interplay between chiral symmetry, topology and disorder is the impact of impurity scattering on pair-breaking and the resulting sub-gap quasiparticle spectrum. Distinct from fully gapped 2D topological states, 3D chiral ground states support symmetry protected nodes of the order parameter which leads to quasiparticle states over the entire energy range from the maximum gap on the Fermi surface down to the Fermi energy. The quasiparticle spectral function defines the angle-resolved quasiparticle density of states is determined by the retarded diagonal propagator,
The local density of states is the Fermi-surface average of the spectral function,
where Nf is the normal-state density of states at the Fermi level. Figures 7, 8 show the effects of impurity induced scattering on the quasiparticle spectrum. The coherence peak at the maximum gap edge is broadened by impurity scattering. The spectral weight is redistributed to sub-gap energies by the formation of sub-gap resonances. The formation of sub-gap impurity bands is clearly visible in the spectral function for positions on the Fermi surface corresponding to the maximum gap as shown in the bottom panel of Figures 7, 8. These resonances correspond to Andreev bound states that hybridize with continuum states near nodal regions of the order parameter (c.f. Ref. [8]). Impurity-induced sub-gap states are formed by multiple Andreev scattering from the combined potential scattering and branch-conversion scattering by the phase-winding of the order parameter on the Fermi surface. The spectrum depends on the structure of the scattering potential as well as the topological winding number of the order parameter. These impurity-induced sub-gap states play a central role in determining the magnitude and temperature dependence of the anomalous thermal Hall conductivity because these states couple to the chiral condensate is the source of broken time-reversal and mirror symmetries.
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Figure 7. The quasiparticle density of states N(ɛ) (top) and spectral function along the direction of gap maximum
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Figure 8. Same description as that in Figure 7, but for kfR =2.5.
7.3 Thermal conductivity tensor for chiral superconductors
The heat current density in Eq. 14 for chiral ground states with embedded impurity disorder reduces to
where
where the spectral resolved transport mean free paths are defined by
with
In the normal state, γ* ∼ Ef ≫ T, and the above formula yields the well known result for the normal-state thermal conductivity with Lxx(0) given by the transport mean-free path. In particular, in the normal state the matrices that determine the anomalous response and vertex corrections are
where σtot and σtr given by Eqs. 99, 100. Then Eq. 90 yields the anomalous response function,
Combining Eqs. 105, 106 yields the normal-state thermal conductivity
is the transport mean free path. Furthermore, the normal state does not break time-reversal and mirror symmetries and thus Lxy vanishes.
In the chiral superconducting phase, γ* is a low energy scale set by the width of the impurity band at the Fermi level, ɛ = 0. When it exists a metallic-like band develops which at very low temperature in the superconducting state gives rise to diffusive heat transport that is again linear in temperature for T < γ*, now for both the longitudinal and Hall conductivities. This regime is shown in Figure 9 for both components of the conductivity tensor for the pairing states E1u and E2g, i.e., the states with
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Figure 9. The components of the thermal conductivity tensor, κxx (top) and κxy (bottom), for T ≪ γ* as a function of impurity density and hard-sphere radii (legend) for the pairing states, E1u (left) and E2g (right) of D6h. The filled circles mark the critical impurity concentrations
The thermal Hall conductivity κxy/T also initially increases with the impurity concentration above the lower threshold density shown in Figure 9. However, κxy/T peaks below
Figure 10 shows the thermal conductivity in the zero-temperature limit for the states E1g and E2u, i.e., with
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Figure 10. Same plots as Figure 9 but for the pairing states E1g (left) and E2u (right). The diamond symbols in the top panels show the values for the “universal limit” for the thermal conductivity in the point-like impurity model [64].
7.4 Comparison with the anomalous thermal Hall conductivity from Berry curvature
Anomalous Hall transport in ultra-clean topological superconductors with broken time-reversal and mirror symmetries was predicted by several authors [44, 46, 50]. In particular, anomalous Hall conductance originating from the gapless edge spectrum confined on the boundary of a topological chiral superconductor is predicted to be quantized,
where
where as before ν is the phase winding of the order parameter about the kz-axis. The resulting Berry phase contribution to the anomalous thermal Hall conductance for isotropic Fermi surfaces in d dimensions is
in the low-temperature limit.
In Figures 9, 10 we compare the our results for the impurity-induced thermal Hall conductivity with the prediction of the edge contribution based on the Berry curvature in the low temperature limit (T → 0) for four different chiral ground states. The comparison is based on a typical coherence length scale kfξ0 = 100. For all four chiral states the Berry phase contribution is dominant at impurity densities below the threshold for impurity-induced transverse transport in the limit T → 0. However, above this threshold the impurity-induced Hall conductivity is comparable to or much larger than the Berry phase contribution. For example, for the chiral E1u state (bottom left panel of Figure 9), the impurity-induced Hall effect yields transverse heat currents in the zero temperature limit which are approximately an order of magnitude larger than the Berry phase contribution for typical impurity dimensions.
Figure 11 depicts the zero-field thermal Hall conductivity as a function of temperature for chiral states belonging to the spin-triplet, odd-parity E1u and E2u representations and the spin-singlet, even-parity E1g and E2g representations of the hexagonal D6h point group, and the Eu and, Eg representations of D4h. Almost all proposed chiral superconductor candidates, including the perovskite Sr2RuO4 and the heavy-fermion superconductor UPt3, belong to one of these representations. The results show that the impurity-induced anomalous Hall effect (solid lines) dominates the Berry curvature contribution [44, 50] (dashed lines) over the full temperature range in all four chiral pairing states for impurities with kfR ≳ 1.5. In this context it is worth reiterating our earlier estimate of the magnitude of the impurity-induced anomalous thermal Hall conductivity for the chiral phase of UPt3 [52]. Namely, for kf = 1 Å−1, ξ0 = 100 Å and Tc = 0.5 K, representative of UPt3 [38] we estimate κxy > 3 × 10−3 WK−1m−1 for T ≃ 0.75Tc for the chiral E2u state with ν = 2 and impurity radius kfR = 1.5 (Figure 11). Compared to the normal-state thermal conductivity at Tc, κN(Tc), one needs sensitivity to transverse heat currents at the level of 0.01–0.03 κN(Tc) as shown in Figure 4.
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Figure 11. The anomalous thermal Hall conductivity versus temperature for hard-sphere radii kfR =0.7,1,1.5,2.0, and for chiral order parameters belonging to the E1u, E2g, E1g and E2u irreducible representations of the hexagonal point group. The Berry phase contribution is shown for comparison (dashed curves). Results are shown for a transport mean free path is LN/ξ0=7.5 and coherence length of
8 Summary and outlook
We presented the theoretical framework for understanding disorder-induced anomalous Hall transport in chiral superconductors, and we reported quantitative predictions for the thermal conductivity and the anomalous thermal Hall conductivity in superconductors with phase winding ν for chiral superconducting ground states belonging to the 2D irreducible representations of the hexagonal and tetragonal point groups. We highlight the role of quasiparticle-impurity scattering by finite-size impurities, i.e., kfR ≳ 1. Our analysis demonstrates that an anomalous thermal Hall effect is obtained for chiral superconductors with winding ν, provided the ionic radius of the impurities satisfies kfR ≳|ν| − 1. Thus, for point-like impurities with kfR ≪ 1 the anomalous thermal Hall current vanishes for all but chiral p-wave ground states. We also discussed the spectrum of impurity-induced Andreev bound states, which are formed via multiple Andreev scattering. The spectrum depends sensitively on the winding number of the chiral order parameter as well as the structure of the impurity potential. Our results also show that the impurity-induced anomalous thermal Hall transport dominates the edge state contribution by an order of magnitude or more over most of the temperature range below Tc. The impurity- and edge contributions to the thermal Hall effect both depend on broken time-reversal and mirror symmetries. Thus, they are equally good signatures of chiral superconductivity. The bulk impurity effect is likely more accessible experimentally; it produces larger Hall currents, and it is insensitive to the quality of the surfaces of a sample. In summary this work provides the theoretical framework for computing and analyzing experiments seeking to identify broken time-reversal and mirror symmetries, as well as non-trivial topology of chiral superconductors, from bulk transport measurements.
8.1 Outlook
There are a number of candidates for chiral superconductivity that have been proposed theoretically and pursued experimentally. The chiral phase of 3He was proven to be chiral p-wave based on the observation of anomalous Hall transport of electrons embedded in superfluid 3He-A [7, 8]. The heavy electron metal UPt3 shows evidence of broken time-reversal symmetry based on Kerr rotation [33], Josephson interferometry [3], μSR [31] and SANS studies of diffraction by the vortex lattice [6]. Observation of an anomalous thermal Hall effect onsetting at the A to B transition would provide a definitive bulk signature of broken time-reversal and mirror symmetries in UPt3. Analysis of the temperature- and impurity-dependences of the Hall conductivity could provide new and quantitative experimental constraints on the symmetry class of E-rep of UPt3. For a number of proposed candidates for chiral superconductivity, e.g., Sr2RuO4, doped graphene, SrPtAs, etc., observation of an anomalous thermal Hall effect would provide confirmation of broken time-reversal and mirror symmetry by the superconducting order parameter. NMR experiments revealed the existence of new superfluid phases of liquid 3He when it is infused into low density, anisotropic, random solids - “aerogels” [80] - or confined into sub-micron cavities [81]. Analysis based on Ginzburg–Landau theory predicts that the ground state of 3He under anisotropic confinement is a chiral phase [82]. Thus, experiments designed to measure the transverse heat current could provide a definitive test of the theory for the ground state of superfluid 3He infused into anisotropic aerogels [83], and similarly for 3He confined in sub-micron cavities [84].
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
VN: Writing–original draft, Writing–review and editing. JS: Writing–original draft, Writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The research of VN was supported through the Center for Applied Physics and Superconducting Technologies at Northwestern University and Fermi National Accelerator Laboratory. The research of JS was supported in part by the National Science Foundation (Grant DMR-1508730), “Nonequilibrium States of Topological Quantum Fluids and Unconventional Superconductors,” and by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Footnotes
2More complex chiral order parameters with large winding numbers are allowed by the point group symmetry, c.f. Ref. [51]; ν = ±1, ±2 are the loweset order harmonics consistent with the E1 and E2 representations, respectively.
3Note that time-reversal and mirror symmetries need not be broken simultaneously. For example, a three-band superconductor may break timereversal symmetry when the order parameter defined on each band has a different phase [60]. Mirror symmetry is however preserved and Hall effects are therefore not expected in this system.
4The sign in Eq. (17) is such that λs(t) > 0 corresponds to an attractive pairing interaction.
5This separation does not apply to superconducting materials without an inversion center, i.e. non-centrosymmetric superconductors.
6The A1 phase of superfluid 3He, which is stabilized by an externally applied magnetic field, is a non-unitary spin-triplet state [66]. The Uranium-based ferromagnetic superconductors are also believed to be non-unitary, spin-polarized, triplet superconductors.
7D∞h, chiral ground states with any integer winding number ν∈ Z are possible. For the discrete point group D6h higher winding numbers with ν = ±1 + mod(6) (E1) or (ν = ±2) + mod(6) (E2) are possible for pairing basis functions exhibiting strong hexagonal anisotropy, but in general the chiral basis functions with higher winding numbers will mix with ν = ±1 ν = ±2.
8Results for heat and charge transport in zero field do not depend on the choice for the direction of
9The summation over m is truncated as a defect with characteristic radius R leads to rapidly decaying phase shifts for |m|≳ kf R.
10Hereafter the retarded (R) and advanced (A) superscripts are not shown for g(ɛ), f(ɛ) etc., but are implied.
11Despite its absence from spectral renormalization, DR,A(ɛ) encodes particle-hole asymmetry, e.g. the difference in scattering lifetimes for particles and holes which could have implications for transport properties [68], especially in thermoelectric responses [70].
12A comprehensive set of tables of basis functions for the tetragonal, hexagonal and cubic point groups is provided in Ref. [69]
13Similar results were derived for the suppression of Tc by non-magnetic disorder in p-wave superconductors and superfluid 3He in aerogel [73, 74].
14The transport and pair-breaking cross sections are different except for |ν| = 1, c.f. Eq. (92).
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Keywords: topological superconductivity, chiral superconductors, broken time-reversal symmetry, broken mirror symmetry, thermal transport, anomalous Hall transport, Hall effects, impurity disorder
Citation: Ngampruetikorn V and Sauls JA (2024) Anomalous Hall effects in chiral superconductors. Front. Phys. 12:1384275. doi: 10.3389/fphy.2024.1384275
Received: 09 February 2024; Accepted: 08 March 2024;
Published: 03 May 2024.
Edited by:
Takeshi Mizushima, Osaka University, JapanReviewed by:
Mario Cuoco, National Research Council (CNR), ItalyPhilip Brydon, University of Otago, New Zealand
Copyright © 2024 Ngampruetikorn and Sauls. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Vudtiwat Ngampruetikorn, vudtiwat.ngampruetikorn@sydney.edu.au; J. A. Sauls, sauls@lsu.edu