Pair Density Waves and Supercurrent Diode Effect in Altermagnets

Introduction.— Collinear magnets are commonly thought to display either ferromagnetic or antiferromagnetic order [1, 2]. Ferromagnets (FM) break time-reversal symmetry with a net magnetic moment leading to spin split electronic bands. On the other hand, traditional antiferromagnets (AFM) show zero net magnetization and are symmetric under the time reversal, which flips the spin, followed by a lattice translation, resulting in Kramer’s spin degenerate bands in momentum space. However, recent investigations have challenged this binary classification, proposing a new subtype of collinear magnetism termed altermagnets (AM), which are characterized by momentum-dependent spin split bands with zero net magnetization [3, 4, 5, 6, 7, 8]. In AM the symmetry that connects the magnetic sublattices with opposite spins are non-trivial rotations  [9], for example a fourfold rotation leads to $d$-wave AM on the square lattice [10, 11, 12].

The interplay between magnetism and superconductivity can lead to qualitatively new phenomena, for example a change of pairing symmetry [13, 14], and has been a central focus in condensed matter physics research [15, 16]. The the spin splitting of electronic bands in FM impedes the formation of conventional $s$-wave spin-singlet superconductivity as the formation of Cooper pairs is energetically suppressed. However, an alternative is to stabilize Cooper pairs with a finite center of mass momentum, as proposed to be realizeable in FM/superconductor junctions [17, 18, 19]. Here, we study the effect of the unique momentum-dependent spin splitting of AM on superconductivity. Theoretical investigations have focused on cases where superconductivity is induced solely through the proximity effect within AM/superconductor junctions [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Nevertheless, the understanding of bulk superconductivity in AM has been limited [31, 32, 33, 34, 35, 36, 37], despite the presence of superconductivity in AM materials, which include RuO₂ thin films with strain [38, 39] and possibly hole-doped La₂CuO₄ [5, 15].

In this work, we explore the bulk superconductivity arising from a conventional BCS-type attractive interaction between electrons in $d$-wave AM metals. We find the emergence of pair density wave (PDW) states even in the absence of an external magnetic field or charge density wave instabilities. Deriving the Ginzburg–Landau theory from a microscopic model, we find the stabilization of inversion symmetry-breaking PDW states like Fulde–Ferrell (FF) and Fulde–Ferrell* (FF*) states. Further investigation into supercurrent properties reveals non-reciprocal supercurrents in the FF state along the $x$ direction and in the FF* state along $x$ and $y$ directions. In a related context, a recent work studied the symmetry properties of possible supercurrent diode effects, e.g. non-reciprocal critical supercurrents [40, 41, 42, 43, 44, 45], in altermagnetic superconductors and related heterostructures [46]. Here we derive the effective Ginzburg-Landau theory and show that non-reciprocal supercurrents may appear for intrinsic inversion-symmetry breaking bulk superconducting states of AM.

Model of a AM metal.— We focus on a two-dimensional $d$-wave AM metal [6, 5] with a low-energy Hamiltonian expanded near the $\Gamma$ point as

with the two-component spinor $\psi_{\bm{k}}^{\dagger}\!\equiv\!(c^{\dagger}_{\bm{k},\uparrow},c^{\dagger}_{\bm{k},\downarrow})$ and the Pauli matrices act on spin space. The first term describes the kinetic energy, while $\lambda$ and $\mu$ denote the strength of AM splitting and the chemical potential, respectively. The AM breaks the time-reversal symmetry $\mathcal{T}$ and the fourfold rotational symmetry $C_{4}^{z}$ about $z$ axis, which is characterized by anisotropic spin splitting vanishing along $k_{x}$ and $k_{y}$ axes, thus forming a $d$-wave magnet. However, the system still preserves a magnetic fourfold rotational symmetry $\mathcal{C}_{4}^{z}\!\equiv\!\mathcal{T}C_{4}^{z}$ such that $\mathcal{C}_{4}^{z}H(\bm{k})(\mathcal{C}_{4}^{z})^{-1}\!=\!H(\bm{k})$. As a result, the net magnetization of the system must be zero although the two bands become spin split. The band structure looks similar to those of RuO₂ and KRu₄O₈ [5, 47]. Throughout this study, we fix $\lambda\!=\!c_{0}\!=\!100$meV.

There are diverse microscopic mechanism of superconducting pairing and in the following we focus on a basic effective BCS-type attractive interaction, most commonly treated for phonon-mediated pairing. The resulting effective interaction is written in momentum space as

Here, $U$ is the strength of onsite interactions and the attraction holds only near the Fermi level within regions bounded by the Debye frequency $\omega_{D}$ as illustrated in Fig. 1. Such interactions typically lead to the condensation of spin-singlet $s$-wave $\bm{q}\!=\!0$ states with the order parameter $\Delta_{0}\equiv\langle c_{\bm{k},\uparrow}c_{-\bm{k},\downarrow}\rangle$. Alternatively, sought-after spin-singlet $s$-wave PDW states with the order parameter $\Delta_{\bm{q}}\equiv\langle c_{\bm{k}+\bm{q}/2,\uparrow}c_{-\bm{k}+\bm{q}/2,\downarrow}\rangle$ carrying a finite center of mass momentum $\bm{q}$ can potentially be realized. Below, we focus on such $s$-wave states ignoring the long-range interactions that can induce spin-singlet $d$-wave or spin-triplet $p$-wave states [35, 48, 37, 49] (see Sec. III of the Supplementary Material for details).

PDW in AM.— The phase diagram of superconducting states can be obtained by minimizing the Ginzburg–Landau free energy. The PDW order parameter can be conveniently written as

with corresponding wavevectors $\{\bm{q},\bar{\bm{q}},-\bm{q},-\bar{\bm{q}}\}$. Here, $\bm{q}$ and $\bar{\bm{q}}$ are connected through the magnetic fourfold rotation such that $\mathcal{C}_{4}^{z}\bm{q}\!=\!-\bar{\bm{q}}$. The free energy is constructed by imposing gauge, translational, inversion, and magnetic fourfold rotational symmetries and is written as [50]

The quadratic and quartic coefficients can be explicitly calculated starting from the microscopic Hamiltonian given in Eq. (1) by integrating out the electronic degrees of freedom [51, 52, 53, 54, 55, 56] (see Sec. I of the Supplementary Material for details). Depending on the coefficients, the free energy stabilizes the $\bm{q}\!=\!0$ state or different types of $\bm{q}\neq 0$ PDW states. First, the free energy stabilizes the PDW states if the minimum of $\alpha(\bm{q})$ is located at $\bm{q}\!\neq\!0$, i.e., $\bm{q}_{\text{min}}\!\neq\!0$. The wavevector $\bm{q}_{\text{min}}$ determines the center of mass momentum that the Cooper pair carries in the ground state. Second, the system selects one of the symmetrically distinct PDW states depending on the quartic coefficients. In Table. 1, we list all possible PDW states which include inversion-breaking ones: the FF and FF* state. For these, the system may display a non-reciprocal superconducting current, since the time-reversal symmetry is broken by the AM and the inversion symmetry is spontaneously broken by the superconducting PDW. It is also worthwhile to note that the free energy contains a $U(1)\times U(1)\times U(1)$ symmetry where two additional $U(1)$ symmetries come from translation invariance along the $x$ and $y$ directions. As a result, the PDW states can induce fractional vortices except for the FF state as pointed out in Ref. 50.

To investigate whether the system stabilizes PDW and which ones, we calculate the coefficients of the free energy, $\alpha(\bm{q})$ and $\beta_{i}$. We obtain the phase diagram for varying $\mu/c_{0}$ as shown in Fig. 2(a). When $\mu/c_{0}\!<\!0.1$, the minimum of $\alpha(\bm{q})$ is located at $\bm{q}_{\text{min}}\!=\!0$ and the system selects the $\bm{q}\!=\!0$ state. The system starts to stabilize the PDW state when $\mu/c_{0}\!\geq\!0.1$. Within such a regime, the ordering wavevector is finite, $\bm{q}_{\text{min}}\!\neq\!0$, and gradually increases as $\mu$ increases, reflecting the size of the normal Fermi surface as shown in Fig. 2(a). In Fig. 2(b), we plot $\alpha(\bm{q})$ with $\mu/c_{0}\!=\!0.2$ which clearly shows that $\bm{q}_{\text{min}}\!\neq\!0$ and the ordering wavevectors lie on the $q_{x}$ and $q_{y}$ axes and are fourfold degenerate. The degeneracy originates from the magnetic fourfold rotational symmetry.

When $0.1\!\leq\!\mu/c_{0}\!<\!0.18$, the quartic coefficients satisfy the following relations given in Table. 1: $\beta_{2}\!>\!0$, $\beta_{2}\!+\!\beta_{3}\!>\!0$, and $3\beta_{2}\!+\!\beta_{3}\!-\!|\beta_{4}|\!>\!0$. Within such a regime, the system selects the FF state with the order parameter $\hat{\Delta}_{\bm{q}}\!=\!\{e^{i\phi_{1}},0,0,0\}$ which spontaneously breaks the inversion symmetry. Here we highlight that the realization of the FF state in AM does not require an external magnetic field unlike the original proposal given in Ref. 57 and Rashba spin-orbit coupled systems [58, 59, 60]. At the same time, the inversion symmetry of the system is spontaneously broken by the superconducting state, unlike the Rashba systems where the inversion is broken by the spin-orbit coupling.

For $0.18\!\leq\!\mu/c_{0}\!<\!0.23$, the system stabilizes the FF* state with $\hat{\Delta}_{\bm{q}}\!=\!\{e^{i\phi_{1}},e^{i\phi_{2}},0,0\}$ which also breaks the inversion symmetry. Unlike the FF state, the order parameter of the FF* state has two phase parameters, $\phi_{1}$ and $\phi_{2}$. As a result, the system can have a half-quantum vortex as a defect though careful comparison between the energy of the conventional vortex and the half-quantum vortex will be needed [50]. When $\mu/c_{0}\!\geq\!0.23$, the system selects the so-called BD-II state with $\hat{\Delta}_{\bm{q}}\!=\!\{e^{i\phi_{1}},ie^{i\phi_{2}},e^{i\phi_{3}},ie^{i(\phi_{1}+\phi_{3}-\phi_{2})}\}$ which preserves the inversion symmetry.

Diode effect in PDW.— In the following, we consider the supercurrent diode effect as a tool to distinguish the different forms of PDW states. Near the superconducting phase transition temperature, the formation of the PDW states can be described phenomenologically by a Ginzburg–Landau functional $F\!=\!\int f(\bm{r})d\bm{r}$, with the free energy density which can be spatially homogeneous given by

Here, ${\bm{D}}=\bm{\nabla}+i(2e)\bm{A}$ is the covariant derivative. The supercurrent can be derived from Eq. (LABEL:eq:free_real) by standard means [61, 62, 63, 64], by varying the free energy concerning the vector potential $\bm{A}$ and subsequently enforcing $\bm{A}=0$, yielding the following result (see Sec. II of the Supplementary Material for details):

Here, we included contributions from the fourth order derivatives as they are crucial for ensuring the stability of the $\bm{q}\neq 0$ ground state [62, 63].

For the FF state with $\Delta_{\bm{q}}(\bm{r})\!=\!|\Delta|e^{iqx}$ and $\Delta_{\bar{\bm{q}}}(\bm{r})\!=\!\Delta_{-\bm{q}}(\bm{r})\!=\!\Delta_{-\bar{\bm{q}}}(\bm{r})\!=\!0$, the supercurrent is given by

For the FF* state with $\Delta_{\bm{q}}(\bm{r})=|\Delta|e^{iqx}$, $\Delta_{\bar{\bm{q}}}(\bm{r})=|\Delta|e^{iqy}$, and $\Delta_{-\bm{q}}(\bm{r})\!=\!\Delta_{-\bar{\bm{q}}}(\bm{r})\!=\!0$, the supercurrent is written as

For both cases, the supercurrent vanishes at $q\!=\!0$ and $q\!=\!\sqrt{-a_{2}/2a_{4}}\!=\!q_{\text{min}}$. We want to emphasize that the FF* state can induce a non-reciprocal supercurrent concurrently in both the $x$ and $y$ directions unlike the FF state, which can induce the non-reciprocal current only in the $x$ direction. Other PDW states, which include UD, BD-I, and BD-II states, cannot induce the non-reciprocal current, since they preserve the inversion symmetry.

In equilibrium $\partial F/\partial\Delta\!=\!0$, where $F$ is the free energy given in Eq. (4), leading to $|\Delta|^{2}=-\alpha(q)/(2\beta_{1})$ for the FF state and $|\Delta|^{2}=-\alpha(q)/(4\beta_{1}+\beta_{2})$ for the FF* state. At the same time, $\alpha(q)\!=\!a_{0}\!+\!a_{2}q^{2}\!+\!a_{4}q^{4}$ which does not contain linear nor cubic terms unlike the Rashba systems with external magnetic field considered in Ref. 40 and Ref. 42. Then one can rewrite the supercurrent as

for the FF state and

for the FF* state. We note that the supercurrent formula in Eq. (9) has the same form as the one given in Ref. 40 for the FF state in Rashba spin-orbit coupled systems.

In Fig. 3(a), we plot the supercurrent along $x$ direction for the FF state stabilized for $\mu/c_{0}\!=\!0.17$, using Eq. (9). It shows the nonreciprocity of the critical current; the maximum $J_{c}^{+}$ and the minimum $-J_{c}^{-}$ are different around the $q_{\text{min}}$, where the supercurrent vanishes. Using Eq. (10), we also plot the current relation for $\mu/c_{0}\!=\!0.2$, where the system stabilizes the FF* state, in Fig. 3(b). Unlike the FF state, the FF* state induces the non-reciprocal current in both the $x$ and $y$ directions.

The non-reciprocal nature of the supercurrent can be conveniently quantified by a diode coefficient [40] which is defined as

In Fig. 3(c), we plot the coefficient $\eta$ as a function of $\mu/c_{0}$. When $\mu/c_{0}\!<\!0.1$, the system selects the $\bm{q}\!=\!0$ state preserving the inversion symmetry with vanishing $\eta$. For $0.1\!\leq\!\mu/c_{0}\!<\!0.23$, one of the inversion breaking states, i.e., FF state and FF* state, is stabilized and $\eta$ becomes finite. When $\mu/c_{0}\!\geq\!0.23$, $\eta$ becomes zero again, since the system stabilizes the BD-II state and preserves the inversion symmetry. Overall, the supercurrent diode effect can be utilized to detect different types of PDW states in future experiments.

Summary and outlook.— We have studied bulk superconductivity in a two-dimensional metallic system with $d$-wave AM spin splitting in momentum space. We focused on the emergence of superconductivity driven by onsite attractive interactions analyzed through a microscopically derived Ginzburg–Landau theory. We find that symmetrically distinct PDW states can be stabilized as a function of varying chemical potential $\mu$, including the inversion-breaking FF and FF* states. We then investigated the supercurrent properties and found a non-zero non-reciprocity; i.e. the FF state induces a non-reciprocal supercurrent in one direction, while the FF* state can induce it in two orthogonal directions.

There have been many theoretical proposals for realizing superconductivity of different types in AM [35, 36]. In the future it would worthwhile to explore PDW formation with different pairing symmetries and potential topological properties, for example with spin-triplet pairing or in the presence of Rashba spin-orbit coupling [33]. Beyond collinear AM other real space non-collinear spin textures might show similar effects [65].

Recent research has identified several promising material candidates for studying the interplay between altermagnetism and superconductivity. RuO₂ thin films, which exhibit d-wave altermagnetic characteristics with momentum-dependent spin splitting and zero net magnetization, demonstrate superconductivity under strain [38, 39]. Hole-doped La₂CuO₄, known for high-temperature superconductivity, offers a rich context for exploring PDW states and the supercurrent diode effect [66]. While other metallic altermagnets, such as KRu4O8 [5], MnTe [67], and CrSb [68], have not yet shown superconductivity, they could be tuned via pressure or proximity effects. These materials, supported by microscopic theoretical modelling, provide exciting opportunities for experimental realisation of non-reciprocal effects of superconductivity.

Acknowledgements.— We thank P. Rao, J. Habel, M. J. Park, V. Leeb, R.-X. Zhang, and L. Šmejkal for insightful discussions related to this work. We acknoweldge M. Scheurer, I. Eremin, R. Fernandes, and D. Agterberg for comments on the manuscript. We especially thank D. Agterberg for pointing out an earlier flaw in the derivation of Eq. (7). G.B.S. is funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No. 771537). J. K. acknowledges support from the Imperial-TUM flagship partnership. The research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus. G.B.S was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) (RS-2024-00453943).