Nonreciprocal transport in $U(1)$ gauge theory of high-$T_{c}$ cuprates

Introduction.— Several decades have passed since the groundbreaking discovery of high-$T_{c}$ superconductors in doped cuprates Bednorz and Müller (1986). The parent phase of undoped cuprates is the antiferromagnetic Mott insulator, and electron or hole doping leads to a complex phase diagram, encompassing the pseudogap, strange metal, Fermi liquid, and $d$-wave superconductor phases. [See Fig. 1(a).] Despite extensive research efforts Scalapino (1995); Kastner et al. (1998); Timusk and Statt (1999); Orenstein and Millis (2000); Damascelli et al. (2003); Campuzano et al. (2004); Lee et al. (2006); Fischer et al. (2007); Alloul et al. (2009); Armitage et al. (2010), achieving a consensus on the origin of the phase diversity in cuprates remains elusive.

Notably, while cuprates are known for their weak spin-orbit coupling (SOC) compared with strong electronic correlations, recent experiments using spin-angle-resolved photoemission spectroscopy Gotlieb et al. (2018) unveiled Rashba-type spin-orbit coupling on the surface of Bi₂Sr₂CaCu₂O_8+d. This arises from local inversion symmetry breaking at the surface of CuO₂ layers. Hence, the potential emergence of nonreciprocal transport in high-$T_{c}$ cuprates is anticipated when subjected to external magnetic fields, owing to both inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetry breaking. Hence, the nonreciprocal transport could be the useful probe to investigate the physical properties of high-$T_{c}$ cuprates.

In metals, nonlinear nonreciprocal transport manifests through current-dependent resistance $R$ Rikken et al. (2001); Tokura and Nagaosa (2018); Wakatsuki et al. (2017); Hoshino et al. (2018), expressed as:

Here, $R_{0}$ represents current-independent resistance, $\gamma$ denotes nonreciprocity, $h$ stands for the magnetic field, and $I$ corresponds to the current. This leads to asymmetric nonlinear $I-V$ curves in Fig. 1(b), a phenomenon known as magnetochiral anisotropy (MCA).

Conversely, in superconductors, nonreciprocal transport encompasses the nonlinear paraconductivity and the superconducting (SC) diode effect. Paraconductivity means a rapid increase in conductivity just above the transition temperature due to SC fluctuations, whose nonlinear part is also measured by $\gamma$ Wakatsuki et al. (2017); Wakatsuki and Nagaosa (2018). The SC diode effect measures the disparity in critical currents depending on the direction of the current flow, quantified by the quality factor $Q$ Cui et al. (2019); Ando et al. (2020); Itahashi et al. (2020); Schumann et al. (2020); Miyasaka et al. (2021); Kawarazaki et al. (2022); Narita et al. (2022); Masuko et al. (2022); Lin et al. (2022); Scammell et al. (2022); He et al. (2022); Du et al. (2023); Nadeem et al. (2023),

Here, $I_{c\pm}$ represent the critical currents for positive and negative current directions in Fig. 1(c), respectively.

In this Letter, we systematically explore the surface nonreciprocal transport across the complete phase diagram by employing the Slave boson $U(1)$ gauge theory of cuprates, where the electrons are considered be fractionalized into particles, i.e., fermionic spinons and bosonic holons. Upon the introduction of a magnetic field ($h$) into the system, we find the rich and nontrivial behaviors of the nonreciprocal transports in the plane of temperature ($T$) and doping ($\delta$). The phase diagram is partitioned into three regions: (I) the normal phases, including antiferromagnet (AF), pseudogap (PG), strange metal (SM), and Fermi liquid (FL) phases observed at high temperatures ($T\gg T_{c}$); (II) the phase marked by superconducting (SC) fluctuations at temperatures around $T\gtrsim T_{c}$; and (III) the $d$-wave superconducting phase existing at $T<T_{c}$ [refer to Fig. 1(a)]. In the AF, FL, and overdoped SC phases, the carriers are predominantly electrons, while in PG, SM, and underdoped SC phases, the carriers manifest as bosonic holons Lee and Nagaosa (1992); Nagaosa and Lee (1992).

Each phase of normal states shows contrasting behavior of nonreciprocal transport, as for magnitude and dependence on $T$ and $\delta$. Moreover, in both the SC fluctuation and $d$-wave SC phases, we find the significant increase of both $\gamma$ for paraconductivity and the quality factor $Q$ for the SC diode effect in the overdoped regions. Based on the observations, we provide a global picture for nonreciprocal transport as well as practical predictions.

These observations are elucidated by the distinct characteristics of the carrier type and its kinetic energy. The escalation of kinetic energy amplifies linear conductivity, diminishing nonreciprocal transport. Additionally, the absence of spin in holons results in an indirect Zeeman effect, and the symmetry breaking is weaker than electron case. Accordingly, the magnitude of nonreciprocal transport in normal states is intricately linked to the kinetic energy of carriers, while its dependence on $T$ and $\delta$ predominantly comes from their symmetry breaking energy scales. Similarly, the overdoped SC and its vicinity shows a sharp increase in nonreciprocal transports compared to the underdoped SC and its vicinity, owing to the carrier type and the fermion pairing energies.

MCA in normal phases.— To simulate the surface of doped cuprates under magnetic fields for normal phases at $T\gg T_{c}$, we employ the extended $t-J$ model $H_{0}=H_{t}+H_{SO}+H_{Z}+H_{S}$ and its associates in a 2D square lattice. The components can be expressed as:

Here, $H_{t}$ represents the spin-independent hopping, $H_{SO}$ includes spin-dependent hopping due to SOC, and $H_{Z}$ accounts for the Zeeman effect of electrons. $H_{S}$ encompasses exchange, Dzyaloshinskii-Moriya (DM), and Zeeman interactions within the spin system. In Fig. 1(d), we illustrate the 2D square lattice with the directions of $\vec{D}_{ij}\parallel\vec{d}_{ij}$ and $\vec{h}\parallel\hat{y}$. The $U(1)$ Slave-boson theory is employed to solve $H_{0}$ for PG and SM phases. The model for AF and FL phases is derived from $H_{0}$. The spin exchanges are replaced with local electron spin couplings to magnetic order, $H_{AF}=-2u\sum_{i}\vec{m_{i}}\cdot\vec{j_{i}}$, where $\vec{m_{i}}$ is the local magnetic moment and $\vec{j}_{i}$ is the electron spin.

To obtain the order parameters in the equilibrium state, we utilize self-consistent mean-field theory. The order parameters in the AF phase are staggered magnetic moments at A and B sublattices ($m_{A}$, $m_{B}$), while in the PG or SM phase, they include the spinonic singlet ($\chi$) and triplet ($\xi_{x},\xi_{y}$) hoppings, singlet pairing ($\Delta$), and spinonic/holonic chemical potentials ($\mu_{F}/\mu_{B}$). Notably, $\vec{h}$ is instrumental in creating the asymmetry of the electronic band in AF and FL phases, while $\xi_{x}$ and $\xi_{y}$ contribute significantly to the asymmetry of the holonic band in PG and SM phases [see Supplementary Information (SI) for details].

For normal phases, we consider the longitudinal current up to the second order in an external electric field $\vec{\mathcal{E}}=Re[\vec{E}e^{i\omega t}]$ in a semiclassical manner. Accounting for SOC and $\vec{h}$, the magnetic space group reduces to $Pma^{\prime}2^{\prime}$, allowing for the second-order current along the $x$-axis. The current is defined as $J_{x}=Re[\sigma_{1}E_{x}e^{i\omega t}+\sigma_{20}|E_{x}|^{2}+\sigma_{22}E_{x}^{2}e^{2i\omega t}]$, and the conductivities can be expressed as follows Nakai and Nagaosa (2019):

Here, $\int_{k}\equiv d^{2}k/(2\pi)^{2}$, $\tau$ represents the relaxation time, $f_{0}$ is the distribution function, $\epsilon_{k}$ is the energy dispersion, and $v_{x}$ is the $x$-direction of group velocity of the carriers. When $\omega=0$, nonreciprocity can be approximated as $\gamma\approx-\frac{\sigma_{22}}{\sigma_{1}^{2}Wh}$, where $W$ represents the sample width, which is assumed to be $\sim 100~{}\mu m$. Remarkably, $\gamma$ is independent of the relaxation time $\tau$. Based on the measured values Zhang and Rice (1988); Kastner et al. (1998); Lee et al. (2006); Gotlieb et al. (2018), the parameters are fixed as $J=1$, $t=2$, $|\mathbf{D}_{ij}|=0.033$, $\lambda_{SO}=0.067$, and $u=10$. The energy unit is $\sim 0.2~{}eV$, and the lattice constant is $4~{}\AA$. For the AF and FL phases, the electronic conductivity is computed while for the PG and SM phases, the holonic conductivity is computed. The magnetic field $|\vec{h}|\sim 7~{}$T is fixed since $\gamma$ is almost constant within $10~{}$T. The range of temperatures is from $170$ to $250~{}$K. According to the phase diagram in Fig. 1(a), we set the doping ranges from $0$ to $0.02$ in AF phase, from $0.02$ to $0.12$ in PG phase, from $0.07$ to $0.27$ in SM phase, and from $0.22$ to $0.3$ in FL phase. The phase transition for AF, PG, and SM phases occurs at $T>300~{}$K in the doping range.

In Fig. 2, we present the change of $\gamma~{}$(T^-1A^-1) in temperatures and dopings for AF, PG, SM, and FL phases, respectively. We observe the followings. $\gamma$ exhibits a sharp contrast for each phase in magnitude and the dependence on $T$ and $\delta$. Specifically, the magnitude of $\gamma$ in PG and SM phases is much larger than that in FL phases, but much smaller than that in AF phases. As $T$ is lowered, $\gamma$ grows in PG, SM, and AF phases while it slightly diminishes in FL phase. As $\delta$ increases, $\gamma$ grows in PG and SM phases while it diminishes in AF and FL phases. Figure 3(a) shows the integrated diagram of $\log_{10}\gamma$ relevant to the phase diagram in Fig. 1(a).

Paraconductivity and SC diode effect.— Meanwhile, to describe the paraconductivity and superconducting diode effect, we must formulate the Ginzburg-Landau (GL) free energy of SC phase. Let us assume that the GL free energy has the form: $F=\int_{k}(\eta(k)|\phi_{k}|^{2}+\nu(k)|\phi_{k}|^{4})$ with the order parameter $\phi_{k}$. From this free energy, we can compute the current due to paraconductivity above $T_{c}$ using the following equation Schmid (1969); Bennemann and Ketterson (2008); Wakatsuki and Nagaosa (2018),

Here, $\Gamma$ represents the coefficient in the time-dependent GL equation, $V$ is the system volume, $\eta(t)=\eta(\vec{k}-2e\vec{E}t)$, and $j_{x}(t)\equiv-\partial\eta(t)/\partial A_{x}$ Wakatsuki and Nagaosa (2018). Contrarily, the supercurrent below $T_{c}$ is computed using the derivative of the integral kernel of the free energy with respect to momentum, denoted as $J(k)=-2e[\eta^{\prime}(k)|\phi_{k0}|^{2}+\nu^{\prime}(k)|\phi_{k0}|^{4}]$ He et al. (2022). Here, $\phi_{k0}$ is the order parameter at the momentum $k$ corresponding to the energy minimum He et al. (2022). The critical currents $I_{c+}$ and $-I_{c-}$ can be achieved at the maximum of $J(k)$ for positive momentum and the minimum for negative momentum, respectively. The quality factor is then computed as per Eq. 2.

The free energy for underdoped and overdoped SC phases should be different due to their distinct carriers and order parameters. In underdoped region, the carriers are predominantly holons near $T_{c}$. Considering that the superconducting phase transition occurs due to holon condensation, the free energy is as follows Nagaosa and Lee (1992):

Here, $\alpha_{1}=\alpha_{10}T_{c}\epsilon^{\prime}$ and $\beta_{1}$ are the conventional Ginzburg-Landau parameters. $\phi_{k}$ is the superconducting order parameter, $\epsilon^{\prime}=(T-T_{c})/T_{c}$, and $\omega_{k}\equiv a_{1}k^{2}-b_{1}(k_{x}-k_{x}^{3}/6)$ is the holonic energy up to third order of momentum. By applying proper algebra, we find that the nonreciprocity of paraconductivity is $\gamma\sim 10-100~{}$T^-1A^-1, and the quality factor of the SC diode effect is $Q\sim 10^{-9}h\sqrt{|\epsilon^{\prime}|}$ [see SI for details].

On the other hand, in overdoped region, the carriers are mostly electrons near $T_{c}$. The superconducting phase transition here occurs due to the condensation of electron pairs. Furthermore in an $s$-wave SC, the SC diode effect can be present due to Rashba SOC He et al. (2022) while the nonlinear paraconductivity mainly arises from even and odd parity mixing by $\mathcal{P}$ symmetry breaking Wakatsuki and Nagaosa (2018). Although cuprates exhibit a $d$-wave SC, the underlying physics remains the same. Thus, we employ two distinct free energies $F_{2}$ and $F_{3}$ to represent $T\gtrsim T_{c}$ and $T<T_{c}$, respectively Mineev et al. (1999); Bauer and Sigrist (2012); Wakatsuki and Nagaosa (2018):

Here, $\psi_{k}$ is the SC order parameter, $A$ and $B$ represent the matrix elements for even and odd parity order parameters, $g$ is the matrix for even and odd-parity electron interactions, $\eta_{3}(k)$ is a function of fourth order in momentum, and $L_{Ak}$ and $\nu_{3}(k)$ are the functions of second order in momentum. The reciprocity of paraconductivity, $\gamma\sim 10^{5}~{}T^{-1}A^{-1}$ and, the quality factor, $Q\sim 10^{-1}h\sqrt{|\epsilon^{\prime}|}$, is obtained [see SI for details]. Compared to the underdoped case, the nonreciprocal transports sharply increase. The results are summarized in Fig. 3(b).

Discussion.— Nonreciprocal transport phenomena in high-$T_{c}$ cuprates are chiefly influenced by two factors: the kinetic energy and the type of carriers. As the kinetic energy increases, the magnitude of linear conductivity becomes more prominent, so the nonreciprocal transport diminishes. On the other hand, considering the spinlessness and indirect Zeeman effect of holons, nonreciprocal transport of holons will be small due to the weak symmetry breaking. Specifically, in normal states, the magnitude of nonreciprocity $\gamma$ is large in AF, intermediate in PG and SM, and small in FL, because the carriers gain more kinetic energy at a higher doping. Upon the kinetic energy, the drop of $\gamma$ at the transition from AF to PG phases can be attributed to the fact that the carrier changes from electron to holon. On the other hand, $\gamma$ drops significantly at the transition from SM to FL despite electron carriers in FL phase, since the kinetic energy increases more rapidly.

Moreover, the disparity in energy scales of symmetry breaking of carriers dictates the dependence of $\gamma$ on $T$ and $\delta$. In FL phase, the symmetry breaking energy scale of $h$ and $\lambda_{SO}$ remains constant with changes in $T$ and $\delta$, leading to a reduction in $\gamma$ as $T$ decreases and $\delta$ increases due to kinetic energy. Similarly, in AF phase, upon $h$ and $\lambda_{SO}$, the symmetry breaking is governed by the magnetization which decreases as $T$ increases and $\delta$ decreases. This results in the enhancement in $\gamma$ as $T$ increases, but $\gamma$ still reduces in increasing $\delta$ because of the kinetic energy. In contrast, in the PG and SM phases, the symmetry breaking is attributed to the energy scale of $\lambda_{SO}\xi_{y}$. This increases as $T$ decreases and $\delta$ increases, resulting in the enhancement of $\gamma$ [see SI for details].

In the superconducting (SC) phase and its vicinity, where the system is regarded as a boson condensate, the energy scale of symmetry breaking assumes paramount importance. The breaking of $\mathcal{T}$ and $\mathcal{P}$ symmetries is attributed to $h$ and $\lambda_{SO}$ in electronic systems, whereas it is linked to $\lambda_{SO}\xi_{y}$ in holonic systems. Notably, since $h\sim 1~{}meV$ and $\lambda_{SO}\sim 10~{}meV$ are much larger than $\lambda_{SO}\xi_{y}\sim 1~{}\mu eV$, the symmetry breaking is strong in the overdoped region. Additionally, as the energy scale is introduced by the electron pairing $T_{c}\lesssim 70~{}K$, both $\gamma$ for paraconductivity and the quality factor for the SC diode effect significantly enhance in the overdoped region.

In light of the theoretical analysis depicted in Figure 3, we propose several predictions for experimental exploration. Discontinuities, suggested by the theoretical computations, might be replaced by a smooth but rapid change in reality, owing to the inherent approximations of the mean-field theory. Accordingly, we envisage distinctive nonmonotonous $\gamma$ profiles, which begins with a $U$-shape in underdoped region and ends with a $\lambda$-shape in overdoped region as a function of $\delta$. Moreover, the spinon pairing reduces the kinetic energy of holons, PG has a slightly larger nonreciprocity than SM. This might result in a weak two-hump shape for $\gamma$ in $\delta$ on optimally doped region. Close to $T_{c}$, the surge in $\gamma$ due to paraconductivity is expected to be more pronounced in the overdoped region compared to the underdoped region. In the SC phase, the rapid escalation of $Q$ in the SC diode effect driven by additional doping, is also foreseen. Lastly, we address a caveat that our argument is not applicable to the bulk 3D system, since we only consider the 2D surface.

Our extensive theoretical examination underscores the intriguing aspects of surface nonreciprocal transport in diverse phases of high-$T_{c}$ cuprates and offers practical insights for future investigations. From the MCA in the AF phase to the nonlinear paraconductivity and SC diode effect in the SC phase, these findings illuminate the intricate behavior of high-$T_{c}$ cuprates. We are confident that these practical expectations will pave the way for pioneering discoveries in cuprates, serving as valuable tools in advancing the field of high-$T_{c}$ superconductors.