Investigation of topological features of Sr₂RuO₄using the dissipationless momentum transport

The 2D topological superconductors with the broken time-reversal symmetry have been extensively studied in recent years due to their potential for topological quantum computations [1,2]. This class of topological materials supports topologically protected edge states and half-flux quantum vortices with Majorana zero modes which obey the non-Abelian statistics [3]. To investigate the topological phases of this category where the charge is not conserved, there are several probes, for example the thermal Hall conductivity [4] and the Hall viscosity [5] that can both be formulated based on geometric framework [6,7]. The Hall viscosity as the momentum dissipationless transport is known as a hot topic in condensed matter physics and high energy physics [7–10]. This quantity is the anti-symmetric part of the fourth-order viscosity tensor and occurs in the gapped topological systems with the broken time-reversal symmetry, for example integer quantum Hall effect [11], fractional quantum Hall effect [12,13], Chern insulator [14], and chiral superconductors [15,16]. The Hall viscosity is the adiabatic response of the electron state to the deformation of the system [11], i.e., a topological response, and can be used to distinguish the topological phases of the topological materials [15,17].

The response to the deformation is the response to the changes in the spatial metric tensor [11] that can be also considered as the stress response to the strain. So the stress tensor plays a key role. For crystals with gapped time-reversal symmetry-breaking electronic states, there are two main methods for calculating this response [5,17]; i) strain is applied as a shift in the lattice momentum (minimal coupling method); ii) electron geometric coupling through lattice distortions. The former method is associated with the definition of a generalized momentum like the discussion of the conventional Maxwell electromagnetism [7,17]. The latter is concerned with the definition of the new hopping energies of the tight binding Hamiltonian [5]. In other words, the strain affects the interatomic distance and the Hamiltonian depends on the strain and the stress operator is defined as the derivative of the Hamiltonian with respect to the strain [5,17].

In this paper, we calculate the dynamical and static Hall viscosity of the superconductor Sr₂RuO₄ (the latter corresponds to a topological characteristic) using minimal coupling method for the variety of the gap functions. Based on the crystal symmetry, the multiband nature of Sr₂RuO₄ and muon spin relaxation and polar Kerr effect experiments that revealed its time reversal symmetry is broken [18,19], we should focus on the two-component order parameters with a phase difference by $\pi/2$, namely the time-reversal symmetry breaking states. Thus, it is expected that the Hall viscosity provides a proper topological probe for distinguishing the topological phases of Sr₂RuO₄. Since the pairing symmetry of Sr₂RuO₄ is a controversial subject, there are numerous chiral state models that are either spin-triplet [20–22] or spin-singlet [23–25]. In the present paper, we focus on the chiral states which have recently received great attention for explaining the experimental data: i) the odd-parity two-dimensional fully gapped chiral states including the isotropic $p_\mathrm{x}+ip_\mathrm{y}$ [26], and the anisotropic $p_\mathrm{x}+ip_\mathrm{y}$ [27] and ii) the even-parity order parameter with chirality, $d_{\mathrm{zx}}+id_{\mathrm{yz}}$ [28,29] which includes the interlayer processes. We also calculate the Hall viscosity of the nonchiral $s+id_{\mathrm{xy}}$ state to investigate the effect of the spectral flow [30,31]. Our investigations may be helpful to present the method for considering the role of the lattice effects and of the interorbital coupling on the Hall response of a multiband chiral superconductor and also the origin of strong electron phonon coupling in Sr₂RuO₄ [32] because the Hall viscosity affects the phononic properties of the system [17].

Sr₂RuO₄ is a ruthenate with a layered perovskite structure whose Fermi surface (FS) consists of three orbitals, one two-dimensional orbital, $d_{\mathrm{xy}}$, and two one-dimensional orbitals, $d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$. This material becomes an unconventional superconductor below $T_c = 1.5\ \text{K}$. There are two main scenarios to consider the non-dissipative properties of Sr₂RuO₄, i.e., the Hall-type currents which come from the chirality characteristic or nonzero angular momentum of the Cooper pairs [33]. First, these features arise from the one-dimensional orbitals $d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$ [34,35]. Second, the Hall-type currents originate from the two-dimensional orbital $d_{\mathrm{xy}}$ [36]. In the former case, the crucial role is played by the interorbital coupling while for the latter, the self-energy due to the impurity gives rise to a Hall response [37].

In the present paper, we use the first scenario to calculate the Hall viscosity of Sr₂RuO₄. There are several experimental and theoretical studies which demonstrate the validity of this model; for example, i) the consistency of STM measurements of DOS spectra in the superconducting state of Sr₂RuO₄ with the model presented in our paper [38,39], ii) amongst orbitals which cross FS, the $d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$ orbitals are highly sensitive to the electron phonon coupling [32], iii) inelastic neutron scattering studies [40] demonstrated that the dominant magnetic correlation in Sr₂RuO₄ is incommensurate antiferromagnetic, and iv) it has been shown that the Kerr effect arises from an intrinsic mechanism [19,35]. Following the method presented in ref. [35] for the calculation of the Hall conductivity, we calculate the Hall viscosity. The stress tensor is known as the key quantity for the viscosity tensor and has a role similar to the current operator in the definition of the electrical conductivity [41].

For better understanding, we investigate the details of the effect of the interorbital coupling on the broadening of the spectral function. An increase in which corresponds to the existence of the Hall currents for a two-orbital model without the impurity. The model Hamiltonian in the presence of the spin-orbit coupling (SOC) is defined as [34]

$$\begin{eqnarray} & H=\sum_{\mathbf{k}\nu\sigma}(\xi_\nu(\mathbf{k})-\mu)c^\dag_{\mathbf{k}\nu\sigma}c_{\mathbf{k}\nu\sigma}+\sum_{\mathbf{k}\sigma} [t(\mathbf{k})c^\dag_{\mathbf{k}\nu\sigma}c_{\mathbf{k}\nu^\prime\sigma}+\mathrm{h.c.}]\nonumber\\ & +\lambda\sum_{\nu\nu^\prime}\sum_{\sigma\sigma^\prime}\sum_{\mathbf{k}}c^\dag_{\mathbf{k}\nu\sigma}c_{\mathbf{k}\nu^\prime\sigma}(\vec{l}_{\nu\nu^\prime}\cdot \vec{\sigma}_{\sigma\sigma^\prime})+U\sum_{i\nu}n_{i\nu\uparrow}n_{i\nu\downarrow}\nonumber\\ & +\frac{V}{2}\sum_{i\nu\neq\nu^\prime}n_{i\nu\sigma}n_{i\nu^\prime\sigma}\qquad\quad(n_{i\nu\sigma}=c^\dag_{i\nu\sigma}c_{i\nu\sigma}), \end{eqnarray} \tag{ 1 }$$

where $\nu=\mathrm{xz},\mathrm{yz}$ and σ denote the orbital and spin indices, respectively. $c_{\mathbf{k}\nu\sigma}$ describes the annihilation of an electron with momentum k and spin σ on the orbital ν. $\xi_\nu$ is the energy of the orbital ν and μ is the chemical potential. The quantity $t(\mathbf{k})$ represents the interorbital coupling, and λ is the SOC parameter. The fourth and fifth terms are the interacting Hamiltonian, $H_{\mathrm{int}}$, where U and V are the intraorbital and interorbital repulsions, respectively.

To get the spectral function, we need the retarded Green's function, $G^R(\nu^{\prime\prime}\nu^\prime;\omega)$, which can be obtained by the equation of motion in frequency [42],

$$\begin{equation} \sum_{\nu^{\prime\prime}}[\delta_{\nu\nu^{\prime\prime}}(\omega+i\eta)-t_{\nu\nu^{\prime\prime}}]G^R(\nu^{\prime\prime}\nu^\prime;\omega)= \delta_{\nu\nu^\prime}+D^R(\nu,\nu^\prime;\omega).\quad \end{equation} \tag{ 2 }$$

Here $\delta_{\nu\nu^\prime}$ and $t_{\nu\nu^\prime}$ are the Kronecker delta and the coupling between two the orbitals ν and $\nu^\prime$, respectively, η is a positive infinitesimal to ensure the correct physical result, and $D^R(\nu,\nu^\prime;\omega)$ has the following definition:

$$\begin{eqnarray} & D^R(\nu,\nu^\prime;\omega)=-i\int_{-\infty}^{\infty}\mathrm{d}{t}\mathrm{e}^{-i(\omega+i\eta)(t-t^\prime)}\theta(t-t^\prime)\nonumber\\ & \times\langle{[[H_{\mathrm{int}},c_\nu](t),c^\dag_{\nu^\prime}(t^\prime)]_{B,F}}\rangle. \end{eqnarray} \tag{ 3 }$$

Letters B and F refer to the boson and fermion statistics, respectively. By following ref. [42] and after doing some calculations, the retarded Green's function will be

$$\begin{eqnarray} &&G^R(\nu\uparrow;\omega)=\frac{1}{\omega+i\eta-\xi_\nu+\mu-U\langle{n_{\nu\downarrow}}\rangle-\Sigma(\omega)}, \end{eqnarray} \tag{ 4 }$$

which is consistent with ref. [43]. $\Sigma(\omega)$ is the self-energy and is defined as

$$\begin{eqnarray} &&\Sigma(\omega)=\frac{\Gamma}{\pi}\ln\left|\frac{W+\omega-\frac{V}{2}\langle{n_{\nu\uparrow}}\rangle+\mu}{W-\omega-\frac{V}{2}\langle{n_{\nu\uparrow}}\rangle+\mu}\right|-i\Gamma. \end{eqnarray} \tag{ 5 }$$

The energy limits are defined based on the orbital width, W, $-W<\xi_{\nu^\prime},\omega<W$. The quantity Γ is defined according to the relation $\pi D(\xi)|g(\mathbf{k})|^2=\Gamma\theta(W-|\xi|)$ which expresses $D(\xi)|g(\mathbf{k})|^2$ is constant within the band limits $(D(\xi)$ is the density of states and $g(\mathbf{k})=t(\mathbf{k})-i\lambda$). As is clear, the origin of the self-energy is the interorbital coupling and SOC. Therefore, based on the relation $A(\nu,\omega)=-2\mathrm{Im} G^R(\nu,\omega)$, the spectral function will be

$$\begin{eqnarray} &&A(\nu,\omega)=\frac{2\Gamma}{(\omega-\xi_\nu+\mu-U\langle{n_{\nu\downarrow}}\rangle)^2+\Gamma^2}. \end{eqnarray} \tag{ 6 }$$

Clearly, the interorbital coupling and SOC in the two-orbital model broaden the spectral function that is equivalent to the creation of the Hall-type currents in the topological materials. Thus, the interorbital coupling and SOC play a role similar to the impurity within the one-orbital approximation. Besides the aforementioned main result, there is a generalization of the energy spectrum due to the Coulomb repulsions which is compatible with ref. [44].

With our current understanding of the model, we now are in the position to consider the Hall viscosity. We start with the matrix form of the Hamiltonian of the two-orbital model of Sr₂RuO₄ in the Nambu space:

$$\begin{equation} H=\sum_{\mathbf{k}}\Psi_\mathbf{k}^\dag\check{H}(\mathbf{k})\Psi_\mathbf{k},\quad\check{H}(\mathbf{k})= \left(\begin{array}{l@{\quad}c}\hat{H}_0(\mathbf{k}) &\hat{\Delta}(\mathbf{k}) \\ \hat{\Delta}^\dag(\mathbf{k}) & -\hat{H}_0(-\mathbf{k}) \end{array}\right)\!,\qquad \end{equation} \tag{ 7 }$$

where $\hat{H}_0(\mathbf{k})$ is the non-interacting Hamiltonian and $\hat{\Delta}(\mathbf{k})$ is the gap function matrix which are defined as follows:

$$\begin{eqnarray} & \hat{H}_0(\mathbf{k})=\left(\begin{array}{l@{\quad}c} \varepsilon_{\mathrm{xz}}(\mathbf{k}) & g(\mathbf{k})\\[2pt] g(\mathbf{k})^* &\varepsilon_{\mathrm{yz}}(\mathbf{k}) \end{array}\right)\!,\nonumber\\ & \hat{\Delta}(\mathbf{k})= \left(\begin{array}{l@{\quad}c} \Delta_{\mathrm{xz}}(\mathbf{k}) & \Delta_{\mathrm{xz},\mathrm{yz}}(\mathbf{k})\\ \Delta_{\mathrm{yz},\mathrm{xz}}(\mathbf{k})& \Delta_{\mathrm{yz}}(\mathbf{k}) \end{array}\right),\nonumber\\ & \varepsilon_{\nu}(\mathbf{k})=\xi_\nu-\mu,\nonumber\\ & \xi_{{\mathrm{xz}}/{\mathrm{yz}}}(\mathbf{k})=2(t_{1/2}\cos \mathrm{k_x}+t_{2/1}\cos \mathrm{k_y} \nonumber\\ & \hspace{4.35pc}+4t^\prime\cos (\mathrm{k_x}/2)\cos (\mathrm{k_y}/2)),\nonumber\\ & t(\mathbf{k})=4g\sin \mathrm{k_x} \sin \mathrm{k_y}+8g^\prime\sin (\mathrm{k_x}/2) \sin (\mathrm{k_y}/2),\nonumber\\ & \Delta_{\mathrm{xz}}(\mathbf{k})=\Delta_1g^{\mathrm{xz}}_1(\mathbf{k})+\Delta_2g^{\mathrm{xz}}_2(\mathbf{k})+\Delta_3g^{\mathrm{xz}}_3(\mathbf{k}),\nonumber\\ & \Delta_{\mathrm{yz}}(\mathbf{k})=-i(\Delta_1g^{\mathrm{yz}}_1(\mathbf{k})+\Delta_2g^{\mathrm{yz}}_2(\mathbf{k})+\Delta_3g^{\mathrm{yz}}_3(\mathbf{k})),\nonumber\\ & g^{\mathrm{xz}}_1(\mathbf{k})=\sin \mathrm{k_x},\quad g^{\mathrm{yz}}_1(\mathbf{k})=\sin \mathrm{k_y},\nonumber\\ & g^{\mathrm{xz}}_2(\mathbf{k})=\sin \mathrm{k_x} \cos \mathrm{k_y},\quad g^{\mathrm{yz}}_2(\mathbf{k})=\sin \mathrm{k_y} \cos \mathrm{k_x},\nonumber\\ & g^{\mathrm{xz}}_3(\mathbf{k})=\sin 3\mathrm{k_x},\quad g^{\mathrm{xz}}_3(\mathbf{k})=\sin 3\mathrm{k_y},\nonumber\\ & \Delta_{\mathrm{xz},\mathrm{yz}}(\mathbf{k})=0. \end{eqnarray} \tag{ 8 }$$

The tight-binding parameters are set as ($\mu,t_1,t_2,t^\prime,g, g^\prime,\lambda)=(0.23,-0.31,-0.045,0.0024t_1,-0.025,0.12g,0.08$) eV [44]. $\Delta_1=0.003\ \text{eV}$, $\Delta_2=0.06\ \text{eV}$, and $\Delta_3=0.3\ \text{eV}$ are the magnitude of nearest-neighbor (NN), next-nearest-neighbor (NNN), and 3rd-nearest-neighbor (3NN) pairing, respectively [45] (we will come back to this important issue later on and investigate other chiral order parameters). We can get the particle spectra by diagonalizing the non-interacting Hamiltonian, $\epsilon_{\alpha/\beta}=(\varepsilon_{\mathrm{xz}}(\mathbf{k})+\varepsilon_{\mathrm{yz}}(\mathbf{k}))/2\pm\sqrt{(\varepsilon_{\mathrm{xz}}(\mathbf{k})+\varepsilon_{\mathrm{yz}}(\mathbf{k})/2)^2+g(\mathbf{k})^2}$. Figure 1(a) illustrates the effect of SOC on these spectra. The bands α and β are created due to the coupling between the $d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$ orbitals and SOC. SOC leads to the splitting at the points Γ, M, and the region between M and Γ. Clearly, the most impact happens in the point M, i.e., in the diagonal region of the FS. Figure 1(b) shows the FS of Sr₂RuO₄ in the two-orbital model.

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To calculate the Hall viscosity, we need to determine the stress tensor. We concentrate on the generalized Peierls phase method which describes the coupling between the strain and the lattice fermions. In this method, the gauge field and the corresponding charge are the distortion field, $\omega_{ij}$, and the momentum, respectively, and the shift in the momentum is defined as $\mathrm{k}_i\rightarrow \mathrm{k}_i-\omega_{ji}\sin \mathrm{k}_j$ [17]. In this framework, using the definition of generalized momentum, the stress tensor can be written as $T_{ij}=\delta {H}/\delta{\omega_{ij}}=\sin \mathrm{k}_iJ_j(\mathbf{k})$, where $J_j(\mathbf{k})$ is the current operator. So, to get the stress tensor, we should have the current operator. According to the two-orbital model, we define the current operator as $\footnotesize{\check{J}_i(\mathbf{k})=\hat{I}\otimes\left(\begin{array}{l@{\quad}c} \partial{\varepsilon_1(\textbf{k})}/\partial{\mathrm{k}_i} & \partial{t(\textbf{k})}/\partial{\mathrm{k}_i} \\ \partial{t(\textbf{k})}/\partial{\mathrm{k}_i} & \partial{\varepsilon_2(\textbf{k})}/\partial{\mathrm{k}_i} \end{array}\right)}$, where $\hat{I}$ represents the $2\times2$ identity matrix. Throughout the present paper, the symbol $\textrm{``}\;\check{}\;\textrm{''}\,(\textrm{``}\;\hat{}\;\textrm{''})$ denotes a $2\times2\,(4\times4)$ matrix in the orbital (orbital-Nambu) space.

To consider the topological phases of Sr₂RuO₄, we need the static Hall viscosity (SHV). First, we calculate the dynamical Hall viscosity (DHV), after that, we set $\omega=0$. The Hall viscosity, $\eta^H$, is the only independent component of the antisymmetric part of the viscosity tensor, $\eta^A_{ijkl}=\eta^H(\delta_{jk}\epsilon_{il}-\delta_{il}\epsilon_{jk})$ where $\epsilon_{ij}=1$ when $i\neq j$ [15], namely $\eta^H=\eta_{\mathrm{xxxy}}=\eta_{\mathrm{xxyx}}=-\eta_{\mathrm{xyxx}}=-\eta_{\mathrm{yxxx}}$ that remain unchanged under $\mathrm{x}\leftrightarrow \mathrm{y}$ [46]. This component describes the nondissipative momentum transport and corresponds to the viscoelastic response to the geometric deformation of the system and can be calculated within the linear response theory; $\eta^H_{\mathrm{xxxy}}(\omega)=-\frac{1}{2i\omega}\displaystyle{\lim_{\mathbf{q} \to 0}}(\Xi_{\mathrm{xxxy}}(\mathbf{q},i\omega)-\Xi_{\mathrm{xyxx}}(\mathbf{q},i\omega))$, where q and ω are the viscoelastic wave vector and energy, respectively, and $\Xi_{\mathrm{xxxy}}(\mathbf{q},i\omega)$ is the stress correlation function:

$$\begin{eqnarray} & \Xi_{\mathrm{xxxy}}(\mathbf{q},i\omega)=\nonumber\\ & \mathrm{T}\sum_{\mathbf{k},i\omega^\prime}\mathrm{Tr} [\check{T}_{\mathrm{xx}}\check{G}_0(\mathbf{k},i\omega^\prime)\check{T}_{\mathrm{xy}}\check{G}_0(\mathbf{k}+\mathbf{q},i\omega+i\omega^\prime)],\quad \end{eqnarray} \tag{ 9 }$$

where $\omega^\prime=2\pi \mathrm{nk_BT}$ is the Fermi Matsubara frequency and $\check{G}_0(\mathbf{k},i\omega)=(i\omega-\check{H}(\mathbf{k}))^{-1}$ is the Green's function.

We are now in a position to calculate DHV of a two-orbital superconductor. After computing the stress correlator and using the definition of the Hall viscosity, DHV becomes

$$\begin{eqnarray} & \eta^H_{\mathrm{xxxy}}(i\omega)=\sum_{\mathbf{k}}[T_{12\mathrm{xx}}(T_{1\mathrm{xy}}(\mathbf{k})-T_{2\mathrm{xy}}(\mathbf{k}))+T_{12\mathrm{xy}}(\mathbf{k})\nonumber\\ & \times(T_{2\mathrm{xx}}(\mathbf{k})-T_{1\mathrm{xx}}(\mathbf{k}))]\frac{\mathrm{Re} g(\mathbf{k})\mathrm{Im}(\Delta^*_{\mathrm{yz}}(\mathbf{k})\Delta_{\mathrm{xz}}(\mathbf{k}))}{E_1(\mathrm{\mathbf{k}})E_2(\mathrm{\mathbf{k}})}\nonumber\\ & \times\biggl[\frac{1-f(E_2(\mathrm{\mathbf{k}}))-f(E_1(\mathrm{\mathbf{k}}))}{(E_2(\mathrm{\mathbf{k}})+E_1(\mathrm{\mathbf{k}}))[(E_2(\mathrm{\mathbf{k}})+E_1(\mathrm{\mathbf{k}}))^2-(i\omega)^2]}\nonumber\\ & +\frac{f(E_2(\mathrm{\mathbf{k}}))-f(E_1(\mathrm{\mathbf{k}}))}{(E_2(\mathrm{\mathbf{k}})-E_1(\mathrm{\mathbf{k}}))[(E_2(\mathrm{\mathbf{k}})-E_1(\mathrm{\mathbf{k}}))^2-(i\omega)^2]}\biggr],\qquad \end{eqnarray} \tag{ 10 }$$

where $f(E(\mathrm{\mathbf{k}}))$ is the Fermi-Dirac distribution function, and $E_1(\mathbf{k})$ and $E_2(\mathbf{k})$ are positive eigenvalues of $\check{H}(\mathbf{k})$ which can be calculated by using $\det[\check{G}^{-1}_0(\mathbf{k},i\omega)]$ or diagonalizing of the Hamiltonian $\check{H}(\mathbf{k})$. As expected, the Hall viscosity depends on the interorbital coupling $\mathrm{Re} g(\mathbf{k})$. The appearance of $\mathrm{Im}(\Delta^*_{\mathrm{yz}}(\mathbf{k})\Delta_{\mathrm{xz}}(\mathbf{k}))$ indicates that there should be a phase difference between the gap functions of two orbitals. That is, we should focus on the orbital order parameters as $\Delta_{\mathrm{xz}}=f_1(\mathbf{k})$ and $\Delta_{\mathrm{yz}}=if_2(\mathbf{k})$ in which the time-reversal symmetry is broken. For this reason, we investigate the various pairing channels NN, NNN, and 3NN pairing of the chiral $p_\mathrm{x}+ip_\mathrm{y}$ state and also the chiral $d_{\mathrm{zx}}+id_{\mathrm{yz}}$ state and the nonchiral $s+id_{\mathrm{xy}}$ state. The isotropic and anisotropic gaps are defined in eq. (8). For the nonchiral $s+id_{\mathrm{xy}}$ state, we define $\Delta_{\mathrm{xz}}=\Delta_1$ and $\Delta_{\mathrm{yz}}=\Delta_1g^{\mathrm{xz}}_1g^{\mathrm{yz}}_1$ in which the gap maxima are along the zone diagonals. The chiral $d_{\mathrm{\mathrm{zx}}}+id_{\mathrm{\mathrm{yz}}}$ state is modelled by the replacement $\{g^{\mathrm{xz}}_1,g^{\mathrm{yz}}_1\}\rightarrow\{g^{\mathrm{xz}}_1,g^{\mathrm{yz}}_1\}\sin \mathrm{k}_\mathrm{z}$ in the isotropic gap $p_\mathrm{\mathrm{x}}+ip_\mathrm{\mathrm{y}}$. By choosing the $d_{\mathrm{\mathrm{zx}}}+id_{\mathrm{\mathrm{yz}}}$ state, indeed we take into account the interlayer processes which lead to the occurrence of the horizontal line node at $\mathrm{k_z} = 0$. In the interlayer processes, the $d_{\mathrm{xz}}$ and the $d_{\mathrm{yz}}$ orbitals play the dominant role, so the choice of this gap function is justifiable for our two-orbital model. It is important to stress that during the calculation of the Hall viscosity for $d_{\mathrm{xz}}+id_{\mathrm{yz}}$ gap function, we have used the normal-state electronic band structure presented in ref. [47] that depends not only on $\mathrm{k_x}$ and $\mathrm{k_y}$ but also on $\mathrm{k_z}$.

After performing an analytic continuation of the viscoelastic frequency, $i\omega\rightarrow \omega+i\gamma$ (where γ is set to 0.01 for numerical calculations) and putting the amounts of the stress tensor in eq. (10), we obtain the following relation for the Hall viscosity at the zero temperature limit, $f(E(\mathrm{\mathbf{k}}))=0$:

$$\begin{eqnarray} & \eta^H_{\mathrm{xxxy}}(\omega)=\frac{1}{\pi^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\mathrm{d} \mathrm{k}_\mathrm{x}\mathrm{d} \mathrm{k}_\mathrm{y}\sin^2\mathrm{k_x}\nonumber\\ & \times\frac{\partial_{\mathrm{k_x}}g(\mathbf{k})\partial_{\mathrm{k_y}}\varepsilon(\mathbf{k})-\partial_{\mathrm{k_y}}g(\mathbf{k})\partial_{\mathrm{k_x}}\varepsilon(\mathbf{k})}{E_1(\mathrm{\mathbf{k}})E_2(\mathrm{\mathbf{k}})}\nonumber\\ & \times\biggl[\frac{\mathrm{Re} g(\mathbf{k})\mathrm{Im}(\Delta^*_{\mathrm{yz}}(\mathbf{k})\Delta_{\mathrm{xz}}(\mathbf{k}))}{(E_1(\mathrm{\mathbf{k}})+E_2(\mathrm{\mathbf{k}}))[(E_1(\mathrm{\mathbf{k}})+E_2(\mathrm{\mathbf{k}}))^2-(\omega+i\gamma)^2]}\biggr],\nonumber\\ \end{eqnarray} \tag{ 11 }$$

where $\varepsilon(\mathbf{k})=\varepsilon_{\mathrm{xz}}(\mathbf{k})-\varepsilon_{\mathrm{yz}}(\mathbf{k})$. e, ℏ , and lattice spacing are set to 1. Using eq. (11), we are able to find the imaginary and real parts of the Hall viscosity. The imaginary part provides information about the breaking of the Cooper pairings, i.e., the quasiparticles, and the static real part helps us to distinguish the topological phases.

Figure 2 shows the imaginary and real parts of DHV for the different pairing channels of the chiral $p_\mathrm{x}+ip_\mathrm{y}$, $d_{\mathrm{zx}}+id_{\mathrm{yz}}$, and $s+id_{\mathrm{xy}}$. This response is known as the viscoelastic response of a two-orbital superconductor with broken time-reversal symmetry and may be helpful to consider the role of the lattice effects and the interlayer processes on the Hall response. For the NN pairing (isotropic gap), $s+id_{\mathrm{xy}}$, and $d_{\mathrm{zx}}+id_{\mathrm{yz}}$, panels (a), (e), (f), the imaginary part takes a nonzero value in the interval $0.2\lesssim\omega\lesssim0.32\ \text{eV}$ and has a sharp singularity which is nearer to the left boundary of the interval. The real part exhibits two maxima almost at the smooth energy boundaries of the interval. The intensity of the first maximum is greater than the second one and behaves like a singularity. Therefore, it is termed the main peak and occurs at $\mathrm{min}[E_1(\mathbf{k})+E_2(\mathbf{k})]$. The second maximum shows the behavior of a quasiparticle quantum gas in the high-frequency region [48]. The maxima are strongly affected by interorbital coupling [35] and SOC. The value of $g^\prime$ affects the position and the intensity of the peaks while λ has only the influence on the intensity. For the NNN pairing, panel (b), DHV does not change in the interval $0<\omega\lesssim\mathrm{min}[E_1(\mathbf{k})+E_2(\mathbf{k})]$ but there is a small shift of the second peak that may be ascribed to the gap minima. SOC affects the process of the creation of the quasiparticles such that gap minima become deeper with a correspondingly smaller gap anisotropy ratio on the electron-like β band. For the 3NN pairing, the situation is completely different, see panel (c). $g^\prime$ and λ do not affect the position of peaks. The former (latter) parameter decreases (increases) the intensity of peaks. There is a pronounced positive feature at 0.58 eV that is attributed to $g^\prime$.

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The behavior of DHV of the nonchiral $s+id_{\mathrm{xy}}$ state is very similar to that of the isotropic gap $p_\mathrm{x}+ip_\mathrm{y}$. For the latter chiral state, there are two contributions to the Hall viscosity, one is associated with net angular momentum of the Cooper pairs and the other comes from a purely nondissipative force like the usual Lorentz force due to the spontaneous surface currents [49], while for the nonchiral $s+id_{\mathrm{xy}}$ state, the Hall viscosity is only created by the latter case because its Cooper pairs do not have a net angular momentum. This behavior is similar to the discussion of spontaneous Hall effect for these pairings [31]. Since the angular momentum is reduced by the effect of spectral flow along the edge states, inside the vortex, and at dislocation [50] and on the other hand, since the $s+id_\mathrm{xy}$ state has the same viscoelastic properties as the isotropic pairing $p_\mathrm{x}+ip_\mathrm{y}$ and does not have gapless edge mode, we can conclude that for isotropic $p_\mathrm{x}+ip_\mathrm{y}$, the spectral flow takes place through the bound states inside vortex and at dislocation. The NNN pairing changes the quantitative amount of the Hall viscosity and the qualitative picture remains unchanged that is consistent with ref. [51].

Also, based on the concept of the spectral flow at the Hall viscosity, no states cross E = 0, unlike the conventional spectral flow [52], and the chiral fermions traveling faster or slower depending on the sign of the Burgers vector [7], we can account for the same plots for NN pairing and NNN pairing. But since the 3NN pairing is closely related to the hybridization between the $d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$ orbitals [45], the Hall viscosity has a relatively different plot. The higher values of the Hall viscosity for NNN and 3NN pairings indicate that these dominate for the dissipationless momentum transport at high viscoelastic energies where Sr₂RuO₄ with the isotropic p-wave gap function behaves as a quantum fluid and the corresponding DHV is independent of the gap magnitude. Increasing the magnitude of this gap just affects DHV at the low energy limit, $\omega\lesssim 0.3\ \text{eV}$. At high energies, the behavior of DHV depends fully on the range of the interaction in the lattice, i.e., NNN and 3NN interactions which lead to NN and 3NN pairings and novel phases [53]. Comparing DHV of the panels (a) and (d) indicates that the range of the interaction affects the dynamical momentum transport.

The higher value of DHV of the $d_{\mathrm{zx}}+id_{\mathrm{yz}}$ pairing, compared with panel (a), may be due to the zero energy peak in the conductance spectra along the c-axis and in the ab-plane and also due to the horizontal line node [54]. The zero bias conductance peak is due to the surface Andreev bound state which stems from the topological properties of the bulk Hamiltonian [55].

We have calculated DHV for the $s+id_{\mathrm{x}^2-\mathrm{y}^2}$ pairing and found that the momentum transport along surfaces normal to the [110] direction is fully suppressed (DHV has a very unusual behavior and its magnitude is of order 10^-18). So the momentum current only flows along surfaces normal to the [100] or [010] direction. For a two-dimensional chiral superconductor with the circular FS, relevant for Sr₂RuO₄ with the dominant γ band and the gap function $s+id_{\mathrm{xy}}$ ($s+id_{\mathrm{x}^2-\mathrm{y}^2}$), the spontaneous momentum transport flows along the [100] and [010] surfaces ([110] surface) [31]. But for the two-orbital model, our calculations demonstrate that for the gap function $s+id_{\mathrm{xy}}$, the momentum transport runs along the [100] and [010] surface, like the one-orbital model, whereas for the gap function $s+id_{\mathrm{x}^2-\mathrm{y}^2}$, it vanishes along the [110] surface, unlike the one-orbital model. This difference is due to the one-dimensional nature of orbitals, their resulting FS, and also its diagonals of the Brillouin zone that strongly affects the transport coefficients [56]. In other words, the absence of spontaneous momentum transport for the gap function $s+id_{\mathrm{x}^2-\mathrm{y}^2}$ originates from the component $d_{\mathrm{x}^2-\mathrm{y}^2}\varpropto\cos \mathrm{k}_\mathrm{x}-\cos {\mathrm{k}_\mathrm{y}}$ and reflects the fact that the interorbital coupling, which occurs at the wavevectors $\mathrm{k_x}\backsimeq \mathrm{k_y}$, plays a key role in determining DHV in the two-orbital model, see fig. 4.

The Hall viscosity is known as a topological response and can be used to investigate the topological phases. In chiral superconductors and superfluids, this quantity is related to the correlation between two particles, i.e., Cooper pair (a change in the correlation length corresponds to the change in the mean orbital spin or Hall viscosity, see fig. 6 of ref. [13]) and determined by the pairing function. In the trivial topological phase (the BEC regime or strong-pairing phase [1]), the pairing function falls exponentially with distance whereas for the nontrivial topological phase (the BCS regime or weak-pairing phase [1]), there is a long tail in the pairing function [12] that leads to an overlap between two pairings. Thus, as the range of the interaction changes, the Hall viscosity will change [13]. The weak-pairing phase describes the chiral states which are bound to the vortex, edge, and dislocation and for the trivial BEC phase, there is no chiral state. Since the Hall viscosity determines the amount of the momentum of the bound states on dislocation, its non-zero value represents the existence of the bound states and the non-trivial topological phase.

Figure 3 illustrates SHV to distinguish the topological and trivial phases for various choices of chiral states with $\lambda=0$ and $\lambda\neq0$. For isotropic gaps, panel (a), the orange and magenta curves of panel (c), as the chemical potential μ is varied, the topological BCS phase appears for $|\mu|\lesssim2.2t_1$ and trivial BEC phase for $\mu\gtrsim2.2t_1$ and $\mu\lesssim-2.2t_1$. SHV is finite in the BCS regime and becomes zero deep in the BEC regime. The rate at which SHV shrinks to zero in the BEC regime, i.e., the behavior of the phase transition between the BCS and BEC, depends on SOC, the kind of the pairing, and the lattice effects. Since during the evolution of the system from the BEC to the BCS the spectral flow occurs, one may conclude that the pairing and the lattice effects affect the spectral flow or the depairing effects which are closely related to the SHV. This gives rise to the topological response for the lattice systems in which the lattice effects cannot be neglected, which differs from the continuum limit, similar to ref. [45].

SHV vanishes at $\mu=0$ where there is a gap closing and its chirality changes sign. This indicates that the Hall viscosity is an odd function of the chemical potential [17].

As one can see from panels (a) and (c), there is a linear SHV around $\mu=0$ for the NN pairing and $d_{\mathrm{zx}}+id_{\mathrm{yz}}$ whereas it does not occurs for the NNN pairing. This difference comes from different ranges of the interaction. For the NNN and 3NN pairing, the BCS regime takes place for $|\mu|<3t_1\,(|\mu|<2t_1)$ and the trivial BEC phase occurs for $\mu>3t_1\,(\mu>2t_1)$ for NNN pairing (3NN pairing) which indicate that the momentum charge or the momentum current is sensitive to the gap anisotropy as microscopic details of the Bogoliubov-de Gennes equations [52]. This difference originates from the spectral flow properties of these anisotropic gap functions that lead to different topological characteristic [52].

Clearly, for all plots, SOC causes a reduction of SHV which may be attributed to the creation of the near-nodes [57,58] and the dependence on $\mathrm{Im}(\Delta^*_{\mathrm{yz}}(\mathbf{k})\Delta_{\mathrm{xz}}(\mathbf{k}))$.

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We showed that the presence of lattice introduces a new length scale and causes that SHV becomes zero "deep" in the trivial BEC phase. In other words, in passing from BCS to BEC phase, there is a nonzero SHV around the critical point in the BEC region. Increasing the chemical potential makes the lattice effects less important and SHV vanishes throughout the trivial phase, as expected in the continuum limit. This finite value of the SHV in the vicinity of the critical point and on the BEC regime side is known as the residual Hall viscosity (RHV) [17]. RHV is depicted in fig. 4 for the isotropic and anisotropic gap functions $p_\mathrm{x}+ip_\mathrm{y}$. The gap function determines how SHV goes to zero in the BEC regime and is closely related to the correlation length [13]. When the presence of lattice affects RHV and leads to a finite value for RHV, the correlation length of the effective bulk gap is comparable with the lattice constant.

As one can see from fig. 4, RHV lies on the diagonal region for all three pairing channels (there is a negative value for RHV in the non-diagonal region for 3NN pairing). For NN pairing, RHV is small, see fig. 3(a), and its dominant contribution comes from the α band. Due to the small RVH in the BEC region, RVH of this pairing channel is not much different from its continuum limit. There is a similar situation for NNN pairing but with stronger intensity which is associated with the kind of the channel and the lattice effects. In this channel, the RVH value differs from its continuum limit value. For 3NN pairing, RHV is completely different from its continuum limit for the same reason as the NNN pairing and its dominant contribution is due to the β band which displays signatures of quasi 1D behavior with a dispersion that reveals a suppression of Fermi velocity and plays an important role in strengthening correlation effects [39]. Another important difference is the existence of positive and negative values of RHV that arise from the spectral flow of the 3rd-nearest-neighbor sites, see the black curve of fig. 3(c).

We have investigated the viscoelastic response and the topological properties of the two-orbital model of chiral superconductor Sr₂RuO₄ using the dissipationless momentum transport for the variety of the pairing channels, the chiral $d_{\mathrm{zx}}+id_{\mathrm{yz}}$ state, and the nonchiral $s+id_{\mathrm{xy}}$ state. We have showed that when lattice effects are strong, the topological response to a viscoelastic field changes and leads to different behaviors of the dynamical Hall viscosities and also different topological regions compared with the continuum limit. By taking account of lattice effects, there is a tiny static Hall viscosity in the strong pairing phase that vanishes in the continuum limit. In other words, the lattice effects lead to a soft phase transition between the weak-pairing phase and the strong-pairing phase. This transition depends on the kind of the pairing and may be associated with the spectral flow which takes place through the BEC-BCS crossover.

The different behaviors of the static Hall viscosity as a function of the chemical potential for NN, NNN, and 3NN pairings lead to different Chern numbers. This result reflects the fact that the topological features depend on the type of the pairing channel and of the gap function. Thus, the topological properties of the NN pairing are similar to those in the continuum limit while for the NNN and 3NN pairings, there are the strong lattice effects, because the lattice constant is comparable with the correlation length, which causes the topological properties of the two latter pairings to differ from those in the continuum limit, especially for 3NN pairing.

The features discussed above for the static Hall viscosity of weak-pairing and strong-pairing phases and also the significant role of the lattice effect for different pairings can help us to predict and analyze the topological regions of Sr₂RuO₄ and the lattice effects including the gap anisotropic on them using the spectral flow.

This study was financially supported by the research council of Ayatollah Boroujerdi University, Iran, Grant No. 15664-171289.