Odd-parity spin-loop-current order mediated by transverse spin fluctuations
in cuprates and related electron systems

Here, we analyze the single-orbital square-lattice Hubbard model

We denote the hopping integrals $(t_{1},t_{2},t_{3})=(-1,1/6,-1/5)$, where $t_{l}$ is the $l$-th nearest hopping integral Kontani-ROP ; Springer . Hereafter, we set the unit of energy as $|t_{1}|=1$, which corresponds to $\sim 4000$ [K] in cuprates, and fix the temperature $T=0.05\ (\sim 200{\rm K})$. The FS at filling $n=0.85$ is given in Fig. 2 (a). The spin susceptibility in the random-phase-approximation (RPA) is $\chi^{s}(q)=\chi^{0}(q)/(1-U\chi^{0}(q))$, where $\chi^{0}(q)$ is the irreducible susceptibility without $U$ and $q\equiv({{\bm{q}}},{\omega}_{l})$. The spin Stoner factor is defined as ${\alpha}_{S}\equiv\max_{q}\{U\chi^{0}(q)\}=U\chi^{0}({{\bm{Q}}}_{s},0)$. Figure 2 (b) shows the obtained $\chi^{s}(q)$ at ${\alpha}_{S}=0.99$ ($U=3.27$). Here, $\chi^{s}({{\bm{Q}}}_{s},0)\sim 30$ [$1/{t_{1}}$] $\sim 80$ [$\mu_{\rm B}^{2}/{\rm eV}$], which is still smaller than Im$\chi^{s}({{\bm{Q}}}_{s},E=31{\rm meV})\sim 200$ [$\mu_{\rm B}^{2}/{\rm eV}$] at $T\sim 200$K in 60K YBCO neutron . Thus, ${\alpha}_{S}>0.99$ in real compounds. Owing to the Mermin-Wagner theorem, the relation ${\alpha}_{S}\lesssim 1$ is naturally satisfied for $U\gg 3.3$ without any fine tuning of $U$ by considering the spin-fluctuation-induced self-energy self-consistently Kontani-ROP .

From now on, we investigate possible exotic density-wave (DW) states for both charge- and spin-channels with general wavevector (${{\bm{q}}}$), which is generally expressed as Nersesyan

where ${{\bm{k}}}_{\pm}\equiv{{\bm{k}}}\pm{{\bm{q}}}/2$, and $d^{c}_{{\bm{q}}}({{\bm{k}}})$ $(\bm{d}^{s}_{{{\bm{q}}}}({{\bm{k}}}))$ is the charge (spin) channel order parameter. It induces the symmetry breaking in the self-energy:

which we call the form factors in this paper. Below, we assume $\bm{g}_{{{\bm{q}}}}({{\bm{k}}})=g_{{{\bm{q}}}}({{\bm{k}}}){\bm{e}}_{z}$ without losing generality. The DW is interpreted as the electron-hole pairing condensation Nersesyan .

Here, $f_{{\bm{q}}}({{\bm{k}}})$ is given by the Fourier transformation of the spin-independent hopping modulation $\sum_{{{\bm{r}}}_{i},{{\bm{r}}}_{j}}\delta t_{i,j}e^{i({{\bm{r}}}_{i}-{{\bm{r}}}_{j})\cdot{{\bm{k}}}}e^{i({{\bm{r}}}_{i}+{{\bm{r}}}_{j})\cdot{{\bm{q}}}/2}$. When $\delta t_{i,j}=\pm\delta t_{j,i}$, the relation $f_{{\bm{q}}}({{\bm{k}}})=\pm f_{{\bm{q}}}(-{{\bm{k}}})$ holds. Also, $g_{{\bm{q}}}({{\bm{k}}})$ is given by the spin-dependent modulation $\delta t_{i,j}^{\uparrow}=-\delta t_{i,j}^{\downarrow}$. The even-parity $f_{{\bm{q}}}({{\bm{k}}})$ and the odd-parity $g_{{\bm{q}}}({{\bm{k}}})$ respectively correspond to the BO state and the sLC state. Both states preserve the time-reversal symmetry.

To find possible DW in an unbiased way, we generalize the DW equation Kawaguchi-CDW for both spin/charge channels:

where $\lambda_{{{\bm{q}}}}$ ($\eta_{{\bm{q}}}$) is the eigenvalue that represents the charge (spin) channel DW instability, $k\equiv({{\bm{k}}},{\epsilon}_{n})$, $p\equiv({{\bm{p}}},{\epsilon}_{m})$ (${\epsilon}_{n}$, ${\epsilon}_{m}$ are fermion Matsubara frequencies). These DW equations are interpreted as the “spin/charge channel electron-hole pairing equations”.

The charge (spin) channel kernel function is $I_{{\bm{q}}}^{c(s)}=I_{{\bm{q}}}^{\uparrow,\uparrow}+(-)I_{{\bm{q}}}^{\uparrow,\downarrow}$; $I_{{\bm{q}}}^{{\sigma},\rho}$ at ${{\bm{q}}}=0$ is given by the Ward identity $-\delta\Sigma_{\sigma}(k)/\delta G_{\rho}(k^{\prime})$, which is composed of one single-magnon exchange term and two double-magnon exchange ones: The former and the latter are called the Maki-Thompson (MT) term and the AL terms; see Fig. 7 in Appendix A. The lowest order Hartree term $-U\delta_{{\sigma},\rho}$ in $I_{{\bm{q}}}^{{\sigma},\rho}$ gives the RPA contribution. while the AL terms are significant for ${\alpha}_{S}\lesssim 1$ in various strongy correlated systems Onari-SCVC ; Kawaguchi-CDW ; Tazai-CeCu2Si2 ; Tazai-CeB6 . The significance of the AL processes have been revealed by the functional-renormalization-group (fRG) study, in which higher-order vertex corrections are produced in an unbiased way Tsuchiizu-CDW ; Tsuchiizu-PRL ; Tazai-FRG ; Tazai-kappa . Note that the MT term is important for the superconducting gap equation, transport phenomena Kontani-ROP , and cLC order Tazai-cLC .

Figure 2 (c) shows the charge-channel eigenvalue $\lambda_{{{\bm{q}}}}$ derived from the DW eq. (4) Kawaguchi-CDW ; Onari-B2g ; Onari-AFB . Hereafter, we put $U$ to satisfy the relation ${\alpha}_{S}=1-0.444p^{2}$ with $p\equiv 1-n$, shown as full line in Fig. 2 (d). The obtained form factor $f_{{\bm{q}}}({{\bm{k}}})$ at ${{\bm{q}}}={\bm{0}},{{{\bm{Q}}}_{\rm d}}$, shown in Fig. 2 (e), belongs to $B_{1g}$ symmetry BO, consistently with previous studies Kawaguchi-CDW ; Tsuchiizu-CDW . As we discuss in Appendix B, the large eigenvalue in Fig. 2 (c) (and Fig. 3 (a)) is strongly suppressed to $O(1)$ by considering the small quasiparticle weight $z=m/m^{*}\sim O(10^{-1})$ due to the self-energy in cuprates Kawaguchi-CDW ; Onari-B2g .

Next, we discuss the spin-fluctuation-driven sLC order, which is the main issue of this manuscript. Figure 3 (a) exhibits the spin-channel eigenvalue $\eta_{{{\bm{q}}}}$ derived from the DW eq. (5). Peaks of $\eta_{{{\bm{q}}}}$ are located at the nesting vectors ${{\bm{q}}}={{\bm{Q}}}_{\rm d}$ (diagonal) and ${{\bm{q}}}={{\bm{Q}}}_{\rm a}$ (axial). The obtained form factor $g_{{\bm{q}}}({{\bm{k}}})$ at ${{\bm{q}}}={{\bm{Q}}}_{\rm d}$ (diagonal sLC) is shown in Fig. 3 (b). The odd-parity solution $g_{{\bm{q}}}({{\bm{k}}})=-g_{{\bm{q}}}(-{{\bm{k}}})$ means the emergence of the sLC order. The reason for large $\eta_{{{\bm{Q}}}_{\bm{d}}}$ is that all hot spots contribute to the diagonal sLC as shown in Fig. 3 (b). Figure 3 (c) shows the form factor in real space ${\rm Im}g_{{{\bm{Q}}}_{\rm a}}({\bm{r}})$ with ${\bm{r}}=(x,y)$. Here, $\delta t_{i,j}^{\sigma}={\sigma}g_{{{\bm{Q}}}_{\rm a}}({\bm{r}}_{i-j})\cos({\bm{r}}_{i+j}\cdot{{\bm{Q}}}_{\rm a}/2)$.

To understand why sLC state is obtained, we simplify Eq. (5) by taking the Matsubara summation analytically by approximating that $I_{{\bm{q}}}^{s}$ and $g_{{\bm{q}}}(k)$ are static:

where $\displaystyle F_{{\bm{q}}}({{\bm{p}}})\equiv-T\sum_{m}G(p_{+})G(p_{-})=\frac{n({\epsilon}_{{{\bm{p}}}_{+}})-n({\epsilon}_{{{\bm{p}}}_{-}})}{{\epsilon}_{{{\bm{p}}}_{-}}-{\epsilon}_{{{\bm{p}}}_{+}}}$ is a positive function, and $n({\epsilon})$ is Fermi distribution function; see Appendix A. In general, the peak positions of $\eta_{{{\bm{q}}}}$ in Eq. (6) are located at ${{\bm{q}}}={\bm{0}}$ and/or nesting vectors with small wavelength (${{\bm{q}}}={{\bm{Q}}}_{\rm a},{{\bm{Q}}}_{\rm d}$ in the present model). The reason is that $I_{{\bm{q}}}\sim T\sum_{p}\chi^{s}({{\bm{p}}}_{+})\chi^{s}({{\bm{p}}}_{-})$ by AL terms is large for small $|{{\bm{q}}}|$, and $F_{{\bm{q}}}({{\bm{p}}})$ is large for wide area of ${{\bm{p}}}$ when ${{\bm{q}}}$ is a nesting vector.

To understand why odd-parity form factor is obtained, we show the spin-channel “electron-hole pairing interaction” $I^{s}_{{{\bm{q}}}={\bm{0}}}({{\bm{k}}},{{\bm{k}}}^{\prime})$ on the FS in Fig. 4 (a). The charge-channel one $I^{c}_{{{\bm{q}}}={\bm{0}}}({{\bm{k}}},{{\bm{k}}}^{\prime})$ is also shown in Fig.4 (b). Here, $\theta$ represents the position of ${{\bm{k}}}$ shown in Figs. 4 (c) and (d). $I_{{\bm{q}}}^{s}({{\bm{k}}},{{\bm{k}}}^{\prime})$ in Fig. 4 (a) gives large attractive interaction for ${{\bm{k}}}\approx{{\bm{k}}}^{\prime}$ and large repulsive one for ${{\bm{k}}}\approx-{{\bm{k}}}^{\prime}$. In this case, we naturally obtain $p$-wave form factor $g_{{\bm{q}}}({{\bm{k}}})$ shown in Fig. 3 (b), as we explain in Fig. 4 (c). Here, red (blue) arrows represent the attractive (repulsive) interaction.

The strong ${{\bm{k}}},{{\bm{k}}}^{\prime}$-dependence of $I^{s}_{{{\bm{q}}}={\bm{0}}}({{\bm{k}}},{{\bm{k}}}^{\prime})$ originates from the AL1 and AL2 terms in Fig. 4 (e), or Fig. 7 (a) in Appendix A. Owing to the spin-conservation law, AL terms in $I^{s}=I^{\uparrow,\uparrow}-I^{\uparrow,\downarrow}$ originates from the spin-flipping processes due to transverse spin fluctuations in Fig. 4 (e), in proportion to $\chi^{s}_{\pm}({{\bm{Q}}}_{s})\chi^{s}_{\pm}({{\bm{Q}}}_{s})$. (In $I^{s}$, the spin non-flipping AL processes in proportion to $\chi^{s}_{z}({{\bm{Q}}}_{s})\chi^{s}_{z}({{\bm{Q}}}_{s})$ are exactly canceled out.) Therefore, $I^{s}=\mbox{[AL1]}-\mbox{[AL2]}$. The AL1 term with the p-h (anti-parallel) pair Green functions causes large attractive interaction for ${{\bm{k}}}\approx{{\bm{k}}}^{\prime}$, and the AL2 term with the p-p (parallel) ones does for ${{\bm{k}}}\approx-{{\bm{k}}}^{\prime}$, as explained in Ref. Onari-B2g in detail. Thus, $\theta,\theta^{\prime}$-dependence in Fig. 4 (a) and resultant odd-parity solution is understood naturally.

In contrast, the charge channel kernel $I_{{\bm{q}}}^{c}({{\bm{k}}},{{\bm{k}}}^{\prime})$ gives an attractive interaction for both ${{\bm{k}}}\approx\pm{{\bm{k}}}^{\prime}$ as shown in Fig. 4 (b), beucase $I^{c}=3(\mbox{[AL1]}+\mbox{[AL2]})/2$. Then, we obtain $d$-wave form factor $f_{{\bm{q}}}({{\bm{k}}})$ in Fig. 2 (e), as we explain in Fig. 4 (d) Kawaguchi-CDW ; Onari-B2g .

In the present transverse spin fluctuation mechanism, the $\bm{g}$-vector will be parallel to $z$-direction when $\chi^{s}_{x(y)}({{\bm{Q}}}_{s})>\chi^{s}_{z}({{\bm{Q}}}_{s})$ (XY-anisotropy) due to the spin-orbit interaction (SOI). When the XY-anisotropy of $\chi^{s}_{\mu}({{\bm{Q}}}_{s})$ is very large, $I^{c}$ due to AL terms is multiplied by $2/3$ whereas $I^{s}$ is unchanged, so it is suitable condition for the sLC order.

The sLC and BO eigenvalues are summarized in Fig. 5 (a). The relation $\eta_{{{\bm{Q}}}_{\rm d}}>\lambda_{{{\bm{Q}}}_{\rm a}}$ around the optimal doping ($p\sim 0.15$) means that the sLC transition temperature $T_{\rm sLC}$ is higher than $T_{\rm CDW}$, as in Fig. 1 (a). The robustness of Fig. 5 (a) is verified in Appendix B. We also verify in Fig. 5 (b) that both $\eta_{{{\bm{Q}}}_{\rm d}}$ and $\lambda_{{{\bm{Q}}}_{\rm a}}$ are larger than the $d_{x^{2}-y^{2}}$-wave superconducting eigenvalue $\lambda_{\rm SC}$ for ${\alpha}_{S}\gtrsim 0.98$. Here, $\lambda_{\rm SC}$ is derived from the gap equation

where $V^{\rm SC}(k,p)=-\frac{3}{2}U^{2}\chi^{s}(k-p)+\frac{1}{2}U^{2}\chi^{c}(k-p)-U$ is the MT-type kernel d-wave . Note that $\eta_{{\bm{q}}},\lambda_{{\bm{q}}}<\lambda_{\rm SC}$ if AL terms are dropped Norman . The large eigenvalues in Fig. 5 are suppressed to $O(1)$ by small $z$; see Refs. Kawaguchi-CDW ; Onari-B2g ; Kontani-ROP and Appendix B. We stress that sLC is not suppressed by the ferro-BO that induces neither Fermi arc nor pseudogap, as explained in Appendix B.

Here, we analyzed the sLC/BO in terms of the electron-hole pairing. Another physical interpretation of the sLC/BO is the “condensation of odd/even parity magnon-pairs”, which is the origin of the nematic order in quantum spin systems Andreev ; Coleman ; Shannon . In fact, the two-magnon propagator shown in Fig. 5 (c) diverges when the eigenvalue of DW equation reaches unity, as we explain in Appendix C. That is, triplet (singlet) magnon-pair condensation occurs at $T=T_{\rm sLC}(T_{\rm CDW})$. Thus, the sLC/BO discussed here and the spin nematic order in quantum spin systems are essentially the same phenomenon.

Now, we discuss the band-folding and hybridization gap due to the diagonal sLC order. Figures 6 (a) and (b) show the Fermi arc structures in the cases of (a) single-${{\bm{q}}}$ and (b) double-${{\bm{q}}}$ orders. We set $g^{\rm max}\equiv\max_{{\bm{k}}}\{g_{{{\bm{Q}}}_{\rm d}}({{\bm{k}}})\}=0.1$. Here, the folded band structure under the sLC order with finite ${{\bm{q}}}_{\rm sLC}$ is “unfolded” into the original Brillouin zone by following Ref. Ku to make a comparison with ARPES results. The Fermi arc due to the single-${{\bm{q}}}$ order in Fig. 6 (a) belongs to $B_{2g}$ symmetry. In contrast, the Fermi arc due to the double-${{\bm{q}}}$ order in Fig. 6 (b) preserves the $C_{4}$ symmetry. The resultant pseudogap in the DOS is shown in Fig. 6 (c). The unfolded band structure in the single-${{\bm{Q}}}_{\rm d}$ sLC order is displayed in Fig. 12 in Appendix B.

Recent magnetic torque measurements revealed the $B_{1g}$ symmetry breaking at $T=T^{*}$ occurs in YBCO Y-Sato , while $B_{2g}$ one appears in Hg-based cuprates Hg-Murayama . To understand different symmetry breakings, we examine the $t_{3}$ dependence of DW equation solution in Appendix B 2: As shown in Figs. 9 (b)-(e), the sLC wavevector ${{\bm{q}}}_{\rm sLC}$ changes from ${{\bm{Q}}}_{\rm d}$ to ${{\bm{Q}}}_{\rm a}$ for larger $t_{3}/t_{1}$. When ${{\bm{q}}}_{\rm sLC}={{\bm{Q}}}_{\rm d}$, the symmetry of the Fermi arc is $B_{2g}$ in Fig. 6 (a). On the other hand, the Fermi arc has $B_{2g}$ symmetry in the axial sLC order in Fig. 1 (d), as we show in Fig. 10 (b).

Since the van Vleck susceptibility becomes anisotropic when $C_{4}$ symmetry of the FS is broken Kawaguchi-B2g , the reported compound-dependent symmetry breaking Y-Sato ; Hg-Murayama would be explained by the sLC order scenario. This is an important future issue.

Next, we investigate the spin current in real space, which is driven by a fictitious Peierls phase due to the “spin-dependent self-energy” $\delta t_{i,j}^{\sigma}={\sigma}g_{i,j}$. As shown in Fig. 3 (c), $\delta t_{i,j}$ is purely imaginary and odd with respect to $i\leftrightarrow j$. The conservation law $\dot{n}_{i}^{\sigma}=\sum_{j}j_{i,j}^{\sigma}$ directly leads to the definition the spin current operator from site $j$ to site $i$ as $j_{i,j}^{\sigma}=-i\sum_{{\sigma}}{\sigma}(h_{i,j}^{\sigma}c_{i{\sigma}}^{\dagger}c_{j{\sigma}}-(i\leftrightarrow j))$, where $h_{i,j}^{\sigma}=t_{i,j}+\delta t_{i,j}^{\sigma}$. Then, the spontaneous spin current from $j$ to $i$ is $J_{i,j}^{s}=\langle j_{i,j}^{s}\rangle_{\hat{h}^{\sigma}}$.

Here, we calculate the spin current for the commensurate sLC order at ${{\bm{q}}}_{\rm sLC}=(\pi/2,\pi/2)$, which is achieved by putting $n=1.0$. Then, the unit cell are composed four sites A-D. In Fig. 14 in Appendix D, we show the obtained spin current $J_{i,j}^{s}$ from the center site ($j={\rm A}$-${\rm B}$) to different site in Fig. 1 (c). The derived spin current pattern between the nearest and second-nearest sites is depicted in Fig. 1 (c).

The Fermi arc structure and the DOS in Fig. 6 are independent of the phase shift $g_{{\bm{q}}}\rightarrow e^{i\psi}g_{{\bm{q}}}$. In contrast, the real space current pattern depends on the phase shift. We discuss other possible sLC patterns in Appendix D. The charge modulation due to the sLC is just $|\Delta n_{i}|\sim 5\times 10^{-4}$ for $g^{\rm max}=0.1$ since $|\Delta n_{i}|\propto(g^{\rm max})^{2}$. Thus, experimental detection of translational symmetry breaking by sLC order may be difficult. However, the cLC is induced by applying uniform magnetic field parallel to ${\bm{g}}_{i,j}$. In the present sLC state, under 10 T magnetic field, the induced cLC gives $\Delta H\sim\pm 0.1$ Oe when $m^{*}/m\sim 10$, which may be measurable by NMR or $\mu$SR study.

Finally, we discuss that the sLC fluctuations can contribute to the $d_{x^{2}-y^{2}}$-wave pairing mechanism. The sLC fluctuations connects the close hot spots at $\bm{P}$ and $\bm{P^{\prime}}=\bm{P}+{{\bm{q}}}$ in Fig. 2 (a). At both points, the $d_{x^{2}-y^{2}}$-wave gap function $\Delta_{{\bm{k}}}$ has the same sign. The pairing interaction mediated by the spin-channel sLC fluctuations between singlet pairs $(\bm{P},-\bm{P})$ and $(\bm{P^{\prime}},-\bm{P^{\prime}})$ is

where ${{\bm{k}}}=(\bm{P}+\bm{P^{\prime}})/2$ and ${{\bm{q}}}=\bm{P}-\bm{P^{\prime}}\sim{{\bm{Q}}}_{\rm a}$. $\chi_{\rm sLC}({{\bm{q}}})\ (>0)$ is the sLC susceptibility and $g_{{{\bm{q}}}}({{\bm{k}}})=-g_{{{\bm{q}}}}(-{{\bm{k}}})$ is its form factor. Thus, Eq. (8) give positive (=attractive) pairing interaction between close hot spots. The derivation of Eq. (8) is given in Appendix A and in SM F of Ref. Onari-AFB . This mechanism may be important for slightly over-doped cuprates with $T_{\rm sLC}\lesssim T_{\rm c}$.