Weak coupling theory of topological Hall effect

Topological Hall effect (THE) caused by a noncoplanar spin texture characterized by a scalar spin chirality is often described by the Berry phase, or the associated effective magnetic field. This picture is appropriate when the coupling $M$ of conduction electrons to the spin texture is strong (strong coupling regime) and the adiabatic condition is satisfied. However, in the weak coupling regime, where the coupling $M$ is smaller than the electrons' scattering rate, the adiabatic condition is not satisfied and the Berry phase picture does not hold. In such a regime, the relation of the effective magnetic field to the spin texture can be “nonlocal,” in contrast to the “local” relation in the strong coupling case. Focusing on continuous but general spin textures, we investigate the THE in various characteristic regions in the weak coupling regime, namely, (1) diffusive and local, (2) diffusive and nonlocal, and (3) ballistic. In the presence of spin relaxation, there arise two more regions in the “weakest coupling” regime: ($1^{\prime }$) diffusive and local and ($2^{\prime }$) diffusive and nonlocal. We derived the analytic expression of the topological Hall conductivity (THC) for each region, and found that the condition for the locality of the effective field is governed by transverse spin diffusion of electrons. In region 1, where the spin relaxation is negligible and the effective field is local, the THC is found to be proportional to $M$, instead of $M^3$ of the weakest coupling regime. In the diffusive, nonlocal regions (2 and $2^{\prime }$), the effective field is given by a spin chirality formed by “effective spins” that the electrons see during their diffusive motion. Applying the results to a skyrmion lattice, we found that the THC decreases as the skyrmion density is increased in region $2^{\prime }$, reflecting the nonlocality of the effective field, and shows a maximum at the boundary to the “local” region. For general spin textures, a scaling-like argument is given to analyze the THC in the nonlocal region. These analytic results are supplemented with numerical calculations for an isolated single skyrmion as well as for periodic textures of skyrmions and merons.

Figure 1

Diagrammatic expression of vertex corrections and self-energy due to random impurities. The blue solid line is the Green function of electrons, and the thin broken line with a cross represents impurity scattering. (a) Diffusion propagator of transverse spin density, ${\mathrm{\Pi }}_{\overline{\sigma }\sigma }(\mathit{q},\omega )$ (red broken line), which we call $M-\mathrm{VC}$. (b) Diffusion propagator of charge density, $\mathrm{\Pi }(\mathit{q},\omega )$ (green double line), which we call $q-\mathrm{VC}$. The upper (lower) lines represent retarded (advanced) Green functions [Eq. (9)]. (c) Vertex correction to transverse spin, which we call $Z-\mathrm{VC}$. The cross with a circle is the (bare) transverse spin vertex. (d) Self-energy in the Born approximation.

Figure 2

Characteristic regions in the weak coupling regime $M\tau <1$ in the plane of $M$ and $q$, where $2M$ is the exchange splitting, $\tau$ is the electron lifetime, $q$ is the wave vector of spin texture, and $\ell =v_{\mathrm{F}}\tau$ is the mean-free path. (a) In the absence of spin relaxation, the weak coupling regime is divided into three regions; (1) Diffusive and local-effective-field region, (2) Diffusive and nonlocal-effective-field region, and (3) Ballistic region. Regions 1 and 2 are separated by the parabola $M\tau ={\left(q\ell \right)}^2$, and regions 2 and 3 by the line $q\ell =1$. (b) In the presence of spin relaxation, the parabola moves to $M\tau ={\left(q\ell \right)}^2+\tau /{\tau }_{\mathrm{s}}$, and there appear two more regions; ($1^{\prime }$) local and ($2^{\prime }$) nonlocal.

Figure 3

Feynman diagrams for ${\sigma }_{xy}$ in the $u$-perturbation method. The blue thick cross (${\tilde{\mathit{u}}}_{\mathit{q}}$) represents the “$u$ vertex,” which include the $Z-\mathrm{VC}$, namely, ${\tilde{\mathit{u}}}_{\mathit{q}}={\mathit{u}}_{\mathit{q}}(1+i\zeta )$. The blue cross with a circle (${\mathit{u}}_{\mathit{q}}$) is the bare $u$ vertex given by the second term of Eq. (24). The red (blue) solid line represents Green functions of electrons with spin projection $\overline{\sigma }$ ($\sigma$). Of the four groups of the diagrams, in the diffusive regime (regions 1, $1^{\prime }$, 2, and $2^{\prime }$), ${\sigma }_{xy}^{\alpha }$ and ${\sigma }_{xy}^{\beta }$ are irrelevant and both ${\sigma }_{xy}^{\gamma }$ and ${\sigma }_{xy}^{\delta }$ are important.

Figure 4

Feynman diagrams for ${\sigma }_{xy}$ in the $M$-perturbation treatment. The thick cross represents the $s-d$ coupling, $H_{sd}$, to $\mathit{n}$, including the effect of $Z-\mathrm{VC}$. The solid lines are electrons' Green functions with $M=0$. The diagrams are classified into four groups; ${\sigma }_{xy}^a$ dominates in region 3, and ${\sigma }_{xy}^c$ and ${\sigma }_{xy}^d$ dominate in regions $1^{\prime }$ and $2^{\prime }$.

Figure 5

Schematic behavior of THC in a skyrmion lattice as a function of the wave number $q$ of the magnetic texture (helix). The THC is larger for lower skyrmion density in region $2^{\prime }$, a behavior opposite to the strong coupling regime, and takes a maximum at the boundary of regions $1^{\prime }$ and $2^{\prime }$.

Figure 6

Physical picture of region 1, where the effective field is local (upper), and region 2, where the effective field is nonlocal (lower). The thick green arrows represent the magnetization, and the (blue) tiny arrow with a sphere represents the spin of the conduction electron. During the single period of precession ${\tau }_{\mathrm{ex}}=\hslash /2\left|M\right|$, electrons diffuse over the distance $\lambda =\sqrt{D{\tau }_{\mathrm{ex}}}$ shorter (longer) than the characteristic texture size $q^{-1}$ for region 1 (for region 2), and see a small portion of (a wider region than) the texture.

Figure 7

Radial profiles of bare ($\mathit{n}$) and effective ($\tilde{\mathit{n}}$) spins [upper eight panels, (a)–(h)] and effective fields [bottom two panels, (i) and (j)] of a single skyrmion. The left five panels are for $q\lambda =0.2,0.4,0.6,1$ and 2 with fixed $q{\ell }_{\mathrm{s}}=10$, and the right panels are for $q{\ell }_{\mathrm{s}}=0.2,0.4,0.6,1$ and 2 with fixed $q\lambda =10$. [(a) and (b)] $n^z [cos\theta (r)$, dashed line] and $\mathrm{Re}{\tilde{n}}^z$ ($\mathrm{Re}C\left(r\right)$, solid lines). [(c) and (d)] In-plane component of $\mathit{n} [sin\theta (r)$, dashed line] and $\mathrm{Re}\tilde{\mathit{n}} \left[\mathrm{Re}S\right(r)$, solid lines]. [(e) and (f)] $\mathrm{Im}C\left(r\right)$, [(g) and (h)] $\mathrm{Im}S\left(r\right)$. [(i) and (j)] Local chirality density $[{\chi }_0\left(r\right)$, dashed line] and the effective chirality density $\left[\chi \right(r)$, solid lines].

Figure 8

(a) Total effective chirality ${\chi }_{\mathrm{tot}}$ (in unit of $4\pi$) for a single skyrmion. (b) Scaling function $\varphi \left(q\right|\zeta |;\eta )$ for finite-density ($\propto q^2$) skyrmions. These are plotted as functions of the scaling variable $q\left|\zeta \right|$ for several choices of $\eta =\lambda /{\ell }_{\mathrm{s}}$. The curves scan regions 1 and 2 for $\eta =0.1,0.5$, and regions $1^{\prime }$ and $2^{\prime }$ for $\eta =2,10$.

Figure 9

[(a)–(c)] Spin and chirality distribution of $\mathit{n}\left(\phi \right)$ [Eq. (105)] and [(d)–(f)] $\mathit{M}\left(\phi \right)$ [Eq. (106)] with $\phi =0$ (a,d), $\pi /4$ (b,e), $\pi /2-\delta$ (c), and $\pi /2$ (f), where $\delta =\pi /180$. (Left) 2D plots of spin structure. The $x,y$ components of $\mathit{n}$ or $\mathit{M}$ are expressed by the green arrows, and the $z$ component ($n^z$ or $M^z$) by the color (red to blue). The dashed diamond in (a) is the SkL unit cell, whose side length is $4\pi /\left(\sqrt{3}q\right)$. (Middle) [(a)–(c)] 2D plots of the local chirality, ${\chi }_0\left(\mathit{r}\right)$ [Eq. (101)] and [(d)–(f)] ${\chi }_M\left(\mathit{r}\right)$. The inset of (c) shows a high-resolution plot. (Right) 1D plots of local chirality (LChi, dashed lines) and effective chirality (EChi, solid lines) along the path AB indicated in the middle panel of (a). Parameters used are $q\lambda =0.5$ for the 2D plots and $q\lambda =0.5$ (region 1), 1 (boundary), and 1.5 (region 2) for the 1D plots, with $\eta =\lambda /{\ell }_{\mathrm{s}}=1/50$ in common. In the 2D plots, the real-space ($x$ and $y$) coordinates are shown in unit of $a_0=2\pi /qL=L_M/L$. In the 1D plots, the LChi for $q\lambda =1.5$ (not shown) can be obtained from that for $q\lambda =1$ by multiplying ${(3/2)}^2=2.25$, since it scales as $q^2$. The 2D (1D) plots are calculated with $L=24$ ($L=48$), except for the inset of (c), which used $L=72$.

Figure 10

The bare ($\mathit{n}$) and effective spin ($\tilde{\mathit{n}}$) for n-trq at $\phi =0$, plotted along the path CDE indicated in Fig. 9. (a) $\mathrm{Re}{\tilde{\mathit{n}}}_{\perp }^{(1,2)}$, (b) $\mathrm{Re}{\tilde{n}}_z^{(1,2)}$, (c) $\mathrm{Re}{\tilde{\mathit{n}}}_{\perp }^{(1^{\prime },2^{\prime })}$, (d) $\mathrm{Re}{\tilde{n}}_z^{(1^{\prime },2^{\prime })}$, (e) $\mathrm{Im}{\tilde{\mathit{n}}}_{\perp }^{(1,2)}$, and (f) $\mathrm{Im}{\tilde{n}}_z^{(1,2)}$. We do not show $\mathrm{Im}{\tilde{\mathit{n}}}^{(1^{\prime },2^{\prime })}$ since they are negligibly small. In (a), (c) and (e), the three solid lines show $n^x$ along CD (where $n^y=0$), and they show $n^y$ along DE (where $n^x=0$). The used parameters are as follows: [(a)–(d)] $q\lambda =0.5$ (region 1), 1 (boundary), and 1.5 (region 2) with $\eta =\lambda /{\ell }_{\mathrm{s}}=1/50$. (e,f) $q{\ell }_{\mathrm{s}}=0.5$ (region $1^{\prime }$), 1 (boundary), and 1.5 (region $2^{\prime }$) with $\eta =50$. Calculated with $L=48$.

Figure 11

Reduced THC, ${\tilde{\sigma }}_{xy}^A$, for n-trq $\mathit{n}$ at $\phi =0$ and for $A=(1,2), (1^{\prime },2^{\prime })$, and $(3*)$. The former two are obtained from $8{\varphi }^{(1,2,1^{\prime },2^{\prime })}\left(q\right|\zeta |;\eta )$ [Eq. (93)] by setting $\eta =1/50$ and 50, respectively. The three curves for ${\tilde{\sigma }}_{xy}^{(3*)}={(\ell /|\zeta \left|\right)}^2{\varphi }^{(3*)}\left(q\ell \right)$ [Eq. (94)] with $\left|\zeta \right|/\ell =1,2$, and 3 are plotted against $q\left|\zeta \right|$, namely, against the same $q$ values as ${\tilde{\sigma }}_{xy}^{(1,2)}$ and ${\tilde{\sigma }}_{xy}^{(1^{\prime },2^{\prime })}$. Calculated with $L=24$.

Figure 12

Reduced THC, ${\tilde{\sigma }}_{xy}={\tilde{\sigma }}_{xy}^{(1,2,1^{\prime },2^{\prime })}+{\tilde{\sigma }}_{xy}^{(3*)}$, for n-trq $\mathit{n}\left(\phi \right)$ (left column) and un-trq $\mathit{M}\left(\phi \right)$ (right column) for $\phi =n\pi /12$ with $n=0,1,2,\cdots ,5$. The upper two panels are for $\eta =1/50$, and the lower two for $\eta =50$. Calculated with $L=24$.

Figure 13

Feynman diagrams for THC in the $u$-perturbation calculation.

Figure 14

Feynman diagrams for THC in the $M$-perturbation calculation.

Figure 15

Real-space and $k$-space diagrams for numerical calculations. (a) Real-space skyrmion lattice (SkL). The yellow circle represents a skyrmion, and the hexagon with a solid outline is the unit cell of SkL, whose size is specified by $L_M$ or $q=2\pi /L_M$. The blue dots inside the unit cell are sampling points that form a lattice (“sampling-point lattice,” SpL), which are specified by the number of division (per direction), $L$, and are generated by the primitive vectors, ${\mathit{a}}_L\equiv (L_M/L)\mathit{a}$ and ${\mathit{b}}_L\equiv (L_M/L)\mathit{b}$. The case of $L=4$ is shown. (b) Reciprocal space. The Brillouin zone dual to the SkL is denoted by ${\mathrm{BZ}}_1$, and the one dual to the SpL with division $L$ by ${\mathrm{BZ}}_L$. The reciprocal lattice vectors for ${\mathrm{BZ}}_L$ are given by ${\mathit{q}}_{L,1}\equiv L{\mathit{q}}_1$ and ${\mathit{q}}_{L,2}\equiv L{\mathit{q}}_2$.