Anisotropic quantum spin Hall effect, spin-orbital textures, and the Mott transition

Tianhan Liu, Benoît Douçot, and Karyn Le Hur

Phys. Rev. B 88, 245119 – Published 17 December 2013

Abstract

We investigate the interplay between topological effects and Mott physics in two dimensions on a graphenelike lattice, via a tight-binding model containing an anisotropic spin-orbit coupling on the next-nearest-neighbor links and the Hubbard interaction. We thoroughly analyze the resulting phases, namely, a topological band insulator phase or anisotropic quantum spin Hall phase until moderate-strength interactions and a Néel and a spiral phase at large interactions in the Mott regime, as well as the formation of a spin-orbital texture in the bulk at the Mott transition. The emergent magnetic orders at large interactions are analyzed through a spin-wave analysis and mathematical arguments. At weak interactions, by analogy with the Kane-Mele model, the system is described through a $Z_{2}$ topological invariant. In addition, we describe how the anisotropic spin-orbit coupling already produces an exotic spin texture at the edges. The physics at the Mott transition is described in terms of a U(1) slave-rotor theory. Taking into account gauge fluctuations around the mean-field saddle-point solution, we show how the spin texture now proliferates into the bulk above the Mott critical point. The latter emerges from the response of the spinons under the insertion of monopoles and this becomes more pronounced as the spin-orbit coupling becomes prevalent. We discuss implications of our predictions for thin films of the iridate compound Na $_{2}$ IrO $_{3}$ and also graphenelike systems.

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Received 26 August 2013

DOI:https://doi.org/10.1103/PhysRevB.88.245119

Authors & Affiliations

Tianhan Liu^1,2, Benoît Douçot¹, and Karyn Le Hur²

¹Laboratoire de Physique Théorique et des Hautes Energies (LPTHE), CNRS et Université Pierre et Marie Curie-Paris 6, 75252 Paris, France
²Centre de Physique Théorique (CPHT), École Polytechnique, CNRS, 91128 Palaiseau Cédex, France

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Vol. 88, Iss. 24 — 15 December 2013

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Figure 1
Our phase diagram. When $U < U_{c}$ (red line), the system is in the class of a $Z_{2}$ two-dimensional topological band insulator. The edge modes have a peculiar spin texture as a result of the anisotropic spin-orbit coupling. We thus refer to this phase as the anisotropic quantum spin Hall (AQSH) phase. At large interactions $U$ , we identify two magnetic phases, the Néel and the spiral phases. Above the Mott critical point $U_{c}$ , we show that the spin texture of the edges present below $U_{c}$ now progressively develops into the bulk when the spin-orbit-coupling strength is increased. This may eventually be reconciled with the spiral phase at large spin-orbit coupling. A nonordered Mott phase at intermediate interaction strengths and intermediate values of $t^{'} / t$ has been found in Ref. 55; this is beyond the scope of our work.Reuse & Permissions
Figure 2
Illustration of the tight-binding model on the honeycomb lattice with complex next-nearest-neighbor spin-orbit couplings entailing hopping of $i t^{'} σ_{x}$ on the $x$ red link, $i t^{'} σ_{y}$ on the $y$ green link, and $i t^{'} σ_{z}$ on the blue $z$ link, in which $σ_{w}$ , $w = x, y, z$ , is the Pauli matrix acting on the space of spins. The anisotropic spin-orbit coupling makes the spin no longer a conserved quantity in the system.Reuse & Permissions
Figure 3
Upper panel: The lower edge of the semi-infinite system with edges parallel to the $x$ -type links. The system consists of layers of one-dimensional chains coupled together, and the edge mode decays exponentially when moving into the bulk. Lower panel: The chiral edge transport corresponding to the boundary configuration. Two helical edge modes with opposite spin polarizations counterpropagate on the boundary of the system.Reuse & Permissions
Figure 4
In the weak-interaction regime, the anisotropic spin-orbit model lies in the phase of a topological band insulator on a cylinder, in which only the wave vector $k_{x}$ is a good quantum number. The two counterpropagating helical edge states protected by the $Z_{2}$ topological invariant of the system have spin polarizations which explicitly depend on the ratio $t^{'} / t$ . Here, we show the spin-polarization components of the edge state with a wave vector $k_{x}$ where $S_{x}$ is maximum, as a function of $t^{'} / t$ (see, for example, Fig. 7). $S_{x}$ prevails over $S_{y}$ and $S_{z}$ at large $t^{'} / t$ .Reuse & Permissions
Figure 5
Spin texture in the intermediate- $U$ regime, in the bulk, induced by the fluctuating gauge field within the U(1) slave-rotor theory [see Eq. (80)]: The spin polarization on site 1 in Fig. 2 as a function of $t^{'} / t$ . When $t^{'} / t ≪ 1$ , the subordinate spin polarization is of the same order as the dominant spin polarization $S_{x}, S_{y} \approx - 0.6 S_{z}$ (see Fig. 14). When $t^{'} / t > 1$ the subordinate spin polarization becomes (much) smaller compared to the dominant polarization: $S_{x}, S_{y} \approx - 0.2 S_{z}$ . The spin texture above the Mott quantum critical point seems to evolve very gradually. Site 1 is facing the $z$ -type links in the system and it acquires a dominant $z$ spin component. The spin texture on other different sites carries a symmetry which is a combination of a $2 π / 3$ rotation around the core of the fluctuating flux and a spin permutation, a symmetry inherent to this anisotropic model.Reuse & Permissions
Figure 6
The magnetic phase diagram for the tight-binding model with anisotropic spin-orbit coupling on the honeycomb lattice in the limit of infinite $U$ described by Eq. (12). The $J_{1} - J_{2}$ model is highly frustrated because of the hexagonal geometry and the anisotropy of the $J_{2}$ coupling. We identify the bipartite Néel phase at $J_{1} > J_{2}$ , the spiral phase with 24 sublattices at $J_{1} < J_{2}$ , and both phases are frustrated at either the classical or the quantum level.Reuse & Permissions
Figure 7
The edge states of the anisotropic spin-orbit-coupling model with zigzag boundary and $x$ links parallel to the boundary. (a) Spectrum of a system with anisotropic spin-orbit coupling on a cylinder at $t^{'} / t = 0.5$ obtained from numerical diagonalization of 70 layers of a one-dimensional system described by the Schrödinger equation (9). The nontrivial $Z_{2}$ topological invariant ensures helical edge states with opposite spin polarizations according to the Kramers theorem. The energy dispersion obtained analytically using the transfer matrix in Appendix fits the numerics well. (b) The different components of the spin polarization measured on the lower edge of the state with the lowest positive energy in the spectrum as a function of momentum obtained from diagonalization of the system. We observe that states with opposite Fermi velocities on both sides of $k_{x} = π / \sqrt{3}$ have opposite spin polarizations, thus implying helical spin transport on the edge, and the dominant spin component corresponds to the type of links parallel to the boundary. (c) The penetration length of the edge modes as a function of the conserved momentum $K_{x}$ .Reuse & Permissions
Figure 8
Spin polarization of the lowest positive-energy state for the exactly diagonalized Hamiltonian on a cylinder with $x$ links parallel to the boundary (see Fig. 3). $k_{x}$ refers to the wave vector along the boundary. Spin polarization at the edge for (a) $t^{'} = 0.2 t$ , (b) $t^{'} = 0.3 t$ , (c) $t^{'} = 0.5 t$ , and (d) $t^{'} = 1.0 t$ . The $x$ component becomes dominant when $t^{'} / t$ increases. The spin polarization at the momentum $k_{x}$ with the maximal dominant component is shown in Fig. 4 as a function of $t^{'} / t$ .Reuse & Permissions
Figure 9
Color topography of the vacuum energy as a function of $θ$ and $ϕ$ in the Néel order phase when $J_{1} > J_{2}$ , in which $θ$ and $ϕ$ indicate the Euler angles describing the quantization axis of the Néel order. The minimum of the vacuum energy is taken when the Néel order parameter coincides with the $x$ , $y$ , and $z$ directions for the quantization axis. The next-nearest-neighbor anisotropic coupling reduces the $SU (2)$ symmetry of the vacuum states for a conventional Néel order to a discrete symmetry of three possible order parameters of this frustrated Néel order.Reuse & Permissions
Figure 10
The global transformation brings the $J_{2}$ anisotropic magnetic model to an antiferromagnetic spin model on triangular lattices with four patterns: $□, ♦, ◯, △$ with black patterns on one sublattice and red patterns on the other. The nearest-neighbor $J_{1}$ Heisenberg coupling locks the angles between two copies of the spiral order and fixes the relative arrangement of the four patterns between the two sublattices as shown in the figure. The sites on which we studied the nearest-neighbor Heisenberg coupling, namely, the local fields ${\vec{h}}_{△}^{1}$ , ${\vec{h}}_{□}^{2}$ , and ${\vec{h}}_{◯}^{3}$ in Eqs. (26) and (27), are colored red with their number indicating the sublattice for the transformed $120^{\circ}$ Néel order. In green, we depict the 12 sublattices (sites) on each triangular sublattice with four patterns. We also represent the wave vectors associated with the spiral phase (in blue) and with the Néel phase (in black). The grey hexagon connects the Dirac points.Reuse & Permissions
Figure 11
Here, we represent the orientations of the 12 black sites in the green unit cell of the spiral phase.Reuse & Permissions
Figure 12
The anisotropic spin texture calculated in Eq. (80) developing into the bulk above the Mott critical point $U_{c}$ as a function of $t^{'} / t$ may be associated with the spin physics on the edge by invoking the $U (1)$ pump argument to a system on a cylinder by Laughlin (Ref. 101): the centered plaquette with flux inserted can be viewed as one edge and the infinity of the system as another. The different sites in Table are labeled in the figure. The spin physics of insertion of flux could be mapped to the edge spin transport on a cylinder under the insertion of flux and the emergent spin texture could be viewed as spin charge. When the gauge field fluctuations insert monopoles (flux in $2 + 1$ dimensions) into the system, the $U (1)$ spin pump induces a spin texture around the edges, namely, the core of the monopoles. The spin texture as a spin response summed over all momenta shared similar configurations with the spin transport in Sec. 1b: when $t^{'} / t ≪ 1$ the three components are comparable while for $t^{'} / t ≫ 1$ one dominant component of spin polarization will appear. The dominant spin polarization depends on the type of links intersected by the line connecting the measured site and the monopole core. It resembles the dependence of the dominant spin-polarization component on the types of links to which the boundary is parallel in the context of edge spin physics in the AQSH phase.Reuse & Permissions
Figure 13
The configuration of ${\vec{r}}_{I}$ , ${\vec{r}}_{J}$ , and ${\vec{R}}_{i}$ related to Eq. (75), in which ${\vec{r}}_{I}$ and ${\vec{r}}_{J}$ are vectors connecting the plaquette centers to its vertices and $| {\vec{r}}_{I} - {\vec{r}}_{J} |$ denotes the first-neighbor link; ${\vec{R}}_{i}$ gives the coordinates of the studied plaquette. We have to pay special attention to the coordinates in Eq. (75): ${\vec{r}}_{i} = {\vec{r}}_{I} + {\vec{R}}_{i}$ and ${\vec{r}}_{j} = {\vec{r}}_{J} + {\vec{R}}_{i}$ . The sum over the coordinates of the studied plaquette at ${\vec{R}}_{i}$ in Eq. (75) will induce the momentum conservation $\vec{k} - {\vec{k}}^{'} = \vec{q}$ of the spinon excitations under the monopole insertion. The vectors ${\vec{d}}_{I}$ and ${\vec{d}}_{J}$ are vectors connecting the plaquettes for the configuration of gauge fields on the honeycomb lattice; see Appendix .Reuse & Permissions
Figure 14
The ratio of $S_{x} / S_{z}$ on site 1 in Fig. 12 as a function of $t^{'} / t$ . This indicates that there are two spin textures when $t^{'} / t$ is varied: (1) The subordinate spin polarizations have opposite components to the dominant polarization $S_{x} = S_{y} = - 0.6 S_{z}$ when $t^{'}$ is small compared to $t$ ; and (2). $S_{z} ≫ (S_{x}, S_{y})$ when $t^{'} > t$ .Reuse & Permissions
Figure 15
Numerical study of spin-polarization magnitude as a function of layer number in the system of 70 layers of one-dimensional chains described by Eq. (A1) at $t^{'} = 0.5 t$ .Reuse & Permissions
Figure 16
The lattice gauge field configuration on the honeycomb lattice. On each triangular and each honeycomb plaquette, a lattice loop variable is defined. The gauge fields on the counterclockwise-oriented links are defined as loop variables on the left-hand side minus loop variables on the right-hand side of the link when traveling parallel to the link orientation. For example, $A_{a 2} = ϕ_{0} - ϕ_{a 1}$ . The loop variable construction satisfies automatically $\nabla \cdot A = 0$ , and $\nabla \times A = Φ_{m} δ_{{\vec{R}}_{o}, \vec{O}}$ is expressed by a Laplace equation in Eq. (B1), in which $Φ_{m}$ is the magnetic flux penetrating the center of the plaquette.Reuse & Permissions
Figure 17
The spin texture under a monopole-antimonople pair (two monopoles) on two adjacent plaquettes: (a) The spin texture on site 1 when the monopole and antimonopole are respectively inserted on plaquettes $O$ and $b$ 1 (Fig. 16). (b) The spin texture on site 3 when the monopole and antimonopole are respectively inserted on plaquettes $O$ and $c$ 1. (c) The spin texture on site 1 when two monopoles are inserted on plaquettes $O$ and $b$ 1. (d) The spin texture on site 3 when two monopoles are inserted on plaquettes $O$ and $c$ 1.Reuse & Permissions
Figure 18
The spin texture on site 1 when the monopole-antimonopole pair (left panel) or the two monopoles (right panel) are respectively inserted on plaquettes $O$ and $b$ 2.Reuse & Permissions

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