Non-Abelian topological spin liquids from arrays of quantum wires or spin chains

We construct two-dimensional non-Abelian topologically ordered states by strongly coupling arrays of one-dimensional quantum wires via interactions. In our scheme, all charge degrees of freedom are gapped, so the construction can use either quantum wires or quantum spin chains as building blocks, with the same end result. The construction gaps the degrees of freedom in the bulk, while leaving decoupled states at the edges that are described by conformal field theories (CFT) in $(1+1)$-dimensional space and time. We consider both the cases where time-reversal symmetry (TRS) is present or absent. When TRS is absent, the edge states are chiral and stable. We prescribe, in particular, how to arrive at all the edge states described by the unitary CFT minimal models with central charges $c<1$. These non-Abelian spin liquid states have vanishing quantum Hall conductivities, but nonzero thermal ones. When TRS is present, we describe scenarios where the bulk state can be a non-Abelian, nonchiral, and gapped quantum spin liquid, or a gapless one. In the former case, we find that the edge states are also gapped. The paper provides a brief review of non-Abelian bosonization and affine current algebras, with the purpose of being self-contained. To illustrate the methods in a warm-up exercise, we recover the tenfold way classification of two-dimensional noninteracting topological insulators using the Majorana representation that naturally arises within non-Abelian bosonization. Within this scheme, the classification reduces to counting the number of null singular values of a mass matrix, with gapless edge modes present when left and right null eigenvectors exist.

Figure 1

Schematic representation of the Hamiltonian for a state with non-Abelian topological order arising from interactions between electronic quantum wires. Each $\otimes$ and $\odot$ represents the right- and left-moving (spinful) electrons of a quantum wire coming out of the plane of the page. Each gray area represents an interaction that gaps out charge fluctuations on the bundle of wires that it encloses. Each line represents a $\mathrm{SU}\left(2\right)$ symmetric Heisenberg interaction between the spin densities of left and right movers that it connects. Lines of the same color represent interaction terms of the same strengths.

Figure 2

Pictorial representation for the selected backscattering in the symmetry class D. Each yellow box represents a quantum wire composed of two Majorana degrees of freedom. The wires are enumerated by $I=1,\cdots ,N$ in ascending order from left to right. For any $I$, the Majorana modes are denoted by ${\chi }_{\mathrm{R},I}^{}$ and ${\chi }_{\mathrm{L},I}^{}$ reading from left to right, respectively.

Figure 3

Pictorial representation for the selected backscattering in the symmetry class DIII. Each yellow box represents a quantum wire composed of four-Majorana degrees of freedom. The wires are enumerated by $I=1,\cdots ,N$ in ascending order from left to right. For any $I$, the Majorana modes are denoted by ${\chi }_{\mathrm{R},+,I}^{}, {\chi }_{\mathrm{L},+,I}^{}, {\chi }_{\mathrm{R},-,I}^{}$, and ${\chi }_{\mathrm{L},-,I}^{}$ reading from left to right, respectively.

Figure 4

Chain of $N$ wires grouped into $n$ bundles of $k+k^{\prime }$ wires with $N=20, n=4, k=4$, and $k^{\prime }=1$. The chain of wires is labeled by $I=1,\cdots ,N$. The bundles of $k+k^{\prime }$ wires are labeled by the teletype font $\mathtt{m}=\mathtt{1},\cdots ,\mathtt{n}$.

Figure 5

(a) A chain of wires is partitioned into bundles. A bundle is depicted by a domino. The pattern in the domino corresponds to a right- and left-moving critical sector with the affine Lie algebra ${\widehat{g}}_{k,k^{\prime }}^{}:=\widehat{su}{\left(2\right)}_k^{}\oplus \widehat{su}{\left(2\right)}_{k^{\prime }}^{}$ represented by a thick vertical like supporting an up arrow for right movers and down arrow for left movers. Its diagonal subalgebra ${\widehat{h}}_{k,k^{\prime }}^{}:=\widehat{su}{\left(2\right)}_{k+k^{\prime }}^{}$ is represented by the forking into a blue solid line. The coset ${\widehat{g}}_{k,k^{\prime }}^{}/{\widehat{h}}_{k,k^{\prime }}^{}$ is represented by the forking into a red dashed line. (b) An arc inside each domino depicts a current-current interaction between the generators of ${\widehat{h}}_{k,k^{\prime }}^{}$, i.e., the second line on the right-hand side of Eq. (3.33d). These arcs gap all the critical modes generated by ${\widehat{h}}_{k,k^{\prime }}^{}:=\widehat{su}{\left(2\right)}_{k+k^{\prime }}^{}$ within a bundle. (c) An arc between two consecutive dominoes depicts the current-current interactions between the generators of ${\widehat{g}}_{k,k^{\prime }}^{}$, i.e., the first line on the right-hand side of Eq. (3.33d). The arrows on these arcs indicate that these interactions break time-reversal symmetry. These arcs gap all remaining critical modes except for the modes generated by the right-moving ${\widehat{g}}_{k,k^{\prime }}^{}/{\widehat{h}}_{k,k^{\prime }}^{}$ on bundle $\mathtt{m}=\mathtt{1}$ and the modes generated by the left-moving ${\widehat{g}}_{k,k^{\prime }}^{}/{\widehat{h}}_{k,k^{\prime }}^{}$ on bundle $\mathtt{m}=\mathtt{n}$. (d) Reversal of time is represented by reversing all arrows.

Figure 6

One possible phase diagram with the interaction (3.44). The time-reversal symmetric point is parametrized by $\kappa =0$. The vertical axis is the many-body gap between the ground state and all excited states when periodic boundary conditions are imposed. The continuous line represents the scenario for an exotic spin-liquid quantum critical point. The dashed line represents the scenario for a first-order quantum phase transition. The signs $-$ and $+$ distinguish the two ground states that evolve adiabatically as $\kappa \ne 0$ changes and cross precisely at $\kappa =0$.

Figure 7

For the case when the phases transitions in the range $-1\le \kappa \le +1$ at strong values of the couplings ${\lambda }_{\mathtt{m}}^{\mathcal{A}}>0$ and $v_{\mathtt{m}}^{\mathcal{B}}>0$ in Eq. (3.44) are continuous, the critical point at $\kappa =0$ is either unstable or stable depending on whether the number of critical points for $-1<\kappa <0$ is even as in (a) or odd as in (b), respectively.

Figure 8

Chain of $N$ wires, each wire supporting four right- and four left-moving flavors, that are coupled by current-current interactions in a way that is explicitly symmetric under reversal of time. A bundle of $8\times (k+k^{\prime })$ right- or left-moving electronic degrees of freedom is represented by a domino. There are $n=N/(k+k^{\prime })$ dominoes. The currents denoted by $\mathcal{J}$ generate the affine Lie algebra $\widehat{su}{\left(2\right)}_k^{}\oplus \widehat{su}{\left(2\right)}_{k^{\prime }}^{}$. The currents denoted by $\mathcal{K}$ generate the affine Lie algebra $\widehat{su}{\left(2\right)}_{k+k^{\prime }}^{}$. The currents denoted by $\tilde{\mathcal{J}}$ generate the affine Lie algebra $\widehat{\widetilde{su}}{\left(2\right)}_k^{}\oplus \widehat{\widetilde{su}}{\left(2\right)}_{k^{\prime }}^{}$. The currents denoted by $\tilde{\mathcal{K}}$ generate the affine Lie algebra $\widehat{\widetilde{su}}{\left(2\right)}_{k+k^{\prime }}^{}$.

Figure 9

The partial gapping of a single domino with local current-current interactions can be done in two ways (a) and (b). The complete gapping of a single domino with local current-current interactions is achieved in (c).

Figure 10

Unfolding a chain of dominoes coupled by time-reversal symmetric interactions.

Figure 11

The partial gapping of a bundle made of $2(k+k^{\prime })$ quantum wires with local current-current interactions can be done in two ways (a) and (b). The complete gapping of a bundle made of $2(k+k^{\prime })$ quantum wires with local current-current interactions is achieved in (c). The dashed vertical line is a mirror axis of symmetry.

Figure 12

A single “hole” in a one-dimensional array of $N$ quantum wires. Electron tunneling between consecutive wires is prohibited across the diameter of the hole.

Figure 13

Pictorial representation for the selected backscattering in the symmetry class C. Each yellow box represents a quantum wire composed of four-Majorana degrees of freedom. The wires are enumerated by $I=1,\cdots ,N$ in ascending order from left to right. For any $I$, the Majorana modes are denoted by ${\chi }_{\mathrm{R},+,I}^{}, {\chi }_{\mathrm{L},+,I}^{}, {\chi }_{\mathrm{R},-,I}^{}$, and ${\chi }_{\mathrm{L},-,I}^{}$ reading from left to right, respectively.

Figure 14

Pictorial representation for the selected backscattering in the symmetry class A. Each yellow box represents a quantum wire composed of four-Majorana degrees of freedom. The wires are enumerated by $I=1,\cdots ,N$ in ascending order from left to right. For any $I$, the Majorana modes are denoted by ${\chi }_{\mathrm{R},1,I}^{}, {\chi }_{\mathrm{L},1,I}^{}, {\chi }_{\mathrm{R},2,I}^{}$, and ${\chi }_{\mathrm{L},2,I}^{}$ reading from left to right, respectively.

Figure 15

Pictorial representation for the selected backscattering in the symmetry class AII. Each yellow box represents a quantum wire composed of four-Majorana degrees of freedom. The wires are enumerated by $I=1,\cdots ,N$ in ascending order from left to right. For any $I$, the Majorana modes are denoted by ${\chi }_{\mathrm{R},+,1,I}^{}, {\chi }_{\mathrm{L},+,1,I}^{}, {\chi }_{\mathrm{R},-,1,I}^{}, {\chi }_{\mathrm{L},-,1,I}^{}, {\chi }_{\mathrm{R},+,2,I}^{}, {\chi }_{\mathrm{L},+,2,I}^{}, {\chi }_{\mathrm{R},-,2,I}^{}$, and ${\chi }_{\mathrm{L},-,2,I}^{}$ reading from left to right, respectively.