Symmetry-protected topological interfaces and entanglement sequences

Luiz H. Santos, Jennifer Cano, Michael Mulligan, and Taylor L. Hughes

Phys. Rev. B 98, 075131 – Published 17 August 2018

Abstract

Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found by J. Cano, T. L. Hughes, and M. Mulligan [Phys. Rev. B 92, 075104 (2015)]. The topological entanglement entropy decomposes as $γ = γ_{a} + γ_{s}$ , where $γ_{a} > 0$ only depends on universal topological properties of the 2D state, while a correction $γ_{s} > 0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of $γ_{s}$ associated with a given interface—which is named the “boundary topological entanglement sequence”—and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.

Received 29 March 2018
Revised 19 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.075131

Physics Subject Headings (PhySH)

Quantum entanglement Symmetry protected topological states Topological phases of matter

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Luiz H. Santos^1,*, Jennifer Cano², Michael Mulligan³, and Taylor L. Hughes¹

¹Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
²Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
³Department of Physics and Astronomy, University of California, Riverside, California 92511, USA

^*Present address: Department of Physics, Emory University, Atlanta, Georgia 30322, USA.

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Vol. 98, Iss. 7 — 15 August 2018

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Images

Figure 1
The interface between the distinct topological phases associated with the K matrices, $K$ and $K_{G} = G K G^{T}$ , runs along the vertical axis. The null vectors are parametrized by $(M_{L}, M_{R}) = (W G, I_{r \times r})$ . This indicates that the local interactions are invariant under the discrete symmetry $G_{L} \times Z_{1}$ , where $| G_{L} | = | \det (G) |,$ and $Z_{1}$ represents the trivial discrete group for the right-hand side of the interface.
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Figure 2
(a) In the limit where the intermediate slab thickness is large, interfaces 1 and 2 can be treated as being decoupled from each other. At each interface, the null vectors are parametrized by $(W_{1} G, I_{r \times r})$ and $(I_{r \times r}, W_{2} G)$ , implying that the local interactions on each interface are invariant under the symmetry $G$ , where $| G | = | \det (G) |$ . (b) As the thickness of the intermediate slab goes to zero, one recovers the homogeneous interface, with null vectors parametrized by $(W_{1} G, W_{2} G)$ , signaling that the local gapping interactions are invariant under the discrete $G \times G$ symmetry. The Smith normal form of the matrix $G$ specifies the discrete symmetry, and the minors of $(W_{1} G, W_{2} G)$ determine whether these interactions are primitive.
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Figure 3
Zero-dimensional domain walls (black dots) separating 1D $Z_{n} \times Z_{n}$ SPT states (red) formed by the interactions in Eq. (43) from the trivial SPT states (blue) formed by the interactions in Eq. (49). Each domain wall supports a pair of parafermions defined in Eqs. (53) and (54).
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Figure 4
An array of 2D topological phases associated with the matrices $K$ (green) and $G K G^{T}$ (red) as defined in Eq. (80). Each bulk phase supports stable edge states represented by thick green and red lines. Each interface running along the vertical direction is gapped by the interaction in Eq. (82), which terminates at a point of transition (black dot) between the stable edge states (running along the horizontal direction) supporting distinct sets of low-energy quasiparticle excitations.
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Figure 5
Points $ℓ \in Z^{2}$ on the integer lattice parametrize local particles $ψ_{μ, ℓ} = exp (i ℓ_{μ}^{T} K_{μ} ϕ_{μ})$ , $μ = L, R$ , of the $ν = 2$ state described by $K_{ν = 2} = I_{2 \times 2}$ . When the interface is gapped by $Z_{5} \times Z_{5}$ interactions, we identify, using Eq. (97) and Eq. (106), the blue vectors $T_{1} = (5, 0)$ and $T_{2} = (- 2, 1)$ as the generators of translations connecting local operators carrying the same $Z_{5}$ charge, with the corresponding unit cell marked by the blue parallelogram. This parallelogram has area $| T_{1} \times T_{2} | = 5$ , corresponding to five distinguishable local particles carrying different $Z_{5}$ charges. The black vectors $m_{1 L} = (1, 2)$ and $m_{2 L} = (2, - 1)$ are defined by the columns of $M_{L} = (\begin{matrix} 1 & 2 \\ 2 & - 1 \end{matrix})$ and the red vectors $m_{1 R} = (1, - 2)$ and $m_{2 R} = (2, 1)$ are defined by the columns of $M_{R} = σ_{z} M_{L} = (\begin{matrix} 1 & 2 \\ - 2 & 1 \end{matrix})$ . The minors of the $2 \times 4$ matrix $(M_{L} M_{R})$ , ${- 5, - 4, - 3, - 3, 4, 5}$ , are given by the determinant of each of the six $2 \times 2$ submatrices, which is interpreted as the (oriented) area of the parallelogram spanned by pairs of distinguished vectors $m_{1 L}, m_{2 L}, m_{1 R}$ , and $m_{2 R}$ . The primitive condition of the gapping interactions reflects the condition that the greatest common divisor of the set of minors is equal to 1. Importantly, the relation between $M_{R} = σ_{z} M_{L}$ implies that $m_{1 R}$ and $m_{2 R}$ are obtained from $m_{1 L}$ and $m_{2 L}$ upon reflection on the horizontal axis. This change in orientation is crucially tied to the primitive condition: had we chosen the case $M_{L} = \pm M_{R}$ , which is a solution of the null equation, the g.c.d. of the minors would have been $| \det (M_{L}) | = | \det (M_{R}) | = 5 > 1$ .
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Figure 6
Points $ℓ \in Z^{2}$ on the integer lattice parametrize local particles $ψ_{μ, ℓ} = exp (i ℓ_{μ}^{T} K_{μ} ϕ_{μ})$ , $μ = L, R$ , of the bosonic state described by $K = σ_{x}$ . When the interface is gapped by $Z_{3} \times Z_{3}$ interactions, we identify, using Eq. (44) and Eq. (106), the blue vectors $T_{1} = (0, 3)$ and $T_{2} = (1, 0)$ as the generators of translations connecting local operators carrying the same $Z_{3}$ charge, with the corresponding unit cell marked by the blue parallelogram. This parallelogram has area $| T_{1} \times T_{2} | = 3$ , corresponding to three distinguishable local particles carrying different $Z_{3}$ charges. The black vectors $m_{1 L} = (1, 0)$ and $m_{2 L} = (0, 3)$ are defined by the columns of $M_{L} = (\begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix})$ and the red vectors $m_{1 R} = (0, 1)$ and $m_{2 R} = (3, 0)$ are defined by the columns of $M_{R} = σ_{x} M_{L} = (\begin{matrix} 0 & 3 \\ 1 & 0 \end{matrix})$ . The minors of the $2 \times 4$ matrix $(M_{L} M_{R})$ , ${3, 1, 0, 0, - 9, - 3}$ , are given by the determinant of each of the six $2 \times 2$ submatrices, which is interpreted as the (oriented) area of the parallelogram spanned by pairs of distinguished vectors $m_{1 L}, m_{2 L}, m_{1 R}$ , and $m_{2 R}$ . The primitive condition of the gapping interactions reflects the condition that the greatest common divisor of the set of minors is equal to 1. Importantly, the relation between $M_{R} = σ_{x} M_{L}$ implies that $m_{1 R}$ and $m_{2 R}$ are obtained from $m_{1 L}$ and $m_{2 L}$ upon a 90 degree rotation. This change in orientation is crucially tied to the primitive condition: had we chosen the case $M_{L} = \pm M_{R}$ , which is a solution of the null equation, the g.c.d. of the minors would have been $| \det (M_{L}) | = | \det (M_{R}) | = 3 > 1$ .
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