Short-range entangled bosonic states with chiral edge modes and $T$ duality of heterotic strings

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Short-range entangled bosonic states with chiral edge modes and $T$ duality of heterotic strings

Eugeniu Plamadeala, Michael Mulligan, and Chetan Nayak

Phys. Rev. B 88, 045131 – Published 26 July 2013

Abstract

We consider states of bosons in two dimensions that do not support anyons in the bulk, but nevertheless have stable chiral edge modes that are protected even without any symmetry. Such states must have edge modes with central charge $c = 8 k$ for integer $k$ . While there is a single such state with $c = 8$ , there are, naively, two such states with $c = 16$ , corresponding to the two distinct even unimodular lattices in 16 dimensions. However, we show that these two phases are the same in the bulk, which is a consequence of the uniqueness of signature $(8 k + n, n)$ even unimodular lattices. The bulk phases are stably equivalent, in a sense that we make precise. However, there are two different phases of the edge corresponding to these two lattices, thereby realizing a novel form of the bulk-edge correspondence. Two distinct fully chiral edge phases are associated with the same bulk phase, which is consistent with the uniqueness of the bulk since the transition between them, which is generically first order, can occur purely at the edge. Our construction is closely related to $T$ duality of toroidally compactified heterotic strings. We discuss generalizations of these results.

Received 30 April 2013

DOI:https://doi.org/10.1103/PhysRevB.88.045131

Authors & Affiliations

Eugeniu Plamadeala¹, Michael Mulligan², and Chetan Nayak^1,2

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, California 93106-6105, USA

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Vol. 88, Iss. 4 — 15 July 2013

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Images

Figure 1
The scaling dimensions of $u_{1, 2}$ (densely dashed and dotted) and ${\tilde{u}}_{1, 2}$ (thick and dashed), plotted as a function of $s$ at $r = 1$ . The $E_{8} \times E_{8}$ phase lives roughly within $0 \leq s < 0.625$ and the $spin (32) / Z_{2}$ phase between $0.625 < s \leq 1$ .Reuse & Permissions
Figure 2
The scaling dimensions of $u_{1}$ (densely dashed) and ${\tilde{u}}_{2}$ (dashed), plotted as a function of $s$ at at $r = 0.2$ . The scaling dimensions of $u_{2}$ and ${\tilde{u}}_{1}$ lie outside the range of the plot and are not displayed. The system is not fully chiral phase between approximately $s = 0.5$ and $0.625$ .Reuse & Permissions
Figure 3
The scaling dimensions of $u_{1, 2}$ (densely dashed and dotted) and ${\tilde{u}}_{1}$ (thick), plotted as a function of $s$ at at $r = 3$ . The scaling dimension of ${\tilde{u}}_{2}$ lies outside the range of the plot and is not displayed. The system is in the not fully chiral phase between approximately $s = 0.425$ and $0.625$ .Reuse & Permissions
Figure 4
Phase diagram of our edge theory as a function of $s$ and $r$ for the theory $S_{4} (s)$ in which the only nonzero perturbations are $u_{1, 2}$ and ${\tilde{u}}_{1, 2}$ . The light region is in the $E_{8} \times E_{8}$ phase. The darkest region is in the $spin (32) / Z_{2}$ phase. The system is not fully chiral in the intermediately shaded region. The dashed phase boundary line indicates a KT transition. The solid lines denote regions where there are two equally relevant couplings; the phase is determined by their ratio.Reuse & Permissions
Figure 5
The small- $r$ region of the phase diagram of our edge theory as a function of $s$ and $r$ for the theory with nonzero $u_{1, 2}$ , ${\tilde{u}}_{1, 2}$ ; $u_{17, \frac{p}{q}}$ , $u_{17, \frac{p}{q}}$ for all $p, q \leq 57$ ; and several $\cos (p_{a} X^{a})$ operators with $p_{a}$ nearly aligned with the direction of the boost $O_{G} (s)$ , as described in the text. The light region is in the $E_{8} \times E_{8}$ phase. The darker region is in the $spin (32) / Z_{2}$ phase. All phase boundary lines denote regions where there are two equally relevant couplings; the phase is determined by the ratio of these couplings. The left panel shows the $r < 0.6$ region of the phase diagram, where we see that regions of the two phases are interspersed with each other along the $s$ axis. In the right panel, we zoom in on the $r < 0.01$ region of the phase diagram and see an even richer intermingling of these two phases as we sweep over $s$ .Reuse & Permissions
Figure 6
Phase diagram of our edge theory as a function of $s$ and $r$ for the theory $S_{4} (s)$ in which the only nonzero perturbations are $u_{1, 2}$ and ${\tilde{u}}_{1, 2}$ ; $u_{17, \frac{p}{q}}$ , $u_{17, \frac{p}{q}}$ for all $p, q \leq 57$ ; and several $\cos (p_{a} X^{a})$ operators with $p_{a}$ nearly aligned with the direction of the boost $O_{G} (s)$ , as described in the text. The latter operators were specifically chosen to remove the remaining points of nonchiral phase at $r = 6$ , $s = p / q$ for $q = 5$ . This set of operators was sufficient to remove all the nonchiral phase displayed previously in Fig. 4. The light region is in the $E_{8} \times E_{8}$ phase. The darker region is in the $spin (32) / Z_{2}$ phase. Solid phase boundary lines denote regions where there are two equally-relevant couplings; the phase is determined by the ratio of these couplings.Reuse & Permissions

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