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arXiv:1410.2894 · Physical Review Letters

Constraints on Topological Order in Mott Insulators

Michael P. Zaletel, Ashvin Vishwanath

Published Feb 18, 2015• 63 citations
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Abstract

We point out certain symmetry induced constraints on topological order in Mott insulators (quantum magnets with an odd number of spin 1/2 moments per unit cell). We show, for example, that the double-semion topological order is incompatible with time reversal and translation symmetry in Mott insulators. This sharpens the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum magnets, which guarantees that a fully symmetric gapped Mott insulator must be topologically ordered, but is silent about which topological order is permitted. Our result applies to the kagome lattice quantum antiferromagnet, where recent numerical calculations of the entanglement entropy indicate a ground state compatible with either toric code or double-semion topological order. Our result rules out the latter possibility.

DOI
10.1103/physrevlett.114.077201

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We point out certain symmetry induced constraints on topological order in Mott insulators (quantum magnets with an odd number of spin 1/2 moments per unit cell). We show, for example, that the double-semion topological order is incompatible with time reversal and translation symmetry in Mott insulators. This sharpens the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum magnets, which guarantees that a fully symmetric gapped Mott insulator must be topologically ordered, but is silent about which topological order is permitted. Our result applies to the kagome lattice quantum antiferromagnet, where recent numerical calculations of the entanglement entropy indicate a ground state compatible with either toric code or double-semion topological order. Our result rules out the latter possibility.
Symmetry-enforced quantum spin Hall insulators in πππ-flux models
We prove a Lieb-Schultz-Mattis theorem for the quantum spin Hall effect (QSHE) in two-dimensional πππ-flux models. In the presence of time reversal, U(1)U(1)U(1) charge conservation and magnetic translation (with πππ-flux per unit cell) symmetries, if a generic interacting Hamiltonian has a unique gapped symmetric ground state at half filling (i.e. an odd number of electrons per unit cell), it can only be a QSH insulator. In other words, a trivial Mott insulator is forbidden by symmetries at half filling. We further show that such a symmetry-enforced QSHE can be realized in cold atoms, by shaking an optical lattice and applying a time-dependent Zeeman field.
The interplay of symmetry and topology has been at the forefront of recent progress in quantum matter.Here, we uncover an unexpected connection between band topology and the description of competing orders in a quantum magnet.Specifically, we show that aspects of band topology protected by crystalline symmetries determine key properties of the Dirac spin liquid (DSL), which can be defined on the honeycomb, square, triangular, and kagome lattices.At low energies, the DSL on all of these lattices is described by an emergent quantum electrodynamics (QED 3 ) with N f ¼ 4 flavors of Dirac fermions coupled to a Uð1Þ gauge field.However, the symmetry properties of the magnetic monopoles, an important class of critical degrees of freedom, behave very differently on different lattices.In particular, we show that the lattice momentum and angular momentum of monopoles can be determined from the charge (or Wannier) centers of the corresponding spinon insulator.We also show that for DSLs on bipartite lattices, there always exists a monopole that transforms trivially under all microscopic symmetries owing to the existence of a parent SUð2Þ gauge theory.We connect our results to generalized Lieb-Schultz-Mattis theorems and also derive the time-reversal and reflection properties of monopoles.Our results indicate that recent insights into free-fermion band topology can also guide the description of strongly correlated quantum matter.
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We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak Z2\mathbb{Z}_2Z2​ symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.
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Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.