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arXiv:1605.00570 · Physical review. B./Physical review. B

Holographic duality between (2+1) -dimensional quantum anomalous Hall state and (3+1) -dimensional topological insulators

Yingfei Gu, Ching Hua Lee, Xueda Wen, Gil Young Cho, Shinsei Ryu, Xiao-Liang Qi

Published Sep 6, 2016• 42 citations
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Abstract

In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states, i.e., band insulators with quantized Hall conductance, using exact holographic mapping. Exact holographic mapping is an approach to holographic duality which maps the quantum anomalous Hall state to a different state living in $(3+1)$-dimensional hyperbolic space. By studying topological response properties and the entanglement spectrum, we demonstrate that the holographic dual theory of a quantum anomalous Hall state is a $(3+1)$-dimensional topological insulator. The dual description enables a characterization of topological properties of a system by the quantum entanglement between degrees of freedom at different length scales.

DOI
10.1103/physrevb.94.125107

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In this paper, we study (2+1)(2+1)(2+1)-dimensional quantum anomalous Hall states, i.e., band insulators with quantized Hall conductance, using exact holographic mapping. Exact holographic mapping is an approach to holographic duality which maps the quantum anomalous Hall state to a different state living in (3+1)(3+1)(3+1)-dimensional hyperbolic space. By studying topological response properties and the entanglement spectrum, we demonstrate that the holographic dual theory of a quantum anomalous Hall state is a (3+1)(3+1)(3+1)-dimensional topological insulator. The dual description enables a characterization of topological properties of a system by the quantum entanglement between degrees of freedom at different length scales.
While many features of topological band insulators are commonly discussed at the level of single-particle electron wave functions, such as the gapless Dirac boundary spectrum, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a U(1)U(1)U(1) gauge field that arises by gauging the continuous symmetry of the target system [the U(1)U(1)U(1) particle number conservation]. Integrating over the U(1)U(1)U(1) gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the U(1)U(1)U(1) gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description of the surface of topological insulators and the implication of its duality are also discussed. We also touch upon the hydrodynamic theory of fractional topological insulators by using the parton construction.
In this paper, starting from a lattice model of topological insulators, we study the quantum phase transitions among different quantum states, including quantum spin Hall state, quantum anomalous Hall state and normal band insulator state by calculating their topological properties (edge states, quantized spin Hall conductivities and the number of zero mode on a Pi-flux). We find that there exist universal features for the topological quantum phase transitions (TQPTs) in different cases : the emergence of nodal fermions at high symmetry points, the non-analytic third derivative of ground state energy and the jumps of the topological "order parameters". In particular, the relation between TQPTs and symmetries of the systems are explored : different TQPTs are protected by different (global) symmetries and then described by different topological "order parameters".
Topological phases are exotic states of matters which do not order locally yet exhibit a global ordering in their entanglement, making their characterization a challenging task. In this work, the authors construct an explicit holographic duality between topological phases in the bulk and the structure of the entanglement spectrum at its boundary, thereby establishing a one-to-one correspondence between different bulk topological orders and symmetry-breaking and symmetry-protected orderings of the entanglement. This allows classification and detection of topological phases holographically through ``entanglement spectroscopy'' at the boundary, and construction of order parameters that measure the condensation and confinement of anyons, and thus permits characterization of the nature of topological phase transitions.
We study the real-space entanglement renormalization group flows of topological band insulators in (2+1) dimensions by using the continuum multiscale entanglement renormalization ansatz (cMERA). Given the ground state of a Chern insulator, we construct and study its cMERA by paying attention, in particular, to how the bulk holographic geometry and the Berry curvature depend on the topological properties of the ground state. It is found that each state defined at different energy scale of cMERA carries a nonzero Berry flux, which is emanated from the UV layer of cMERA, and flows towards the IR. Hence, a topologically nontrivial UV state flows under the renormalization group to an IR state, which is also topologically nontrivial. On the other hand, we found that there is an obstruction to construct the exact ground state of a topological insulator with a topologically trivial IR state. That is, if we try to construct a cMERA for the ground state of a Chern insulator by taking a topologically trivial IR state, the resulting cMERA does not faithfully reproduce the exact ground state at all length scales.
This dissertation studies quantum entanglement in relation to the geometric response known as Hall viscosity. We begin by reviewing geometric response in the quantum Hall effect in comparison to its well-known electromagnetic response, Hall conductivity. We develop an understanding of momentum transport due to Hall viscosity in analogy to charge transport under Hall conductivity. We apply our momentum transport argument to continuum and lattice models of the quantum Hall effect. We also leverage this insight to reveal a previously-unrecognized manifestation of Hall viscosity: the acoustic Faraday effect in superfluid Helium-3 B. We suggest that the acoustic Faraday effect is a new platform for the direct observation of Hall viscosity in Helium-3 B and other systems. We then turn our focus to the entanglement spectrum. We calculate the momentum polarization of lattice models of the quantum Hall effect to determine the Hall viscosity based on the entanglement spectrum, revealing the close connection between geometric response and entanglement. Finally, we turn away from the quantum Hall effect to consider other topological insulators; we use the entanglement spectrum to develop a topological classification of composite systems comprising components of topological phases in different dimensions. Our composite topological index generalizes classification of weak and antiferromagnetic topological insulators. We predict the presence of topological bound states localized to defects in systems that are trivial under all other topological classifications.