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Topics in topologically ordered phases of matter

Alexander Selem

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

An outstanding problem in the study of topologically ordered phases is to find methods to distinguish topologically ordered from unordered phases. The topological entanglement entropy, a negative shift to the entanglement entropy scaling, has emerged as one possible distinguishing probe. The n=2 Rényi topological entanglement entropy of the triangular quantum dimer model is studied by classical Monte Carlo sampling of Rokhsar-Kivelson-like ground state wave functions, and using quantum Monte Carlo sampling for Hamiltonian parameter values away from the Rokhsar-Kivelson point. Both results demonstrate the universality of topological entanglement entropy over a wide range of parameter values within a topologically ordered phase. For systems sizes that are accessible numerically, it is found that the quantization of topological entanglement entropy depends sensitively on correlations. At the Rokhsar-Kivelson point, competing corner contributions to the entanglement entropy are also characterized.Another class of topologically ordered materials existing in three dimensions are topological insulators, which exhibit Dirac modes on its surface. The anomalous currents arising from these modes are investigated. The Dirac energy eigenstates are computed on a flat torus (genus one topology) and closed cap cylinder (genus zero topology) for various mass-term geometries. From the resulting properties of the surface spectra, a potential application for a flux-charge qubit is presented.

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An outstanding problem in the study of topologically ordered phases is to find methods to distinguish topologically ordered from unordered phases. The topological entanglement entropy, a negative shift to the entanglement entropy scaling, has emerged as one possible distinguishing probe. The n=2 Rényi topological entanglement entropy of the triangular quantum dimer model is studied by classical Monte Carlo sampling of Rokhsar-Kivelson-like ground state wave functions, and using quantum Monte Carlo sampling for Hamiltonian parameter values away from the Rokhsar-Kivelson point. Both results demonstrate the universality of topological entanglement entropy over a wide range of parameter values within a topologically ordered phase. For systems sizes that are accessible numerically, it is found that the quantization of topological entanglement entropy depends sensitively on correlations. At the Rokhsar-Kivelson point, competing corner contributions to the entanglement entropy are also characterized.Another class of topologically ordered materials existing in three dimensions are topological insulators, which exhibit Dirac modes on its surface. The anomalous currents arising from these modes are investigated. The Dirac energy eigenstates are computed on a flat torus (genus one topology) and closed cap cylinder (genus zero topology) for various mass-term geometries. From the resulting properties of the surface spectra, a potential application for a flux-charge qubit is presented.
Topological phenomena in physical systems are determined by topological structures and are thus universal and protected against perturbations. We theoretically establish exotic topological phenomena and their consequences for experiments in crystalline systems such as topological insulators and topologically ordered phases. We show that protected one-dimensional fermionic modes may be associated with the line defects such as dislocations in three-dimensional topological insulators, and strong electron-electron repulsion may lead to topological Mott insulators via spontaneous spin-orbit correlations in three dimensions. We also predict anomalous Aharonov Bohm conductance oscillations maximized at half integer multiples of a flux quantum in a topological insulator nanowire with strong surface disorder, arising from surface curvature induced Berry phase. In addition, we classify three-dimensional inversion symmetric insulators and their quantized responses.Quantum entanglement provides a promising probe to the properties of many-body systems, especially topological phases not captured by local order parameters. We present a characterization of topological insulators using entanglement spectrum based only on bulk ground-state wave function. Further, by studying entanglement of trivial partitions, we establish topological order in candidate Gutzwiller projected wave functions for gapped spin liquids and Laughlin states; and with entanglement's dependence on the ground states for bipartition of a torus into two cylinders, we demonstrate a method to extract the modular matrices and statistics and braiding of quasiparticle excitations. Our method helps to determine the topological order with only the set of ground-state wave functions on a torus. Our variational Monte Carlo calculations of topological entanglement entropy agree well with theory. We also find a violation of the boundary law for a critical spin liquid of Gutzwiller projected Fermi sea on the triangular lattice, where the entanglement entropy's enhancement by a logarithmic factor reflects the presence of emergent fermions in a bosonic wave function.
We study the n=2n=2n=2 R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.
Topological entanglement entropy (TEE) provides a universal measure of long-range quantum correlations in topologically ordered systems, offering critical insights into exotic phases of matter. While theoretical frameworks, including exactly solvable lattice models, tensor networks, and field-theoretic approaches, enable precise computation of TEE in idealized settings, practical implementation faces multiple challenges. Finite-size and geometry effects, including boundary corners and lattice discretization, introduce significant corrections that can obscure the small topological contribution. Gapless edges in chiral topological phases further complicate the isolation of bulk entanglement, requiring careful separation of edge and bulk contributions. Symmetry-protected and symmetry-enriched phases necessitate additional diagnostics, such as symmetry-resolved entanglement and entanglement negativity, to fully characterize topological properties. Experimental measurement remains challenging, with current approaches limited to small systems using Rényi entropies via swap operations and interferometric methods. This work synthesizes these practical considerations, outlines mitigation strategies, and highlights ongoing directions for bridging theoretical, numerical, and experimental approaches to robustly quantify TEE. Understanding these limitations is essential for accurate characterization of topological order and for guiding the development of quantum technologies leveraging topological protection.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice ( t = 1) and the square lattice ( t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term—proportional to the length of the boundary—and a universal subleading constant s n ( t ). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s n with a very high numerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find s n ( t > 0) = − ln2, independent of the fugacity t and the Rényi parameter n . At t = 0 we recover a previously known result, for n < 1 and s n ( t = 0) = − ln(2)/( n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropy s n KP ( t 0) = − ln2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit are discussed.
A characterization of topological order in terms of bi-partite entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid. 96, 110405 (2006)]. It was argued that in a topological phase there is a universal additive constant in the entanglement entropy, called the topological entanglement entropy, which reflects the underlying gauge theory for the topological order. In the present paper, we evaluate numerically the topological entanglement entropy in the ground states of a quantum dimer model on the triangular lattice, which is known to have a dimer liquid phase with Z2{\mathbb{Z}}_{2}Z2​ topological order. We examine the two original constructions to measure the topological entropy by combining entropies on plural areas, and we observe that in the large-area limit they both approach the value expected for Z2{\mathbb{Z}}_{2}Z2​ topological order. We also consider the entanglement entropy on a topologically nontrivial ``zigzag'' area and propose to use it as another way to measure the topological entropy.