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arXiv:1306.4763 · Journal of the Korean Physical Society

Topological entanglement entropy in bilayer quantum hall systems

Myung-Hoon Chung

Published Apr 1, 2014• 0 citations
arXivHTMLPDF

Abstract

We calculate the topological entanglement entropy in bilayer quantum Hall systems, dividing the set of quantum numbers into four parts. This topological entanglement entropy allows us to draw a phase diagram in the parameter space of layer separation and tunneling amplitude. We perform a finite-size scaling analysis of the topological entanglement entropy in order to see the quantum phase transition clearly.

DOI
10.3938/jkps.64.1154

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We calculate the topological entanglement entropy in bilayer quantum Hall systems, dividing the set of quantum numbers into four parts. This topological entanglement entropy allows us to draw a phase diagram in the parameter space of layer separation and tunneling amplitude. We perform a finite-size scaling analysis of the topological entanglement entropy in order to see the quantum phase transition clearly.
We study the entanglement spectra of bilayer quantum Hall systems at total filling factor $\ensuremath{\nu}=1$. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half-filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels. Finally, we describe the connection between those two types of spectra in terms of an effective temperature leading to relations for the entanglement entropy which are in full analogy to canonical thermodynamics.
We study quantum phase transitions involving fractional quantum Hall states using numerical calculations of entanglements and related quantities. We tune finite-size wave functions on spherical geometries by varying the interaction potential away from the Coulomb interaction. We uncover signatures of quantum phase transitions contained in the scaling behavior of the entropy of entanglement between two parts of the sphere. In addition to the entanglement entropy, we show that signatures of quantum phase transitions also appear in other aspects of the reduced density matrix of one part of the sphere.
The entanglement entropy of the ν=1/3 and ν=5/2 quantum Hall states in the presence of short-range random disorder has been calculated using direct diagonalization. A microscopic model of electron–electron interaction is used and spin-polarized electrons are confined to a single Landau level and interact with long-range Coulomb interaction. In the case of very weak disorder, the values of topological entanglement entropy are roughly consistent with the expected theoretical results. By considering a broad range of disorder strengths, entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a signature of the transition as the function of the disorder strength for the ν=5/2 state. Prospects of using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.
We consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a subleading term, which scales as ${L}^{\ensuremath{-}1/\ensuremath{\alpha}}$ with the size of the subsystem LLL, here $\ensuremath{\alpha}$ is the R\'enyi index. This term reveals the scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length, which is sensitive to the topological index. The scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$ is independent of model parameters, suggesting some degree of its universality.
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.