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arXiv:1003.5656 · New Journal of Physics

Entanglement scaling of fractional quantum Hall states through geometric deformations

Andreas Martin Läuchli, Emil J. Bergholtz, Masudul Haque

Published Jul 7, 2010• 42 citations
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Abstract

We present a new approach for obtaining the scaling behavior of the entanglement entropy in fractional quantum Hall (FQH) states from finite-size wavefunctions. By employing the torus geometry and the fact that the torus aspect ratio can be readily varied, we can extract the entanglement entropy of a spatial block as a continuous function of the block boundary length. This approach allows us to extract the topological entanglement entropy with an accuracy superior to that possible for the spherical or disc geometry, where no natural continuously variable parameter is available. Other than the topological information, the study of entanglement scaling is also useful as an indicator of the difficulty posed by FQH states for various numerical techniques.

DOI
10.1088/1367-2630/12/7/075004

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We present a new approach for obtaining the scaling behavior of the entanglement entropy in fractional quantum Hall (FQH) states from finite-size wavefunctions. By employing the torus geometry and the fact that the torus aspect ratio can be readily varied, we can extract the entanglement entropy of a spatial block as a continuous function of the block boundary length. This approach allows us to extract the topological entanglement entropy with an accuracy superior to that possible for the spherical or disc geometry, where no natural continuously variable parameter is available. Other than the topological information, the study of entanglement scaling is also useful as an indicator of the difficulty posed by FQH states for various numerical techniques.
We present a detailed analysis of bipartite entanglement entropies in fractional quantum Hall (FQH) states, considering both Abelian (Laughlin) and non-Abelian (Moore-Read) states. We derive upper bounds for the entanglement between two subsets of the particles making up the state. We also consider the entanglement between spatial regions supporting a FQH state. Using the latter, we show how the so-called topological entanglement entropy $\ensuremath{\gamma}$ of a FQH state can be extracted from wave functions for a limited number of particles.
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 topological entanglement entropy is a robust measurement of the quantum many-body state with topological order. In fractional quantum Hall (FQH) state, it has a connection to the quantum dimension of the state itself and its quasihole excitations from the conformal field theory (CFT) description. We systematically study the entanglement entropy in the Laughlin, Moore-Read, and Read-Rezayi FQH states. The Abelian nature of the quasihole manifests in the invariant of the entanglement entropy and spectrum while it is created in the subsystem. Whereas the non-Abelian quasihole excitation induces an extra correction of the topological entanglement entropy and the changing of the structure of the entanglement spectrum. The quantum dimension of the quasihole can be obtained from the entropy difference before and after inserting the quasihole. On the other hand, the entanglement entropy behaviors similarly to the density profile and has less oscillations far away from the center of quasihole. It gives us a better definition of the quasihole boundary and the measurement of the size of the quasihole.
Using the newly developed matrix product state formalism for non-Abelian fractional quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a nonunitary conformal field theory (CFT) can describe the bulk of a gapped FQH phase. We show that the nonunitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of a Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has an infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-Abelian states with and without quasiholes. For the first time in the FQH effect the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the nonunitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the composite fermion Jain state at equal filling.
We give a complete definition of the entanglement gap separating low-energy, topological levels from high-energy, generic ones, in the "entanglement spectrum" of fractional quantum Hall (FQH) states. This is accomplished by removing the magnetic length inherent in the FQH problem--a procedure which we call taking the conformal limit. The counting of the low-lying entanglement levels starts off as the counting of modes of the edge theory of the FQH state, but quickly develops finite-size effects which we find to serve as a fingerprint of the FQH state. As the sphere manifold where the FQH resides grows, the level spacing of the states at the same angular momentum goes to zero, suggestive of the presence of relativistic gapless edge states. By using the adiabatic continuity of the low-entanglement energy levels, we investigate whether two states are topologically connected.
Once the fractional quantum Hall (FQH) state for a finite-sized system is put on the surface of a cylinder, the distance between the two ends with open boundary conditions can be tuned by varying the aspect ratio γ . It scales linearly with increasing the system size and therefore has a larger adjustable range than that on a disk. The previous study of the quasi-hole tunneling amplitude on a disk by Hu et al (2011 New J. Phys. 13 035020 ) indicates that the tunneling amplitudes have a scaling behavior as a function of the tunneling distance and the scaling exponents are related to the scaling dimension and the charge of the transported quasiparticles. However, the scaling behaves poorly due to the narrow range of the tunneling distance on the disk. Here we systematically study the quasiparticle tunneling amplitudes of the Laughlin state in the cylinder geometry which shows a much better scaling behavior. In particular, there are some crossover behaviors at the two length scales when the two open edges are close to each other. These lengths are also reflected in the bipartite entanglement and the electron Green's function as either a singularity or a crossover. These two critical length scales of the edge–edge distance, and are found to be related to the dimension reduction and back scattering point respectively.