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arXiv:1007.5356 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy of integer quantum Hall states in polygonal domains

Ivan Dario Jimenez Rodriguez, Germán Sierra

Published Dec 23, 2010• 30 citations
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Abstract

The entanglement entropy of the integer Quantum Hall states satisfies the area law for smooth domains with a vanishing topological term. In this paper we consider polygonal domains for which the area law acquires a constant term that only depends on the angles of the vertices and we give a general expression for it. We study also the dependence of the entanglement spectrum on the geometry and give it a simple physical interpretation.

DOI
10.1088/1742-5468/2010/12/p12033

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The entanglement entropy of the integer Quantum Hall states satisfies the area law for smooth domains with a vanishing topological term. In this paper we consider polygonal domains for which the area law acquires a constant term that only depends on the angles of the vertices and we give a general expression for it. We study also the dependence of the entanglement spectrum on the geometry and give it a simple physical interpretation.
Generally speaking, entanglement entropy (EE) between two subregions of a gapped quantum many-body state is proportional to the area/length of their interface due to the short-range quantum correlation. However, the so-called area law is violated logarithmically in a quantum critical phase. Moreover, the subleading correction exists in long-range entangled topological phases. It is referred to as topological EE which is related to the quantum dimension of the collective excitation in the bulk. Furthermore, if a non-smooth sharp angle is in the presence of the subsystem boundary, a universal angle dependent geometric contribution is expected to appear in the subleading correction. In this work, we simultaneously explore the geometric and edge contributions in the integer quantum Hall (IQH) state and its edge reconstruction in a unified bipartite method. Their scaling is found to be consistent with conformal field theory (CFT) predictions and recent results of particle number fluctuation calculations.
We study the quantum entanglement structure of integer quantum Hall states via the reduced density matrix of spatial subregions. In particular, we examine the eigenstates, spectrum, and entanglement entropy (EE) of the density matrix for various ground and excited states, with or without mass anisotropy. We focus on an important class of regions that contain sharp corners or cusps, leading to a geometric angle-dependent contribution to the EE. We unravel surprising relations by comparing this corner term at different fillings. We further find that the corner term, when properly normalized, has nearly the same angle dependence as numerous conformal field theories (CFTs) in two spatial dimensions, which hints at a broader structure. In fact, the Hall corner term is found to obey bounds that were previously obtained for CFTs. In addition, the low-lying entanglement spectrum and the corresponding eigenfunctions reveal ``excitations'' localized near corners. Finally, we present an outlook for fractional quantum Hall states.
Boundaries constitute a rich playground for quantum many-body systems because they can lead to novel degrees of freedom such as protected boundary states in topological phases. Here, we study the ground state of integer quantum Hall systems in the presence of boundaries through the reduced density matrix of a spatial region. We work in the lowest Landau level and choose our region to intersect the boundary at arbitrary angles. The entanglement entropy (EE) contains a logarithmic contribution coming from the chiral edge modes, and matches the corresponding conformal field theory prediction. We uncover an additional contribution due to the boundary corners. We characterize the angle dependence of this boundary corner term, and compare it to the bulk corner EE. We further analyze the spatial structure of entanglement via the eigenstates associated with the reduced density matrix. The influence of the physical boundary and the region's geometry on the reduced density matrix is thus clarified. Finally, we test a bulk-boundary correspondence for the EE originally obtained for quantum critical systems such as conformal field theories in two spatial dimensions, and discuss the implications of our findings for other topological phases.
Quantum geometry, which encompasses both Berry curvature and the quantum metric, plays a key role in multiband interacting electron systems. We study the entanglement entropy of a region of linear size $\ensuremath{\ell}$ in fermion systems with nontrivial quantum geometry, i.e., whose Bloch states have nontrivial kkk dependence. We show it scales as $S=\ensuremath{\alpha}{\ensuremath{\ell}}^{d\ensuremath{-}1}ln\ensuremath{\ell}+\ensuremath{\beta}{\ensuremath{\ell}}^{d\ensuremath{-}1}+\ensuremath{\cdots}$, where the first term is the well-known area-law violating term from the Fermi surface and $\ensuremath{\beta}$ contains the leading contribution from quantum geometry. We compute this for the case of uniform quantum geometry and cubic domains and provide numerical results for the Su-Schrieffer-Heeger model, two-dimensional (2D) massive Dirac cone, and 2D Chern bands. An experimental probe of the quantum geometric entanglement entropy is proposed using particle number fluctuations. We offer an intuitive account of the area-law entanglement related to the spread of maximally localized Wannier functions.
The entanglement entropy of the incompressible states of a realistic quantum Hall system is studied by direct diagonalization. The subdominant term to the area law, the topological entanglement entropy (which is believed to carry information about topologic order in the ground state), was extracted for filling factors $\ensuremath{\nu}=1/3$, $\ensuremath{\nu}=1/5$, and $\ensuremath{\nu}=5/2$. The results for $\ensuremath{\nu}=1/3$ and $\ensuremath{\nu}=1/5$ are consistent with the topological entanglement entropy for the Laughlin wave function. The $\ensuremath{\nu}=5/2$ state exhibits a topological entanglement entropy consistent with the Moore-Read wave function.