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Matrix Product States and the Fractional Quantum Hall Effect

Nicolas Regnault

Published Mar 1, 2013• 0 citations
OpenAlex

Abstract

he understanding and simulation of quantum many-body states in one space dimension has experienced revolutionary progress with the advent of the density matrix renormalization g roup. In modern language, this method can be viewed as a variational optimization over the set of matrix product states (MPS). Due to their perimeter law entanglement, 2-D system s such as the fractional quantum Hall effect are harder to simulate by MPS. Two recent articles[1,2] have proposed an exact MPS representation of a large series of fractional quantum Hall states in various geometries. The model states include all paired $k=2$ Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi $k=3$ state. We will discuss how a controlled truncation can be performed on this r epresentation and we will give a natural interpretation from the entanglement spectrum perspective. We will discuss the numerical accuracy of this truncated prescription for vari ous geometries. [1] M.P. Zaletel and R.S. K. Mong, Phys. Rev. B 86, 245305 (2012). [2] B. Estienne, Z. Papic, N. Regnault, B. A. Bernevig, arXiv:1211.3353.

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he understanding and simulation of quantum many-body states in one space dimension has experienced revolutionary progress with the advent of the density matrix renormalization g roup. In modern language, this method can be viewed as a variational optimization over the set of matrix product states (MPS). Due to their perimeter law entanglement, 2-D system s such as the fractional quantum Hall effect are harder to simulate by MPS. Two recent articles[1,2] have proposed an exact MPS representation of a large series of fractional quantum Hall states in various geometries. The model states include all paired k=2k=2k=2 Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3k=3k=3 state. We will discuss how a controlled truncation can be performed on this r epresentation and we will give a natural interpretation from the entanglement spectrum perspective. We will discuss the numerical accuracy of this truncated prescription for vari ous geometries. [1] M.P. Zaletel and R.S. K. Mong, Phys. Rev. B 86, 245305 (2012). [2] B. Estienne, Z. Papic, N. Regnault, B. A. Bernevig, arXiv:1211.3353.
We obtain an exact matrix-product-state (MPS) representation of a large series of fractional quantum Hall (FQH) states in various geometries of genus 0. The states in question include all paired k=2k=2k=2 Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3k=3k=3 state. We also outline the procedures through which the MPSs of other model FQH states can be obtained, provided their wave function can be written as a correlator in a 1+11+11+1 conformal field theory (CFT). The auxiliary Hilbert space of the MPS, which gives the counting of the entanglement spectrum, is then simply the Hilbert space of the underlying CFT. This formalism enlightens the link between entanglement spectrum and edge modes. Properties of model wave functions such as the thin-torus root partitions and squeezing are recast in the MPS form, and numerical benchmarks for the accuracy of the new MPS prescription in various geometries are provided.
Using truncated conformal field theory (CFT), we present the formalism necessary to obtain exact matrix product state (MPS) representations for any fractional quantum hall model state which can be written as an expectation value of primary fields in some conformal field theory. The auxiliary bond space is the Hilbert space of the descendants of primary fields and matrix elements can be calculated using well-known CFT commutation relations. We obtain a site-independent MPS representation on the thin annulus and cylinder, while a site-dependent representation is possible on the disk and sphere geometries. We discuss the complications that arise due to the presence of null vectors in the theory's Hilbert space, and present an array of procedures which optimize the numerical calculation of matrix elements. By truncating the Hilbert space of the CFT, we then build an approximate MPS representation for Laughlin, Moore-Read, Read-Rezayi, Gaffnian, minimal M(3,r+2)M(3,r+2)M(3,r+2) models, and a series of superconformal minimal models. We show how to obtain, in the MPS representation, the Orbital, Particle, and Real Space entanglement spectrum in both the finite and infinite systems.
We show that the model wave functions used to describe the fractional quantum Hall effect have exact representations as matrix product states (MPS). These MPS can be implemented numerically in the orbital basis of both finite and infinite cylinders, which provides an efficient way of calculating arbitrary observables. We extend this approach to the charged excitations and numerically compute their Berry phases. Finally, we present an algorithm for numerically computing the real-space entanglement spectrum starting from an arbitrary orbital basis MPS, which allows us to study the scaling properties of the real-space entanglement spectra on infinite cylinders. The real-space entanglement spectrum obeys a scaling form dictated by the edge conformal field theory, allowing us to accurately extract the two entanglement velocities of the Moore-Read state. In contrast, the orbital space spectrum is observed to scale according to a complex set of power laws that rule out a similar collapse.
In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\ensuremath{\nu}=1/3$ and 5/25/25/2. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\ensuremath{\nu}=5/2$ for large systems with electron numbers up to Ne=34{N}_{e}=34Ne​=34, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.
We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.
Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical representation as an exact matrix product state (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional quantum Hall wave functions, the Halperin states. These can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In particular, we deal with the caveat of the full wave-function symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation lengths, which are compared to plasma analogy arguments.