Correlation Lengths and Topological Entanglement Entropies of Unitary and Nonunitary Fractional Quantum Hall Wave Functions

Abstract

Using the newly developed Matrix Product State (MPS) 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 non-unitary Conformal Field Theory (CFT) can describe the bulk of a gapped FQH phase. We show that the non-unitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has 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 FQH the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the non-unitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the Composite Fermion Jain state at equal filling.PACS numbers: 03.67. Mn, 05.30.Pr, Our understanding of the Fractional Quantum Hall (FQH) effect has benefited substantially from the use of model wavefunctions [1][2][3][4]. These wavefunctions, although not ground-states of realistic hamiltonians, are nonetheless supposed to capture the universal behavior of the state such as quasiparticle charge, statistics, braiding in the gapped bulk, as well as electron and quasihole exponents on the gapless edge. In a seminal paper [4] Moore and Read proposed to use conformal blocks, i.e. correlation functions in a Conformal Field Theory (CFT), as a building block to write down bulk model wavefunctions for the ground state and its quasihole excitations. This construction relies on a number of conjectures, the most important being that such a model bulk wavefunction describes a gapped topological state. Another assumption is that the universality class of the fractional quantum Hall state -most notably the braiding and fusion properties of the excitations -can be read off directly from the bulk CFT. Finally the bulk-edge correspondence is usually assumed. It states that (most) properties of the physical gapless edge states should be described by the same CFT that was used to build the bulk wavefunctions. Despite the nontrivial nature of these conjectures, there is a large body of (mostly exact diagonalization) evidence that supports the Moore-Read construction.However, this program has been observed recently to break down for non-unitary CFTs. While large sets of bulk trial wavefunctions can be written as correlation functions in a non-unitary CFT [5][6][7][8][9], the bulk and edge CFT can no longer match. Indeed the edge CFT is a low-energy effective theory describing the physical edge states, and as any proper quantum field theory it has to be unitary [10]. In that case one of the aforementioned hypothesis has to break down : either the edge CFT is different from the one used to write the bulk statewhich was shown to ...

Variational Ansatz for an Abelian to Non-Abelian Topological Phase Transition in ν=1/2+1/2 Bilayers

“…They identified a large class of model WFs and their quasihole excitations with Conformal Field Theory (CFT) correlators from which the topological content of the phase may be read off (under the generalized screening assumption [58]). It furthermore allows for an exact Matrix Product State (MPS) description of these strongly correlated phases of matter [59][60][61], allowing for large scale numerical study of their relevance and properties [62,63].…”

“…2) which couples to an excited state of the transfer matrix. The correlation length governing the decay of Ψ θ |c † e (z)c e (w)|Ψ θ is then related to the corresponding eigenvalue of the transfer matrix [62,77]. While bulk excitations generically couple to the first excited state of the transfer matrix, the P z symmetry, which translates into φ s → −φ s in the CFT, imposes further selection rules.…”

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

“…. We compute the momentum polarization using the entanglement spectrum of the exact Gaffnian state on the infinite cylinder from Ref 48…”

Matrix product state description and gaplessness of the Haldane-Rezayi state

“…This feature is reminiscent of the Gaffnian state [23], also built on a nonunitary CFT, as shown in Ref. [13]. The second group consists of two states which appear as a Jordan block in the transfer matrix.…”

“…Both states are non-Abelian and built from non-unitary CFT. In both cases, the TEE seems to only capture the Abelian part of the phase [13].…”