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arXiv:0902.1719 · Physical Review B

Modified Coulomb gas construction of quantum Hall states from nonunitary conformal field theories

Milica V. Milovanović, Th. Jolicœur, Ivana Vidanović

Published Oct 23, 2009• 15 citations
arXivHTMLPDF

Abstract

Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties such as the Moore-Read Pfaffian for filling factor $\ensuremath{\nu}=5/2$. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate fractional quantum Hall states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompressible, i.e., gapped liquid state. We show that by changing the way one derives the wavefunction from its parent CFT it is possible to obtain an incompressible candidate state when starting from a nonunitary parent. The construction mimics a global change in parameters in the phase space of the electron system. We explicit our construction in the case of the so-called Gaffnian state (a state for filling factor 2/5) and also for the Haldane-Rezayi state (a spin-singlet state at filling 1/2). We note that there are obstructions along this path in the case of the permanent spin-singlet state of Read and Rezayi which can be characterized as a robust gapless state.

DOI
10.1103/physrevb.80.155324

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Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties such as the Moore-Read Pfaffian for filling factor \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate fractional quantum Hall states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompressible, i.e., gapped liquid state. We show that by changing the way one derives the wavefunction from its parent CFT it is possible to obtain an incompressible candidate state when starting from a nonunitary parent. The construction mimics a global change in parameters in the phase space of the electron system. We explicit our construction in the case of the so-called Gaffnian state (a state for filling factor 2/5) and also for the Haldane-Rezayi state (a spin-singlet state at filling 1/2). We note that there are obstructions along this path in the case of the permanent spin-singlet state of Read and Rezayi which can be characterized as a robust gapless state.
After almost half a century of Laughlin's celebrated study of the wavefunctions of integer and fractional quantum Hall effects, there have still existed difficulties to prove whether the given wavefunction can describe gapped phase or not in general. In this work, we show the FQH states constructed from nonunitary conformal field theories (CFTs), such as Gaffiinian and Haldane-Rezayi states have a difficulty gapping out under preserving bulk-edge correspondence in the cylinder geometry. Contrary to the common understandings of the condensed matter communities, the gaplessness for these systems seems not to come from the negative conformal dimensions of nonunitary CFTs in this setting at least directly. We propose the difficulty is coming from the mismatch of monodromy charge and simple charge of underlying CFTs, known as Galois shuffle. In the Haldane-Rezayi state, this corresponds to the conjugate operation of the Neveu-Schwartz and Ramond sectors for unitary Weyl fermion and symplectic fermion. In the Gaffinian state, besides Galois shuffle structure, the anomalous conformal dimension of the Z2Z_{2}Z2​ simple current results in the cylinder partition functions outside of the existing local quantum field theory. This indicates the existing gapless fractional quantum Hall states have similar nonlocal structures, similar to deconfined quantum criticality. Our work opens up a new paradigm which gives a criterion to predict whether the candidate of topological ordered states are gapped or not, and local or nonlocal, by revisiting the problem of anomaly and the duality of symplectic and Dirac fermion.
We motivate a close look on the usefulness of the Gaffnian and Haffnian quasihole manifold (null spaces of the respective model Hamiltonians) for well-known gapped fractional quantum Hall phases. The conformal invariance of these subspaces is derived explicitly from microscopic many-body states. The resultant conformal field theory (CFT) description leads to an intriguing emergent primary field with h=2,c=0h=2,\phantom{\rule{0.28em}{0ex}}c=0h=2,c=0, and we argue the quasihole manifolds are quantum mechanically well defined and well behaved. Focusing on the incompressible phases at $\ensuremath{\nu}=\frac{1}{3}$ and 25\frac{2}{5}52​, we show the low-lying excitations of the Laughlin phase are quantum fluids of Gaffnian and Haffnian quasiholes, and give a microscopic argument showing that the Haffnian model Hamiltonian is gapless against Laughlin quasielectrons. We discuss the thermal Hall conductance and shot-noise measurements at $\ensuremath{\nu}=\frac{2}{5}$, and argue that the experimental observations can be understood from the dynamics within the Gaffnian quasihole manifold. A number of detailed predictions on these experimental measurements are proposed, and we discuss their relationships to the conventional CFT arguments and the composite fermion descriptions.
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.
The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N\ensuremath{-}1)$-singlet FQH states at filling fraction $\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]$ labeled by integers kkk and mmm, which are potentially realized in multicomponent quantum Hall systems or SU(N)SU(N)SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N\ensuremath{-}1)$-component free-boson CFT and the $SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N)SU(N)SU(N) Heisenberg model.
We propose to describe bulk wave functions of fractional quantum Hall states in terms of correlators of nonunitary b/cb/cb/c-spin systems. These yield a promising conformal field-theory analogon of the composite fermion picture of Jain. Fractional statistics are described by twist fields which naturally appear in the b/cb/cb/c-spin systems. We provide a geometrical interpretation of our approach in which bulk wave functions are seen as holomorphic functions over a ramified covering of the complex plane, where the ramification precisely resembles the fractional statistics of the quasiparticle excitations in terms of branch points on the complex plane. To extend Jain's main series, we use the concept of composite fermions pairing to spin singlets, which enjoys a natural description in terms of the particular $c=\ensuremath{-}2$ b/cb/cb/c-spin system as known from the Haldane-Rezayi state. In this way we derive conformal field-theory proposals for lowest Landau-level bulk wave functions for more general filling fractions. We obtain a natural classification of the experimentally confirmed filling fractions, which does not contain prominent unobserved fillings. Furthermore, our scheme fits together with classifications in terms of K matrices of effective multilayer theories leading to striking restrictions of these coupling matrices.
In the theory of the fractional quantum Hall effect, much attention is paid to the correspondence between fractional quantum Hall wave functions and conformal blocks in certain rational conformal field theories (CFT). This correspondence is powerful, enabling the calculation of the fractional statistics of a state that would be difficult if not impossible to calculate directly from the wave functions. But it is, in general, conjectural, remaining without microscopic justification in many cases of interest, and involves heavy mathematical machinery. We detail an alternative method to calculate Abelian and non-Abelian fractional statistics, the coherent state method. The method relies on assumptions which are independent of those underlying the CFT correspondence, so it serves as an independent check of results for the statistics where they exist, and an alternative source of results when a CFT is not known. We show how the coherent state method can be used to derive the statistics of several increasingly complicated trial wave functions: ν=1/2 Laughlin, Moore-Read, and k=3 Read-Rezayi. We discuss implications of our method for a possible notion of braiding statistics of the Gaffnian state, a "nonunitary" quantum Hall state for which the ramifications of the CFT correspondence are less well understood. We go on to derive formulas for the counting of zero modes of all these states on the torus.