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The Haldane-Rezayi Quantum Hall State and Conformal Field Theory

Victor Gurarie, Michael Flohr, Chetan Nayak

Published Jan 1, 1997• 81 citations
OpenAlex

Abstract

We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the c = −2 conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the c = −2 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the c = 1 chiral Dirac fermion, which is related in a simple way to the c = −2 theory of the bulk. This theory is reformulated as a truncated version of a

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Edge excitations of paired fractional quantum Hall states

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The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely, the Pfaffian, Haldane-Rezayi, and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero-energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The wave functions for each case have a similar form, related to Slater determinants, and the edge states satisfy a ``projection rule,'' that the parity of the number of fermions added to the edge equals the parity of the charge added. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a noninteracting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system, we exhibit a chiral Lagrangian that has manifest SU(2) symmetry, but breaks Lorentz invariance, because of the breakdown of the spin-statistics connection implied by the scalar nature of the field and the positive-definite norm on the Hilbert space. Finally, we consider systems on a cylinder, where the fluid has two edges, and construct the sectors of zero-energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. The corresponding theory for the hierarchy and its generalizations is also given. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions. Two appendixes contain technical details. \textcopyright{} 1996 The American Physical Society.

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Gaplessness protected by bulk-edge correspondence

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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 $Z_{2}$ 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.

arXiv:2212.12996•Dec 26, 2022•0 citations
We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the c = −2 conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the c = −2 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the c = 1 chiral Dirac fermion, which is related in a simple way to the c = −2 theory of the bulk. This theory is reformulated as a truncated version of a
The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely, the Pfaffian, Haldane-Rezayi, and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero-energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The wave functions for each case have a similar form, related to Slater determinants, and the edge states satisfy a ``projection rule,'' that the parity of the number of fermions added to the edge equals the parity of the charge added. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a noninteracting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system, we exhibit a chiral Lagrangian that has manifest SU(2) symmetry, but breaks Lorentz invariance, because of the breakdown of the spin-statistics connection implied by the scalar nature of the field and the positive-definite norm on the Hilbert space. Finally, we consider systems on a cylinder, where the fluid has two edges, and construct the sectors of zero-energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. The corresponding theory for the hierarchy and its generalizations is also given. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions. Two appendixes contain technical details. \textcopyright{} 1996 The American Physical Society.
The low-energy effective quantum-field theory of the edge excitations of a fully gapped bulk topological phase corresponding to a local Hamiltonian must be local and unitary. Here it is shown that whenever all the edge excitations propagate in the same direction with the same velocity, it is a conformal field theory. In particular, this is the case in the quantum Hall effect for model ``special Hamiltonians'' for which the ground-state, quasihole, and edge excitations can be found exactly as zero-energy eigenstates, provided the spectrum in the interior of the system is fully gapped. In addition, other conserved quantities in the bulk, such as particle number and spin, lead to affine Lie algebra symmetries in the edge theory. Applying the arguments to some trial wave functions related to nonunitary conformal field theories, it is argued that the Gaffnian state and an infinite number of others cannot describe a gapped topological phase because the numbers of edge excitations do not match any unitary conformal field theory.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.
The structure of ground states of generic FQH states on a torus is studied by using both effective theory and electron wavefunction. The relation between the effective theory and the wavefunction becomes transparent when one considers the ground state structure. We find that the non-abelian Berry’s phases of the abelian Hall states generated by twisting the mass matrix are identical to the modular transformation matrix for the characters of Gaussian conformal field theory. We also show that the Haldane-Rezayi spin singlet state has a ten fold ground state degeneracy on a torus which indicates such a state is a non-abelian Hall 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.