Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2010.13799 · Physical review. B./Physical review. B

Gaffnian and Haffnian: Physical relevance of nonunitary conformal field theory for the incompressible fractional quantum Hall effect

Bo Yang

Published Mar 2, 2021• 14 citations
arXivHTMLPDF

Abstract

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 $h=2,\phantom{\rule{0.28em}{0ex}}c=0$, 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 $\frac{2}{5}$, 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.

DOI
10.1103/physrevb.103.115102

Related papers

Microscopic geometric theory for gapped excitations in fractional topological fluids

Yuzhu Wang, Bo Yang

We propose a geometric description of all gapped excitations in fractional quantum Hall phases that reveals several fundamental understandings with experimental consequences. These include a duality between the Hilbert space of multiple gapped ``graviton-like" spin 2 excitations at $ν=1/3$ Laughlin phase, and that of non-Abelian quasiholes of an irrational Haffnian conformal field theory. This leads us to construct microscopic wave functions for multiple higher-spin gapped neutral modes in the Laughlin phase. Carrying spin $s \ge 2$, they emerge from higher-order geometric deformations of the topological ground state, and live within the Gaffnian conformal Hilbert space asymptotically. We show that the full many-body Hilbert space of an FQH phase can be generated from superpositions of such geometric deformations, supporting a concrete geometric interpretation of all gapped excitations. We analyse the scattering between these higher spin modes and conjecture that they can have a long lifetime, and propose methods for their experimental detection.

arXiv:2603.13489•Mar 13, 2026•0 citations

Graviton modes in fractional quantum hall liquids

Yuzhu Wang

The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.

•Jan 1, 2023•1 citation

Fractionalization and Dynamics of Anyons and Their Experimental Signatures in the ν=n+1/3 Fractional Quantum Hall State

Ha Quang Trung, Bo Yang

We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.

arXiv:2009.14214•Physical Review Letters•Jul 23, 2021•13 citations

Analytic exposition of the graviton modes in fractional quantum Hall effects and its physical implications

Yuzhu Wang, Bo Yang

Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this paper, we derive a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and the thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin $\ensuremath{\nu}=1/5$ and the Gaffnian $\ensuremath{\nu}=2/5$ phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a ``phase transition'' even when the ground state is invariant. We discuss the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the ``hollow-core'' modes, as well as their experimental consequences.

arXiv:2109.08816•Physical review. B./Physical review. B•Jan 26, 2022•18 citations

Elementary excitations in fractional quantum Hall effect from classical constraints

Bo Yang, Ajit Coimbatore Balram

Abstract Classical constraints on the reduced density matrix of quantum fluids in a single Landau level termed as local exclusion conditions (LECs) (Yang 2019 Phys. Rev. B 100 241302), have recently been shown to characterize the ground state of many fractional quantum Hall (FQH) phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed quasielectron excitations of the Laughlin state obtained from LECs are topologically equivalent to those obtained from the composite fermion theory. Intriguingly, the LEC construction unveils interesting connections between different FQH phases and offers a novel perspective on exotic states such as the Gaffnian and the Fibonacci state.

New Journal of Physics•Dec 17, 2020•23 citations

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

Bo Yang, Ying-Hai Wu, Zlatko Papić

Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the $\ensuremath{\nu}=2/5$ fractional quantum Hall effect, prominent theoretical approaches---the composite fermion theory and conformal field theory---have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground-state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different; e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighboring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range.

arXiv:1907.12572•Physical review. B./Physical review. B•Dec 19, 2019•16 citations
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 \ensuremathν=13\ensuremath{\nu}=\frac{1}{3}\ensuremathν=31​ 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 \ensuremathν=25\ensuremath{\nu}=\frac{2}{5}\ensuremathν=52​, 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.
We propose a geometric description of all gapped excitations in fractional quantum Hall phases that reveals several fundamental understandings with experimental consequences. These include a duality between the Hilbert space of multiple gapped ``graviton-like" spin 2 excitations at ν=1/3ν=1/3ν=1/3 Laughlin phase, and that of non-Abelian quasiholes of an irrational Haffnian conformal field theory. This leads us to construct microscopic wave functions for multiple higher-spin gapped neutral modes in the Laughlin phase. Carrying spin s≥2s \ge 2s≥2, they emerge from higher-order geometric deformations of the topological ground state, and live within the Gaffnian conformal Hilbert space asymptotically. We show that the full many-body Hilbert space of an FQH phase can be generated from superpositions of such geometric deformations, supporting a concrete geometric interpretation of all gapped excitations. We analyse the scattering between these higher spin modes and conjecture that they can have a long lifetime, and propose methods for their experimental detection.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.
Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this paper, we derive a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and the thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin $\ensuremath{\nu}=1/5$ and the Gaffnian $\ensuremath{\nu}=2/5$ phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a ``phase transition'' even when the ground state is invariant. We discuss the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the ``hollow-core'' modes, as well as their experimental consequences.
Abstract Classical constraints on the reduced density matrix of quantum fluids in a single Landau level termed as local exclusion conditions (LECs) (Yang 2019 Phys. Rev. B 100 241302), have recently been shown to characterize the ground state of many fractional quantum Hall (FQH) phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed quasielectron excitations of the Laughlin state obtained from LECs are topologically equivalent to those obtained from the composite fermion theory. Intriguingly, the LEC construction unveils interesting connections between different FQH phases and offers a novel perspective on exotic states such as the Gaffnian and the Fibonacci state.
Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the $\ensuremath{\nu}=2/5$ fractional quantum Hall effect, prominent theoretical approaches---the composite fermion theory and conformal field theory---have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground-state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different; e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighboring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range.