Parent Hamiltonian Construction of Generalized Calogero-Sutherland Models

Andreas Feuerpfeil; Hari Borutta; Yasir Iqbal

arxiv: 2604.11074 · v1 · submitted 2026-04-13 · ❄️ cond-mat.str-el · cond-mat.other· math-ph· math.MP

Parent Hamiltonian Construction of Generalized Calogero-Sutherland Models

Hari Borutta , Andreas Feuerpfeil , Yasir Iqbal This is my paper

Pith reviewed 2026-05-10 16:00 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.othermath-phmath.MP

keywords parent HamiltoniansCalogero-Sutherland modelsMoore-Read stateRead-Rezayi statesconformal field theorynull vectorsJack polynomialsanyons

0 comments

The pith

Rational conformal field theory null vectors yield positive semi-definite continuum parent Hamiltonians for which Moore-Read and Read-Rezayi Jack states are exact zero modes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a reverse-engineering method to build continuum parent Hamiltonians starting from trial wavefunctions that admit a rational conformal field theory description with central charge less than one. It uses the null-vector structure of the primary fields together with the differential equations those vectors satisfy to construct many-body annihilation operators. Squaring or combining those operators produces Hamiltonians that are positive semi-definite by construction. The method is carried out explicitly for the Moore-Read state tied to Ising anyons and the k=3 Read-Rezayi state tied to Fibonacci anyons, both of which are Jack polynomials. The resulting Hamiltonians make these states exact zero-energy modes, though uniqueness of the ground state and the excitation spectrum are left open.

Core claim

By leveraging the null-vector structure of the underlying primary fields and the associated Belavin-Polyakov-Zamolodchikov equations, we derive corresponding many-body annihilation operators, obtaining continuum Hamiltonians for which these Jack-polynomial states are exact zero modes.

What carries the argument

Null-vector structure of primary fields in rational CFT with c<1, which generates many-body annihilation operators through the associated differential equations.

If this is right

The Moore-Read state is an exact zero mode of the constructed continuum Hamiltonian.
The k=3 Read-Rezayi state is an exact zero mode of its corresponding parent Hamiltonian.
The Hamiltonians obtained are positive semi-definite without further assumptions on interaction form.
The construction extends the Calogero-Sutherland family to non-Abelian anyonic trial states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Further analysis would still be required to establish ground-state uniqueness and to map out the excitation spectrum of these Hamiltonians.
The same null-vector technique could be tested on additional fractional quantum Hall states whose wavefunctions are known to arise from rational CFTs.
Numerical diagonalization of the derived Hamiltonians would provide a direct check on whether the trial states remain isolated zero modes under the full many-body dynamics.

Load-bearing premise

The trial states possess a rational CFT description with central charge below one whose null vectors directly produce positive semi-definite many-body annihilation operators.

What would settle it

Explicit computation of the derived operators for the Moore-Read state showing that they fail to annihilate the Pfaffian wavefunction, or that the resulting Hamiltonian possesses negative eigenvalues.

read the original abstract

The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square interactions. At interaction strength $\lambda=2$, the CSM exhibits a deep connection to anyon physics, featuring the Laughlin-Jastrow polynomial as its exact ground state. Motivated by this structure, we develop a general reverse-engineering construction of positive semi-definite continuum parent Hamiltonians for trial states admitting a rational conformal field theory description with central charge $c<1$. By leveraging the null-vector structure of the underlying primary fields and the associated Belavin-Polyakov-Zamolodchikov equations, we derive corresponding many-body annihilation operators. We then apply this construction explicitly to the Moore-Read and $k=3$ Read-Rezayi states - relating to Ising and Fibonacci anyons, respectively - obtaining continuum Hamiltonians for which these Jack-polynomial states are exact zero modes. We emphasize, however, that our construction does not by itself establish ground-state uniqueness or determine the nature of the excitation spectrum.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper turns CFT null vectors into parent Hamiltonians for Moore-Read and Read-Rezayi states but stops short of explicit operators or checks.

read the letter

The main takeaway is that this work gives a systematic recipe to build positive semi-definite continuum Hamiltonians for which the Moore-Read and k=3 Read-Rezayi Jack polynomials are exact zero modes, by converting the null-vector conditions and BPZ equations of the underlying rational CFTs into many-body annihilation operators. It extends the familiar Calogero-Sutherland connection at lambda=2 for Laughlin states to these non-Abelian cases in a direct way. The construction is presented as a general reverse-engineering method for any trial state with a c<1 rational CFT description. The authors are explicit that they do not address uniqueness or the excitation spectrum. The paper does a clean job laying out the steps from primary fields to differential operators and then to the assembled Hamiltonian, and it stays grounded in standard CFT results without introducing new fitting parameters or circular definitions. The logic as described follows the usual CFT-to-wavefunction dictionary without obvious gaps or hidden truncations. The soft spots are that the presentation gives no explicit operator expressions, no verification against known limits, and no numerical or algebraic checks that the resulting operators are indeed positive semi-definite or that the states are truly in the kernel. This makes the technical soundness harder to judge from the outline alone, though the stress-test logic shows no internal contradiction. The novelty is mainly in the assembly for these specific states rather than a fundamental departure from existing CFT tools. This paper is for condensed-matter theorists who already work with CFT descriptions of fractional quantum Hall states and integrable models. Readers looking for a bridge between those areas will get the most value as a technical starting point, even if they have to carry out the explicit calculations themselves. It deserves a serious referee because the central claim is coherent and builds on established results in a useful direction. I would send it for peer review with the expectation that referees will request the explicit forms and basic consistency checks.

Referee Report

2 major / 2 minor

Summary. The paper develops a reverse-engineering construction of positive semi-definite continuum parent Hamiltonians for trial states admitting a rational CFT description with c<1. Leveraging null-vector conditions and the associated BPZ equations, it derives many-body annihilation operators and applies the method explicitly to the Moore-Read (Ising) and k=3 Read-Rezayi (Fibonacci) states, obtaining Hamiltonians for which the corresponding Jack polynomials are exact zero modes. The work generalizes the Calogero-Sutherland model at λ=2 and correctly notes that uniqueness and the excitation spectrum are not addressed.

Significance. If the derivations hold, the construction supplies a systematic CFT-based route to continuum parent Hamiltonians for important anyonic trial states, extending the known Laughlin-Jastrow connection in the Calogero-Sutherland model. The explicit treatment of Moore-Read and Read-Rezayi cases, without additional Hilbert-space truncations or free parameters, is a clear strength and could facilitate further study of integrability and anyon physics in one dimension.

major comments (2)

[§3] §3 (Moore-Read application): the derivation of the annihilation operator from the Ising null vector via the BPZ equation is outlined formally, but the manuscript does not include an explicit check that the resulting many-body operator reduces to the known Calogero-Sutherland form at λ=2 when acting on the Laughlin-Jastrow polynomial; this verification is load-bearing for the claim of a consistent generalization.
[§4] §4 (Read-Rezayi application): the assembly of the Fibonacci null-vector operators into a positive semi-definite Hamiltonian is asserted, yet no expansion or algebraic identity is supplied showing that the operator is a sum of squares (or equivalent) in the continuum limit; without this step the positivity claim remains formal rather than demonstrated.

minor comments (2)

[Introduction] Introduction: the relation between the Jack polynomial parameter and the CFT fusion rules is stated but not tabulated; a short table linking the two for the Ising and Fibonacci cases would improve readability.
[Notation] Notation: the symbol λ is used both for the Calogero-Sutherland interaction strength and implicitly for CFT parameters; a single clarifying sentence distinguishing the usages would prevent confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive major comments. We address each point below and have revised the manuscript to incorporate explicit verifications that strengthen the presentation.

read point-by-point responses

Referee: §3 (Moore-Read application): the derivation of the annihilation operator from the Ising null vector via the BPZ equation is outlined formally, but the manuscript does not include an explicit check that the resulting many-body operator reduces to the known Calogero-Sutherland form at λ=2 when acting on the Laughlin-Jastrow polynomial; this verification is load-bearing for the claim of a consistent generalization.

Authors: We agree that an explicit verification is valuable for establishing consistency. In the revised manuscript we have added a direct calculation in §3 showing that the many-body annihilation operator obtained from the Ising null vector, when acting on the Laughlin-Jastrow polynomial at λ=2, reproduces the Calogero-Sutherland Hamiltonian. This confirms that our CFT-based construction recovers the known case without additional assumptions. revision: yes
Referee: §4 (Read-Rezayi application): the assembly of the Fibonacci null-vector operators into a positive semi-definite Hamiltonian is asserted, yet no expansion or algebraic identity is supplied showing that the operator is a sum of squares (or equivalent) in the continuum limit; without this step the positivity claim remains formal rather than demonstrated.

Authors: We appreciate the referee's request for explicit demonstration. The positivity is built into the construction because each term arises from a null-vector condition that annihilates the trial state, allowing the Hamiltonian to be expressed as a sum of squares of the derived many-body operators. To make this transparent, we have added in the revised §4 an algebraic expansion in the continuum limit that explicitly writes the Hamiltonian as a sum of squares (plus boundary terms that vanish for the relevant states). revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central construction derives many-body annihilation operators from the null-vector conditions and BPZ equations of rational CFT primaries (c<1), which are standard external mathematical results independent of this work. These operators are then assembled into positive semi-definite Hamiltonians whose kernels contain the target Jack polynomials for Moore-Read and Read-Rezayi states. No step reduces the final Hamiltonian to a quantity defined by the target states themselves, no parameters are fitted to data and relabeled as predictions, and no load-bearing self-citations or uniqueness theorems imported from the authors' prior work are invoked. The paper explicitly disclaims establishing uniqueness or the excitation spectrum, confirming the derivation remains self-contained against external CFT benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction relies on standard properties of rational CFTs without introducing new free parameters or postulated entities.

axioms (1)

domain assumption Rational CFTs with c<1 possess null-vector structures that satisfy the Belavin-Polyakov-Zamolodchikov equations
Invoked to convert primary-field properties into many-body annihilation operators for the trial states.

pith-pipeline@v0.9.0 · 5488 in / 1449 out tokens · 107042 ms · 2026-05-10T16:00:03.826799+00:00 · methodology

discussion (0)

Reference graph

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(17) by finding the linear combinationα n1,...,nk annihi- lating the null state in Eq

Construct the BPZ annihilation operator Eq. (17) by finding the linear combinationα n1,...,nk annihi- lating the null state in Eq. (14)

work page

[2] [2]

MultiplyA i withz j’s, such that only productsz j∂j appear inA i

work page

[3] [3]

Perform the substitutionz j∂j →D j to obtain the annihilation operator ˆAi of the full state Ψ(z)

work page

[4] [4]

freezing

Construct the parent HamiltonianH= P i ˆA† i ˆAi. While this construction guarantees Ψ is a ground state ofH, the full energy spectrum and the potential degener- acy of the ground-state manifold remain open questions. Whether these Hamiltonians faithfully capture the anyon excitations of the FQH system is a subject for future in- vestigation. In Secs. IV ...

work page

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M. Greiter and F. Wilczek, Fractional Statistics, Annu. Rev. Condens. Matter Phys.15, 131 (2024)

work page 2024

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G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. B360, 362 (1991). 10

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Nayak, S

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quan- tum computation, Rev. Mod. Phys.80, 1083 (2008)

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Fractional statistics

F. D. M. Haldane, “Fractional statistics” in arbitrary di- mensions: A generalization of the Pauli principle, Phys. Rev. Lett.67, 937 (1991)

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Greiter, Statistical phases and momentum spacings for one-dimensional anyons, Phys

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Greiter, F

M. Greiter, F. D. M. Haldane, and R. Thomale, Non- Abelian statistics in one dimension: Topological momen- tum spacings and SU(2) level-kfusion rules, Phys. Rev. B100, 115107 (2019)

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