Calogero-Sutherland Interaction

• Calogero-Sutherland-type interaction is defined by pairwise inverse-square potentials that yield exactly solvable classical and quantum many-body systems.
• It employs integrability tools like Lax pairs, Dunkl operators, and Jack polynomials to construct commuting operators and explicit eigenfunctions.
• Extensions include effective field theories, hydrodynamic limits, and parent-Hamiltonian frameworks that generalize the canonical inverse-square structure.

Calogero-Sutherland-type interaction denotes a class of one-dimensional many-body interactions whose canonical realization is a pairwise inverse-square-type potential and whose broader contemporary usage includes Hamiltonians, effective field theories, and parent-Hamiltonian constructions that retain the characteristic Jastrow, Lax/Dunkl, conformal-block, or algebraic structure of the Calogero-Sutherland model. In the standard setting, these interactions define exactly solvable classical and quantum systems on the line or circle; in more recent work, the same interaction type also appears as a nonlinear hydrodynamic term, a $W_{1+\infty}$-organized effective Hamiltonian, a conformal-block differential operator, or a positive-semidefinite many-body annihilation framework rather than only as a bare two-body force (Hallnäs, 2023, Bottesi et al., 2024, Borutta et al., 13 Apr 2026).

1. Canonical forms and geometric realizations

In the classical Calogero-Moser-Sutherland family, the interaction enters through the Hamiltonian

$$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$

with $V$ one of four standard inverse-square-type potentials (Hallnäs, 2023).

Type	Potential $V(x)$	Geometry
Rational	$\frac{2}{x^2}$	line
Hyperbolic	$\frac{a^2}{4\sinh^2(ax/2)}$	line
Trigonometric / Sutherland	$\frac{a^2}{4\sin^2(ax/2)}$	circle
Elliptic	$\wp(x;\omega_1,\omega_2)$	doubly periodic

The elliptic interaction is the master case: the trigonometric, hyperbolic, and rational models arise as degenerations obtained by sending one or both periods to infinity (Hallnäs, 2023). In the trigonometric or Sutherland realization, particles move on a circle and interact through the inverse square of the chord distance. A standard microscopic ring Hamiltonian is

$$H_{\mathrm{CS}}=\sum_{j=1}^N \left(-\partial_{x_j}^2\right)+g\sum_{i<j}\frac{\pi^2/L^2}{\sin^2\!\big(\pi(x_i-x_j)/L\big)},$$

with stability requiring $g\ge -\tfrac12$ in one convention, while other papers use parameterizations such as $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$0, $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$1, or $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$2 depending on normalization (Bottesi et al., 7 Jan 2025, Bottesi et al., 2024, Wu et al., 2011).

A central feature of the interaction is that it is long-ranged and singular, yet still exactly tractable. For $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$3 in the rational model the interaction is repulsive and singular, preventing collisions and preserving particle order; on the circle it gives the standard Sutherland interaction, while in elliptic form it becomes doubly periodic and finite-gap-like (Hallnäs, 2023). This canonical inverse-square structure remains the benchmark against which later “Calogero-Sutherland-type” generalizations are defined.

2. Integrability, commuting structures, and exact eigenfunctions

The defining structural property of the Calogero-Sutherland-type interaction is integrability. Classically, Liouville integrability means the existence of $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$4 functionally independent conserved quantities $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$5 in involution,

$$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$6

while quantum integrability means that the Schrödinger operator admits $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$7 algebraically independent commuting differential operators (Hallnäs, 2023).

For the rational model, Moser’s proof uses a Lax pair $$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$8 obeying

$$H=\frac{1}{2m}\sum_{i=1}^{N} p_i^2+\frac{g}{m}\sum_{1\le i<j\le N}V(x_i-x_j),$$9

so the isospectral invariants

$V$0

are conserved. In the elliptic case a spectral-parameter-dependent Lax matrix is required, and its spectral curve encodes the integrals of motion (Hallnäs, 2023). On quantization,

$V$1

and one obtains the quantum Hamiltonian with the shifted coupling $V$2, together with commuting quantum integrals constructed through quantum Lax pairs, Dunkl operators, Cherednik/Heckman operators, or elliptic Dunkl operators (Hallnäs, 2023).

The spectral theory reflects the geometry of the interaction. In the trigonometric case, the eigenfunctions are related to Heckman-Opdam hypergeometric functions and, in polynomial sectors, to Jack polynomials. The ground-state factor is

$V$3

and eigenfunctions have the form $V$4, where $V$5 are Jack polynomials (Hallnäs, 2023). In the rational case, special couplings admit Baker-Akhiezer functions, while generic $V$6 yields multivariable Bessel functions of $V$7-type (Hallnäs, 2023).

A complementary second-quantized viewpoint rewrites the trigonometric model in a fermionic basis in which the interaction becomes triangular. In that formulation, Jack states are obtained from free-fermion Schur states by a finite-order similarity transformation, and the strictly triangular part only connects a state to squeezed Young diagrams below it in dominance order (Wu et al., 2011). This makes explicit that the Calogero-Sutherland interaction is not merely solvable; it is organized by an algebraic partial order on partitions.

3. Effective-field-theory and hydrodynamic realizations

In the thermodynamic limit, the interaction admits an effective description in which the microscopic inverse-square force is replaced by an interacting $V$8-organized dynamics of density modes. For the spinless fermion model on a circle, the limit $V$9 at fixed density produces a Fermi sea with two Fermi points, and the low-energy theory becomes a $V(x)$0 conformal field theory with independent right- and left-moving sectors. The $V(x)$1 generators encode the gapless particle-hole excitations, and the interaction is incorporated by changing the representation rather than the algebra itself (Bottesi et al., 2024).

After a Bogoliubov transformation mixing chiralities, the interacting fluid is recast as a free compact boson with modified compactification radius

$V(x)$2

while the exact Fermi velocity remains

$V(x)$3

The density field then obeys the quantum Benjamin-Ono equation,

$V(x)$4

so the Calogero-Sutherland interaction survives the continuum limit not as an explicit pair potential but as a nonlinear dispersive hydrodynamic term (Bottesi et al., 2024). In this sense, the effective theory describes a one-dimensional quantum compressible fluid with two chiralities rather than a purely chiral CFT.

A related short-distance analysis of the same thermodynamic-limit theory identifies an attractive interaction between opposite-moving particle-hole bilinears in the repulsive regime. In momentum space the coefficient is

$V(x)$5

and its Fourier transform gives

$V(x)$6

That interaction is proposed as responsible for destabilizing the free-fermion vacuum and favoring a correlated bosonic ground state, thereby supplying a dynamical mechanism for charge and statistics fractionalization in the low-energy theory (Bottesi et al., 2024). The proposal does not alter the microscopic repulsive character of the original particles; it concerns an effective attraction in the bilinear collective description.

4. Dynamic structure factor and the problem of sharp versus finite-width response

The dynamic structure factor $V(x)$7 is the principal response observable for Calogero-Sutherland fluids. In the effective chiral theory, the density-density response is concentrated in a single mode. For the free chiral boson, one obtains

$V(x)$8

while the interacting effective theory gives

$V(x)$9

with the same one-peak structure but an interaction-renormalized resonant frequency. The calculation proceeds by evolving the density operator in the Heisenberg picture and inserting projectors onto neutral descendants, with only the one-phonon neutral subspace contributing (Bottesi et al., 7 Jan 2025). The result is a sharp response with no continuum in the low-energy effective description.

The same paper compares this effective result with the first-quantized calculation in which the support is an approximately narrow rectangle between

$\frac{2}{x^2}$0

with width

$\frac{2}{x^2}$1

In the narrow-width limit, the rectangle collapses to the delta peak, and in the common domain of validity the two descriptions agree (Bottesi et al., 7 Jan 2025). The effective-field-theory treatment explicitly notes that it does not reproduce the full finite-width continuum away from the strict infrared, nor the high-energy backward-scattering processes integrated out in bosonization.

At the microscopic level, the full model exhibits a much richer dynamical response. A stochastic method combined with exact sum rules computes $\frac{2}{x^2}$2 across weak, intermediate, and strong coupling, with the interaction strength $\frac{2}{x^2}$3 related to the Luttinger parameter by

$\frac{2}{x^2}$4

and sound velocity

$\frac{2}{x^2}$5

For $\frac{2}{x^2}$6, the response has finite support

$\frac{2}{x^2}$7

with $\frac{2}{x^2}$8. Weak coupling yields a Bogoliubov-type upper branch, $\frac{2}{x^2}$9 gives a flat response within the allowed interval, and strong coupling produces a lower-edge singularity together with umklapp-driven Brillouin-zone structure characteristic of a quasicrystal or crystal-like regime (Motis et al., 23 Jul 2025). The apparent tension between a sharp EFT peak and a finite-width microscopic continuum is therefore a question of regime: the former isolates the strict low-energy chiral response, whereas the latter resolves beyond-Luttinger-liquid excitations.

5. Conformal-block, symmetric-function, and parent-Hamiltonian generalizations

Calogero-Sutherland-type interaction has a precise conformal-field-theoretic meaning when conformal blocks contain only second-order degenerate fields. In that case the blocks satisfy second-order differential equations and can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality relates the two second-order degenerate fields, excited states are characterized by two partitions in the Virasoro case and by $\frac{a^2}{4\sinh^2(ax/2)}$0 partitions in $\frac{a^2}{4\sinh^2(ax/2)}$1 theories, and after extending the theory by a $\frac{a^2}{4\sinh^2(ax/2)}$2 field the resulting Hilbert-space integrals of motion become identical to the AFLT integrals known from Liouville theory. Upon bosonization they are expressed as sums of two, or in general $\frac{a^2}{4\sinh^2(ax/2)}$3, bosonic Calogero-Sutherland Hamiltonians coupled by a triangular interaction term (Estienne et al., 2011).

A different generalization replaces the inverse-square potential by a positive-semidefinite parent-Hamiltonian construction inspired by the factorized Calogero-Sutherland operator. Starting from a trial state

$\frac{a^2}{4\sinh^2(ax/2)}$4

with $\frac{a^2}{4\sinh^2(ax/2)}$5 a primary field in an RCFT with $\frac{a^2}{4\sinh^2(ax/2)}$6, one uses a null vector and the corresponding BPZ equation to obtain many-body annihilation operators $\frac{a^2}{4\sinh^2(ax/2)}$7, and then defines

$\frac{a^2}{4\sinh^2(ax/2)}$8

This yields continuum parent Hamiltonians for the Moore-Read and $\frac{a^2}{4\sinh^2(ax/2)}$9 Read-Rezayi states, for which those Jack-polynomial states are exact zero modes (Borutta et al., 13 Apr 2026). The construction is explicitly not a new simple two-body inverse-square potential; it is a Calogero-Sutherland-inspired framework for non-Abelian trial states. The paper also stresses its limits: it does not prove ground-state uniqueness, determine the excitation spectrum, or establish integrability (Borutta et al., 13 Apr 2026).

At the operator-theoretic level, infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland type act on the ring of symmetric functions, where Hermite and Laguerre symmetric functions arise as natural eigenfunctions and deformed CMS operators emerge upon restriction to suitable quotient rings (Desrosiers et al., 2011). Recent work further organizes circular, Hermite, Laguerre, Jacobi, and Bessel cases through a generalized Lassalle-Nekrasov correspondence, commuting $\frac{a^2}{4\sin^2(ax/2)}$0-operators and $\frac{a^2}{4\sin^2(ax/2)}$1-representations, generalized hypergeometric functions, and superintegrable integral chains (Liu et al., 26 Feb 2025). In this broader usage, “Calogero-Sutherland-type” no longer denotes only a specific pair potential; it denotes an operator class with a characteristic many-body singular structure and Jack-polynomial spectral organization.

6. Truncations, deformations, spinful systems, and applications

Several exact and partially exact models retain the characteristic interaction while modifying its range, analyticity, or internal degrees of freedom. Truncated Calogero-Sutherland models on the line introduce harmonic confinement together with finite-range two-body and three-body inverse-square interactions, truncated by neighbor index rather than by metric distance, and interpolate between the full Calogero-Sutherland model, the Jain-Khare nearest-/next-nearest-neighbor model, the Tonks-Girardeau gas, and a truncated Tonks-Girardeau extension (Pittman et al., 2016). On a circle, the truncated Hamiltonian contains inverse-square two-body terms and cotangent-cotangent three-body terms; it reduces to the Sutherland model when the range reaches all particles, and otherwise retains an exact Jastrow ground state together with a partially known excitation spectrum (Tummuru et al., 2016). A rational extension of the truncated model preserves the spectrum while replacing ordinary Laguerre radial functions by exceptional $\frac{a^2}{4\sin^2(ax/2)}$2 and $\frac{a^2}{4\sin^2(ax/2)}$3 Laguerre polynomials (Yadav et al., 2018).

Non-Hermitian and supersymmetric deformations preserve the interaction structure in other directions. A $\frac{a^2}{4\sin^2(ax/2)}$4-symmetric deformation shifts the coordinates into the complex plane, completely regularizes the inverse-square singularities, preserves integrability, and enlarges the spectrum by adding an infinite tower of previously non-normalizable states; for integral couplings it also yields a nonlinear conserved supersymmetry charge (Correa et al., 2019). Supersymmetric hyperbolic models derived by gauging matrix superfield theories have as their bosonic core the standard $\frac{a^2}{4\sin^2(ax/2)}$5 hyperbolic Calogero-Sutherland system, while the $\frac{a^2}{4\sin^2(ax/2)}$6 version introduces semi-dynamical $\frac{a^2}{4\sin^2(ax/2)}$7 spin variables and realizes a spin Calogero-Sutherland hyperbolic system (Fedoruk et al., 2019).

Spinful and Yangian-based formulations extend the same interaction to internal symmetry sectors. A classical hyperbolic model with two independent spin matrices $\frac{a^2}{4\sin^2(ax/2)}$8 and $\frac{a^2}{4\sin^2(ax/2)}$9 has Hamiltonian

$\wp(x;\omega_1,\omega_2)$0

and is completely integrable through an explicit Lax pair and classical $\wp(x;\omega_1,\omega_2)$1-matrix (Kharchev et al., 2017). The trigonometric quantum spin-Calogero-Sutherland model carries a Yangian $\wp(x;\omega_1,\omega_2)$2, admits Heisenberg-style Bethe-algebra conserved charges, and supports a Bethe-ansatz eigenbasis inside each irreducible Yangian representation; in the freezing limit this structure reduces to the Haldane-Shastry chain (Ferrando et al., 2023).

Applications and derivations outside the traditional many-body setting confirm the breadth of the interaction type. Exactly solvable Kondo models place a localized spin-$\wp(x;\omega_1,\omega_2)$3 impurity at the boundary of a two-component Calogero-Sutherland/Luttinger-liquid bath with inverse-square bulk and impurity couplings, producing Jastrow-product singlet ground states and a chiral-correlator description in which the bath particles are fractionalized to screen the impurity (Zhang et al., 2022). A gauge-theoretic construction starts from 2D Yang-Mills theory on a cylinder coupled to a colored particle and, after solving Gauss’ law and reducing to holonomy variables, yields an $\wp(x;\omega_1,\omega_2)$4 many-body Hamiltonian with

$\wp(x;\omega_1,\omega_2)$5

that is, an $\wp(x;\omega_1,\omega_2)$6 inverse-sine-square Calogero-Sutherland-type interaction derived from non-Abelian gauge dynamics rather than postulated directly (Tenser et al., 11 Jun 2026).

In this enlarged landscape, the phrase “Calogero-Sutherland-type interaction” has a stable core and a flexible perimeter. The stable core is the inverse-square many-body structure together with its integrable algebraic apparatus; the flexible perimeter consists of the many settings in which that structure persists after truncation, bosonization, gauging, supersymmetrization, conformal-block reformulation, or parent-Hamiltonian reconstruction.