Calogero-Sutherland Model Overview

• Calogero-Sutherland model is a 1D quantum many-body system with inverse-square interactions, formulated in both rational (line) and trigonometric (circle) forms.
• Its exact solvability is underpinned by closed-form ground states and Jack-polynomial eigenfunctions, linking the system to conformal field theory and integrable hydrodynamics.
• Extensions such as finite-range, spin, supersymmetric, and PT-symmetric variants broaden its applications, connecting the model to fractional quantum Hall effects and Luttinger-liquid behavior.

The Calogero–Sutherland model is a class of integrable one-dimensional quantum many-body systems with inverse-square interactions, realized most prominently in a rational form on the line and a trigonometric form on the circle. In a standard circle formulation with $N$ particles on a circumference $L$, the Hamiltonian is

$$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$

while a line version uses the pair potential $1/(x_i-x_j)^2$ directly. Across these realizations, the model is distinguished by exact Jastrow-type ground states, solvability in terms of Jack polynomials and related algebraic structures, effective descriptions by extended conformal field theory and quantum hydrodynamics, and a large family of deformations including finite-range, spin, supersymmetric, PT-symmetric, and two-species variants (Bottesi et al., 7 Jan 2025, Motis et al., 23 Jul 2025, Bottesi et al., 2024).

1. Canonical formulations and coupling structures

The trigonometric Calogero–Sutherland Hamiltonian on the circle describes particles interacting through the periodic inverse-square potential $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$. In one formulation the model is written for $N$ spinless fermions on a circle of circumference $L$, with coupling $g=\lambda(\lambda-1)$ and thermodynamic density $n_0=N/L$ (Bottesi et al., 7 Jan 2025). A line formulation replaces the chord distance by the direct separation,

$$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$

and on a ring one may write the interaction in terms of

$L$0

In this language the dimensionless parameter $L$1 governs the physics and is related to the Dyson index by $L$2 in random-matrix theory (Motis et al., 23 Jul 2025).

A broader formulation uses root systems. For a positive-root set $L$3, the bosonic hyperbolic/trigonometric Calogero–Sutherland Hamiltonian is

$L$4

with

$L$5

For $L$6 this becomes

$L$7

while $L$8, $L$9, and $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$0 add $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$1 terms and, where applicable, single-particle $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$2 terms with a second coupling $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$3 (2002.03929).

This multiplicity of forms is not merely a change of notation. It organizes the model into rational, trigonometric, and hyperbolic branches, and into root-system families whose algebraic structures differ while preserving the inverse-square character of the interactions.

2. Exact ground states, spectra, and Jack-polynomial structure

One of the defining properties of the model is the existence of closed-form ground states. On the infinite line,

$$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$4

and on a ring of length $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$5,

$$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$6

For the ring model the ground-state energy per particle is

$$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$7

These formulas also exhibit the model’s direct connection to $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$8-ensemble joint densities through $$H_{\rm CS} \;=\; \sum_{j=1}^N\!-\frac{\partial^2}{\partial x_j^2} \;+\; g\sum_{i<j}\frac{(\pi/L)^2}{\sin^2\!\bigl[\pi(x_i-x_j)/L\bigr]}, \qquad g=\lambda(\lambda-1),$$9 (Motis et al., 23 Jul 2025).

In the trigonometric $1/(x_i-x_j)^2$0 system, eigenfunctions may be written as

$1/(x_i-x_j)^2$1

where

$1/(x_i-x_j)^2$2

and $1/(x_i-x_j)^2$3 are symmetric Jack polynomials with $1/(x_i-x_j)^2$4. The corresponding spectrum is

$1/(x_i-x_j)^2$5

subject to an overall shift depending on $1/(x_i-x_j)^2$6 (2002.03929).

The Jack-polynomial formulation becomes especially explicit after factoring out the ground state. For $1/(x_i-x_j)^2$7 and $1/(x_i-x_j)^2$8, one defines

$1/(x_i-x_j)^2$9

and the Jack polynomial $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$0 satisfies

$\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$1

A hidden Virasoro structure permits a recursive construction of Jack states from singular vectors, with rectangular partitions playing a fundamental role in the skew hierarchy (Wu et al., 2011).

A recurrent theme in later work is that the Jack description is not only a solution method but also a bridge to conformal blocks, Yangian symmetry, and fractional-quantum-Hall wavefunctions.

3. Effective field theory, $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$2, and hydrodynamics

In the thermodynamic limit $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$3 at fixed density $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$4, the low-energy theory reorganizes into chiral and anti-chiral sectors. One formulation identifies the effective theory as an extended conformal field theory with

$\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$5

and plasma velocity

$\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$6

In this second-quantized description the Hamiltonian becomes a bilinear combination of $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$7 generators, and the chiral density operator is proportional to the spin-one current (Bottesi et al., 7 Jan 2025).

A closely related effective Hamiltonian is written as

$\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$8

with $\sin^{-2}\!\bigl[\pi(x_i-x_j)/L\bigr]$9, $N$0, and $N$1 cubic in boson modes. Highest-weight states are labelled by the particle-number shift $N$2 and the pumped chiral charge $N$3, with

$N$4

and exact finite-size energy and momentum

$N$5

(Bottesi et al., 2023).

The chiral density dynamics is governed by the quantum Benjamin–Ono equation. In one form, for the right-moving density field $N$6,

$N$7

where $N$8 is the Hilbert transform (Bottesi et al., 2023). An operator-level derivation shows that this result does not rely on a semiclassical assumption. In that approach, the chiral density satisfies

$N$9

which the source identifies as precisely the quantum Benjamin–Ono equation (Bottesi et al., 2024).

At the classical hydrodynamic level, the continuum limit yields a continuity equation, an Euler equation with a nonlocal enthalpy, and a bidirectional Benjamin–Ono equation. The latter appears as a real reduction of the modified KP hierarchy, and a chiral reduction gives a Chiral Non-linear Equation whose large-density degeneration is the conventional Benjamin–Ono equation (Abanov et al., 2008). A common simplification is to identify the thermodynamic Calogero–Sutherland theory with an ordinary free compactified boson; the sources instead emphasize that the Hilbert spaces may be isomorphic while the time evolution is deformed by the interaction-dependent effective Hamiltonian (Bottesi et al., 7 Jan 2025).

4. Dynamic structure factor and correlation physics

The dynamic structure factor provides one of the sharpest diagnostics of the model’s integrable dynamics. In the chiral effective theory and in the repulsive regime $L$0, the zero-temperature structure factor takes the form

$L$1

with $L$2. In the strict thermodynamic limit $L$3, the interaction-induced shift vanishes and the resonance sits at $L$4. The corresponding free compactified boson has

$L$5

so the Hilbert spaces coincide while the resonance position differs away from the strict thermodynamic limit (Bottesi et al., 7 Jan 2025).

In the first-quantized treatment, the same response is described by a narrow rectangle,

$L$6

with approximate form

$L$7

In the overlap regime $L$8, $L$9, this rectangle collapses to the same delta peak at $g=\lambda(\lambda-1)$0, which the source presents as a precise matching between first- and second-quantized descriptions (Bottesi et al., 7 Jan 2025).

A broader account of dynamic correlations goes beyond the conventional Luttinger-liquid regime by combining sum rules with Monte Carlo sampling of the exact phase-space representation. At small $g=\lambda(\lambda-1)$1, the excitation is always phononic,

$g=\lambda(\lambda-1)$2

For weak coupling $g=\lambda(\lambda-1)$3, only the upper branch $g=\lambda(\lambda-1)$4 is populated and a Bogoliubov-type spectrum emerges. At $g=\lambda(\lambda-1)$5, the Tonks–Girardeau or ideal Fermi limit yields

$g=\lambda(\lambda-1)$6

For strong coupling $g=\lambda(\lambda-1)$7, the support again lies in $g=\lambda(\lambda-1)$8, but the dominant behavior shifts to a singular lower edge and, in the $g=\lambda(\lambda-1)$9 limit, to a quasi-crystalline Brillouin-zone structure (Motis et al., 23 Jul 2025).

The same source makes explicit that the exact ground state reproduces the universal Luttinger-liquid form with

$n_0=N/L$0

This suggests an exact solvable realization of the full Luttinger-liquid universality class rather than a merely asymptotic low-energy approximation (Motis et al., 23 Jul 2025).

5. Finite-range, truncated, and two-species generalizations

A finite-range generalization on the line introduces particles in a harmonic trap with inverse-square two-body and three-body interactions truncated to $n_0=N/L$1 neighbors. The Hamiltonian is

$n_0=N/L$2

Its exact ground state is

$n_0=N/L$3

with ground-state energy

$n_0=N/L$4

The limits $n_0=N/L$5, $n_0=N/L$6, and $n_0=N/L$7 recover, respectively, the full Calogero–Sutherland model, the Jain–Khare nearest- plus next-nearest-neighbor model, and the Tonks–Girardeau gas (Pittman et al., 2016).

A circle version truncates the inverse-square interaction to an $n_0=N/L$8-neighbor shell on each side and introduces an attractive three-body term. Its exact ground state is

$n_0=N/L$9

with an exact ground-state energy that reduces to the full-range Sutherland result for $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$0 and to the Jain–Khare case at $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$1. A part of the excitation spectrum can be constructed in terms of elementary symmetric functions $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$2 (Tummuru et al., 2016).

The finite-range models alter correlation properties in a controlled way. Numerically, increasing either $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$3 or the interaction range $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$4 enhances spatial antibunching, broadens the density cloud, suppresses off-diagonal long-range order in the one-body reduced density matrix, and broadens the momentum distribution tails (Pittman et al., 2016). The source explicitly interprets truncation as a screening of the inverse-square force; although full long-range integrability is broken, the model remains quasi-exactly solvable because the ground state and a large class of collective excitations are still known in closed form (Pittman et al., 2016).

A distinct extension is the two-species or deformed Calogero–Sutherland Hamiltonian,

$$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$5

whose exact eigenfunctions are built from super-Jack polynomials $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$6. In the interpretation given by the source, the $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$7-particles and $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$8-particles may be viewed as electrons and quasi-holes (Atai et al., 2016).

Finally, a rational extension of the truncated model adds a radial term $$H \;=\; -\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} \;+\;\frac{\hbar^2\,\lambda(\lambda-1)}{m}\sum_{1\le i<j\le N}\frac{1}{(x_i - x_j)^2},$$9 while leaving the energy eigenvalues unchanged: $L$00 The eigenfunctions are modified from classical Laguerre to exceptional $L$01 or, more generally, $L$02 Laguerre polynomials. The cases $L$03, $L$04, and $L$05 recover, respectively, the conventional truncated model, the extended Jain–Khare model, and the extended full-range Calogero–Sutherland model (Yadav et al., 2018).

6. Spin, supersymmetric, and PT-symmetric variants

Spin degrees of freedom admit several nontrivial integrable realizations. A classical hyperbolic model with two interacting spin variables $L$06 and $L$07 has Hamiltonian

$L$08

with $L$09, $L$10. The source constructs a Lax pair, a classical $L$11-matrix with spectral parameter, and a Hitchin-system realization, and proves complete integrability (Kharchev et al., 2017).

At the quantum level, the trigonometric spin-Calogero–Sutherland model can be analyzed by Bethe ansatz using Yangian symmetry. In additive coordinates,

$L$12

with $L$13 the spin permutation operator. From a transfer matrix with diagonal twist one obtains a Bethe algebra that refines the usual commuting Hamiltonians and yields a new eigenbasis generalizing the Yangian Gelfand–Tsetlin basis of Takemura–Uglov (Ferrando et al., 2023).

Supersymmetric extensions are likewise extensive. For root systems $L$14, $L$15, $L$16, and $L$17, $L$18 and $L$19 hyperbolic/trigonometric supersymmetric models can be constructed so that the bosonic core is the standard Calogero–Sutherland system. In the $L$20 construction, fermionic bilinears $L$21 enter the supercharges and Hamiltonian, and the supercharges satisfy

$L$22

The source emphasizes that the full supersymmetric Hamiltonians are determined explicitly, while the bosonic limit preserves the standard integrable model (2002.03929).

A gauged-matrix approach yields $L$23 and $L$24 hyperbolic Calogero–Sutherland systems whose bosonic sectors reduce to

$L$25

and to a $L$26-spin hyperbolic model in the $L$27 case (Fedoruk et al., 2019). The $L$28 $L$29-spin model admits explicit supercharges, a Lax representation, and a consistent reduction to the spinless hyperbolic Calogero–Sutherland model (Fedoruk, 2020). The $L$30 hyperbolic model has both classical and quantum supercharges; the source stresses that the quantum supercharges, unlike the classical ones, can be restricted to an invariant subsector without off-diagonal fermion operators (Fedoruk, 2019).

A non-Hermitian but PT-symmetric deformation acts by a constant complex shift $L$31, equivalently $L$32, and regularizes all two-body singularities while preserving PT symmetry. The deformed Hamiltonian is isospectral to the undeformed one, but the regularization adds an infinite tower of previously non-normalizable states. For integer coupling, additional degeneracy appears and a nonlinear conserved supersymmetry charge enlarges the ring of Liouville charges, while the Dunkl-operator integrability structure is maintained (Correa et al., 2019).

7. Conformal blocks, fractional quantum Hall structures, and universality

The Calogero–Sutherland model has a deep conformal-field-theoretic realization. Conformal blocks with only second-order degenerate Virasoro or $L$33-algebra fields satisfy second-order differential equations that, after $L$34 dressing and multiplication by appropriate Jastrow factors, become the Schrödinger equation of the trigonometric Calogero–Sutherland Hamiltonian,

$L$35

The same framework exhibits a duality $L$36, organizes excited states by two partitions in the Virasoro case and by $L$37 partitions in $L$38, and maps the Calogero–Sutherland integrals of motion to the operators $L$39 acting in $L$40Virasoro or $L$41-algebra modules (Estienne et al., 2011).

At special rational couplings

$L$42

the relevant conformal blocks reduce to Jack polynomials with $L$43-clustering. The cited cases include the Laughlin state for $L$44, $L$45, the Moore–Read state for $L$46, $L$47, and the Read–Rezayi series for general $L$48, $L$49 (Estienne et al., 2011). This is a direct instance in which Calogero–Sutherland eigenfunctions become fractional-quantum-Hall trial states rather than merely analogous structures.

The deformed two-species model makes this relation more explicit. Its collective-field description is formulated in terms of a chiral boson CFT with anyonic vertex operators, and the source identifies the two species naturally with electrons and quasi-hole excitations in Wen’s effective field theory of the fractional quantum Hall effect. Super-Jack polynomials then provide simple explicit formulas for an orthonormal CFT basis originally proposed in that context (Atai et al., 2016).

The model’s broader significance extends beyond conformal theory. One source describes the effective field theory of the Calogero–Sutherland model as a universality class of one-dimensional quantum hydrodynamic fluids and develops a ribbon geometry in which the current profile becomes

$L$50

a Poiseuille-type parabola that is nonetheless dissipationless because all one-dimensional layers drift with the same velocity (Bottesi et al., 2023). Another source places the model simultaneously in cold-atom waveguides, disordered metallic grains, and random-matrix theory, emphasizing that its exact ground state, sound velocity, Luttinger parameter, excitation spectrum, and dynamic correlations can be obtained analytically or by stochastic sampling of an exact phase-space integral (Motis et al., 23 Jul 2025).

Taken together, these developments establish the Calogero–Sutherland model not simply as a solvable inverse-square many-body problem, but as a structural node linking Jack and super-Jack polynomials, $L$51 symmetry, hydrodynamic Benjamin–Ono dynamics, Yangian and Dunkl-operator integrability, and fractional-quantum-Hall conformal blocks.