Parent Hamiltonians of Jastrow wavefunctions

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.

arXiv:2107.02869SciPost Physics CoreNov 18, 20210 citations

Exact ground states of quantum many-body systems under confinement

Adolfo del Campo

Knowledge of the ground state of a homogeneous quantum many-body system can be used to find the exact ground state of a dual inhomogeneous system with a confining potential.We first identify the complete family of one-dimensional parent Hamiltonians with a ground state of Bijl-Jastrow form in free space.For each instance in this family, the dual trapped system is shown to include a one-body harmonic potential and two-body long-range interactions.The extension to anharmonic potentials and quantum solids with Nosanov-Jastrow ground-state wave functions is also presented.We apply this exact mapping to construct eigenstates of trapped systems from free-space solutions for different choices of the pair correlation functions entering the Bijl-Jastrow form.This construction is user friendly and allows one to readily rederive known instances of quasisolvable and integrable models, as well as to introduce unexplored examples.Among the latter, we describe a Lieb-Liniger-Coulomb model whose exact ground state is a McGuire soliton solution embedded in a harmonic trap.Knowledge of the ground state of a homogeneous quantum many-body system can be used to find the exact ground state of a dual inhomogeneous system with a confining potential.We first identify the complete family of one-dimensional parent Hamiltonians with a ground state of Bijl-Jastrow form in free space.For each instance in this family, the dual trapped system is shown to include a one-body harmonic potential and two-body long-range interactions.The extension to anharmonic potentials and quantum solids with Nosanov-Jastrow ground-state wave functions is also presented.We apply this exact mapping to construct eigenstates of trapped systems from free-space solutions for different choices of the pair correlation functions entering the Bijl-Jastrow form.This construction is user friendly and allows one to readily rederive known instances of quasisolvable and integrable models, as well as to introduce unexplored examples.Among the latter, we describe a Lieb-Liniger-Coulomb model whose exact ground state is a McGuire soliton solution embedded in a harmonic trap.

arXiv:2005.03904Physical Review ResearchOct 21, 202019 citations

Taxonomy of integrable and ground-state solvable models: Jastrow wave functions on graphs and parent Hamiltonians

Nilanjan Sasmal, Adolfo del Campo

We introduce a family of many-body systems of distinguishable continuous-variable particles in which interparticle interactions are set by the adjacency matrix of a graph. The ground-state wave function of such systems is of a generalized Jastrow form involving the product of pair-correlation functions over the edge set of the graph. These systems describe quantum fluids when the graph is complete, and the pair function has a well-defined permutation symmetry. In general, they provide the continuous-variable generalization of spin systems on graphs, with broken permutation symmetry. The corresponding parent Hamiltonian is shown to include (a) two-body interactions determined by the graph adjacency matrix and (b) three-body interactions over all possible 2-paths on the graph. Employing elements of graph theory, we chart the landscape of models, recovering known instances in the literature and providing numerous new examples of ground-state solvable models for which the system Hamiltonian, ground-state wave function, and corresponding energy eigenvalue are specified.We introduce a family of many-body systems of distinguishable continuous-variable particles in which interparticle interactions are set by the adjacency matrix of a graph. The ground-state wave function of such systems is of a generalized Jastrow form involving the product of pair-correlation functions over the edge set of the graph. These systems describe quantum fluids when the graph is complete, and the pair function has a well-defined permutation symmetry. In general, they provide the continuous-variable generalization of spin systems on graphs, with broken permutation symmetry. The corresponding parent Hamiltonian is shown to include (a) two-body interactions determined by the graph adjacency matrix and (b) three-body interactions over all possible 2-paths on the graph. Employing elements of graph theory, we chart the landscape of models, recovering known instances in the literature and providing numerous new examples of ground-state solvable models for which the system Hamiltonian, ground-state wave function, and corresponding energy eigenvalue are specified.

quant-pharXiv:2602.22315Physical review. B./Physical review. BFeb 27, 20261 citation

One-Dimensional Quantum Systems with Ground State of Jastrow Form Are Integrable

Jing Yang, Adolfo del Campo

Exchange operator formalism describes many-body integrable systems using phase-space variables involving an exchange operator that acts on any pair of particles. We establish an equivalence between models described by exchange operator formalism and the complete infinite family of parent Hamiltonians describing quantum many-body models with ground states of Jastrow form. This makes it possible to identify the invariants of motion for any model in the family and establish its integrability, even in the presence of an external potential. Using this construction we establish the integrability of the long-range Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact and Coulomb interactions in one dimension. We further identify a variety of models exemplifying the integrability of Hamiltonians in this family.Exchange operator formalism describes many-body integrable systems using phase-space variables involving an exchange operator that acts on any pair of particles. We establish an equivalence between models described by exchange operator formalism and the complete infinite family of parent Hamiltonians describing quantum many-body models with ground states of Jastrow form. This makes it possible to identify the invariants of motion for any model in the family and establish its integrability, even in the presence of an external potential. Using this construction we establish the integrability of the long-range Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact and Coulomb interactions in one dimension. We further identify a variety of models exemplifying the integrability of Hamiltonians in this family.

arXiv:2201.08282Physical Review LettersOct 3, 20227 citations

Long-range interactions in the quantum many-body problem in one dimension: Ground state

Saugata Ghosh

We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.

arXiv:quant-ph/0401116Physical Review EMar 30, 20042 citations

Parent Hamiltonian construction of generalized Calogero-Sutherland models

Anonymous, Andreas Feuerpfeil, Yasir Iqbal

The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square-type interactions. At interaction strength $λ=2$ , 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 semidefinite 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, viewed as one-dimensional chiral spin liquid wave functions, 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.The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square-type interactions. At interaction strength $λ=2$, 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 semidefinite 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, viewed as one-dimensional chiral spin liquid wave functions, 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.

arXiv:2604.11074Physical review. B./Physical review. BJul 16, 20260 citations

The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square-type interactions. At interaction strength

λ=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 semidefinite 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, viewed as one-dimensional chiral spin liquid wave functions, 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.