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:2310.16576 · Progress of Theoretical and Experimental Physics

Topologically nontrivial three-body contact interaction in one dimension

Satoshi Ohya

Published Dec 7, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.

DOI
10.1093/ptep/ptad149

Related papers

Quantum Mechanics of One-Dimensional Three-Body Contact Interactions

The quantum mechanical description of topologically nontrivial three-body contact interactions in one dimension is not well understood. This study explores the Hamiltonian description of these interactions using the path-integral formalism.

JPS Hot Topics•Jan 1, 2024•0 citations

Twisted gauge theory model of topological phases in three dimensions

Yidun Wan, Juven Wang, Huan He

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a 4-cocycle $\ensuremath{\omega}$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the $(3+1)$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.

arXiv:1409.3216•Physical Review B•Jul 1, 2015•126 citations

Discrete Scale Invariance and U(2) Family of Two-Body Contact Interactions in One Dimension

Satoshi Ohya

Abstract Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U (2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U (2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.

arXiv:2404.00275•Few-Body Systems•Mar 29, 2024•1 citation

The Tenfold Way: Fermionic Systems with N-body interactions

Vijay B. Shenoy, Adhip Agarwala, Arijit Haldar

We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.

Bulletin of the American Physical Society•Mar 14, 2017•0 citations

The three-body problem in dimension one: From short-range to contact interactions

Giulia Basti, Claudio Cacciapuoti, Domenico Finco +1 more

We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.

arXiv:1803.08358•Journal of Mathematical Physics•Jul 1, 2018•16 citations

Singular Cluster Interactions in Few-Body Problems

Sergio A Albeverio, Pavel Kurasov

The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.

CRC Press eBooks•Apr 24, 2019•0 citations
Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+13+13+1 dimensions, based on a generic finite group GGG and a 4-cocycle $\ensuremath{\omega}$ over GGG. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z)SL(3,\mathbb{Z})SL(3,Z) generators as the modular SSS and TTT matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)(3+1)(3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group GGG. This work is presented to be accessible for a wide range of physicists and mathematicians.
Abstract Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U (2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U (2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.
We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.