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arXiv:2510.00110 · Physical review. B./Physical review. B

Continuum fractons: Quantization and the few-body problem

Ylias Sadki, Abhishodh Prakash, Shivaji L. Sondhi

Published Jun 9, 2026• 0 citations
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Abstract

We formulate a continuum quantum mechanics for nonrelativistic, dipole-conserving fractons. Imposing symmetries and locality results in novel phenomena absent in ordinary quantum mechanical systems. A single fracton has a vanishing Hamiltonian, and thus its spectrum is entirely composed of zero modes. For the two-body problem, the Hamiltonian is perfectly described by Sturm-Liouville (SL) theory. The effective two-body Hamiltonian is an SL operator on ( − 1 , 1 ) whose spectral type is set by the edge behavior of the pair inertia function K ( x ) ∼ | x − x edge | θ . We identify a sharp transition at θ = 2 : for θ < 2 the spectrum is discrete and wave packets reflect from the edges, whereas for θ > 2 the spectrum is continuous and wave packets slow down and, dominantly, squeeze into asymptotically narrow regions at the edges. For three particles, the differential operator corresponding to the Hamiltonian is piecewise defined, requiring several “matching conditions” which cannot be analyzed as easily. We proceed with a lattice regularization that preserves dipole conservation and implicitly selects a particular continuum Hamiltonian that we analyze numerically. We find a spectral transition in the three-body spectrum and find evidence for quantum analogs of fracton attractors in both eigenstates and in the time evolution of wave packets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.

DOI
10.1103/2pt1-d469

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1. Schroedinger operators : spectra, scattering and semiclassical behaviour.- Negative discrete spectrum of the Schroedinger operator with large coupling constant : qualitative discussion.- Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit.- On the asymptotics of discrete spectrum for the Schroedinger operator in electric and homogeneous magnetic fields.- Asymptotics of the discrete spectrum of Hamiltonians of quantum systems with a homogeneous magnetic field.- Asymptotics of eigenvalues for many-particle Hamiltonians at symmetry subspaces.- Spectral asymptotics with highly accurate remainder estimates.- Bound states and resonances in quantum wires.- Spectral properties of the operators H? = ??XX + p(x)? + v(?x), p is periodic.- Discrete spectrum for a periodic Schroedinger operator perturbed by a decreasing potential.- On a complete description of the principal discrete series of spectral invariants of the Hill operator.- WKB-approximations from the perturbation theory viewpoint.- Moving potentials and the completeness of wave operators. Part II : Propagating observables on scattering states.- Representation of the three-body S-matrix in terms of effective amplitudes.- A three-body one-dimensional system with increasing interaction : wave function asymptotics.- Surprises of quantum tunneling (something about the velocity of sub-barrier motion).- 2. Point and contact interactions, self-adjoint extensions.- Lifshitz tails and non-Lifshitz tails for one-dimensional random point interactions.- Point interactions with an internal structure as limits of nonlocal point interactions.- Green's function for the Aharonov-Bohm effect with a non-Abelian gauge group.- A model of zero-width slits and the real diffraction problem.- Construction of an inelastic scatterer in nanoelectronics by the extension-theory methods.- On boundary theory for Schroedinger operators and stochastic processes.- Dirac Hamiltonian with Coulomb potential and contact intraction on a sphere.- Self-adjoint extensions of Schroedinger operators with singular potentials.- Currents and the extension theory.- 3. Quantum chaos.- Chaotic quantum billiards.- Relevance of the localization to quasienergy statistics in quantum chaotic systems.- Singular continuous quasi-energy spectrum in the kicked rotator with separable perturbation : onset of quantum chaos ?.- Relation between correlation functions and spectrum statistics in the region of quantum chaos.- 4. Other topics.- Localization effects in non-homogeneous dielectrics.- A self-adjoint dilation of the linear Boltzmann operator.- Kato problem for functional-differential equations and difference Schroedinger operators.- An existence theorem for some nonlinear nonlocal Schroedinger operators and the solitonlike dynamics for the corresponding dynamical systems.- The Sturm-Liouville problem with a potential linear in spectral parameter.- Stochastic model of tree growth.- Critical phenomenon in the stationary model of random growth.- On a quantum-classical connection, hidden symmetries and a model of Josephson junction.- Open multiquantum systems. Method of a generating functional.- Generalized functions and their applications.- List of unpublished contributions.- List of participants.

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We formulate a continuum quantum mechanics for nonrelativistic, dipole-conserving fractons. Imposing symmetries and locality results in novel phenomena absent in ordinary quantum mechanical systems. A single fracton has a vanishing Hamiltonian, and thus its spectrum is entirely composed of zero modes. For the two-body problem, the Hamiltonian is perfectly described by Sturm-Liouville (SL) theory. The effective two-body Hamiltonian is an SL operator on ( − 1 , 1 ) whose spectral type is set by the edge behavior of the pair inertia function K ( x ) ∼ | x − x edge | θ . We identify a sharp transition at θ = 2 : for θ < 2 the spectrum is discrete and wave packets reflect from the edges, whereas for θ > 2 the spectrum is continuous and wave packets slow down and, dominantly, squeeze into asymptotically narrow regions at the edges. For three particles, the differential operator corresponding to the Hamiltonian is piecewise defined, requiring several “matching conditions” which cannot be analyzed as easily. We proceed with a lattice regularization that preserves dipole conservation and implicitly selects a particular continuum Hamiltonian that we analyze numerically. We find a spectral transition in the three-body spectrum and find evidence for quantum analogs of fracton attractors in both eigenstates and in the time evolution of wave packets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.
We initiate the study of the classical mechanics of nonrelativistic fractons in its simplest setting—that of identical one-dimensional particles with local Hamiltonians characterized by a conserved dipole moment in addition to the usual symmetries of space and time translation invariance. We introduce a family of models and study the N -body problem for them. We find that locality leads to a “Machian” dynamics in which a given particle exhibits finite inertia only if within a specified distance of another particle. For well-separated particles, this dynamics leads to immobility, much as for quantum models of fractons discussed before. For two or more particles within inertial reach of each other at the start of motion, we obtain an interesting interplay of inertia and interactions. Specifically, for a solvable “inertia only” model of fractons, we find that two particles always become immobile at long times. Remarkably, three particles generically evolve to a late time state with one immobile particle and two oscillating about a common center of mass with generalizations of such “Machian clusters” for N > 3 . Interestingly, these Machian clusters exhibit physical limit cycles in a Hamiltonian system even though mathematical limit cycles are forbidden by Liouville's theorem. Published by the American Physical Society 2024
We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N , despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of “order-by-disorder” borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics. Published by the American Physical Society 2024
One-dimensional fracton systems can exhibit perfect localization, failing to reach thermal equilibrium under arbitrary local unitary time evolution. We investigate how this nonergodic behavior manifests in the dynamics of a driven fracton system, specifically a one-dimensional Floquet quantum circuit model featuring conservation of a U(1) charge and its dipole moment. For a typical basis of initial conditions, a majority of states heat up to a thermal state at near-infinite temperature. In contrast, a small number of states flow to a localized steady state under the Floquet time evolution. We refer to these athermal steady states as ``dynamical scars, in analogy with the scar states observed in the spectra of certain many-body Hamiltonians. Despite their small number, these dynamical scars are experimentally relevant due to their high overlap with easily-prepared product states. Each scar state displays a single agglomerated fracton peak, in agreement with the steady-state configurations of fractonic random circuits. The details of these scars are insensitive to the precise form of the Floquet operator, which is constructed from random unitary matrices. Rather, dynamical scar states arise directly from fracton conservation laws, providing a concrete mechanism for the appearance of scars in systems with constrained quantum dynamics.
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.
We present the path-integral approach to treat quantum phase transitions and critical phenomena for few-body quantum systems. Allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. Imposing the constraint that any change in space time must preserve the scaling invariance of Brownian paths, we show that the mapped classical lattice system has a known scaling behavior when the particle is free, which breaks down when the strength of the interaction potential reaches a certain value. In principle, any quantity with known scaling behavior may be used to determine the transition point. We illustrate the method by numerically evaluating the correlation length and the radial mean distance for a system composed of a single particle in the presence of an attractive Pöschl-Teller potential in one and three dimensions. The method is general and has potential applicability for large systems.
1. Schroedinger operators : spectra, scattering and semiclassical behaviour.- Negative discrete spectrum of the Schroedinger operator with large coupling constant : qualitative discussion.- Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit.- On the asymptotics of discrete spectrum for the Schroedinger operator in electric and homogeneous magnetic fields.- Asymptotics of the discrete spectrum of Hamiltonians of quantum systems with a homogeneous magnetic field.- Asymptotics of eigenvalues for many-particle Hamiltonians at symmetry subspaces.- Spectral asymptotics with highly accurate remainder estimates.- Bound states and resonances in quantum wires.- Spectral properties of the operators H? = ??XX + p(x)? + v(?x), p is periodic.- Discrete spectrum for a periodic Schroedinger operator perturbed by a decreasing potential.- On a complete description of the principal discrete series of spectral invariants of the Hill operator.- WKB-approximations from the perturbation theory viewpoint.- Moving potentials and the completeness of wave operators. Part II : Propagating observables on scattering states.- Representation of the three-body S-matrix in terms of effective amplitudes.- A three-body one-dimensional system with increasing interaction : wave function asymptotics.- Surprises of quantum tunneling (something about the velocity of sub-barrier motion).- 2. Point and contact interactions, self-adjoint extensions.- Lifshitz tails and non-Lifshitz tails for one-dimensional random point interactions.- Point interactions with an internal structure as limits of nonlocal point interactions.- Green's function for the Aharonov-Bohm effect with a non-Abelian gauge group.- A model of zero-width slits and the real diffraction problem.- Construction of an inelastic scatterer in nanoelectronics by the extension-theory methods.- On boundary theory for Schroedinger operators and stochastic processes.- Dirac Hamiltonian with Coulomb potential and contact intraction on a sphere.- Self-adjoint extensions of Schroedinger operators with singular potentials.- Currents and the extension theory.- 3. Quantum chaos.- Chaotic quantum billiards.- Relevance of the localization to quasienergy statistics in quantum chaotic systems.- Singular continuous quasi-energy spectrum in the kicked rotator with separable perturbation : onset of quantum chaos ?.- Relation between correlation functions and spectrum statistics in the region of quantum chaos.- 4. Other topics.- Localization effects in non-homogeneous dielectrics.- A self-adjoint dilation of the linear Boltzmann operator.- Kato problem for functional-differential equations and difference Schroedinger operators.- An existence theorem for some nonlinear nonlocal Schroedinger operators and the solitonlike dynamics for the corresponding dynamical systems.- The Sturm-Liouville problem with a potential linear in spectral parameter.- Stochastic model of tree growth.- Critical phenomenon in the stationary model of random growth.- On a quantum-classical connection, hidden symmetries and a model of Josephson junction.- Open multiquantum systems. Method of a generating functional.- Generalized functions and their applications.- List of unpublished contributions.- List of participants.