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arXiv:2610.32054 · Communications in Theoretical Physics

Real-space construction and selection rules of center-of-mass momentum eigenstates in two-dimensional many-body lattices

Ling Lin, Yongguan Ke, Chaohong Lee

quant-phSubmitted Oct 23, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

We present a comprehensive framework for the real-space construction of center-of-mass (c.m.) momentum eigenstates in two-dimensional (2D) quantum many-body systems. Unlike standard momentum-space approaches, our real-space formalism explicitly captures the non-uniformity of cyclic subspace dimensions, which are fundamentally governed by the spatial symmetries of the many-body configurations and particle statistics. We demonstrate that performing 2D real-space translations on multi-particle states naturally induces non-trivial translation phase factors associated with the characters of these stabilizer subgroups, leading to a renormalization of permissible momentum values. Notably, we identify an anomalous cross-dimensional momentum coupling in highly symmetric particle distributions, where the permissible momentum values in one direction are strictly constrained by those in the perpendicular direction. We emphasize the algorithmic necessity of this coupling, as neglecting it leads to missing or duplicated basis states in numerical implementations. Concrete examples are provided to demonstrate the practical implementation and physical consequences of this framework.

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We present a comprehensive framework for the real-space construction of center-of-mass (c.m.) momentum eigenstates in two-dimensional (2D) quantum many-body systems. Unlike standard momentum-space approaches, our real-space formalism explicitly captures the non-uniformity of cyclic subspace dimensions, which are fundamentally governed by the spatial symmetries of the many-body configurations and particle statistics. We demonstrate that performing 2D real-space translations on multi-particle states naturally induces non-trivial translation phase factors associated with the characters of these stabilizer subgroups, leading to a renormalization of permissible momentum values. Notably, we identify an anomalous cross-dimensional momentum coupling in highly symmetric particle distributions, where the permissible momentum values in one direction are strictly constrained by those in the perpendicular direction. We emphasize the algorithmic necessity of this coupling, as neglecting it leads to missing or duplicated basis states in numerical implementations. Concrete examples are provided to demonstrate the practical implementation and physical consequences of this framework.
The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.
A method is presented for the calculation of the one-body and two-body density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance, in the case of a nucleus (a finite self-bound system). We restore translational invariance by using the so-called fixed center-of-mass approximation for constructing an intrinsic nuclear ground state wavefunction by starting from a non-translationally invariant wavefunction and applying a projection prescription. We discuss results for the one-body and two-body momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator orbitals, as the initial non-translationally invariant wavefunction. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.
We present a systematic framework to construct model Hamiltonians that have unconventional superconducting pairing states as exact energy eigenstates, by incorporating multibody interactions (i.e., interactions among more than two particles). The multibody interactions are introduced in a form of the local density-density coupling in such a way that any pair configuration in real space has a constant interaction energy by canceling the two-body and multibody interactions. Our approach is applicable to both spinless and spinful models in any spatial dimensions and on any bipartite lattices, facilitating an exhaustive extension of Yang's s − wave η − pairing state to various other unconventional pairing symmetries ( p − wave , d − wave , f − wave , etc.). Particularly, the constructed eigenstates have off-site pairs with finite center-of-mass momentum, which leads to superconducting states with either even-parity and spin-triplet or odd-parity and spin-singlet symmetry. We verify that the two-dimensional spinful Hubbard model on a square lattice with the multibody interactions has the spin-triplet d − wave pairing state as an energy eigenstate, which can be regarded as a quantum many-body scar state as evidenced from the numerical analysis of the pair correlation function, entanglement entropy, and level statistics. We also discuss other examples, including spin-triplet f − wave pairing states on a honeycomb lattice and spin-singlet p − wave pairing states in a one-dimensional chain. These findings open up the possibility of realizing nonequilibrium unconventional superconductivity in a long-lived manner protected against thermalization.
A method is presented for the calculation of the one‐body (1DM) and two‐body (2DM) density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance (TI), in the case of a nucleus (a finite self‐bound system). We restore TI by using the so‐called fixed center‐of‐mass (CM) approximation for constructing an intrinsic nuclear ground state wavefunction (WF) by starting from a non‐translationally invariant (nTI) WF and applying a projection prescription. We discuss results for the one‐body (OBMD) and two‐body (TBMD) momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator (HO) orbitals, as the initial nTI WF. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.
A class of quantum many-body models of arbitrary dimension and arbitrary statistics of particles, for which exact eigenstates may be obtained, is found. It is assumed that: (i) models contain two (or 2m) kinds of particles with 'symmetric' matrix elements of pairwise interaction (all potentials coincide with each other to within a sign and wavefunctions of free particles of two components coincide to within a phase factor; pairwise interactions are otherwise arbitrary); (ii) there exists the degeneracy of (the sum) of free-particle spectra. Exact many-body eigenstates correspond to a condensation of non-interacting composite particles ('excitons') which are not exactly bosons, into a single quantum state, and to excitations over the condensate. The origin of the possibility of exact solution is the symmetry under the continuous rotations in the isospin space of two components, to which Bogolubov canonical transformations with parameters u, nu independent of momentum correspond. The class of such models comprises, in particular, two-dimensional electron-hole systems in a strong magnetic field.
Building on the development of momentum state lattices (MSLs) over the past decade, we introduce a simple extension of this technique to higher dimensions. Based on the selective addressing of unique Bragg resonances in matter-wave systems, MSLs have enabled the realization of tight-binding models with tunable disorder, gauge fields, non-Hermiticity, and other features. Here, we examine and outline an experimental approach to building scalable and tunable tight-binding models in two dimensions describing the laser-driven dynamics of atoms in momentum space. Using numerical simulations, we highlight some of the simplest models and types of phenomena this system is well-suited to address, including flat-band models with kinetic frustration and flux lattices supporting topological boundary states. Finally, we discuss many of the direct extensions to this model, including the introduction of disorder and non-Hermiticity, which will enable the exploration of new transport and localization phenomena in higher dimensions.