Topological exchange statistics in one dimension

Harshman, N. L.; Knapp, Ádám

doi:10.1103/physreva.105.052214

Cited by 10 publications

(15 citation statements)

References 100 publications

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“…See also refs. [20,30] for similar results in continuous spaces. 6 The sign representation can also be written as [−] ( ) = sgn( ), where sgn( ) stands for the signature of .…”

Section: Identical Walkers In One Dimensionmentioning

confidence: 62%

“…The key to this problem is the indistinguishability of identical particles, where physical observables must be invariant under permutations of multiparticle coordinates. As is well known, this indistinguishability always makes the multiparticle configuration space an orbit space [13,[17][18][19][20]. The basic idea behind this is to regard the permutation invariance as a gauge symmetry (i.e., redundancy in description).…”

Section: Identical Walkers In One Dimensionmentioning

confidence: 99%

“…The first regards the configuration space of identical particles as the orbit space ( − Δ )/ , where is the -fold Cartesian product of a single-particle configuration space and Δ ⊂ is the set of fixed points under the action of the symmetric group [13,[17][18][19]. On the other hand, the second includes the fixed points and regards the configuration space as the orbit space / [20]. The difference between these two formulations is very subtle (especially in lattices) and beyond the scope of this note.…”

Section: Identical Walkers In One Dimensionmentioning

confidence: 99%

“…A typical example for such configuration spaces is that for a single walker on a periodic lattice. Another typical example is the configuration space for identical walkers on an arbitrary lattice, where the indistinguishability of identical particles makes their configuration space an orbit space [13,[17][18][19][20]. We show that the time-evolution kernel on the orbit space Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Walk on Orbit Spaces

Ohya¹

2023

Preprint

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Inspired by the covering-space method in path integral on multiply-connected spaces, we here present a universal formula of time-evolution kernels for continuous-and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. We show that the time-evolution kernel on Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulae of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on Λ/Γ, all of which are constructed in exactly the same way as for the timeevolution kernel.

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“…See also refs. [20,30] for similar results in continuous spaces. 6 The sign representation can also be written as [−] ( ) = sgn( ), where sgn( ) stands for the signature of .…”

Section: Identical Walkers In One Dimensionmentioning

confidence: 62%

Section: Identical Walkers In One Dimensionmentioning

confidence: 99%

Section: Identical Walkers In One Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum Walk on Orbit Spaces

Ohya¹

2023

Preprint

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“…However, it was recently demonstrated that a different form of anyonic exchange statistics intrinsic to 1D is possible if the two-body hard-core relation is relaxed but a three-body hardcore constraint is enforced [16,17]. Hard-core three-body constraints in one-dimension also make the configuration space not simply-connected, and this allows for multi-valued wave functions and non-trivial exchange phases.…”

Section: Introductionmentioning

confidence: 99%

Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension

Nagies,

Wang,

Knapp

et al. 2024

SciPost Phys.

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Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.

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Topologically nontrivial three-body contact interaction in one dimension

Ohya

2023

Progress of Theoretical and Experimental Physics

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It is known that three-body contact interactions in one-dimensional 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 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 one-dimensional 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.

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Topological exchange statistics in one dimension

Cited by 10 publications

References 100 publications

Quantum Walk on Orbit Spaces

Quantum Walk on Orbit Spaces

Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension

Topologically nontrivial three-body contact interaction in one dimension

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