Topologically Nontrivial Three-Body Contact Interaction in One Dimension

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Introduction
One of the most fundamental classes of interactions in nature would be contact interaction: in the low-energy regime where particles' wavelengths are longer than a characteristic range of interactions, particles cannot resolve their microscopic details so that any short-range interaction is expected to be described by a contact (i.e., zero-range or pointlike) interaction.For interparticle interactions in quantum many-body problems, there are two main approaches to study such contact interactions.
The first approach is to consider a potential whose support is the set of particle coincidence points Δ.In this approach, contact interactions are simply described by .The second approach, on the other hand, is to remove Δ from the many-body configuration space and to consider the subtracted space − Δ as a new configuration space, because the coincidence points generally become singularities-such as branch points-at which wavefunctions are not well-defined.In this approach, contact interactions are described by boundary (or connection) conditions around Δ.These two approaches basically yield the same results.However, there exists a case in which the latter has a big advantage over the former: it is the case where the fundamental group 1 ( − Δ) becomes nontrivial and topology comes into play.In local quantum theory of nonidentical particles, there are just two examples of such topologically nontrivial contact interactions [1][2][3]]. 1he first example is two-body contact interaction in two dimensions, where the set of two-body coincidence points Δ becomes codimension-two singularities in the -body configuration space.In this case, the fundamental group is given by the pure braid group [4]-a group of pure braids on strands with no double-crossings-and all the topologically nontrivial two-body contact interactions are classified by unitary irreducible representations of .Just as in Wilczek's charge-flux picture of anyons [5,6], these singularities can be viewed as infinitely-thin magnetic fluxes and described by background gauge fields, which enter into the theory through covariant derivatives rather than through a potential .
The second example is three-body contact interaction in one dimension, where the set of threebody coincidence points Δ also becomes codimension-two singularities in the -body configuration space.In this case, the fundamental group is given by the pure twin group [7]-a group of planar pure braids on strands with no triple-crossings-whose unitary irreducible representations classify three-body contact interactions for nonidentical particles in one dimension.Topological and grouptheoretical aspects of such interactions have recently been studied by Harshman and Knapp [1][2][3].However, the operator realization of such interactions is still missing.
The purpose of this paper is to identify the Hamiltonian operator for topologically nontrivial threebody contact interactions in one dimension.Focusing on spinless particles, we will show that topologically nontrivial three-body contact interactions are associated with background Abelian gauge fields in the configuration space.Just as in two-body contact interactions in two dimensions, these gauge fields describe infinitely-thin magnetic fluxes penetrating through the codimension-two singularities in the configuration space.
The paper is organized as follows.Section 2 is devoted to a detailed study of three-body problems of nonidentical spinless particles in one dimension.Since 3 is isomorphic to the additive integer group Z [8,9], all the three-body contact interactions are just classified by unitary representations of Z.We will show that, by using the path-integral formalism, there exist two equivalent descriptions of topologically nontrivial three-body contact interactions: the one is to impose a twisted boundary condition around the three-body coincidence point and the other is to introduce a background gauge field that describes an infinitely-thin magnetic flux at the three-body coincidence point.We will see that these two descriptions are related through a gauge transformation and hence physically equivalent.Section 3 studies a generalization to (≥ 3)-body problems.By generalizing the gauge-field description, we will introduce an ( − 1)( − 2)/3!-parameter family of -body Hamiltonians that corresponds to one particular one-dimensional unitary representations of .We conclude in section 4. Appendix A presents a brief review of the pure twin group .
2 Gauge-field description of three-body contact interaction In general, topologically nontrivial contact interactions are originated from nontrivial topology of many-body configuration space and dictated solely in terms of representation theory of the fundamental group.This interplay between topology and representation theory is best described by Feynman's path integral.In this section, we illustrate this idea by focusing on three-body problems of nonidentical spinless particles on the line R with three-body contact interaction.

Coordinate systems for the three-body configuration space
To begin with, let us define some notation and coordinate systems.Let ∈ R be the coordinate of the th particle of mass , where = 1, 2, 3.The set of three-body coincidence points is given by the following codimension-two locus in R 3 : The three-body configuration space of nonidentical particles in one dimension is then defined by the following subtracted space: In the next subsection, we will see that Δ 3 corresponds to the support of a fictitious infinitely-thin magnetic flux and becomes a branch-point singularity of three-body wavefunctions.Now, suppose that the three-body system is invariant under the spatial translation ( 1 , 2 , 3 ) ↦ ( 1 + , 2 + , 3 + ), where is an arbitrary real.In this case, the total momentum is conserved so that we can separate the center-of-mass motion from the system.The most convenient coordinate system for such a system is the Jacobi coordinate system ( 1 , 2 , 3 ) given by which satisfy the following identities: where Physically, 1 is the relative coordinate between the first and second particles; 2 is the relative coordinate between the center-of-mass of the first and second particles and the third particle; and 3 describes the center-of-mass of the three particles.( = 1, 2) is the reduced mass with respect to and 3 is the total mass.Noting that the condition 1 = 2 = 3 is equivalent to the condition 1 = 2 = 0, we see that the three-body configuration space ( 2) is factorized as follows: where R = { 3 ∶ −∞ < 3 < ∞} is the one-dimensional space of center-of-mass motion and R2 = {( 1 , 2 ) ∈ R2 ∶ ( 1 , 2 ) ≠ (0, 0)} is the one-punctured plane of relative motion.It is now obvious that the three-body configuration space is a multiply-connected space because there is a hole in the space of relative motion.
For the following discussion, it is convenient to introduce a polar coordinate system ( , ) in the one-punctured plane R2 .We write where ≔ arctan = arctan Here 0 (> 0) is an arbitrary reference mass scale introduced to assign the dimension of length to the radius .Now it is obvious that the three-body configuration space (6) can be decomposed as where R + = { ∶ 0 < < ∞} and 1 = { ∶ 0 ≤ < 2 (mod 2 )}.Physically, describes the distance from the three-body coincidence point; describes the ratio between the center-of-mass coordinate

Path integral on multiply-connected spaces: General theory
Path integrals on multiply-connected spaces are best described by the Dowker's covering-space method [10].In this section, we briefly review this method by following (and generalizing) [11,12].Let be a multiply-connected space of the form = ̃ / , where ̃ is the universal covering space of and ≅ 1 ( ) ⊂ Isom( ̃ ) is a discrete subgroup of the isometry of ̃ that is isomorphic to the fundamental group of .The time-evolution kernel (the integral kernel of time-evolution operator) for quantum particles on is given by where stands for the action of on ∈ ̃ .Here ∶ → (1) is a one-dimensional unitary representation of and satisfies the following conditions for any , ′ ∈ : where the overline ( ) stands for the complex conjugate.̃ ( , ) is the time-evolution kernel on the universal covering space ̃ ; it is given by the Feynman's path integral: where is a Lagrangian of classical mechanics on the universal covering space ̃ .Once given the kernel , the time-evolution of a wavefunction is given by the map ( ) ↦ ( ), where Since ̃ is defined on the universal covering space, the domain of the function (⋅, ⋅) defined by (10) can be naturally extended from × to ̃ × ̃ .In particular, it satisfies the following identity: Correspondingly, the domain of the wavefunction (13) can also be extended from to ̃ and satisfies the following identity: This identity turns out to give boundary conditions for quantum systems on .

Path-integral description of three-body contact interaction
Now let us turn to the problem of three nonidentical particles in one dimension with a three-body contact interaction.First, the fundamental group of the three-body configuration space ( 9) is isomorphic to 1 ( 1 ), which is given by the additive group of integers: Physically, this fundamental group describes winding numbers of three-particle trajectories around the codimension-two singularity Δ 3 .Next, we have to find one-dimensional unitary representations of Z. Since Z is a free group generated by a single generator , there exists a one-parameter family of one-dimensional unitary representations of Z labeled by a real parameter .It is given by the map [ ] ∶ Z → (1), where Figure 1: A three-particle trajectory with the winding number = 1.In the configuration-space picture on the left-hand side, such a trajectory is described by a closed loop encircling the codimension-two singularity Δ 3 .In the space-time picture on the right-hand side, such a trajectory is described by a planar pure braid on three strands.For planar pure braids, see appendix A.
Correspondingly, for three nonidentical particles in one dimension, there exists a one-parameter family of time-evolution kernels given by which satisfies the following twisted boundary condition (see ( 14)): Note that, in the polar coordinate system introduced in section 2.1, the action of ∈ Z is written as ( , , 3 ) ↦ ( , + 2 , 3 ).It is now clear that the one-dimensional unitary representation [ ] describes the Aharonov-Bohm phase acquired by a three-particle state encircling the codimensiontwo singularity; see figure 1.It is also clear from (15) that the three-body wavefunction becomes a multi-valued function of with the branch-point singularity at = 0. Now, there exists an alternative, unitary-equivalent description to this three-body system.To see this, let us consider a gauge transformation ( ) ↦ ′ ( ) = ( , 0 ) −1 ( ), where ( , 0 ) is the Wilson line given by Here , 0 = { ( ) ∈ 3-body ∶ (0) = 0 , ( ) = , ∈ [0, ]} is a path from 0 to with 0 being an arbitrary reference point.( ) stands for the angle defined by the last line of (8b). is a gauge-field one-form given by = ℏ where Under this gauge transformation, eq. ( 18) is transformed as [ ] ( , ) ↦ [ ]′ ( , ), where Here the last line follows from exp(−( /ℏ ) = e − ( ( ( ))− ( (0))) and ( ( )) − ( (0)) = ( ) − ( ) = ( ) − ( ) − 2 for a path ( ) that satisfies the conditions (0) = and ( ) = .Notice that, in this new gauge, the time-evolution kernel (23) satisfies the periodic boundary condition Note also that the gauge-field one-form ( 21) describes an infinitely-thin magnetic flux penetrating through the three-body coincidence point 1 = 2 = 3 (or 1 = 2 = 0) in 3-body .In fact, the field-strength two-form = takes the following form: which describes the infinitely-thin magnetic field along the codimension-two singularity Δ 3 .Hence, there exist two equivalent descriptions for topologically nontrivial three-body contact interactions: the first is the twisted boundary condition (19) with no gauge field, and the second is the periodic boundary condition (24) with the background Abelian gauge field (22).These two descriptions are unitarily equivalent so that they describe the same physical system.

Hamiltonian description of three-body contact interaction
Let us finally study Hamiltonian descriptions of the system.Under the twisted boundary condition (19), the classical Hamiltonian is just given by the Legendre transform of the classical Lagrangian ( , ̇ ) in the path integral (18).If is of the form ), then the corresponding Hamiltonian operator in the quantum three-body problem is just given by On the other hand, under the periodic boundary condition (24), the classical Hamiltonian is given by the Legendre transform of the classical Lagrangian ( , ̇ ) − =1 ̇ in the path integral (23).The corresponding Hamiltonian operator then becomes Note that these two operators are unitarily equivalent and mutually related through the unitary transformation periodic = ( , 0 ) −1 twisted ( , 0 ).

A generalization to -body problems
Now we wish to generalize the results in the previous section to (≥ 3)-body problems of nonidentical particles.Obviously, the key is the fundamental group of -body configuration space.Under threebody contact interactions, the -body configuration space of nonidentical particles on R is given by where Δ is the set of three-body coincidence points defined by The fundamental group of this space has been studied in the mathematical literature: in the late 1990s, Khovanov [7] showed that the fundamental group of ( 28) is isomorphic to the pure twin group -a group of planar pure braids on strands without triple-intersection points.Thus we find In appendix A, we give a brief review of and (30).Note that 3 is isomorphic to Z [8,9].Hence, to identify topologically nontrivial contact interactions, we have to classify all possible one-dimensional unitary representations of .This is, however, a very complicated mathematical problem and in fact still unsolved.In the following, we would like to construct a family of -body Hamiltonians that corresponds to one particular, yet physically reasonable, unitary representation of by using physical intuition.First, in the three-body problem, the topologically nontrivial three-body contact interaction was described by a single infinitely-thin magnetic flux penetrating through the three-body coincidence point 1 = 2 = 3 .In the -body problem, there exist 3 = ( − 1)( − 2)/3! distinct three-body coincidence points given by = = with 1 ≤ < < ≤ .Hence, it would be natural to expect that topologically nontrivial three-body contact interactions in the -body problem could be described by ( − 1)( − 2)/3! distinct infinitely-thin magnetic fluxes penetrating through these three-body coincidence points.The field-strength two-form describing this situation is easily found to be = 2 ℏ where is a real parameter.This two-form can be written as = , where = ℏ Here is a natural generalization of (8b) given by Notice that, in the Cartesian coordinate system ( 1 , ⋯ , ), the gauge-field one-form (32) is written as = ∑ =1 , where The -body Hamiltonian operator in the presence of the fictitious magnetic fluxes (31) is then given by This is the simplest one-dimensional -body model of nonidentical particles with topologically nontrivial three-body contact interactions.Notice that, in this gauge-field description, -body wavefunctions satisfy the periodic boundary condition around the three-body coincidence points.Finally, we note that the model (35) corresponds to the ( − 1)( − 2)/3!-parameter family of one-dimensional unitary representations [ ] ∶ → (1) given by the following Wilson loop: This Wilson loop clearly satisfies the group multiplication law ) for any planar pure braids , ′ ∈ (which are in one-to-one correspondence with closed loops , ′ in -body ), thus giving a one-dimensional unitary representation of .Note, however, that eq. ( 36) is just one particular example: for instance, 4 is known to be isomorphic to the free group 7 [8,9] so that its one-dimensional unitary representation contains up to seven independent parameters [2], while (36) possesses just four parameters for = 4.Our simple magnetic fluxes (31) are therefore just a small subset of all possible three-body contact interactions.Future studies should investigate the most general gauge-field one-form that describes the entire one-dimensional unitary representation of .

Conclusion
As was shown by Harshman and Knapp [1][2][3], three-body contact interactions in -body problems of nonidentical particles on R can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group -the fundamental group of the -body configuration space -body = R − Δ .It was, however, unknown how those topologically nontrivial three-body contact interactions are described by Hamiltonian operators.
This paper studied Hamiltonian descriptions for the topologically nontrivial three-body contact interactions by using the path-integral formalism.In the three-body problem, we showed that all the three-body contact interactions corresponding to one-dimensional unitary representations of the fundamental group 1 ( 3-body ) ≅ 3 ≅ Z are realized by background Abelian gauge fields.These gauge fields describe an infinitely-thin magnetic flux penetrating through the three-body coincidence point in 3-body .By generalizing this result, we constructed the ( − 1)( − 2)/3!-parameter family of (≥ 3)-body Hamiltonians for nonidentical particles on R.This family is made up of background Abelian gauge fields that describe ( − 1)( − 2)/3! infinitely-thin magnetic fluxes in -body and corresponds to one-dimensional unitary representations of 1 ( -body ) ≅ given by (36).There remain several issues still to be addressed.Examples include the classification of unitary irreducible representations of , the construction of field-theory description for topologically nontrivial three-body contact interactions, and experimental realizations.We hope to address these issues in the future.