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A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalsim

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A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalsim A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalism Satoshi Ohya Institute of Quantum Science, Nihon University, Kanda-Surugadai 1-8-14, Chiyoda, Tokyo 101-8308, Japan [email protected] (Dated: May 11, 2021) Abstract We study boson-fermion dualities in one-dimensional many-body problems of identical parti- cles interacting only through two-body contacts. By using the path-integral formalism as well as the configuration-space approach to indistinguishable particles, we find a generalization of the boson-fermion duality between the Lieb-Liniger model and the Cheon-Shigehara model. We present an explicit construction of n-boson and n-fermion models which are dual to each other and characterized by n−1 distinct (coordinate-dependent) coupling constants. These models enjoy the spectral equivalence, the boson-fermion mapping, and the strong-weak duality. We also discuss a scale-invariant generalization of the boson-fermion duality. arXiv:2105.04288v1 [quant-ph] 10 May 2021 1 1 Introduction Inhisseminalpaper[1] in 1960, Girardeau proved the one-to-one correspondence—the duality—between one-dimensional spinless bosons and fermions with hard-core interparticle interactions. By using this duality, he presented a celebrated example of the spectral equivalence between impenetrable bosons and free fermions. Since then, the one-dimensional boson-fermion duality has been a testing ground for studying strongly-interacting many-body problems, especially in the field of integrable models. So far there have been proposed several generalizations of the Girardeau’s finding, the most promi- nent of which was given by Cheon and Shigehara in 1998 [2]: they discovered the fermionic dual of the Lieb-Liniger model [3] by using the generalized pointlike interactions. The duality between the Lieb-Liniger model and the Cheon-Shigehara model is a natural generalization of [1] and consists of (i) the spectral equivalence between the bosonic and fermionic systems, (ii) the one-to-one mapping between bosonic and fermionic wavefunctions, and (iii) the one-to-one correspondence between a strong-coupling regime in one system and a weak-coupling regime in the other (i.e., the strong-weak duality). The purpose of the present paper is to derive and further generalize this duality by using the path-integral formalism. Before going into details, however, let us first briefly recall the basics of the boson-fermion duality by simplifying the argument of [2]. Let us consider n identical particles which have no internal structures (i.e., spinless), move on the whole line R, and interact through two-body contact interactionsbut otherwise freely propagate in the bulk. The Lieb-Liniger model and the Cheon-Shigehara model are particular examples of such n-body systems and described by the following Hamiltonians, respectively: 2 n 2 2 ℏ ) ℏ 1 (1a) HB = − É 2 + É (xj − xk; a ) 2m j=1 )xj m 1≤j<k≤n 2 n 2 2 , (1b) F = − É 2 + É ( j − k; ) 2 j=1 j 1≤j<k≤n ℏ ) ℏ where is the mass of the particlesH and is the coordinate" ofx thextha particle., Here 1 1 is m j )x m ( ; a ) = a ( ) the -function potential with being a real constant that has the dimension of length and ( ; ) is the so-calledm -function potential defined byx a limit of a particular linearj combination of thex -functions x [4]. To see the duality, however,a there is no need to know the precise definition of ( ; ) because" x a both the - and" -function potentials are prescribed by connection conditions for wavefunctions. In other words, the -body systems described by the Hamiltonians (1a) and (1b) are equivalently" x a described by the bulk free Hamiltonian ℏ n ) with some connection conditions at the coincidence " 0 = − 2m ∑j=1 )x 2 2 points j = nk. For example, the -body Hamiltonian2 (1a) is described by the bulk free Hamiltonian 0 with the following connection conditions for thej -function potential 1 : H ( j − k; a ) x x ó n ó H B ó B ó 1 ó ó (2a) ó − ó − Bóx =0 + Bóx =0x= 0x jk óx =0 jk óx =0 ) ) jkó + jkó − (2b) jk + jk − Bóx =0 − Bóx =0 = 0, )x )x a where and ) ) ) . On the other hand,jk the+ -bodyjk − Hamiltonian (1b) is described jk = j − k )x = )x − )x , by the bulk free Hamiltonian 0 with the following connection conditions for the -function potential 1 jk j k ( j − xk; ) [x2, 4]:x n H " F F " x x a ó ó (3a) Fóx =0 − Fóx =0 − + = 0 jk x =0 jk x =0 jk + jk − ) ó ) ó a óFjk + óFjk − , (3b) 0 ó − ó 1 = 0 )x jkó x =0 )x jkó x =0 ) ó ) ó 1 ó jk + ó jk − Yet another pseudo-potential realization of the connection conditionsó (3a) andó (3b) was discussed. in [5]. Note that the contact interaction described by (3a) and (3b) is also widely called)x theó “ -interaction”)x ó in the literature; see, e.g., [6]. ¨ 2 Notice that B and its derivatives become continuous in the limit 1/ 0, which indicates that 1/ describes the deviation from the smooth continuous theory (i.e., the free theory)and plays the role of a coupling constant in the Lieb-Liniger model (1a). In contrast, F and itsa derivatives → become continuousa in the limit 0, which indicates that describes the deviation from the free theory and plays the role of a coupling constant in the Cheon-Shigehara model (1b ). This inverse relation of the coupling constants is already incorporated into the notations 1 and and the heart of the strong-weak a → a ( ; a ) ( ; ) duality. Now, all the above connection conditions are valid x for generic" x waavefunctions without any symme- try. However, further simplifications occur if the wavefunctions are totally symmetric (antisymmetric) under the exchange of coordinates, which must hold for identical spinless bosons (fermions) due to the indistinguishability of identical particles in quantum mechanics. For example, if B is the totally symmetric function and satisfies the identity B( j k ) = B( k j ), there automat- ically hold the additional conditions and ) B ) B , which reduce (2a) B x =0 = B x =0 )x x =0 = − )x x =0 to ) B 1 . Similarly, if is⋯ the , x totally,⋯,x antisymmetric,⋯ ⋯ , x function,⋯,x ,⋯ and satisfies the )x x =0 − a B x =0 = 0 jk + F jk − jk jk + jk jk − | | ó ó identity F( j ) = − F( j ), there automaticallyó hold theó additional conditions jk jk + jk k + k ) F ) F ) F ó | and , which reduce (3a) to . Putting F x =0ó = − F x =0 )x x =0 = )x x =0 F x =0 − )x x =0 = 0 all these things⋯ , x together,,⋯,x ,⋯ one immediately ⋯ , x ,⋯,x sees,⋯ that the systems of identical bosons and fermions de- jk + jk − jk + jk ó jk + jk ó jk − jk ó jk + scribed| by (1a )| and (1b) are bothó describedó by the bulk free Hamiltonian | 0 togethera ó with the following Robin boundary conditions at the coincidence points: n H B/F 1 (4) − B/F x =0 = 0 jk x =0 ) ó jk + Since both systems are described by the sameó jk + bulk Hamiltonianó and the same boundary conditions, ó ó . they automatically become isospectral.)x Thisó is thea boson-fermion duality in one dimension, which consists of the spectral equivalence between B and F, the equivalence between the strong coupling regime of B/F and the weak coupling regime of F/B, and, as we will see in section 2.3, the one-to-one mapping between B and F. Note that the extremeH H case 0 corresponds to the simplest duality between theH impenetrable bosons and the free fermions.H Now it is obvious from the above discussion that the one-dimensa → ional boson-fermion duality just follows from simple connection condition arguments. However, it took more than thirty years to arrive at the above findings since the discovery of the simplest duality by Girardeau. One reason for this would be the lack of a systematic derivation of the dual contact interactions for identical bosons and fermions. The purpose of the present paper is to fill this gap and to present a systematic machinery for deriving (and generalizing) the boson-fermion duality. As we will see in the rest of the paper, this purpose is achieved by using the path-integral formalism as well as the configuration-space approach to identical particles [7–10]. The paper is organizedas follows. In section 2 we first present the basics of the configuration-space approach to identical particles following the argument of Leinaas and Myrheim [10]. In this approach, the indistinguishability of identical particles is incorporated into the configuration space rather than the permutation symmetry of multiparticle wavefunctions. We first see that the configuration space R Rn of identical particles on is given by the orbit space n = ( − Δn)/ n, where Δn is the set of coincidence points of two or more particles and n the symmetric group of order . We see that thisn space has a number of nontrivial boundaries, where -body contact interactionsS take place at codimension-( − 1) boundaries. We then see that,S irrespective of the particle statistics,n two-body contact interactions are generally described by the Robin bounk dary conditions at the codimension-1 boundaries of k n. We also discuss that the boson-fermion mapping holds irrespective of the boundary conditions. In section 3 we study the Feynman kernel for identical particles from the viewpoint of path integral. We will show that, by using the Feynman kernel on n, the boson-fermion duality between (1a) and (1b) is generalized to the duality between the models described by the Hamiltonians (49a) and (49b).
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