何淼和江云峰2023年8月在JHEP期刊发表论文.pdf

Published for SISSA by Springer Received: June 6, 2023 Accepted: August 3, 2023 Published: August 16, 2023 Miao He and Yunfeng Jiang School of Physics, Southeast University, Nanjing 211189, China Shing-Tung Yau Center, Southeast University, Nanjing 210096, China E-mail: hemiao@seu.edu.cn, jinagyf2008@seu.edu.cn Abstract: The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit. Keywords: Bethe Ansatz, Integrable Field Theories, Lattice Integrable Models ArXiv ePrint: 2305.16046 c The Authors. Open Access, ⃝ Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2023)079 JHEP08(2023)079 Integrable crosscap states: from spin chains to 1D Bose gas Contents 1 2 Crosscap states of integrable spin chains 2.1 Integrability of the crosscap state 2.2 Overlaps formula for compact chains 2.3 Overlap formula for the non-compact chain 3 4 6 12 3 Crosscap state of Lieb-Liniger model 3.1 Crosscap state in Lieb-Liniger model 3.2 Exact overlap formula 3.3 Crosscap partition function 14 14 17 20 4 Dynamical correlation functions in crosscap state 4.1 Two-point function 4.2 Four-point function 22 23 25 5 Conclusion and discussion 26 A Proof of equation (2.56) 27 B Deriving the crosscap state in Lieb-Liniger model 28 C Lattice computation of correlation functions 29 1 Introduction An integrable quantum field theory (IQFT) in 1+1 dimension has infinitely many local conserved charges. In the presence of boundaries, some of these charges are no longer conserved. Nevertheless, there are special boundary conditions which preserves an infinite subset of the conserved charges. These boundary conditions are called integrable. For a Lorentz invariant theory, one can equivalently place the boundary in the temporal direction, in which case the boundary condition becomes a special state in the Hilbert space called an integrable boundary state [1]. A characteristic feature of integrable boundary states is that they are annihilated by infinitely many odd charges of the model. In recent years, it has become clear that the notion of integrable boundary states can be generalized to a broader class of theories such as integrable lattice models [2, 3]. Interests on integrable boundary states stem from both statistical mechanics and AdS/CFT correspondence. In statistical mechanics, these states can serve as initial states for the investigation of out-of-equilibrium dynamics [4–8]. Due to their integrability, one –1– JHEP08(2023)079 1 Introduction –2– JHEP08(2023)079 can have more analytic control for the calculations. In AdS/CFT correspondence, it turns out that various kinds of correlation functions in N = 4 SYM theory and ABJM theory at weak coupling can be computed by the overlap of an on-shell Bethe state and an integrable boundary state. These include the one-point functions in defect CFT [9–15], three-point functions of two giant gravitons and one non-BPS single-trace operator [16–18], and correlation functions of involving circular Wilson loops [19, 20] and ’t Hooft loops [21]. In all these cases, the exact overlap formulae play an important role. For integrable boundary states, only the Bethe states with parity even rapidities lead to non-vanishing overlaps. Moreover, the overlap formula has very nice analytic structure [22–27]. In all known cases, it can be written as a the product of a prefactor and a ratio of Gaudin-like determinants. The former is state dependent while the latter is universal and only depends on the symmetry. The exact formulae have been proven in a number of cases using both the coordinate Bethe ansatz [28, 29] and algebraic Bethe ansatz [22–27, 30, 31], while for other cases they remain conjectures with extensive numerical evidence. Very recently, a new type of integrable boundary states called crosscap states have been investigated. These states first arise in 2d CFT, which are special conformal boundary states [32]. Geometrically, they correspond to non-orientable surfaces such as RP2 and the Klein bottle, which cannot be described by a local boundary condition. In [33], by generalizing the geometric intuition from CFT, the authors defined crosscap states for 2d massive IQFTs and the integrable spin chains. Remarkably, they discovered that crosscap states as they defined are integrable. The crosscap states have several new features which make them rather special. First of all, most known integrable boundary states such as the two-site states and matrix product states are short-range entangled while crosscap states are long-range entangled by construction. In addition, the overlap formula for the crosscap state and an on-shell Bethe state has a trivial prefactor [33], and is given simply by the ratio of Gaudin-like determinants. In this sense, crosscap states are probably the ‘cleanest’ integrable boundary states. The unique features of the crosscap states are intimately related to their geometric origin. Therefore we expect it should be possible to define such states for a wide class of models. Indeed, generalizations to gl(N ) spin chains [34] and classical sigma models [35] have been studied recently. Apart from relativistic IQFTs and spin chains, there is yet another important class of integrable models which are continuous but non-relativistic. The prototype of these models is the Lieb-Liniger model [36], which describes one dimensional bosonic particles interacting with a pairwise δ-function potential. Apart from theoretical interests, the Lieb-Liniger model has direct relevance to cold atom experiments (see for examples the reviews [37–39]). It is particularly interesting to compute its dynamical observables since they can be measured in the laboratory. However, so far the only known integrable boundary state for the Lieb-Liniger model is the so-called Bose-Einstein Condensate (BEC) state. It was found that the overlap of the BEC state and on-shell Bethe states obey parity even selection rules and the overlap formula exhibit the same structure as spin chains [29, 40, 41]. Since both for IQFTs and spin chains, one can construct many integrable boundary states, it is a natural question whether we can construct more integrable boundary states for the Lieb-Liniger model. In view of its close relation to experiments, such integrable boundary |C⟩ = exp Z ℓ/2 0 ! † † dx Φ (x)Φ (x + ℓ/2) |Ω⟩ , (1.1) where Φ† (x) is the bosonic operator which creates a particle at position x. The geometric meaning of (1.1) is quite clear — particles are created in pairs at antipodal points. As a result, this state is long-range entangled by construction. One might notice that this state is similar to the integrable boundary states constructed by Ghoshal and Zamolodhikov in IQFT [1] where one replaces Φ† (x)Φ† (x + 2ℓ ) by K ab (θ)A†a (θ)A†b (−θ). However, we want to emphasis two important differences. First, the Faddeev-Zamolodchikov operator A†a (θ) is a creation operator in momentum space while Φ† (x) is the creation operator in position space. Second, the Ghoshal-Zamolodchikov construction describes boundary states in the infinite volume while for the crosscap state finite volume is necessary to define antipodal points. The rest of this paper is organized as follows. In section 2, we generalize the proof of integrability of crosscap states to anisotropic inhomogeneous Heisenberg spin chains, and derive the exact overlap formula using the coordinate Bethe ansatz. This section can be seen as a useful technical preparation for similar calculations in the Lieb-Liniger model. In section 3, we propose the crosscap state in the Lieb-Liniger model by taking the continuum limit from spin chain crosscap state. We prove its integrability and derive the exact overlap formula of the crosscap state and an on-shell Bethe state. The dynamical correlation function in crosscap states are studied in section 4. We conclude and discuss future directions in section 5. Some details of the calculations are given in the appendices. 2 Crosscap states of integrable spin chains The crosscap state is constructed by identifying the states at antipodal sites. For XXX spin chain, the entangled pair of states at sites j and j + L2 is   + |c⟩⟩j = 1 + Sj+ Sj+L/2 | ↓⟩j ⊗ | ↓⟩j+L/2 , –3– (2.1) JHEP08(2023)079 states might be even more interesting. The crosscap state seems to be a natural candidate, as its geometric origin gives us a clear guidance for its construction. As we will show below this indeed turns out to be the case. Now let us sketch our strategy for the construction. The Lieb-Liniger model sits somewhere between IQFTs and spin chains. On the one hand, it can be obtained as the non-relativistic limit of the sinh-Gordon model [42, 43]; on the other hand, it corresponds to special continuum limits of certain spin chain models [44, 45]. Therefore we can start with crosscap states in either IQFTs or spin chains and then take the proper limit. It turns out that the second option is more feasible. We will comment on its relation to the first option later. We consider two methods to obtain crosscap state in the Lieb-Liniger model from spin chains: the first one involves taking a special continuum limit of the XXZ spin chain [44], while the second one involves discretizing the Lieb-Liniger model as a generalized XXX spin chain [46]. It turns out that the two methods lead to the same result, given by where we denote the generators of the SU(2) algebra at site-j by Sj± and Sjz . The crosscap state is then defined by taking the tensor product of such entangled pairs L/2 |C⟩SU(2) ≡ ⊗ (|c⟩⟩j ) = Y L/2  Y j=1 j=1 + 1 + Sj+ Sj+ L  |Ω⟩ , (2.2) 2 |c⟩⟩j = ∞ X 1  n=0 n! + Sj+ Sj+L/2 n (2.3) |0⟩j ⊗ |0⟩j+L/2 , where |0⟩ represents the lowest-weight state of SL(2, R). The crosscap state is given by L/2 |C⟩SL(2) ≡ Y L/2 " ∞ Y X 1  ⊗ (|c⟩⟩j ) = j=1 n! j=1 n=0 n + Sj+ Sj+L/2 # |Ω⟩. (2.4) Both (2.2) and (2.4) has been studied in [33]. In this section, we derive two new results which are useful for the Lieb-Liniger model. One is defining crosscap states for the inhomogeneous XXZ spin chain and proving their integrability. This constitutes a slight generalization of the results in [33]. The other is deriving the exact overlaps between crosscap states and on-shell Bethe states for both compact and non-compact spin chains. The overlap formula was first conjectured in [33] and later proven in [34] using algebraic Bethe ansatz for XXX spin chain (as a special case of gl(N ) spin chain). Here we give an alternative proof using CBA, which works for both compact and non-compact spin chains and can be generalized to the Lieb-Liniger model. 2.1 Integrability of the crosscap state The integrable boundary states |Ψ0 ⟩ are the states which are annihilated by the odd charges Q2n+1 |Ψ0 ⟩ = 0. (2.5) For spin chains, it is more convenient to work with the transfer matrix, which is the generating functional of the conserved charges T (λ) = exp ∞ X λn n=1 n! ! Qn+1 . (2.6) In [2, 3], the authors propose to define integrable boundary states as the states satisfying T (λ)|Ψ0 ⟩ = T (−λ)|Ψ0 ⟩. We shall adopt this definition here. –4– (2.7) JHEP08(2023)079 where the |Ω⟩ = | ↓L ⟩ is the pseudovacuum. The crosscap state can also be defined for the non-compact SL(2, R) spin chain [33]. The main difference is that more than one particles can be excited on the same site. In this case, the entangled pair at sites j and j + L2 reads Let us consider the inhomogeneous XXZ spin chain defined Inhomogeneous XXZ chain. by the R-matrix   sinh(λ + η) sinh(λ) sinh(η) sinh(η) sinh(λ)   R(λ) =    sinh(λ + η)   .   (2.8) For later convenience, we can also write it in the tensor product form    1 sinh(λ + η) + sinh(λ) 1 ⊗ 1 + sinh(λ + η) − sinh(λ) S z ⊗ σ z 2   + sinh(η) S + ⊗ σ − + S − ⊗ σ + , (2.9) where σ α are Pauli matrices and S α = σ α /2. The Lax operator at site-j is defined as Lj (λ) = Raj (λ − ξj − η/2) , (2.10) where ξj is the inhomogeneity. The inhomogeneous XXZ spin chain is defined by the following transfer matrix TXXZ (u) = tra (L1 (λ) . . . LL (λ)) . (2.11) Crosscap state and integrability. For the inhomogeneous XXZ spin chain, we define the crosscap state to be the same state (2.2). We now prove its integrability. Using the relation ∓ Sj± |c⟩⟩j = Sj+L/2 |c⟩⟩j , (2.12) σ2 Lj (λ)σ2 |c⟩⟩j = −Lj+L/2 (−λ)|c⟩⟩j , (2.13) we find that if the inhomogeneities at site j and j + L/2 are also identified as [47] ξj+L/2 = −ξj , j = 1, 2, . . . , L/2 . (2.14) By employing (2.13), the action of the transfer matrix on the crosscap state reads TXXZ (λ)|C⟩ = tra h L1 (λ) · · · LL/2 (λ) = (−1)L/2 tra h = (−1)L/2 tra h = (−1)L tra h  LL/2+1 (λ) · · · LL (λ) L1 (λ) · · · LL/2 (λ) i |C⟩  σ2 L1 (−λ) · · · LL/2 (−λ)σ2 i |C⟩  i |C⟩ i |C⟩ L1 (−λ) · · · LL/2 (−λ) L1 (−λ) · · · LL/2 (−λ) σ2 L1 (λ) · · · LL/2 (λ)σ2  LL/2+1 (−λ) · · · LL (−λ) = TXXZ (−λ)|C⟩. Therefore the crosscap state is integrable if the inhomogeneities at antipodal sites have paired structure (2.14). –5– JHEP08(2023)079 R(λ) = 2.2 Overlaps formula for compact chains Owing to the condition (2.7), the overlap of an on-shell Bethe state |λN ⟩ and an integrable boundary state |Ψ0 ⟩ is only non-zero if the Bethe roots are parity even, namely {λN } = {−λN }. For N being even, this implies {λN } = {λ1 , −λ1 , λ2 , −λ2 . . . , λN/2 , −λN/2 } . (2.15) For parity even Bethe roots, it is known that the overlap take the following form 1 ⟨λN |Ψ0 ⟩ = ⟨λN |λN ⟩ j=1 Y p v u u det G+ N/2 F (λ ) × t , j det G− N/2 (2.16) where F (λ) is a state-dependent function and det G± N/2 are the Gaudin-like determinants. We will prove that for the crosscap states the overlaps indeed take the form (2.16) with F (λ) = 1. We follow the method proposed in [28] with important modifications. Coordinate Bethe ansatz. An N -particle eigenstate |λN ⟩ for both the XXX and XXZ spin chains take the following form |λN ⟩ = X χ (nN , λN ) |n1 , n2 , · · · , nN ⟩ , (2.17) {nN } where |n1 , n2 , . . . , nN ⟩ = Sn+1 Sn+2 · · · Sn+N |Ω⟩. (2.18) The summation over {nN } means summing over all the possible particle positions with the following constraint 0 ≤ n1 < n2 < . . . < nN ≤ L − 1. (2.19) The wave function χ (nN , λN ) is given by χ (nN , λN ) =  X Y f λ σj − λ σk N Y eip(λσj )nj , (2.20) j=1 σ∈SN j>k where the explicit form of p(λ) and f (λ) depend on the model. The rapidities λN satisfy the Bethe ansatz equations e ip(λj )L N Y f (λj − λk ) k=1 k̸=j f (λk − λj ) = 1, j = 1, 2, . . . , N. (2.21) For the XXX and XXZ spin chains, the two functions are given by 1 For some boundary states such as matrix product states with higher bond dimensions, the prefactor can take a more complicated form. –6– JHEP08(2023)079 N/2 • XXX spin chain: λ − i/2 , λ + i/2 f (λ) = λ+i . λ (2.22) sinh(λ − iη/2) , sinh(λ + iη/2) f (λ) = sinh(λ + iη) , sinh(λ) (2.23) eip(λ) = • XXZ spin chain: eip(λ) = where η is related to the anisotropy by ∆ = cosh η. H |λN ⟩ = EN (λN ) |λN ⟩ . (2.24) For the XXX spin chain, the eigenvalue reads EN (λN ) = − N X 2 (2.25) . λ2 + 1/4 j=1 j For the XXZ spin chain, the eigenvalue reads EN (λN ) = Norm of Bethe states. form [48] ⟨λN |λN ⟩ = N X 4 sinh2 η . cos(2λ ) − cosh η j j=1 (2.26) The norm of the on-shell Bethe states takes the following N Y N 1 Y f (λj − λk ) f (λk − λj ) × det GN , p′ (λj ) jk and the Bethe equations can be written as N Y f (lk , lj ) k=1 k̸=j f (lj , lk ) , Crosscap overlaps. Now we turn to the overlaps. We first consider the overlaps between the crosscap state and the N -particle basis states (2.18). From the definition of the crosscap state (2.2), we find N/2 ⟨C|n1 , n2 , . . . , nN ⟩ = Y (2.36) δni ,ni+N/2 −L/2 , i=1 which implies that the particles must appear in pairs with distance L/2. Therefore both L and N should be even. We can split the valid N -particle positions into two parts {nN }C := {nN/2 } ∪ {nN/2 + L2 }, (2.37) {nN/2 } := {n1 , n2 , . . . , nN/2 |0 ≤ n1 < n2 < . . . < nN/2 ≤ L2 − 1}, (2.38) where the second part is completely determined by the first part. Using (2.36), the overlap of the crosscap state and a Bethe state reads ⟨C|λN ⟩ = SN (lN , aN ) = X Y  f l σ j , l σk  X N/2 Y nj nj +L/2 lσj lσ(j+N/2) . (2.39) {nN/2 } j=1 σ∈SN j>k We introduce the summation function X N/2 Y nj nj +L/2 BN (lN |L) = (2.40) lj lj+N/2 , {nN/2 } j=1 for later convenience. To see how the method works, let us first consider the simplest 2-particle state. 2-particle states. The overlap can be calculated straightforwardly L/2 ⟨C|λ2 ⟩ = f (l2 , l1 ) l2  1 − (l1 l2 )L/2  1 − l1 l2 –8– L/2 + f (l1 , l2 ) l1  1 − (l1 l2 )L/2 1 − l1 l2  . (2.41) JHEP08(2023)079 aj = ljL = The Bethe equations read a1 = l1L = f (l2 , l1 ) , f (l1 , l2 ) a2 = l2L = f (l1 , l2 ) . f (l2 , l1 ) (2.42) For an on-shell Bethe state, the Bethe equation implies that (l1 l2 )L/2 = 1. If we naively substituting this into (2.41), we find that the overlap is vanishing for any on-shell Bethe state. The key point here is to notice that we need to first take the paired rapidities limit l1 l2 → 1 and then impose Bethe ansatz equations, which leads to q ⟨C|λ2 ⟩ = L f (2λ1 )f (−2λ1 ). ′ det G+ 1 = p (λ1 )L. (2.44) From this simple example, we learned that the non-vanishing overlap is obtained by taking the paired rapidities limit before imposing Bethe ansatz equations. N -particle states. be written as For N -particle states, the summation function defined in (2.40) can L−N +1 L−N +2 2 2 N/2 BN (lN |L) = Y L/2 X lj+N/2 L X ... n1 =0 n2 =n1 +1 j=1 2 X N/2  Y lj lj+N/2 n j . (2.45) nN/2 =nN/2−1 +1 j=1 If we ignore the overall factor in (2.45), the summation function basically becomes a summation over half of the spin chain without constraints, except that we have lj lj+N/2 instead of lj in the summand. The latter summation function can be computed by using a recursion relation [9, 28]. This allows us to obtain an explicit albeit slightly involved expression for BN (lN |L) N/2 BN (lN |L) = X BN,j (lN |L), (2.46) j=0 where Qj+N/2 Qj 1/2 QN k−1 k=j+N/2+1 ak k=2 (lk lk+N/2 ) k=j+1 (ak ) Q Q . BN,j (lN |L) = QN/2 Q j j k l l − 1 l l − 1 i=j+1 i i+N/2 k=j+1 k=1 i=k i i+N/2 (−1)j (2.47) The summation function is a rational function of lj . In order to take the paired rapidity limit, we first consider the behavior of BN (lN |L) near the pole at lm lm+N/2 = 1. There are two terms BN,m−1 (lN |L) and BN,m (lN |L) that contain this pole. Taking the sum of these two terms and using lm lm+N/2 = 1 for the regular part, we obtain BN,m−1 (lN |L) + BN,m (lN |L)  = am am+N/2 1/2  −1 am+N/2 1/2 lm lm+N/2 − 1 × Qm+N/2−1 Q Qm−1 k−1 (ak )1/2 N k=m+N/2+1 ak k=2 (lk lk+N/2 ) k=m+1     . QN/2 Qk Qm−1 Qm−1 l l − 1 l l − 1 i=m+1 i i+N/2 k=m+1 k=1 i=k i i+N/2 (−1)m−1 –9– (2.48) JHEP08(2023)079 The overlap can be written in Gaudin determinant form 1 q ⟨C|λ2 ⟩ = ′ f (2λ1 )f (−2λ1 ) × det G+ 1, p (λ1 ) (2.43) Notice that the second line is nothing but BN,m with two particles at m and m + N/2 removed. Therefore near the pole lm lm+N/2 = 1, we have  BN (lN |L) ∼ am am+N/2 1/2 −1 lm lm+N/2 − 1 am+N/2 1/2 ✘ . . . , N }|L). BN −2,m−1 ({1, . . . ✚ m m✘ +✘ N/2 ✚. . .✘ ✘ (2.49) SN (lN , aN ) =   f lσj , lσk BN (σlN |L) X Y σ∈SN j>k  ∼ am am+N/2 1/2 −1  lm lm+N/2 −1    ✘ ✘✘✘ . . . , N }|L), f lσj , lσk BN −2,m−1 (σ{1, . . .✚ m m+N/2 ✚. . .✘ X Y σ∈SN −2 j>k j,k̸=m,m+N/2 × Fm +Fm+N/2 (2.50) where the last line does not depend on lm and lm+N/2 . The sum of Fm and Fm+N/2 reads m+N/2−1 Fm + Fm+N/2 =  f (lj , lm ) f lj , lm+N/2   1/2    Y j=m+1 f (lm , lj ) f lm+N/2 , lj N Y ×   N Y f (lj , lm ) f lj , lm+N/2     j=m+N/2+1 f (lm , lj ) f lm+N/2 , lj f (lm , lj ) f lm+N/2 , lj × F(lm , lm+N/2 ), (2.51) j=1 j̸=m,m+N/2 where m+N/2−1 F(lm , lm+N/2 ) =   f (lj , lm ) f lm+N/2 , lj  1/2    Y  ×f (lm+N/2 , lm ) am+N/2 j=m+1 f (lm , lj ) f lj , lm+N/2 m+N/2−1    1/2     + Y j=m+1 f lj , lm+N/2 f (lm , lj ) f lm+N/2 , lj f (lj , lm ) 1/2 ×f (lm , lm+N/2 ) (am )1/2 . (2.52) The main new feature of the crosscap state is that the function F(lm , lm+N/2 ) depends on not only lm , lm+N/2 but also lj for m < j < m + N/2, while for the integrable boundary states considered in [28], the poles appear at neighboring lm lm+1 = 1 and the function F just depends on lm and lm+1 . We introduce the modified parameters amod = j     aj , f (lj , lm ) f lj , lm+N/2 f (lm , lj ) f lm+N/2 , lj – 10 – 1≤j≤N, (2.53) JHEP08(2023)079 Plugging back to the overlap formula (2.39), we also need to multiply a factor in front of BN (lN |L) and then sum over all the permutations. Since we focus on the pole lm lm+N/2 = 1, we also need to separate out the lm and lm+N/2 dependent part for the multiplying factor. Note that the exchange of lm and lm+N/2 preserve the relative position of m and m + L/2, which gives the same pole. After factorizing the lm lm+N/2 = 1 pole, we find so that the first line on the right hand side in (2.51) can be absorbed in BN −1,m−1 by making the replacement aj → amod . Then the recursion relation (2.50) can be written as j  SN (lN |L) ∼ am am+N/2 1/2 −1 lm lm+N/2 − 1 N/2 F(lm , lm+N/2 ) f¯ (lm , lj ) Y j=1 j̸=m ✘ . . . , N }|L) mod × SN m m✘ +✘ N/2 ✚. . .✘ −2 ({1, . . . ✚ ✘ (2.54) j = 1, 2, . . . , N/2. (2.55) Now we consider the paired rapidity limit In this limit, F(lm , lm+N/2 ) simplifies drastically q F(lm , lm+N/2 ) → 2 f (−2λm )f (2λm ) . (2.56) The details can be found   in appendix A. Let us denote the paired rapidity limit of SN (lN |L) by D λN/2 , mN/2 |L , in which we have introduced the new parameter mj = −i d log(aj ) = p′ (λj )L. dλj (2.57) The recursion relation (2.54) implies that ∂D(λN/2 , mN/2 |L) = ∂mm N/2 f (−2λm )f (2λm ) Y ¯ f (λm , λj ) × D(λN/2−1 , mmod N/2−1 |L), (2.58) p′ (λm ) j=1 p j̸=m where the modified parameter is given by mmod = −i j   d log amod = mj + φ+ (λj , λm ). j dλj  (2.59)  To write the recursion relation in a nicer form, let us define D̃ λN/2 , mN/2 |L via  D λN/2 , mN/2 |L = q Y f (−2λj )f (2λj ) Y j=1 p′ (λj ) 1≤jk Similarly, it is convenient to introduce the summation function BN (lN |L) = N/2 L/2−1 L/2−1 Y L/2 X j=1 n1 =0 n2 =0 lj+N/2 X ... L/2−1 N/2  X Y lj lj+N/2 n j (2.72) . nN/2 =0 j=1 N/2 BN (lN |L) = X BN,j (lN |L), (2.73) j=0 where BN,j (L) = QN/2 1/2 QN N/2−k k=j+N/2+1 ak k=j+1 (lk lk+N/2 ) k=j+1 (ak )     . QN/2 Qk Qj Qj i=j+1 li li+N/2 − 1 k=j+1 k=1 i=k li li+N/2 − 1 (−1)j Qj+N/2 (2.74) The main observation is that, although the summation functions are different, their behavior near the pole lm lm+N/2 are the same  BN (lN |L) ∼ am am+N/2 1/2 −1 lm lm+N/2 − 1 am+N/2 1/2 ✘ . . . , N }|L). BN −2,m−1 ({1, . . . ✚ m m✘ +✘ N/2 ✚. . .✘ ✘ (2.75) This leads to the same F(lm , lm+N/2 ) function and the same recursion relation for the overlap. As a consequence, we get the crosscap overlap formula ⟨C|λN ⟩ ⟨λN |λN ⟩ p v u u det G+ N/2 =t . det G− N/2 (2.76) Before ending the section, let us comment on the universality of the crosscap overlaps. In fact, the key point is the overlaps between crosscap states and N -particle basis gives a product of the Kronecker deltas, which imposes strong constraints on the particle positions. Such constraints reflects the geometric origin of the crosscap state. It effectively reduces the Bethe wave function of N particles to be the one with N/2 particles, except for the replacements lj → lj lj+N/2 , see (2.45) and (2.72). The overlaps are the sum of Bethe wave functions which satisfy the constraints on the particle positions. For the crosscap constaints, the overlaps turn to be the sum of reduced Bethe wave function of N/2 particles without constraints on the particle positions. After taking the paired rapidities limit, the sum of the reduced Bethe wave functions is actually the norm of the Bethe states except for the + replacement of Gaudin-like determinant det G− N/2 → det GN/2 , which hence leads to the trivial prefactor for the overlaps. – 13 – JHEP08(2023)079 For the non-compact chain, BN can also be computed by a recursion relation, leading to 3 Crosscap state of Lieb-Liniger model In this section, we present the derivations of our proposal for the crosscap state of the LiebLiniger model (1.1). We will then prove its integrability and derive the exact overlap formula. 3.1 Crosscap state in Lieb-Liniger model In the second quantized form, the Hamiltonian of the Lieb-Liniger model is given by H= Z ℓ i (3.1) where the bosonic fields satisfy the usual commutation relations [Φ(x), Φ† (y)] = δ(x − y), [Φ(x), Φ(y)] = [Φ† (x), Φ† (y)] = 0. (3.2) We consider the periodic boundary condition with system size ℓ. This model can be solved by coordinate Bethe ansatz as well as the Quantum Inverse Scattering Method (QISM) [49]. We shall make use of both approaches in what follows. For the proof of integrability, it is more convenient to use QISM, but it is necessary to first define the model on the lattice and then take the continuum limit. For the derivation of exact overlap formula, we make use of coordinate Bethe ansatz. As mentioned before, Lieb-Liniger model can be obtained by taking continuum limit of integrable lattice models. There are at least two ways to achieve this. The first one is by taking the special continuum limit of the XXZ spin chain after performing the Dyson-Maleev transformation [44]. The second one is taking the continuum limit of a generalized XXX spin chain [46]. Our strategy is staring from the crosscap state in spin chains, and then taking the continuum limit. We consider both approaches, and it turns out that they lead to the same crosscap state in the Lieb-Liniger model. Method I: Dyson-Maleev transformation. The Dyson-Maleev transformation maps the local spin operators to the bosonic operators Si+ = a†i (1 − a†i ai ), Si− = ai , 1 Siz = − + a†i ai , 2 (3.3) where the bosonic operators satisfy the canonical commutation relations [ai , a†j ] = δij , [ai , aj ] = [a†i , a†j ] = 0. (3.4) Applying the transformation to the crosscap state of the XXZ spin chain (2.2), we obtain L/2   (3.5) ai |Ω⟩ = Si− |Ω⟩ = 0. (3.6) |C⟩ = Y 1 + a†j a†j+L/2 |Ω⟩, j=1 where we have used the fact – 14 – JHEP08(2023)079 0 h dx ∂x Φ† (x)∂x Φ(x) + c Φ† (x)Φ† (x)Φ(x)Φ(x) , Let us consider a XXZ spin chain of length L. We denote the lattice spacing by δ. The system size of the continuum model is ℓ = Lδ. The bosonic operators can be written in terms of Fourier modes 1 X −ikxn an = √ e ãk , L k xn = nδ. (3.7) In this representation, the continuum limit is obtained by X Z  dk, ãk → 2π ℓ 1/2 Φ̃k . (3.8) Then one transforms back to real space by the inverse Fourier transformation 1 Φ̃k = √ 2π Z dx eikx Φ(x), k= 2πm . ℓ (3.9) Applying above procedure to (3.5), we arrive at the crosscap state |C⟩ = exp Z ℓ/2 0 ! dxΦ† (x)Φ† (x + ℓ/2) |Ω⟩, (3.10) where we assume the pseudovacuum in spin chain corresponds to the Fock vacuum |Ω⟩ of the Lieb-Liniger model in the continuum limit. Details of the derivation can be found in appendix B. Method II: generalized XXX model. In this approach, we first discretize the LiebLiniger model by picking L points on the interval [0, ℓ] located at xn = ∆n, xL = ℓ [46]. We then define the operators 1 ψn = √ ∆ Z xn xn−1 Φ(x) dx, 1 ψn† = √ ∆ Z xn xn−1 Φ† (x) dx. (3.11) One can check the operators satisfy † [ψn , ψm ] = δmn , † [ψn , ψm ] = [ψn† , ψm ] = 0. (3.12) In the lattice model, the pseudovacuum is identified with the Fock vacuum which satisfies Φ(x)|0⟩ = ψn |0⟩ = 0. (3.13) The quantum Lax operator takes the form Ln (u) = ! √ c∆ † † ρ+ 1 − iu∆ + ψ ψ −i c∆ψ n n 2 2 n n √ , iu∆ c∆ † i c∆ρ− ψ 1 + + n n 2 2 ψn ψn (3.14) where the operator ρ± n satisfy two constraints:   ± † ρ± n = ρn ψn ψn , − ρ+ n ρn = 1 + – 15 – c∆ † ψ ψn . 4 n (3.15) JHEP08(2023)079 k ℓ → 2π For example, we can take ρ− n = 1, ρ+ n =1+ c∆ † ψ ψn . 4 n (3.16)  (3.17) The transfer matrix is defined as usual  T (u) = tr L1 (u)L2 (u) . . . LL (u) .  i i∆ c∆ h z u1 ⊗ 1 + S̃n ⊗ σ 3 − S̃n+ ⊗ σ − + S̃n− ⊗ σ + , 2 2 L̃n (u) = σ3 σ2 Ln (u)σ2 = where we have introduced 2 S̃nz = + ψn† ψn , c∆ 2i S̃n− = √ ρ− n ψn , c∆ 2i S̃n+ = √ ψn† ρ+ n. c∆ (3.18) (3.19) One can verify that they satisfy the standard SU(2) algebra h i S̃nz , S̃n± = ±S̃n± , h i S̃n+ , S̃n− = 2S̃nz . (3.20) The local spin operators act on the vacuum as following S̃n− |Ω⟩ = 0, S̃n+ |Ω⟩ = ψn† |Ω⟩, S̃nz |Ω⟩ = 2 |Ω⟩. c∆ (3.21) For the generalized XXX spin chain, we can define the crosscap state using the local spin operators S̃ |C⟩ = L/2  Y n=1 + 1 + S̃n+ S̃n+ L  (3.22) |Ω⟩. 2 We then show that the crosscap state so defined is integrable in the sense of (2.7). Firstly, one can verify σ2 L̃n (u)σ2 |c⟩⟩n = −L̃n+L/2 (−u)|c⟩⟩n . (3.23) σ2 Ln (u)σ2 |c⟩⟩n = Ln+L/2 (−u)|c⟩⟩n . (3.24) It follows that Then we have T (u)|C⟩ = ⟨C|tr h L1 (u) . . . LL/2 (u)  LL/2+1 (u) . . . LL (u) = ⟨C|tr h L1 (u) . . . LL/2 (u)  σ2 L1 (−u) . . . LL/2 (−u)σ2 i = ⟨C|tr h L1 (−u) . . . LL/2 (−u)  i = ⟨C|tr h L1 (−u) . . . LL/2 (−u)  = T (−u)|C⟩. i σ2 L1 (u) . . . LL/2 (u)σ2 LL/2+1 (−u) . . . LL (−u) i (3.25) – 16 – JHEP08(2023)079 In order to define the crosscap state for the discretized Lieb-Liniger model, we make use of the fact that it is closely related to the generalized XXX spin chain [46, 50]. This can be seen easily by the following transformation of the Lax operator In the continuum limit, we expect the crosscap state becomes an integrable boundary state for the Lieb-Liniger model. The continuum version of the crosscap states can be obtained L/2  |C⟩ ≡ Y n=1  + 1 + S̃n+ S̃n+ |Ω⟩ → exp L Z ℓ/2 0 2 ! dx Φ† (x)Φ† (x + ℓ/2) |Ω⟩. (3.26) Details about taking the continuum limit can be found in appendix B. The resulting state is indeed the same as the result from the first method. Two comments about the crosscap state are in order. ∞ X |C⟩ = (3.27) |C2N ⟩, N =0 where the 2N -particle state is given by |C2N ⟩ = Z T dN x N Y Φ† (xj )Φ† (xj + ℓ/2)|Ω⟩. (3.28) j=1 The factor 1/N ! from the exponential is dropped because we have fixed the order of particle position T : 0 ≤ x1 < x2 < . . . < xN ≤ ℓ/2. The crosscap state is a superposition state of even number of particles. Therefore, the crosscap state can have non-vanishing overlaps with Bethe states with any even number of particles. • The crosscap state for Lieb-Liniger model can also be written in momentum space by performing a Fourier transformation ! |C⟩ = exp i X † † Kmn ξm ξn |Ω⟩, 2π m,n Kmn = 1 X iqx Φ(x) = √ e ξq , ℓ q q= (−1)m − (−1)n , m+n (3.29) where we used 2πm . ℓ (3.30) This formula is similar to the boundary state in integrable quantum field theory, and the coefficient Kmn plays the role of two-particle boundary amplitudes [1]. 3.2 Exact overlap formula In this subsection, we will derive the overlap of the crosscap state and an on-shell Bethe state using the coordinate Bethe ansatz. This method has been applied in the calculating the overlap of the BEC state and Bethe state in Lieb-Liniger model [29]. Using coordinate Bethe ansatz, the eigenstate is given by 1 |λN ⟩ = √ N! Z ℓ 0 dN x χN (xN |λN ) Φ† (x1 ) . . . Φ† (xN ) |Ω⟩, – 17 – (3.31) JHEP08(2023)079 • By expanding the exponential function, we can write the crosscap state as where |Ω⟩ is the Fock vacuum of the Lieb-Liniger model. The wave function is N X 1 X Y λσj − λσk − icϵ (xj − xk ) χN (xN |λN ) = √ exp i x n λ σn , λ σj − λ σk N ! σ∈SN j>k n=1 " # ! (3.32) where ϵ is the sign function. We consider the configuration space T : 0 ≤ x1 < x 2 < . . . < x N ≤ ℓ . (3.33) Introducing λ − ic , λ lj = eiλj , (3.34) we can write the wave function as N Y 1 X Y χN (xN |λN ) = √ f (λσj − λσk ) lσxnn . N ! σ∈SN j>k n=1 (3.35) Similar to the spin chain, the norm of an on-shell Bethe state is given by the Gaudin determinant ⟨λN |λN ⟩ = f (λj − λk ) f (λk − λj ) det GN , Y (3.36) j>k where the Gaudin matrix elements are N X Gjk = δjk ℓ + ! φ (λj − λl ) − φ (λj − λk ) , (3.37) l=1 with φ(λ) = 2c λ2 + c 2 (3.38) . If the rapidities are paired, the norm factorizes N/2 ⟨λN |λN ⟩ = Y f (2λj ) f (−2λj ) j=1 Y h f¯ (λj , λk ) i2 − det G+ N/2 det GN/2 , (3.39) 1≤jk where the integral region T is defined in (3.33), the function BN (lN |ℓ) is given by  N/2  BN (lN |ℓ) =  Y lnℓ/2  Z ℓ/2 n=1 Z xN/2 dxN/2 0 0 dxN/2−1 · · · N/2 Z x2 dx1 0 (lj lj+N/2 )xj . Y (3.46) j=1 Following the strategy used in spin chain, we can calculate the overlap. The mainly difference is that the particle position is continuous and the summations become integrals. For the 2-particle states, one can compute the integral exactly 1/2 (a1 a2 ) B2 (l2 |ℓ) = ia2 1/2 − 1 λ 1 + λ2 (3.47) . The overlap reads 1/2 (a1 a2 ) ⟨C|λ2 ⟩ = if (l2 , l1 )a2 1/2 − 1 λ1 + λ2 1/2 (a1 a2 ) + if (l1 , l2 )a1 1/2 − 1 λ1 + λ2 . (3.48) Similar to the spin chain case, we first take the paired rapidity limit (3.49) λ2 → −λ1 , which leads to the result q ⟨C|λ2 ⟩ = −ℓ f (−2λ1 )f (2λ1 ) = q f (−2λ1 )f (2λ1 ) det G+ 1. (3.50) For the N -particle states, we have N/2 BN (lN |ℓ) = X BN,j (lN |ℓ), (3.51) j=0 where BN,j (lN |ℓ) = (−1) j QN/2 k=j+1 Qj+N/2 1/2 QN a k=j+N/2+1 ak k=j+1 k  Q P i=j+1 i λi +λi+N/2 Pk – 19 – j k=1 j i=k i  λi +λi+N/2  . (3.52) JHEP08(2023)079 ⟨C|λN ⟩ = By investigating the behavior of BN (lN |ℓ) near the pole λm + λm+N/2 = 0, we find that  BN (lN |ℓ) ∼ am am+N/2 1/2 −1 i(λm + λm+N/2 ) am+N/2 1/2 ✘ . . . , N }|ℓ). BN −2,m−1 ({1, . . . ✚ m m✘ +✘ N/2 ✚. . .✘ ✘ (3.53) Interestingly, this relation is exactly the same as the one for Heisenberg spin chain (2.49). Following the same steps in spin chain model, one can obtain the overlap N/2 q Y f (2λj ) f (−2λj ) j=1 f¯ (λj , λk ) × det G+ N/2 Y (3.54) 1≤j