The ground state-vector of the XY Heisenberg chain and the Gauss decomposition
TThe ground state-vector of the
X Y
Heisenberg chain and the Gauss decomposition
Nikolay Bogoliubov, Cyril Malyshev
St.-Petersburg Department of Steklov Institute of Mathematics, RASFontanka 27, St.-Petersburg, RUSSIA
Abstract
The XY Heisenberg spin chain is considered in the fermion representation.The construction of the ground state-vector is based on the group-theoreticalapproach. The exact expression for the ground state-vector will allow to studythe combinatorics of the correlation functions of the model. Key words: XY Heisenberg spin chain, ground state-vector, Gauss decomposition a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Introduction
The correlation functions of certain quantum integrable models demonstrate connec-tion with enumerative combinatorics and with the theory of symmetric functions [1–5].For instance, random lattice walks and boxed plane partitions, as subjects of enumera-tive combinatorics [6], are related to the correlation functions of the XX model [7–11].Various spin lattice models [12], including the XY Heisenberg chain model, as wellas its isotropic limit, the XX model, provide a base for such actively developingsubjects in the theory of quantum information [13] as random lattice walks [14] andentanglement entropy [15].Interest in the study of the correlation functions for the XY spin chain still existsafter the pioneer works [16–18]. The determinantal representation for the the equal-time correlation functions for the model was obtained in the paper [19]. The approachbased on the application of the coherent states to the problem of time and temperaturedependent correlation functions was developed in [20, 21]. In the present paper weconsider a group theoretical approach to the XY model and derive its ground state-vector using the Gauss decomposition [22]. The representation of the obtained groundstate wave function would be of importance in the combinatorial interpretation of thecorrelation functions of the model. Namely, the scalar products of the state-vectorsmay be described as a linear combination of nests of self-avoiding lattice paths, so-called, watermelons (see Figure 1 and Ref. [2]). С С С С B B B B Figure 1: A nest of self-avoiding lattice paths called watermelon
The paper is organised as follows. Section 1 is introductory. The fermion represen-tation of the model is provided in Section 2, and diagonalization of the Hamiltonianby means of the Bogoliubov transformation is performed. The ground state wavefunction is derived in Section 3. Section 4 concludes the paper.2
Outline of the model
The XY Heisenberg spin chain is described by the Hamiltonian, [16, 17]: H = H xx + γH anis , H xx ≡ − M (cid:88) n,m =1 ∆ (+) nm σ + n σ − m , (1) H anis ≡ − M (cid:88) n,m =1 ∆ (+) nm ( σ + n σ + m + σ − n σ − m ) , (2)where γ is anisotropy parameter, and M = 0 (mod 2) is the number of sites. Thelocal spin operators σ ± n = ( σ xn ± iσ yn ) and σ zn depend on the lattice argument n ∈M ≡ { , , . . . , M } and satisfy the commutation relations: [ σ + k , σ − l ] = δ kl σ zl , [ σ zk , σ ± l ] = ± δ kl σ ± l . (3)The entries of the hopping matrix ∆ ( s ) in (1), (2) are expressed as follows, [9]: ∆ ( s ) nm ≡
12 ( δ | n − m | , + sδ | n − m | ,M − ) , (4)where δ n,l ( ≡ δ nl ) is the Kronecker symbol, and s is either ± or zero. The periodicboundary conditions σ αn + M = σ αn , ∀ n ∈ M , are imposed. The Hamiltonian H xx (1)(i.e., H at γ = 0 ) is that of the periodic XX chain.Let us pass from the spin operators σ αk to the canonical fermion operators c k , c † k subjected to the algebra { c k , c n } = { c † k , c † n } = 0 , { c k , c † n } = δ kn , (5)where the brackets { , } imply anti-commutation. We use the Jordan-Wigner trans-formation [23]: c k = exp (cid:16) iπ k − (cid:88) n =1 σ − n σ + n (cid:17) σ + k , c † k = σ − k exp (cid:16) − iπ k − (cid:88) n =1 σ − n σ + n (cid:17) . (6)Inversion of (6) takes the form: σ − n = c † n n − (cid:89) j =1 (1 − c † j c j ) , σ + n = n − (cid:89) j =1 (1 − c † j c j ) c n . (7)The periodic boundary conditions for the spin variables are equivalent to the followingboundary conditions for the fermion variables: c M +1 = ( − N c , c † M +1 = c † ( − N , (8)where N ≡ Q ( M ) = (cid:80) Mk =1 c † k c k is the total number of particles.3he transformations (6), (7) enable to represent H (1) as follows [16, 17]: H = H + P + + H − P − , (9) H ± = − M (cid:88) k =1 (cid:2) c † k c k +1 + c † k +1 c k + γ ( c k +1 c k + c † k c † k +1 ) (cid:3) , (10)where P ± are the projectors onto the states with even ( + )/odd ( − ) number offermions: P ± = (1 ± ( − N ) . The indices s = ± point out the correspondencebetween the operators H s (10) and appropriately specified boundary conditions (8): c M +1 = − sc , c † M +1 = − sc † . (11)The operator N commutes with H xx (1). The parity operator ( − N commutes with H and anti-commutes with c † k and c k .The requirements (11) suggest to use the Fourier series: c k = e − iπ/ √ M (cid:88) q ∈ S ± e iqk c q , c † k = e iπ/ √ M (cid:88) q ∈ S ± e − iqk c † q , (12)where (cid:80) q ∈ S ± implies summation over quasi-momenta q ∈ S ± respecting cos M q = ∓ : S + = { q : q = − π + π (2 l − /M , l ∈ M} , S − = { q : q = − π + 2 πl/M , l ∈ M} . Substitution of (12) into (10) yields the XY Hamiltonian in the momentum represen-tation: H ± = 12 (cid:88) q ∈ S ± ( c † q , c − q ) H q (cid:18) c q c †− q (cid:19) , (13) H q = (cid:15) q σ z + Γ q ( σ + + σ − ) , (14)where (cid:15) q ≡ − cos q and Γ q ≡ γ sin q .It appropriate to introduce three quadratic operators J ± q , J q expressed throughthe fermion operators c q and c † q : J − q = c − q c q , J + q = c † q c †− q , (15) J q = 12 ( c † q c q + c †− q c − q − . (16)The operators (15), (16) are related to the algebra su (2) since satisfy the commutationrelations of the form (compare with (3)): [ J + q , J − p ] = 2 J q δ pq , [ J q , J ± p ] = ±J ± q δ pq . (17)The definitions (15) and (16) allow us to express H ± (13) as follows: H ± = (cid:88) q ∈ S ± (cid:15) q J q + Γ q J + q + J − q ) . (18)4et us relate the canonical operators c † q , c q to the new fermionic operators A † q , A q by means of the unitary matrix g † θ ∈ SU (2 , R ) , (cid:18) c q c †− q (cid:19) = g † θ (cid:18) A q A †− q (cid:19) , (19) g † θ = e − θ q ( σ − − σ + ) = e iθ q σ y = cos θ q σ + sin θ q ( σ + − σ − ) . (20)The transformation (19) and its conjugated are used in (13), and it enables to diago-nalize the matrix H q (14) as follows: g θ H q g † θ = E q σ z . (21)The relation (21) is equivalent to the following equations: (cid:15) q cos 2 θ q − Γ q sin 2 θ q = E q , (22) (cid:15) q sin 2 θ q + Γ q cos 2 θ q = 0 , (23)where E q = ( (cid:15) q + Γ q ) / . It follows from (23) that θ q respects tan 2 θ q = − Γ q /(cid:15) q .Let us introduce, analogously to (15), (16), the appropriate operators ¯ J ± q , ¯ J q interms of the fermion operators A q and A † q : ¯ J − q = A − q A q , ¯ J + q = A † q A †− q , (24) ¯ J q = 12 ( A † q A q + A †− q A − q − . (25)The following transformation takes place: J − q + J + q = ( ¯ J − q + ¯ J + q ) cos 2 θ q − J q sin 2 θ q , (26) J q = ( ¯ J − q + ¯ J + q ) sin 2 θ q + 2 ¯ J q cos 2 θ q . (27)Applying the transformations (26) and (27) allows us to express the Hamiltonians H ± (18) as follows: H ± = (cid:88) q ∈ S ± E q ¯ J q = (cid:88) q ∈ S ± E q A † q A q + E ± gr , (28)provided that the relations (22) and (23) hold, and E ± gr is expressed as E ± gr = − (cid:88) q ∈ S ± E q . The canonical operators c k , c † k , as well as c q , c † q , characterized by the relations (5)possess the Fock vacuum | (cid:105) (and its conjugate (cid:104) | ): c k | (cid:105) = 0 , (cid:104) | c † k = 0 , k ∈ M , (29) c q | (cid:105) = 0 , (cid:104) | c † q = 0 , q ∈ S ± . (30)5he vacuum | (cid:105) is normalized, (cid:104) | (cid:105) = 1 , and | (cid:105) is the same for both Hamiltonians H ± .The ground-state vector | (cid:105)(cid:105) of the Hamiltonian (28) have to satisfy the relations: A q | (cid:105)(cid:105) = 0 , (cid:104)(cid:104) | A † q = 0 , q ∈ S ± . (31)We introduce the unitary operator U θ , U θ ≡ exp (cid:16) (cid:88) q ∈ S ± θ q ( J + q − J − q ) (cid:17) = (cid:89) q ∈ S ± exp (cid:0) θ q ( J + q − J − q ) (cid:1) , (32)where J − q and J + q are defined by (15) and 17) and formulate the following Proposition 1
The relations (31) take place provided that the state | (cid:105)(cid:105) is definedas: | (cid:105)(cid:105) = U θ/ | (cid:105) . (33) Proof
The commutation relations are valid: [ c q , (cid:88) p ∈ S ± θ p J + p ] = 2 θ q c †− q , [ c † q , (cid:88) p ∈ S ± θ p J + p ] = 0 , (34) [ c †− q , (cid:88) p ∈ S ± θ p J − p ] = 2 θ q c q , [ c q , (cid:88) p ∈ S ± θ p J − p ] = 0 , (35)where the property θ − q = − θ q is used. Taking into account (34) and (35) we obtain: U θ/ c q U † θ/ = A q , (36) U θ/ c †− q U † θ/ = A †− q . (37)The state | (cid:105)(cid:105) (33) is annihilated by A q (36) since c q annihilates the Fock vacuum | (cid:105) ,Eq. (30), and U † θ/ U θ/ = 1 . The introduced ground state-vector (33) is normalized tounity, (cid:104)(cid:104) | (cid:105)(cid:105) = 1 .The alternative derivation of Proposition 1 is based on the equivalence of therelations (36), (37) and the transformation (19). The action of operator A q on theground state (33) may be written as A q | (cid:105)(cid:105) = (cos θ q c q − sin θ q c †− q ) U θ/ | (cid:105) . (38)The commutation relations c q U θ/ = U θ/ (cos θ q c q + sin θ q c †− q ) , (39) c †− q U θ/ = U θ/ ( − sin θ q c q + cos θ q c †− q ) , (40)ensure that A q | (cid:105)(cid:105) = U θ/ c q | (cid:105) = 0 . (41) (cid:3) g θ ≡ exp( − iθσ y ) = exp (cid:0) θ ( σ − − σ + ) (cid:1) (20): e θ ( σ − − σ + ) = e − tan θ σ + e ¯ γσ z e tan θ σ − , (42)where e ¯ γ = 1 / cos θ . With regard to (42) we arrive at the decomposition for theelements of the operator U θ (32): exp (cid:0) θ q ( J + q − J − q ) (cid:1) = exp (cid:0) tan θ q J + q (cid:1) exp (cid:0) γ q J q (cid:1) exp (cid:0) − tan θ q J − q (cid:1) , (43)where e ¯ γ q = 1 / cos θ q . Equation (43) suggests to formulate the following Proposition 2
Provided that the representation (43) holds, the state | (cid:105)(cid:105) (33) ac-quires the equivalent representation : | (cid:105)(cid:105) = (cid:16) (cid:89) q ∈ S ± cos / θ q (cid:17) exp (cid:16) (cid:88) q ∈ S ± tan θ q J + q (cid:17) | (cid:105) . (44) Proof
First of all, the following note is of importance: U θ/ = exp (cid:16) (cid:88) q ∈ S ± θ q J + q − J − q ) (cid:17) = exp (cid:16)(cid:88) q + θ q ( J + q − J − q ) (cid:17) (45) = exp (cid:16) (cid:88) q ∈ S ± tan θ q J + q (cid:17) exp (cid:16) (cid:88) q ∈ S ± ¯ γ q J q (cid:17) exp (cid:16) − (cid:88) q ∈ S ± tan θ q J − q (cid:17) , (46)where (cid:80) q + ≡ (cid:80) { ( q ∈ S ± ) ∩ ( q ∈ R + ) } , and antisymmetry of θ q , J + q , J − q with respect to thereflection of q is taken into account in (45). The Gauss decomposition (43) is used topass from (45) to (46).From definitions of operators J − q (15) and J q (16) it follows that J − q | (cid:105) = 0 , while J q | (cid:105) = − | (cid:105) . Therefore, exp (cid:16) (cid:88) q ∈ S ± tan θ q J − q (cid:17) | (cid:105) = | (cid:105) , (47)and exp (cid:16) (cid:88) q ∈ S ± ¯ γ q J q (cid:17) | (cid:105) = (cid:89) q ∈ S ± e − ¯ γ q / = (cid:89) q ∈ S ± cos / θ q . (48)Thus, we obtain from (46), (47) and (48) that (44) is valid. (cid:3) The representation (44) of the ground state | (cid:105)(cid:105) coincides with that proposedin [20]: | (cid:105)(cid:105) = N − / Ω + | (cid:105) , Ω + = exp (cid:32) (cid:88) p ∈ S ± tan θ p c † p c †− p (cid:33) . The normalizing factor N − / , where N = (cid:104) | Ω Ω + | (cid:105) was calculated in [20] as theintegral over the Grassmann coherent states, is equal to N − / = (cid:89) p + cos θ p , (49)7nd coincides with the coefficient in (44). The statement of Proposition 2 clarifiesthe origin of this coefficient from the group theoretical viewpoint.For the sake of completeness we shall give the direct proof that the state | (cid:105)(cid:105) expressed by (44) is annihilated by the operator A q . Really, since A q is given by (36),it is enough to show that the state U † θ/ exp (cid:16) (cid:88) q ∈ S ± tan θ q J + q (cid:17) | (cid:105) (50)is annihilated by c q . The Gauss decomposition (46) admits the “antinormal” fromlooking as follows (see [22]): U θ/ = exp (cid:16) − (cid:88) q ∈ S ± tan θ q J − q (cid:17) exp (cid:16) − (cid:88) q ∈ S ± ¯ γ q J q (cid:17) exp (cid:16) (cid:88) q ∈ S ± tan θ q J + q (cid:17) . (51)Substituting the conjugated form of (51) into (50) we obtain the state annihilated by c q : (cid:16) (cid:89) q ∈ S ± cos − / θ q (cid:17) exp (cid:16)(cid:88) q + tan θ q J − q (cid:17) | (cid:105) = (cid:16) (cid:89) q ∈ S ± cos − / θ q (cid:17) | (cid:105) . Let us turn to the state | (cid:105)(cid:105) (44) and consider the following representation: (cid:16) (cid:89) q ∈ S ± cos − / θ q (cid:17) | (cid:105)(cid:105) = exp (cid:16) (cid:88) q ∈ S ± tan θ q J + q (cid:17) | (cid:105) = | (cid:105) + M/ (cid:88) n =1 n ! (cid:88) { q + l } ≤ l ≤ n n (cid:89) l =1 (cid:0) T q l J + q l (cid:1) | (cid:105) , T q l ≡ tan θ q l , (52)where (cid:80) { q + l } ≤ l ≤ n ≡ (cid:80) q +1 , q +2 ,..., q + n , and the sum over n is finite since J + q squared iszero. The expression in right-hand side of (52) is similar to that derived in [19] as theground state wave function of XY chain. Recall that tan 2 θ q = − Γ q /(cid:15) q (see (23)) isknown, and therefore tan θ q is found in the form: T q = (cid:15) q ± (cid:112) (cid:15) q + Γ q Γ q = (cid:15) q ± E q Γ q , (53)where T q is defined in (52). The answer (53) fulfils the quadratic equation as thecorresponding parameter does in [19]. With regard to (53) it can be argued thatEq. (52) just coincides (up to irrelevant factor) with the ground state wave functionfound in [19]. The expression for the ground ground state-vector (33) of the XY spin chain obtainedby the group theoretical approach was written in the form (44) with the help of8aussian decomposition. The representation (44) brought to the form (52) revealsthe connection between state-vectors studied in [19] and [20].The approach discussed in the present paper will allow to spread the combinatorialinterpretation of the correlation functions developed in [2] for the XX model on the XY case. Acknowledgement
This work was supported by the Russian Science Foundation (grant no. 18-11-00297).
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