A symmetry formula for correlation functions in the superintegrable chiral Potts spin chain

A symmetry formula for correlation functions in the superintegrable chiral Potts spin chain Haoran Zhu a a School of Physical and Mathematical Sciences, Nanyang T echnological Univer sity , 21 Nanyang Link, Singapor e , 637371, Singapor e Abstract W e prov e an exact finite-volume symmetry formula for two-point functions in the periodic N -state superinte grable chiral Potts spin chain. W e show that, for e very chain length L and e very simultaneous eigen vector of the Hamiltonian and the one-site translation operator , the correlations satisfy ⟨ Z r 0 Z † r R ⟩ ∗ = ⟨ Z r 0 Z † r L − R ⟩ for 1 ⩽ r ⩽ N − 1. Hence, whenever L is ev en, the midpoint correlation ⟨ Z r 0 Z † r L / 2 ⟩ is real. Then we generalise the three-state chain case to arbitrary N and to ev ery translation eigensector . This resolves a conjecture of Fabricius and McCoy . K e ywor ds: superintegrable chiral Potts spin chain, Onsager algebra, correlation functions, translation symmetry 2020 MSC: 82B20, 82B23, 81R12 1. Introduction The superintegrable chiral Potts spin chain furnishes an N - state generalisation of the Ising chain with the same under - lying Onsager algebra. Onsager’ s 1944 solution of the two- dimensional Ising model introduced the algebraic structure now bearing his name [22]. In 1985, von Gehlen and Rittenberg discov ered a family of Z N -symmetric quantum chains with in- finitely many conserved char ges [29], and Baxter identified the superintegrable chiral Potts case as a particularly tractable rep- resentativ e [6]. Explicit finite-size computations revealed the characteristic Ising-like square-root form of the spectrum [1], and this form was later sho wn to follow directly from Onsager’ s algebra [10, 11, 12]. Important progress on eigenv ectors, On- sager sectors, and related algebraic identities was subsequently made by Au-Y ang and Perk, by Nishino and Deguchi, and by Roan [2, 3, 5, 4, 21, 25]. Progress on spin-operator matrix el- ements and on the reconstruction and form factors of local op- erators was later obtained by Iorgo v et al. and by Grosjean, Maillet, and Niccoli [14, 15]. For Onsager -type algebraic per- spectiv es, please see, for example, [26, 27, 28, 19]. For a periodic chain of length L , the basic observables are the two-point functions ⟨ Z r 0 Z † r R ⟩ , 1 ⩽ r ⩽ N − 1 , R ∈ { 0 , 1 , . . . , L − 1 } . By contrast with the Ising case, where free-fermion and deter- minant methods gi ve detailed control of correlations and spon- taneous magnetisation [17, 18, 20, 30] (see also free-fermion approach in [8, 9, 16]), finite-distance correlation functions in the chiral Potts setting remain far less explicit. Fabricius and McCoy computed these ground-state correlations for the three- state superintegrable chain at lengths L = 3 , 4 , 5 [13]. From Email addr ess: zhuh0031@e.ntu.edu.sg (Haoran Zhu) their finite-size data they proposed a conjectural form for the nearest-neighbour correlation, and in their concluding discus- sion, they pointed to a second striking phenomenon: for e ven chain length the half-chain correlation ⟨ Z 0 Z † L / 2 ⟩ appears to be real [13]. The main aim of the present paper is to sho w that the finite- volume reality statement is governed by a symmetry principle. W e pro ve an exact finite-volume identity for two-point func- tions in ev ery translation eigensector of the periodic superinte- grable chiral Potts chain. W e write N for the number of spin states and L for the chain length (thus our L corresponds to the symbol N in [13]). Let ω = e 2 π i / N , and let Z , X ∈ M N ( C ) be the standard W eyl operators, Z ab = ω a δ ab , X ab = δ a , b + 1 , Z X = ω X Z , Z N = X N = id . On the periodic chain ( C N ) ⊗ L we write Z j and X j for the cor- responding local operators at site j , with all indices understood modulo L . The periodic superintegrable chiral Potts Hamilto- nian [29, 6, 13] is H = A 0 + λ A 1 , (1.1) where A 0 = − L − 1 X j = 0 N − 1 X r = 1 e i π (2 r − N ) 2 N sin( π r / N ) Z r j Z † r j + 1 , A 1 = − L − 1 X j = 0 N − 1 X r = 1 e i π (2 r − N ) 2 N sin( π r / N ) X r j . (1.2) For real λ , this is the finite-volume Hamiltonian considered in [13]. Let T denote the unitary one-site translation operator . Since H is a periodic sum, it commutes with T ; hence each eigenspace of H admits a basis of simultaneous eigen vectors of H and T . Our main result is the following theorem. Theorem 1.1. Let | ψ ⟩ be a normalised simultaneous eigen vec- tor of the Hamiltonian H in (1.1) – (1.2) and the one-site trans- lation operator T . F or 1 ⩽ r ⩽ N − 1 and R ∈ Z / L Z , define ρ r ( R ) : = ⟨ ψ | Z r 0 Z † r R | ψ ⟩ . Then ρ r ( R ) ∗ = ρ r ( − R ) (1.3) for every R ∈ Z / L Z . In particular , if L is even, then ρ r ( L / 2) ∈ R . (1.4) 2. Symmetry formula and r eality conjectur e W e begin with the one-site translation operator . Let T be the unitary operator on ( C N ) ⊗ L characterised by T Z j T − 1 = Z j + 1 , T X j T − 1 = X j + 1 , T L = id , (2.1) with all site labels understood modulo L . Since the Hamiltonian (1.1)–(1.2) is a periodic sum ov er the sites, we have T H T − 1 = H , equiv alently [ H , T ] = 0 . (2.2) Hence, each eigenspace of H admits an orthonormal basis of simultaneous eigen vectors of H and T . The follo wing elementary lemma is needed for the proof of the main theorem. Lemma 2.1. Let | ψ ⟩ be a normalised eigen vector of T .Then, for every oper ator O on ( C N ) ⊗ L and every inte ger m, ⟨ ψ | T − m O T m | ψ ⟩ = ⟨ ψ |O| ψ ⟩ . (2.3) Pr oof. Since T | ψ ⟩ = τ | ψ ⟩ , we hav e T m | ψ ⟩ = τ m | ψ ⟩ and, by taking adjoints, ⟨ ψ | T − m = τ m ⟨ ψ | . Therefore ⟨ ψ | T − m O T m | ψ ⟩ = τ m τ m ⟨ ψ |O| ψ ⟩ = ⟨ ψ |O| ψ ⟩ , as required. W e now prov e the symmetry relation (1.3). The key point is that comple x conjugation rev erses the order of the local oper- ators, while translation transports the resulting separation to its opposite on the ring. Pr oof of Theor em 1.1. Fix r ∈ { 1 , . . . , N − 1 } and R ∈ Z . By definition, ρ r ( R ) = ⟨ ψ | Z r 0 Z † r R | ψ ⟩ , where the site label R is read modulo L . T aking complex con- jugates and using ( A B ) † = B † A † , we obtain ρ r ( R ) ∗ = ⟨ ψ | Z r R Z † r 0 | ψ ⟩ . (2.4) Apply Lemma 2.1 with O = Z r R Z † r 0 and m = R . Then ρ r ( R ) ∗ = ⟨ ψ | T − R ( Z r R Z † r 0 ) T R | ψ ⟩ . (2.5) Using (2.1), we compute T − R Z r R T R = Z r 0 , T − R Z † r 0 T R = Z † r − R , again with site labels understood modulo L . Substituting this into (2.5) giv es ρ r ( R ) ∗ = ⟨ ψ | Z r 0 Z † r − R | ψ ⟩ = ρ r ( − R ) . This prov es (1.3). If L is e ven and R = L / 2, then − R ≡ R (mod L ). Hence ρ r ( L / 2) ∗ = ρ r ( L / 2) , so ρ r ( L / 2) is real. This establishes (1.4). It is con venient to record the original three-state statement in the form conjectured by Fabricius and McCoy . Corollary 2.2 (Fabricius–McCoy conjecture) . Assume N = 3 and let L be even. Let | ψ 0 ⟩ be a normalised finite-volume gr ound state c hosen to be an eigen vector of the one-site translation op- erator . Then ⟨ ψ 0 | Z 0 Z † L / 2 | ψ 0 ⟩ ∈ R . Pr oof. Since [ H , T ] = 0, the finite-volume ground-state eigenspace contains an orthonormal basis of simultaneous eigen vectors of H and T . The claim is the case N = 3 and r = 1 of Theo- rem 1.1. Acknowledgements W e would like to thank the support of the NTU Research Scholarship (RSS). The author also benefits from the discussion with K. W ang. References [1] G. Albertini, B. M. McCoy , J. H. H. Perk, and S. T ang, Ex- citation spectrum and order parameter for the integr able N -state chiral P otts model , Nucl. Phys. B 314 (1989) 741– 763. 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