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U Mixed-state form factors of (1) twist fields in the Dirac theory Yixiong Chen Department of Mathematics, King’s College London, Strand WC2R 2LS, UK 6 Abstract 1 0 2 Using the “Liouville space” (the space of operators) of the massive Dirac theory, we l define mixed-state form factors of U(1) twist fields. We consider mixed states with density u J matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state 1 as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specializedtoathermalGibbsstate,usingaRiemann-Hilbertproblemandlow-temperature ] expansion, we obtain finite-temperature form factors of U(1) twist fields. We then propose h c theexpressionforformfactorsofU(1)twistfieldsingeneraldiagonalmixedstates. Weverify e that these form factors satisfy a system of nonlinear functional differential equations, which m isderivedfromthe tracedefinitionofmixed-stateformfactors. Atlast,underweakanalytic - conditions on the eigenvalues of the density matrix, we write down the large distance form t a factor expansions of two-point correlation functions of these twist fields. Using the relation t between the Dirac and Ising models, this providesthe large-distanceexpansionof the Rényi s . entropy (for integer Rényi parameter) in the Ising model in diagonal mixed states. t a m - d n o c [ 3 v 2 0 7 3 0 . 1 0 6 1 : v i X r a March 2016 Contents 1 Introduction 1 2 Dirac theory at zero temperature 3 2.1 Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Bosonic primary twist fields and their form factors . . . . . . . . . . . . . . . . . 5 2.3 Fermionic primary twist fields and their form factors . . . . . . . . . . . . . . . . 6 3 Liouville space and mixed-state form factors 7 4 Calculations and main results 9 4.1 Finite-temperature form factors of U(1) twist fields . . . . . . . . . . . . . . . . 10 4.1.1 Riemann-Hilbert problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.1.2 Low-temperature expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Mixed-state form factors of U(1) twist fields . . . . . . . . . . . . . . . . . . . . . 14 4.2.1 Exact mixed-state form factors of U(1) twist fields . . . . . . . . . . . . . 14 4.2.2 Non-linear functional differential system of equations . . . . . . . . . . . . 15 4.2.3 General solution as integral-operator kernel . . . . . . . . . . . . . . . . . 18 4.3 Mixed-state two-point correlation functions of twist fields . . . . . . . . . . . . . 22 4.3.1 Three modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3.2 Two-point correlation functions . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Application: Reńyi entropy for integer n in the Ising model 25 5.1 Branch-point twist fields in the n-copy Ising model . . . . . . . . . . . . . . . . . 25 5.2 Explicit representation of the branch-point twist fields in the n-copy Ising model in terms of the U(1) twist fields in the n-copy Dirac theory . . . . . . . . . . . . 26 5.3 Rényi entropy for integer n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Conclusion 29 A Proof of relation (4.50) 30 B Recursion relation for the normalization of mixed-state form factors of U(1) twist fields 31 1 Introduction Correlation functions of local fields play an important role in quantum field theory (QFT) be- causetheyyieldallphysicalinformationofthemodelandtheyaredirectlyrelatedtoexperiment results in condensed matter systems near criticality. However, in general, the evaluation of cor- relationfunctionsinQFTisaquitenon-trivialwork. Fortunately, in1+1dimensionalintegrable models of QFT [1, 2, 3, 4, 5], the existence of an infinite number of conserved charges makes it possible to exactly evaluate many quantities such as form factors, which are matrix elements of local fields in eigenstates of the Hamiltonian. Successes for these models have been achieved 1 over the last two decades on the determination of vacuum correlation functions via form factor bootstrap approach [1, 2, 3, 4, 5]. For instance, large distance expansions of two-point correlation functions, which are hardly accessible by perturbation theory, can be obtained by form factor expansion (Kallen-Lehmann expansions) [1, 2, 3, 4, 5]under the factorized scattering theory and both their large-distance and short-distance asymptotic behaviors agree with general QFT expectations (see for instance, [6, 7]). In recent years, correlation functions in general mixed states have attracted growing interest and triggered an enormous amount of work, because of their wide scope of applications both of theoretical and experimental interest. For instance, correlation functions in thermal Gibbs state, which can be related to correlation functions on an infinite cylindrical geometry [8], have been the subject of intense study in massive integrable QFT (see for instance the review [9]). In particular, the Ising model at finite temperature has been widely investigated by employing severalapproachesincludingformfactorexpansions[10,11,12,13,14,15], integrabledifferential equations[16,17],semi-classicmethods[18],andthefinitevolumeregularizationmethod[19,20, 21, 22, 23, 24]. On the other hand, more mixed states have been explored, including generalized Gibbsensembles(GGEs)whichhavebeenpredictedtooccurafterquantumquenchinintegrable models[25,26, 27, 28,29], non-equilibriumsteady state [30,31,32,33,34]and others. However, further development still needs to be made in order to clarify how the structure of correlation functions depend on the mixed states in general situations. In this paper, we concentrate on correlationfunctionsofU(1)twistfieldsinfreemassiveDiractheoryingeneralmixedstateswith diagonal density matrix. These density matrices include thermal Gibbs states and generalized Gibbs ensembles. Twist fields are local fields equipped with non-trivial exchange relations with respect to the fundamental boson fields or fermion fields, associated with a symmetry of the model. The concept of twist fields was first introduced in [35], as Z monodromy field of the Majorana 2 fermion, corresponding to the spin field of the Ising model. In the massive Dirac model, there exist a family of bosonic primary twist fields and two families of fermionic primary twit fields, which are associated with the U(1) symmetry transformation. These U(1) twist fields are interacting fields, due to their semi-locality with respect to the Dirac fermions, even though the model we are considering in this paper is a free massive model with a trivial scattering matrix. Vacuumformfactors of thesefieldsarewell known[36,37,38]andithasbeenclear thatvacuum correlation functionsofU(1) twistfieldscan beparameterized in termsof solutions tonon-linear differential equations [39, 40, 41, 42]. However, there is relatively little known about correlation functions of U(1) twist fields in general mixed states even though we expect same non-linear differential equations for them. In this paper, we obtain for the first time large distance expansions of two-point correlation functionsofU(1)twistfieldsingeneraldiagonalmixedstates, usingthemethodofthe“Liouville space”. This method was initially established in [12, 15] to derive finite-temperature spin-spin correlation functions, and then further developed in [43] to obtain general diagonal mixed-state spin-spincorrelation functions,bothintheIsingmodelofQFT.TheLiouvillespaceconstruction [44, 45] is based on the GNS construction of C∗-algebras [46] and it has applications in thermal and non-equilibrium physics. In the present paper, we apply this method to the free Dirac theory. We define and evaluate the associated mixed-state form factors of U(1) twist fields, and then formulate mixed-state two-point functions of these fields using form factor expansion with 2 respect to the vacuum in the Liouville space. At zero temperature, the ordinary form factors are mostly obtained by solving a Riemann- Hilbert problem in terms of form factors as functions of rapidities. However, when it comes to the case of general mixed states, this method exhibits some difficulties due to the inability to know the analytic structure of the general density matrix. In this paper, we first evaluate finite-temperature form factors of U(1) twist fields by deriving low-temperature expansions of thermal form factors and setting up a similar Riemann-Hilbert problem as the one in [12, 15] for finite-temperature factors in the Ising model. Considering the way how these finite-temperature form factors rely on the eigenvalues of density matrix, we then conjecture an explicit expression forgeneraldiagonalmixed-stateformfactorsofU(1)twistfields. Theseconjecturedformfactors areproventobethesolutionofasystemofnon-linearfirst-orderfunctionaldifferentialequations for mixed-state form factors. In the derivation of finite-temperature form factors, we employ imaginary-time formalism [8] to relate to quantization on the circle and to derive Kubo-Martin- Schwinger (KMS) identity leading to the Riemann-Hilbert problem. But, in the case of general mixed states, we perform real-time manipulation since there is no clear quantization scheme related in imaginary time. This paper is organized as follows. In section 2, we review the basic concepts of the free massiveDiractheory, andprovideasummaryofU(1)twistfieldsandtheirvacuumformfactors. Insection3,weconstructtheLiouvillespaceanddefinethemixed-stateformfactorsintheDirac model. In section 4, we present calculations leading to our main results which include form factors of U(1) twist fields in general diagonal mixed states and the corresponding two-point correlation functions. In section 5, we apply our result of the mixed-state two-point correlation function of U(1) twist fields to compute the Rényi entropy for integer n in the Ising model. Finally, we conclude in Section 6. 2 Dirac theory at zero temperature 2.1 Dirac fermions In the free massive Dirac theory, fermion operators with evolution in real time t are given in † terms of mode operators D (θ) and D (θ): ± ± Ψ (x,t) = √m dθeθ/2 D† (θ)eitEθ−ixpθ iD (θ)e−itEθ+ixpθ R + − − Z (cid:16) (cid:17) Ψ (x,t) = √m dθe−θ/2 iD† (θ)eitEθ−ixpθ D (θ)e−itEθ+ixpθ (2.1) L + − − Z (cid:16) (cid:17) where E = mcoshθ , θ p = msinhθ . θ 3 The Hermitian Conjugate of fermion operators can be obtained by directly taking Hermitian Conjugate of (2.1): Ψ† (x,t) = √m dθeθ/2 iD† (θ)eitEθ−ixpθ +D (θ)e−itEθ+ixpθ R − + Z (cid:16) (cid:17) Ψ†(x,t) = √m dθe−θ/2 D†(θ)eitEθ−ixpθ iD (θ)e−itEθ+ixpθ . (2.2) L − − − + Z (cid:16) (cid:17) The creation and annihilation operators satisfy canonical anti-commutation relations: † D (θ ),D (θ ) = δ(θ θ ) { + 1 + 2 } 1− 2 † D (θ ),D (θ ) = δ(θ θ ) (2.3) { − 1 − 2 } 1− 2 with other anti-commutators vanishing. The fermion operators satisfy the equations of motion ∂¯Ψ = mΨ , ∂Ψ = mΨ R L L R ∂¯Ψ† = mΨ†, ∂Ψ† = mΨ† (2.4) R L L R where we define notations ∂ := ∂ ∂ and ∂¯:= ∂ +∂ , and their anti-commutation relations x t x t − are † Ψ (x ),Ψ (x ) = 4πδ(x x ) { R 1 R 2 } 1 − 2 † Ψ (x ),Ψ (x ) = 4πδ(x x ) (2.5) { L 1 L 2 } 1 − 2 with other anti-commutators vanishing. The Hilbert space is simply the Fock space over H algebra (2.3) with vacuum state defined by D vac = 0 and with multi-particle states denoted ± | i by θ ,...,θ := D† D† vac , θ > > θ (2.6) | 1 Niν1,...,νN ν1··· νN| i 1 ··· N where ν are signs( ), corresponding with particle type. Multi-particle states with different i ± ordering can be obtained upon exchange of two particles (θ ,ν ) and (θ ,ν ), in agreement with i i j j (2.3). The inner products are normalized as follows: N θ ,...,θ θ′,...,θ′ = δ δ(θ θ′). (2.7) ν1,...,νNh 1 N| 1 Niν1′,...,νN′ νi,νi′ i − i i=1 Y Then the resolution of the identity are written as: ∞ 1 ∞ ∞ 1 = dθ dθ θ ,...,θ θ ,...,θ (2.8) N! 1··· N | 1 Niν1,...,νN ν1,...,νNh 1 N| NX=0 ν1X,...,νNZ−∞ Z−∞ where N! in the denominator comes from overcounting the same state with different orderings of rapidities. 4 2.2 Bosonic primary twist fields and their form factors The Dirac theory possesses a U(1) internal symmetry Ψ e2πiαΨ where 0 α < 1 and R,L R,L 7→ ≤ there exists a family of primary twist fields σ (x,t) associated with this symmetry, which are α local, Lorentz spinless, and U(1) neutral, with dimension α2 [40]. These fields generate even number of fermions and hence are of bosonic statistics. The bosonic primary twist fields with negative index can be defined by Hermitian conjugation: σ† = σ , 0 α < 1. (2.9) α −α ≤ Twist fields σ are associated with branch cuts through which other fields are affected by the α U(1) symmetry transformation. These branch cuts, in principle, can be taken arbitrarily. Here, for convention, we denote by σ+ the twist fields with branch cuts running towards the right α direction, while by σ− the ones with branch cuts running towards the left direction. These two α types of twist fields are related to each other by an unitary operator Z := exp 2πiνα dθD†(θ)D (θ) (2.10) ν ν " # ν Z X which implements the U(1) symmetry transformation and we have σ−(x,t) = σ+(x,t)Z. (2.11) α α η Twist fields σ with η = are semi-local with respect to the Dirac fermion fields, and are α ± characterized by equal-time exchange relations δ e2πiηα +δ ση(0)Ψ (x) (x < 0) Ψ (x)ση(0) = η,− η,+ α R,L (2.12) R,L α ( (cid:0)δη,+e2πiηα +δη,−(cid:1)σαη(0)ΨR,L(x) (x > 0) and (cid:0) (cid:1) δ e−2πiηα +δ ση(0)Ψ† (x) (x < 0) Ψ† (x)ση(0) = η,− η,+ α R,L (2.13) R,L α ( (cid:0)δη,+e−2πiηα +δη,−(cid:1)σαη(0)Ψ†R,L(x) (x > 0) . η Thanks to these twist conditions,(cid:0)two-particle form fa(cid:1)ctors of twist fields σ for 1 < α <1 can α − be fixed, up to normalization, [35, 36, 37, 38](see also appendix A of [39]): sin(πα)eν1α(θ1−θ2) vac ση(0)θ ,θ = δ ν σ (2.14) h | α | 1 2iν1,ν2 ν1,−ν2 1 2πi cosh θ1−θ2 h αi 2 where σ := vac ση vac = c mα2 is the vacuum expectation value. The dimensionless α α α h i h | | i constants c are computed in [47, 48]. All other higher-particle form factors can be obtained α η by Wick’s theorem due to the fact that twist fields σ can be expressed as normal-ordered α exponentials of bilinear expressions in Dirac fermion operators. Other matrix elements can be η evaluated using crossing symmetry. Note that twist fields σ have non-zero form factors only α for even particle numbers since they are U(1) neutral. Finally, it is natural for us to define twist fields σ for all α R Z∗ by noticing that their form factors for fixed rapidities are analytic α ∈ \ functions of α on α C Z∗, with general poles on Z∗ := Z 0 . ∈ \ \{ } 5 2.3 Fermionic primary twist fields and their form factors In the U(1) Dirac theory, there also exist two families of primary twist fields of fermionic statistics, which can be obtained as the coefficients occuring in the operator product expansions (OPEs) of primary twist fields σ with the Dirac fields Ψ and Ψ† , for all α R Z∗: α R R ∈ \ σ (x,t) = lim(z w)αΨ† (x′,t′)σ (x,t) (2.15) α+1,α z→w − R α σ (x,t) = lim(z w)−αΨ (x′,t′)σ (x,t) (2.16) α−1,α R α z→w − where z = 1(x′ t′) and w = 1(x t) with the time ordering t′ > t. The factors (z w)α −2 − −2 − − and (z w)−α are taken on the principal branch. Twist fields σ have charges 1 , spins α±1,α − ∓ α+1/2, and dimensions α2 α+1/2. Their Hermitian conjugations are given by ± ± † σ = σ . (2.17) α±1,α −α∓1,−α Again, we define two types of fermionic primary twist fields: σ+ with branch cuts on the α±1,α right and σ− with branch cuts on the left, which are related to each other by the unitary α±1,α operator Z σ− = σ+ Z. (2.18) α±1,α α±1,α η From the definitions (2.15) (2.16) and twist properties of σ (2.12) (2.13), these twist fields α should obey non-trivial equal-time exchange relations with the Dirac fermion fields δ e2πiηα +δ ση (0)Ψ (x) (x < 0) Ψ (x)ση (0) = − η,− η,+ α±1,α R,L (2.19) R,L α±1,α ( −(cid:0)δη,+e2πiηα +δη,−(cid:1)σαη±1,α(0)ΨR,L(x) (x > 0) (cid:0) (cid:1) and δ e−2πiηα +δ ση (0)Ψ† (x) (x < 0) Ψ† (x)ση (0) = − η,− η,+ α±1,α R,L (2.20) R,L α±1,α ( −(cid:0)δη,+e−2πiηα +δη,−(cid:1)σαη±1,α(0)Ψ†R,L(x) (x > 0) . (cid:0) (cid:1) One-particle form factors of these twist fields can be deduced in the OPEs [41]: e−iπα/2e2πiναδη,− vac ση (0)θ = δ mα+1/2e(α+1/2)θ σ (2.21) h | α+1,α | iν ν,+ Γ(1+α) h αi eiπα/2e2πiναδη,− vac ση (0)θ = iδ m−α+1/2e(−α+1/2)θ σ . (2.22) h | α−1,α | iν − ν,− Γ(1 α) h αi − Any higher-particle form factors can be factorised into a product of the associated one-particle form factor and two-particle form factors due to Wick’s theorem. Other matrix elements can be obtained by crossing symmetry. It is worth noting that we can obtain the same families of fermionic primary twist fields σ and σ [41, 42] by shifting α α 1 in σ and shifting α α+1 in σ α,α−1 α,α+1 α+1,α α−1,α 7→ − 7→ respectively. These fields are just a relabelling of the same fermionic primary twist fields. 6 3 Liouville space and mixed-state form factors The Liouville space for the U(1) Dirac theory is the space of the operators End( ), with ρ L H inner product specified by the density matrix ρ Tr ρA†B ρ AB ρ = (3.1) h | i Tr(ρ) (cid:0) (cid:1) where A ρ and B ρ arethecorrespondingLiouville states, withA,B End( ). Inthepresent | i | i ∈ H paper, we confine ourselves to the density matrices ρ that are diagonal on the asymptotic state basis: ρ= exp dθ W (θ)D†(θ)D (θ) (3.2) − ν ν ν " # Z ν X where functions W (θ) with ν = are integrable on the real line and make the density matrices ν ± well-defined. We consider two cases: the untwisted and twisted cases. In the untwisted case, we consider the density matrix (3.2). In the twisted case, we consider the density matrix ρ♯ with the presence of two extra unitary operators e2πiνα dθDν†(θ)Dν(θ) which implement the U(1) R symmetry: ρ♯ = exp dθ W♯(θ)D†(θ)D (θ) (3.3) − ν ν ν " # Z ν X ♯ where W (θ) = W (θ)+2πiνα. ν ν The Liouville space is spanned by a set of products of creation and annihilation operators in Hilbert space with some particular normalization: vac ρ 1, θ ,...,θ ρ Qρ (θ ,...,θ )Dǫ1(θ ) DǫN(θ ), | i ≡ | 1 Ni(ν1,ǫ1)...(νN,ǫN) ≡ (ν1,ǫ1)...(νN,ǫN) 1 N ν1 1 ··· νN N (3.4) with the ordering θ > > θ , where the normalization factors are chosen as 1 N ··· N Qρ(ν1,ǫ1)...(νN,ǫN)(θ1,...,θN) := 1+e−ǫiWνi(θi) , (3.5) Yi=1(cid:16) (cid:17) andwherewedenotebyD− theannihilationoperatorsD andbyD+ thecreation operatorsD†. ν ν ν ν Here, we refer to a doublet (ν,ǫ) as representing the type of a “Liouville particle” of rapidity θ. Inthissense,ourLiouvillespacecanbeinterpretedasthespaceofparticlesandholesexcitations from the Liouville vacuum which consists of a number of different types of particles with density ρ. Usingthecyclic propertyof thetraceandthecanonical anti-commutation relations, theinner product of basis states can be deduced as N (ν1,ǫ1)...(νN,ǫNρ)hθ1,...,θN|θ1′,...,θN′ iρ(ν1′,ǫ′1)...(νN′ ,ǫ′N) = 1+e−ǫiWνi(θi) δνi,νi′δǫi,ǫ′iδ(θi −θi′) Yi=1h(cid:16) (cid:17) i (3.6) with the ordering θ > > θ and θ′ > > θ′ . 1 ··· N 1 ··· N 7 Every operator A End( ) can be mapped to an operator Aℓ End( ) by a linear ρ ∈ H ∈ L Liouville left-action Aℓ End( ) : Aℓ B ρ = AB ρ. (3.7) ρ ∈ L | i | i To Dǫ(θ), their associated Liouville operators are ν Z† (θ) Z (θ) [Dǫ(θ)]ℓ = ν,ǫ + ν,−ǫ (3.8) ν 1+e−ǫWν(θ) 1+eǫWν(θ) † whereZ (θ)andZ (θ)arebothdefinedasLiouvillemodeoperatorssatisfyinganti-commutation ν,ǫ ν,ǫ relations Z (θ),Z† (θ′) = 1+e−ǫWν(θ) δ δ δ(θ θ′) (3.9) { ν,ǫ ν′,ǫ′ } ν,ν′ ǫ,ǫ′ − Z (θ),Z (θ′) = (cid:16)Z† (θ),Z† (cid:17)(θ′) = 0 . (3.10) { ν,ǫ ν′,ǫ′ } { ν,ǫ ν′,ǫ′ } The Liouville space can be seen as the Fock space over this algebra, Z (θ)vac ρ = 0, θ ,...,θ ρ = Z† (θ ) Z† (θ )vac ρ. (3.11) ν,ǫ | i | 1 Ni(ν1,ǫ1)...(νN,ǫN) ν1,ǫ1 1 ··· νN,ǫN N | i With the definitions above, it is obvious to see that the mixed-state averages of operators on are vacuum expectation values on : ρ H L A = ρ vac Aℓ vac ρ. (3.12) ρ h i h | | i Using the resolution of the identity in the Liouville space ∞ ∞ dθ dθ 1ℓ = 1··· N NX=0ν1X,...,νNǫ1X,...,ǫNZ−∞N! Nj=1 1+e−ǫjWνi(θj)  (cid:16) (cid:17) θ ,...,θ ρ Q ρ θ ,...,θ (3.13) ×| 1 Ni(ν1,ǫ1)...(νN,ǫN) (ν1,ǫ1)...(νN,ǫN)h 1 N| i whereN!inthedenominatorcomesfromovercounting thesamebasisstate, two-pointfunctions, such as (x,τ) †(0) = ρ vac (x,τ)ℓ †(0,0)ℓ vac ρ ρ hO O i h |O O | i should have a spectral decomposition on , where we define the matrix elements of left-action ρ L operators in the Liouville space as mixed-state form factors fρ;O (θ ,...,θ ) := ρ vac (0,0)ℓ θ ,...,θ ρ . (3.14) (ν1,ǫ1)...(νN,ǫN) 1 N h |O | 1 Ni(ν1,ǫ1)...(νN,ǫN) With the definition (3.14) and anti-commutation relations (3.10), the mixed-state form factors satisfy the relations ρ;O f (θ ,...,θ ,θ ,...,θ ) (ν1,ǫ1)···(νj,ǫj)(νj+1,ǫj+1)···(νN,ǫN) 1 j j+1 N ρ;O = f (θ ,...,θ ,θ ,...,θ ) . (3.15) − (ν1,ǫ1)···(νj+1,ǫj+1)(νj,ǫj)···(νN,ǫN) 1 j+1 j N 8 The cyclicity of traces leads to the relation ρ θ ,...,θ ℓ vac ρ = fρ;O (θ ,...,θ ). (3.16) (ν1,ǫ1)···(νN,ǫN)h 1 N|O | i (νN,−ǫN)···(ν1,−ǫ1) N 1 Accordingto(3.12),themixed-stateformfactorsareessentiallytraceswithinsertionofoperators Dǫ(θ), up to an overall factor Qρ (θ ,...,θ ) and up to the subtraction of contact ν (ν1,ǫ1)···(νN,ǫN) 1 N terms at colliding rapidities: ρ;O f (θ ,...,θ ) (ν1,ǫ1)···(νN,ǫN) 1 N = Qρ (θ ,...,θ ) Dǫ1(θ ) DǫN(θ ) . (3.17) (ν1,ǫ1)···(νN,ǫN) 1 N hO ν1 1 ··· νN N iρ connected h i For example, two-particle mixed-state form factors can be written as fρ;O (θ ,θ ) = Qρ (θ ,θ ) Dǫ1(θ )Dǫ2(θ ) ρ θ θ ρ . (3.18) (ν1,ǫ1)(ν2,ǫ2) 1 2 (ν1,ǫ1)(ν2,ǫ2) 1 2 hO ν1 1 ν2 2 iρ−ν2,−ǫ2h 2| 1iν1,ǫ1hOiρ Again, using cyclicity of the trace, we have ρ θ ℓ θ ρ = fρ;O (θ ,θ )+ ρ θ θ ρ , (3.19) ν2,ǫ2h 2|O | 1iν1,ǫ1 (ν1,ǫ1)(ν2,−ǫ2) 1 2 hOiρ ν2,ǫ2h 2| 1iν1,ǫ1 ρ θ ,θ ℓ vac ρ = fρ;O (θ ,θ )+ ρ θ θ ρ . (3.20) (ν2,ǫ2)(ν1,ǫ1)h 2 1|O | i (ν1,−ǫ1)(ν2,−ǫ2) 1 2 hOiρ ν2,ǫ2h 2| 1iν1,−ǫ1 Similar equations for higher numbers of particles can be obtained in the same fashion. 4 Calculations and main results Before we start, let us define the normalized mixed-state form factors fη (θ) := ση −1fρ;µη(θ), (4.1) ν,ǫ h αiρ ν,ǫ fη (θ ,θ ) := ση −1fρ;σαη (θ ,θ ), (4.2) (ν1,ǫ1)(ν2,ǫ2) 1 2 h αiρ (ν1,ǫ1)(ν2,ǫ2) 1 2 where fρ;µη(θ) := δ fρ;σαη−1,α(θ)+δ fρ;σαη+1,α(θ) and ση is the normalization, and their ν,ǫ ν,−ǫ ν,ǫ ν,ǫ ν,ǫ α ρ h i pure-state limits f(0)η(θ) := lim fη (θ), (4.3) ν,ǫ ν,ǫ W±→∞ (0)η η f (θ ,θ ) := lim f (θ ,θ ). (4.4) (ν1,ǫ1)(ν2,ǫ2) 1 2 W±→∞ (ν1,ǫ1)(ν2,ǫ2) 1 2 Using the trace definition of mixed-state form factors and the ordinary form factors of U(1) twist fields in Hilbert space, we have e−iπνα/2 f(0)+(θ) = ( iǫδ +δ ) mνǫα+1/2e(νǫα+1/2)θ, ν,ǫ − ν,− ν,+ Γ(1+νǫα) f(0)−(θ) = f(0)+(θ)e2πiναδǫ,+ (4.5) ν,ǫ ν,ǫ 9