Finite temperature form factors in the free Majorana theory

Transcription

1 Finite temperature form factors in the free Majorana theory Benjamin Doyon Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK supported by EPSRC postdoctoral fellowship hep-th/ Annecy, September 2005

2 Correlation functions and form factors Euclidean correlation functions at zero temperature: insert a complete set of eigenstates of the Hamiltonian between operators in vacuum expectation values dθ1 dθ k O(x, τ)o(0, 0) = e Mr j cosh(θj) k! k=0 vac O(0, 0) θ 1,...,θ k in in θ 1,...,θ k O(0, 0) vac where r = x 2 + τ 2. Useful representation in 2-d integrable models because: form factors vac O(0, 0) θ 1,...,θ k in of local fields can be evaluated (up to normalization) by solving a Riemann-Hilbert problem in rapidity space; the first few terms (k =0, 1, 2,...) in the expression above give a good description of correlation functions at large (and also not too large) distances; in condensed matter applications, for instance, the most relevant region is the large-distance one (low energy).

3 Finite temperature Trace expression: must take statistical average: O(x, τ)o(0, 0) L = Tr ( e LH O(x, τ)o(0, 0) ) Tr (e LH ). Geometrically, the trace gives a theory on a cylinder of circumference L where time is around the cylinder and space is along it. Vacuum expectation value on the circle: different quantization scheme for the same theory, take time along the cylinder and space around the cylinder, O(x, τ) L = L vac e iπs/2 O L ( τ,x) vac L where s is the spin of O and O L is the operator on the cylinder associated to O.

4 Goal: Find a computable large-distance expansion of finite-temperature correlation functions.

5 From vacuum expectation values on the circle Form factor decomposition on the circle (integrable models), by inserting a complete set of eigenstates of the Hamiltonian: L vac O L (x, τ)o L (0, 0) vac L = e j n j 2πix L E n 1,...,n τ k k=0 n 1,...,n k L vac O L (0, 0) n 1,...,n k n 1,...,n k O L (0, 0) vac L Problems: Spectrum E {n} is complicated in interacting models; Form factors L vac O L (0, 0) n 1,...,n k depend on discrete variables: no obvious analytical way of evaluating them. What has been done: Form factors in Ising field theory: A. I. Bugrij: hep-th/ , hep-th/ A. B. Zamolodchikov and P. Fonseca: J. Stat. Phys. 110 (2003) Spectrum in interacting integrable models by various numerical means (TBA, NLIE,...)

6 From trace expression Finite temperature form factor expansion approach: using known form factors at zero temperature and partially performing the trace to evaluate one-point and two-point functions (some problems, some success...) A. Leclair, F. Lesage, S. Sachdev, H. Saleur: Nucl. Phys. B482 [FS] 1996, 579. A. Leclair, G. Mussardo: Nucl. Phys. B , 624. H. Saleur: Nucl. Phys. B , 602. G. Delfino: J. Phys. A 34, 2001, L161. G. Mussardo: J. Phys. A , O. A. Castro-Alvaredo and A. Fring: Nucl. Phys. B636 [FS] 2002, 611. Free energy: deriving TBA from zero-temperature form factors J. Balog: Nucl. Phys. B419 (1994)

7 New techniques from the trace expression A general way of developing the trace expression into a large-distance expansion: precise definition of finite-temperature form factors; A temperature-dependent Riemann-Hilbert problem fixing finite-temperature form factors of order and disorder fields in Ising field theory; A way of evaluating form factors on the circle by analytical continuation of finite-temperature form factors.

8 General idea Find a construction where the trace Tr ( e LH O(x, τ)o(0, 0) ) Tr (e LH ) is a vacuum expectation value in some vector space and where eigenvalues of the momentum operator are described by continuous variables (Ising field theory: rapidities θ j and states {θ} {ɛ} ). [Ising field theory] Find a measure ρ({θ}) in 1 = {dθ} ρ({θ}, {ɛ}) {θ} {ɛ} {ɛ} {θ} n! {ɛ} such that finite-temperature form factors of non-interacting local fields, vac O(0, 0) {θ} {ɛ}, are entire functions of the rapidities {θ}; order and disorder field: simple Riemann-Hilbert problem; Calculate form factors on the cylinder by analytical continuation in the rapidity variables to the positions of the poles of the measure ρ: L vac O L (0, 0) {n} Res ρ vac O(0, 0) {α n ±iπ/2}

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10 Ising field theory (free Majorana theory) Free Majorana fermion operators on the line: ψ(x, τ) = 1 m 2 π ψ(x, τ) = i m 2 π dθ e θ/2 ( a(θ) e ip θx E θ τ + a (θ) e ip θx+e θ τ ) dθ e θ/2 ( a(θ) e ip θx E θ τ a (θ) e ip θx+e θ τ ) {a (θ),a(θ )} = δ(θ θ ), p θ = m sinh θ, E θ = m cosh θ. Equations of motion and equal-time canonical anti-commutation relations ψ(x, τ) 1 2 ( x + i τ ) ψ = m 2 ψ ψ(x, τ) 1 2 ( x i τ ) ψ = m 2 ψ {ψ(x),ψ(x )} = δ(x x ), { ψ(x), ψ(x )} = δ(x x ). Hilbert space H: Fock space over mode algebra. Hamiltonian: H = m dθ cosh(θ)a (θ)a(θ).

11 Space of operators: Liouville space (Thermofield dynamics) Consider the vector space L of operators of the theory: Vacuum: vac L 1 H Complete basis: θ 1,...,θ N ɛ 1,...,ɛ N a ɛ 1 (θ 1 ) a ɛ N (θ N ) θ 1 >θ 2 > >θ N where ɛ j are signs (± : particles / holes ) and a + (θ) =a (θ), a (θ) =a(θ). Inner product on L u v U V L, if u U, v V. Operators O on the Hilbert space H can be seen also as operators on L: acting by left-action Two-point function is a vacuum expectation value on L: O(x)O(0) L = vac L O(x)O(0) vac L

12 Finite-temperature form factors Normalized set of states on L: θ 1,...,θ N ɛ1,...,ɛ N N (1+e ɛ j LE θj )a ɛ 1 (θ 1 ) a ɛ N (θ N ) j=1 with θ 1,...,θ N ɛ1,...,ɛ N =0if θ i = θ j for some i j. Finite-temperature form factors: f O ɛ 1,...,ɛ N (θ 1,...,θ N )= vac L O(0) θ 1,...,θ N ɛ1,...,ɛ N They are entire functions for non-interacting fields due to the general representation fɛ O 1,...,ɛ N (θ 1,...,θ N )= = {a ɛ 1 (θ 1 ), [a ɛ 2 (θ 2 ), {, O(0, 0) }]} L because commutators with fermion fields are of the form [ψ(x), O i (0)] = j c j i δ( d i d j 1 2) (x)oj (0). and because the modes in terms of local fermi fields are a ± (θ) = 1 2 m π dx e ±ip θx (e θ/2 ψ(x) ie θ/2 ψ(x))

13 Resolution of the identity Note that and a(θ)a (θ ) L = δ(θ θ ) 1+e LE θ a (θ)a(θ ) L = δ(θ θ ) 1+e LE θ Then, orthogonality relation: ɛ θ θ ɛ =(1+e ɛle θ )δ(θ θ )δ ɛ,ɛ As in zero temperature, we have a resolution of the identity: 1 = {ɛ} {dθ} N! N j=1 1 1+e ɛ jle θj {θ} {ɛ} {ɛ} {θ}

14 Finite-temperature Large-distance expansion Can write two-point function as (for Hermitian fields) O(x)O(0) L = {dθ} N e iɛ j p θj x f O N! 1+e ɛ j LE θj {ɛ}({θ}) f{ɛ}({θ}) O {ɛ} j=1 In the limit L (zero temperature): The finite-temperature form factors become the usual form factors: lim L f O +,...,+,,..., (θ 1,...,θ N+,θ N+ +1,...,θ N )= θ N,...,θ N+ +1 O(0) θ 1,...,θ N+ (θ i θ j i {1,...,N + },j {N + +1,...,N}); The expansion above becomes the usual form factor expansion. But, as in the zero-temperature case, we need to do analytical continuation in θ in order to get workable large-x expansion.

15 Relation with previous work The previous finite-temperature form factor expansion appeared some time ago in: A. Leclair, F. Lesage, S. Sachdev, H. Saleur: Nucl. Phys. B482 [FS] 1996, 579. But: They did not construct the Liouville space: expansion deduced by explicitly partially preforming the trace (but neglecting some singularities...) In place of our finite-temperature form factors, they had the usual zero-temperature form factors; They did not have a clear way of taking care of the kinematical poles in the finite-temperature form factor expansion of order-order correlation functions. Other works on finite-temperature form factor expansions of two-point functions in more complicated models were essentially based on this. Other people (Saleur, Castro Alvaredo Fring, Fonseca Zamolodchikov) pointed out some possible problems in this finite-temperature form factor expansion approach.

16 Finite-temperature form factors of uninteracting fields: mixing At zero temperature: form factors of uninteracting fields are also entire functions of rapidities. Are they the same as the finite-temperature form factors? No, in general: f{+} O ({θ}) = vac (O(0) +...) {θ} (on the right-hand side: zero-temperature form factors are involved) where...contains local fields at x =0of lower dimension than that of O. For instance: Hamiltonian density h(x) gets a vacuum expectation value E vac L h(x) vac L = vac (h(x)+e1) vac.

17 Mixing map In general, the mixing can be described by a mixing map U acting on L: O(0) +...= U O(0) This is a generalization of the CFT situation, where L H and U is written in terms of Virasoro generators and makes a transformation to the cylinder. Using creation and annihilation operators A ± (θ) on L with {A ɛ(θ),a ɛ (θ )} =(1+e ɛle θ ) δ(θ θ ) δ ɛ,ɛ we can write [ U =exp dθ A ] (θ)a + (θ) 1+e LE θ

18 Spin (or twist) operators The Majorarana theory has Z 2 global symmetry ψ ψ, ψ ψ. There are two associated spin fields (twist fields): σ and µ (with non-zero form factors for even and odd particle number respectively). On the geometry of the cylinder, every spin operator has two realizations: σ ± and µ ±, with branch cut on the right (+) or on the left ( ) of the position of the operator.

19 Finite-temperature form factors of spin operators: Riemann-Hilbert problem Consider f(θ 1,...,θ k )=f σ +,µ + {+} (θ 1,...,θ k ). Then, 1. f(...,θ i,...,θ j,...)= f(...,θ j,...,θ i,...) 2. f(θ 1,...,θ k ) has poles at θ j = α n + iπ 2,n Z and has zeroes at θ j = α n + iπ 2,n Z f(θ 1,...,θ k +2iπ) = f(θ 1,...,θ k ) 4. f(θ 1,...,θ k ) ( 1) k 1 1+e LE θ k 1 f(θ 1,...,θ k 2 ) π 1 e LE θ k 1 θ k θ k 1 iπ 5. f(θ 1,...,θ k ) does not have poles for Im(θ j ) [ iπ, iπ] except those mentionned above. Also, we have in general f O ɛ 1,...,ɛ k (θ 1,...,θ k + iπ) =if O ɛ 1,..., ɛ k (θ 1,...,θ k ).

20 Solution to the Riemann-Hilbert problem One-particle: f µ + ± (θ) iπ =e± 4 C [ i0 + ( dθ 1 1+e LE exp i0 + 2πi sinh(θ θ ) ln θ 1 e LE θ ) ] with C = σ L 2π where the average σ L was calculated in S. Sachdev: Nucl. Phys. B464 (1996) Multi-particle: f O + +,...,+ (θ 1,...,θ k )=i [ k 2 ] σ L ( k f µ + + (θ ) j) ( ) θj θ i tanh σ L 2 j=1 1 i<j k

21 Form factors on the cylinder from finite-temperature form factors e iπs 2 L ñ 1,...,ñ l O L (0) n 1,...,n k L = ( ) k+l 2π 2 ml l j=1 f O +,...,,... 1 cosh(αñj ) k 1 j=1 cosh(α nj ) ( α n1 + iπ 2,...,α n k + iπ 2,α ñ l + iπ 2,...,α ñ 1 + iπ 2 ) with sinh α n = 2πn ml (n Z ). This formula is valid for any excited states in the NS (anti-periodic) sector. When O is a twist field, it can then be applied only if one of the bra or the ket is the vacuum, and if the branch associated to this twist field is chosen so that this vacuum is in the R sector and the excited state is in the NS sector.

22 Perspectives Compute explicitly form factors of descendant spin operators, including mixing; Generalize to interacting integrable models of quantum field theory: Riemann-Hilbert problem for finite-temperature form factors? Relation with energy spectrum on the cylinder? Apply some of the ideas to integrable spin chain models: Göhmann et. al. s results are obtained using the spin-chain counterpart of quantization on the circle, can we calculate the spin-chain counterpart of finite-temperature form factors?