Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results
aa r X i v : . [ m a t h - ph ] N ov Quantum Quench dynamics in Non-local LuttingerModel: Rigorous Results
Zhituo Wang
Institute for Advanced Study in Mathematics, Harbin Institute of Technology,150006, Harbin, China,Research Center for Operator Algebras, East China Normal University,
Email:[email protected]
We investigate, in the Luttinger model with fixed box potential, the time evolutionof an inhomogeneous state prepared as a localized fermion added to the noninter-acting ground state. We proved that, if the state is evolved with the interactingHamiltonian, the averaged density has two peaks moving in opposite directions,with a constant but renormalized velocity. We also proved that a dynamical ‘Lan-dau quasi-particle weight’ appears in the oscillating part of the averaged density,asymptotically vanishing with large time. The results are proved with the Mattis-Lieb diagonalization method. A simpler proof with the exact Bosonization formulasis also provided.
1. INTRODUCTION
Recent experiments on cold atoms [1] have motivated increasing interest in the dynam-ical properties of many body quantum systems which are closed and isolated from anyreservoir or environment [2]. Nonequilibrium properties can be investigated by quantumquenches , in which the system is prepared in an eigenstate of the non-interacting Hamilto-nian and its subsequent time evolution driven by an interacting many-body Hamiltonianis observed. As the resulting dynamical behavior is the cumulative effect of the inter-actions between an infinite or very large number of particles, the computation of localobservables averaged over time-evolved states poses typically great analytical difficulties;therefore, apart for some analysis in two dimensions (see, for instance [3, 4]), the problemis mainly studied in one dimension [5]-[30]. A major difference with respect to the equilib- rium case relies on the fact that in such a case a form of universality holds, ensuring thata number of properties are essentially insensitive to the model details. At non-equilibriumthe behavior depends instead on model details; for instance integrability in spin chainsdramatically affects the non equilibrium behavior [13], [40],[41] while it does not alter the T = 0 equilibrium properties [43]. This extreme sensitivity to the details or approxima-tions asks for a certain number of analytical exact results at non-equilibrium, to providea benchmark for experiments or approximate computations.One of the interacting Fermionic system where non-equilibrium properties can be inves-tigated is the Luttinger model [32, 33] (see also [34–36]), which provides a great number ofinformation in the equilibrium case. In the Luttinger model model the quadratic disper-sion relation of the non relativistic fermions is replaced with a linear dispersion relation,leading to the ”anomaly” in the distribution of the ground states density. This anomalyis proved to be universal for a large class of one dimensional Fermionic system, called theLuttinger liquid [31]. Luttinger model became of great interest in mathematical physicsever since the exact solutions founded by Mattis-Lieb [34] and is a key to investigate themathematical properties of condensed matter physics.It is important to stress that there exist two versions of this model, the local Luttingermodel (LLN) and the non local Luttinger model (NLLM); in the former a local delta-likeinteraction is present while in the latter the interaction is short ranged but non local.The finite range of the interaction plays as an ultraviolet cut-off. At equilibrium suchtwo models are often confused as they have similar behavior, due to the above mentionedinsensitivity to model details; there is however no reason to expect that this is true alsoat non equilibrium. It should be also stressed that the LLM is plagued by ultravioletdivergences typical of a QFT and an ad-hoc regularization is necessary to get physicalpredictions; the short time or distance behavior depends on the chosen regularization.In this paper we study the evolution of inhomogeneous states in the non-local LuttingerModel with a fixed box potential , with the Mattis-Lieb diagonalization method, which wasproved to be mathematically rigorous ([35, 36]). Then we perform rigorous analysis of theasymptotic behavior in the infinite volume limit. The main result shows that (see Theorem2.2), when the interaction is turned on, the dynamics is ballistic with a constant butrenormalized velocity, and the interaction produces a dynamical ‘Landau quasi-particleweight’ in the oscillating part, asymptotically vanishing with time. The expressions weget do not require any ultraviolet regularization, and correctly capture also the short timedynamics. We also invite the physically oriented reader to read this article along with ashort letter [18], in which we studied the quench dynamics of non-local Luttinger modelbut without giving full details of the proof. In the current article we put full details ofthe proof and specialize to the box potential, for which the change of velocity due to themany-body interaction is more transparent; we provide also a simpler proof of the maintheorem with the exact Bosonization formulas.The quantum quench of homogeneous states in the NLLM was derived in [20],[21], inwhich steady states were found. However mathematical rigor is lacking in these work.The quenched evolution of the NLLM prepared in domain wall initial state was studiedin [42] and the universality of the quantum Landauer conductance for the final states wasproved, in a mathematically rigorous way.The plan of the paper is the following. We introduce the NLLM with box potentialin § II. In § III we prove Theorem 2.2 with the Mattis-Lieb diagonalization method. Somedetails of the proof are presented in the Appendix. The proof of Theorem 2.2 based onthe Bosonization method is given in § IV.
2. THE LUTTINGER MODEL AND MAIN RESULTSA. The Luttinger model with box potential
The non-local Luttinger model (NLLM) is defined by the Hamiltonian: H λ = Z L/ − L/ dx i v F (: ψ + x, ∂ x ψ − x, : − : ψ + x, ∂ x ψ − x, :)+ λ Z L − L dxdy v ( x − y ) : ψ + x, ψ − x, :: ψ + y, ψ − y, : (2.1)where ψ ± x,ω = √ L P k a k,ω e ± ikx , ω = 1 , k = πnL , n ∈ N are fermionic creation orannihilation operators, :: denotes Wick ordering and v F is the Fermi velocity. We arechoosing units so that v F = 1. The two-body interaction potential v ( x − y ) is given by: v ( x − y ) = sin( x − y ) x − y , (2.2)whose fourier transform reads: v ( p ) = (cid:26) v f or p ≤ , f or p > . (2.3)The potential v ( x ) or v ( p ) is also called the box potential and v is called the strength of v ( p ). Equilibrium Luttinger model with box potential was first considered in [44].In the Fourier space the Luttinger Hamiltonian can be written as H = H + V = X k> k [( a + k, a − k, + a −− k, a + − k, ) + ( a + − k, a −− k, + a − k, a + k, )+ λL X p> v ( p )[ ρ ( p ) ρ ( − p ) + ρ ( − p ) ρ ( p )] + λL v (0) N N (2.4)where, for p > ρ ω ( p ) = X k a + k + p,ω a − k,ω , N ω = X k> ( a + k,ω a − k,ω − a −− k,ω a + − k,ω ) . (2.5)It is well known that Fock space canonical commutation relations don’t have a uniquerepresentation in a system with infinite degree of freedom. So one has to introduce a cutofffunction χ Λ ( k ) with Λ a large positive number such that χ Λ ( k ) = 1 for | k | ≤ Λ and equals 0otherwise and the regularized operators ρ ω ( p ) must be thought as lim Λ →∞ P k χ Λ ( k ) χ Λ ( k + p ) a + k + p,ω a − k,ω .The Hamiltonian H as well as ρ ω ( p ) can be regarded as operators acting on the Hilbertspace H constructed as follows. Let H be the linear span of vectors obtained by applyingfinitely many times creation or annihilation operators on | > = Y k ≤ a + k, a + − k, | vac > . (2.6)In this way we get an abstract linear space to which we introduced scalar products betweenany pair of vectors. H is defined as the completion of H in the scalar product justintroduced. Moreover the operators H and ρ ω ( p ), regarded as operators on H with domain H , are self adjoint.The basic property of the Luttinger model is the validity of the following anomalouscommutation relations, first proved in [34], for p, p ′ > ρ ( − p ) , ρ ( p ′ )] = [ ρ ( p ) , ρ ( − p ′ )] = pL π δ p,p ′ . (2.7)Remark that this commutator acting on the Fock space is not precise due to theinfinitely many degrees of freedom of the system. So one should introduce a cutoff Λ sothat the commutator: − Λ X k =Λ+ p a + k,ω a − k,ω + Λ − p X k = − Λ a + k,ω a − k,ω = − Λ+ p X k = − Λ a + k,ω a − k,ω − Λ X k =Λ − p a + k,ω a − k,ω . (2.8)on any state of H is equal, in the limit Λ → ∞ , to pL π .Moreover one can verify that ρ ( p ) | > = 0 , ρ ( − p ) | > = 0 . (2.9)Other important commutation relations (see [34, 45] for proofs) are as follows:[ H , ρ ω ( ± p )] = ± ε ω pρ p ( ± p ) , [ ρ ω , ψ ± ω,x ] = e ipx ψ ± ω,x (2.10)where ω = 1 , ε ω = 1 for ω = 1 and ε ω = − ω = 2. B. The Mattis-Lieb diagonalization
The Hamiltonian (2.4) can be diagonalized with the method of Lieb-Mattis [34], asfollows. First of all we introduce an operator T = 1 L X p> [ ρ ( p ) ρ ( − p ) + ρ ( − p ) ρ ( p )] (2.11)and write H = ( H − T ) + ( V + T ) = H + H . Note that H is already diagonalized inthat it commutes with ρ ω . The key for the diagonalization of H is the introduction of abounded operator S acting on the Hilbert space H : S = 2 πL X p =0 φ ( p ) p − ρ ( p ) ρ ( − p ) , tanh φ ( p ) = − λv ( p )2 π . (2.12)Using the following Bogolyubov transformations for the operators ρ ω ( ± p ): e iS ρ , ( ± p ) e − iS = ρ , ( ± p ) cosh φ ( p ) + ρ , ( ± p ) sinh φ, (2.13)we can easily prove that H can be written in diagonal form: e iS H e − iS = ˜ H := 2 πL X p sech2 φ ( p )[ ρ ( p ) ρ ( − p ) + ρ ( − p ) ρ ( p )] + E . (2.14)By Formula (2.12) we can easily find that the operator S hence the transformation in(2.14) is well defined only for | λv ( p ) | < π ; The model is instable for | λv ( p ) | > π .Define D = ˜ H − T = 2 πL X p σ ( p )[ ρ ( p ) ρ ( − p ) + ρ ( − p ) ρ ( p )] + E , (2.15)we have [ H , D ] = 0. The diagonalization formula for the Hamiltonian reads: e iS e iHt e − iS = e i ( H + D ) t . (2.16) C. The time evolution of the one particle state and the main theorem
Define ψ ± x,δ = e iH t ψ ± ω,x e − iH t = 1 √ L X k a ± ω,k e ± i ( kx − ε ω kt ) − δ | k | , (2.17)where δ → + , ε = + , ε = − . By direct calculation we find that: < | ψ ε ω ω,x,δ ψ − ε ω ω,y,δ | > = (2 π ) − iε ω ( x − y ) − i ( t − s ) + δ . (2.18)The relation between the creation or annihilation Fermionic operators and the quasi-particle operators is ψ x = e ip F x ψ x, + e − ip F x ψ x, , (2.19)where p F is the Fermi momentum and we call e ip F x ψ x, = ˜ ψ x, and e − ip F x ψ x, − = ˜ ψ x, . Inmomentum space this simply means that the momentum k is measured from the Fermipoints, that is c k,ω = ˜ c k + ε ω p F ,ω . The ground state of H is | GS > = e iS | > , where | > isthe ground state of H and the inhomogeneous one particle initial state is given by: | I t > = e iH λ t ( ˜ ψ +1 ,x + ˜ ψ +2 ,x ) | > . (2.20)Let n ( z ) be the density operator, which is defined as the limit δ → , ε → X ρ = ± ( ˜ ψ +1 ,z + ρε ˜ ψ − ,z, + ˜ ψ +2 ,z + ρε ψ − ,z + ˜ ψ +2 ,z + ρε ˜ ψ − ,z (2.21)+ ˜ ψ +1 ,z + ρε ˜ ψ − ,z + ˜ ψ +1 ,z + ρε ˜ ψ − ,z + ˜ ψ +2 ,z + ρε ˜ ψ − ,z ) . Note that summing over ρ = ± is the point spitting regularization, which plays the samerole as the Wick ordering for avoiding divergences. We are interested in the average valueof the density operator w. r. t. the 1-particle initial state (2.20), formally defined by: G ( x, z, t, δ ) := < I t | n ( z ) | I t > (2.22):= X ω,ω ′ =1 , (cid:2) h | ˜ ψ − ω,x e iHt ˜ ψ + ω,z + ρε ˜ ψ − ω ′ ,z e − iHt ˜ ψ + ω ′ ,x | i + h | ˜ ψ − ω,x e iHt ˜ ψ + ω ′ ,z + ρε ˜ ψ − ω ′ ,z e − iHt ˜ ψ + ω,x | i (cid:3) As a first step we consider the non-interacting case. Let | I ,t > := e iH t ( ˜ ψ +1 ,x + ˜ ψ +2 ,x ) | > ,we have: Theorem 2.1
When λ = 0 , H = H , we have lim L →∞ < I ,t | n ( z ) | I ,t > (2.23)= 12 π cos 2 p F ( x − y )( x − z ) − t + 14 π [ 1(( x − z ) − t ) + 1(( x − z ) + t ) ] . Proof 2.1
We consider first the term with ω = 1 , ω ′ = 2 . Using the explicit expressionsof the Fermionic operators and taking the limit ε → , we can easily find that this termis equal to e ip F ( x − y ) (4 π ) − [( x − z ) − t ] − ; a similar result is found for the second term.The third and fourth terms are vanishing as P ρ ρε = 0 ; similarly the last two term give (4 π ) − [( x − z ) ± t ] − . Combine all these terms we can derive Formula (2.23) , henceproved this theorem. Remark 2.1
The physical meaning of Theorem 2.1 is quite clear: when the interactionis turned off, the average of the density is sum of two terms, an oscillating and a nonoscillating part (when the particle is added to the vacuum there are no oscillations p F = 0 ).At t = 0 the density is peaked at z = x , where the average is singular. With the timeincreasing the particle peaks move in the left and right directions with constant velocity v F = 1 (ballistic motion); that is, the average of the density is singular at z = x ± t anda ”light cone dynamics” is found. When we turn on the interaction and let the system driven by the full interacting Hamil-tonian, the ground states and the dynamics will be significantly changed. The explicitexpression of (2.22) can be derived with the Mattis-Lieb diagonalization method followedby a rigorous analysis of the asymptotic behavior for L → ∞ and large t . We have Theorem 2.2
Let the interacting box potential (see (2.3) ) be turned on in the Hamilto-nian, let γ = v and ω = q − (cid:0) v π (cid:1) . The average of the density operator with respectto the one particle initial state | I λ,t > in the limit L → ∞ reads: lim L →∞ < I λ,t | n ( z ) | I λ,t > = 14 π [ 1(( x − z ) − t ) + 1(( x − z ) + t ) ] + 12 π cos 2 p F ( x − z ) e Z ( t ) ( x − z ) − ( ω t ) . (2.24) where Z ( t ) = γ Z dpp (cos 2 ω pt −
1) (2.25) is the Landau quasi particle factor, such that Z (0) = 1 and exp Z ( t ) ∼ cst ( 12 ω t ) γ , (2.26) for t ≥ .
3. PROOF OF THEOREM 2.2
We consider first the term: h | ψ − ,x e iHt ψ +1 ,z ψ − ,z e − iHt ψ +2 ,x | i , (3.27)and forget the phase factor e ± ip F x for the moment for simplicity; these factors are veryeasy to restore. The rest of this subsection is devoted to the calculation of (3.27).Let I be an identity operator in H . Using the fact that e − iεS e iεS = I and e − iHt e iHs | t = s = I , we can write (3.27) as h | ψ − ,x e − iεS ( e iεS e iHt e − iεS )( e iεS ψ +1 ,z e − iεS ) · (3.28) · ( e iεS ψ − ,z e − iεS )( e iεS e − iHs e − iεS ) e iεS ψ +2 ,x | i| ε =1 ,s = t Lemma 3.1
Let ˆ I be an operator valued function of ρ ( ± p ) and ψ ± and ˆ I be an operatorvalued function of ρ ( ± p ) and ψ ± , then we have the following factorization Formula for (3.27) : G = I I , (3.29) where I = h | ˆ I | i and I = h | ˆ I | i . Proof 3.1
We shall prove this lemma by deriving the explicit expressions of ˆ I and ˆ I .Using the diagonalization formula (2.16) , formula (3.28) can be written as: h | ψ − ,x e − iεS e i ( H + D ) t e iεS ψ +1 ,z e − iεS e − i ( H + D ) t e − iεS · (3.30) · e iεS e i ( H + D ) s e iS ψ − ,z e − iS e − i ( H + D ) s e iεS ψ +2 ,x | i| ε =1 , s = t . Now we consider the term of e iεS ψ +1 ,z e − iεS . It is a well known result [34] that: e iεS ψ ∓ ,z e − iεS = ψ ∓ ,z W ± ,z R ± ,z , (3.31) where W ± ,z = exp {∓ πL X p> p [ ρ ( p ) e − ipz − ρ ( − p ) e ipz ](cosh εφ − } R ± ,z = exp {± πL X p> p [ ρ ( p ) e − ipz − ρ ( − p ) e ipz ] sinh εφ } . (3.32) Similarly one has e iεS ψ ∓ ,z e − iεS = ψ ∓ ,z W ± ,z R ± ,z (3.33) where W ± ,z = exp {∓ πL X p> p [ ρ ( p ) e − ipz − ρ ( − p ) e ipz ] sinh εφ } R ± ,z = exp {± πL X p> p [ ρ ( p ) e − ipz − ρ ( − p ) e ipz ](cosh εφ − } . (3.34) Then we consider the term e − iεS e i ( H + D ) t W − ,z R − ,z e − i ( H + D ) t e iεS , (3.35) which, after inserting the identity operator I = e iεS e − iεS and I = e − i ( H + D ) t e i ( H + D ) t , isequal to [ e − iεS e i ( H + D ) t W − ,z e − i ( H + D ) t e iεS ] · [ e − iεS e i ( H + D ) t R − ,z e − i ( H + D ) t e iεS ] . (3.36) Let f ( p, t ) be an arbitrary regular function, define σ ( p ) = sech2 φ − and ω ( p ) = σ ( p ) + 1 = sech2 φ , we have the following commutation relation [ H + D, ρ ω ( ± p )] = ± ε ω p ( σ ( p ) + 1) ρ ω ( ± p ) , ω = 1 , , ε = + , ε = − , (3.37)0 which implies that e i ( H + D ) t e f ( p,t ) ρ ω ( ± p ) e − i ( H + D ) t = e e ± εωi ( σ +1) pt f ( p,t ) ρ ω ( ± p ) . (3.38) Combining the above formula with (2.13) and (2.17) we find that (3.36) can be written asa product of e − iεS e i ( H + D ) t W ± ,z e − i ( H + D ) t e iεS = exp ± πL X p (cosh φ − p [ ( ρ ( − p ) cosh εφ − ρ ( − p ) sinh εφ ) e ipx − ipt ( σ +1) − ( ρ ( p ) cosh εφ − ρ ( p ) sinh εφ ) e − ipx + ipt ( σ +1) ] := ¯ W ± ,z . (3.39) and e − iεS e i ( H + D ) t R − ,z e − i ( H + D ) t e iεS = exp ± πL X p sinh φp [ ( ρ ( − p ) cosh εφ − ρ ( − p ) sinh εφ ) e ipy + ips ( σ +1) − ( ρ ( p ) cosh εφ − ρ ( p ) sinh εφ ) e − ipy − ips ( σ +1) ] := ¯ R ± ,z . (3.40) Using again (3.31) , (3.33) and (3.46) , we have: e − iεS e i ( H + D ) t e iS ψ +1 ,z e − iS e − i ( H + D ) t e iεS = z a A A − A A − ψ +1 ,zt,δ ˜ W − t ˜ R − t W − tε R − tε ˆ W − tε ˆ R − tε , (3.41) and e − iS e iS e iHs e − iS e iS ψ − ,z e − iS e iS e − iHs e − iS e iS = z b ¯ W sε ¯ R sε ˆ W sε ˆ R sε ˜ W ˜ R ψ ,z,s,δ B − B B − B , (3.42) where ˜ W − , ,t,ε , ˜ R − , ,t,ε and ˆ W − , ,t,ε , ˆ R − , ,t,ε are operators depending on ρ , ( ± p ) , respectivelyand z a , z b are functions of p . The explicit expressions of the above factors are given inthe Appendix.Then we can easily find that the terms depending on ρ ( ± p ) and ψ ± are factorized withrespect to the terms depending on ρ ( ± p ) and ψ ± . Let I := h | ˆ I | i := h | ψ x A A − ψ +1 ,zt ˜ W − ¯ W − t ˆ W − t ˜ W t ¯ W t ˆ W t B B − | i , (3.43)1 and I := h | ˆ I | i := h | A A − ˜ R − ¯ R − ˆ R − ¯ R ˆ R ˜ R ψ ,zt B B − ψ † x | i , (3.44) and using the fact that z a = z − b we have G = I I , (3.45) So we proved Lemma 3.1.
A. Calculation of I and I In this part we derive the explicit expressions for I and I . It is also useful to introducethe following proposition, which can be easily proved using (2.10): Proposition 3.1
Let f ( p, t ) is an arbitrary regular function. Then we have: e iH t e f ( p,t ) ρ ω ( ± p ) e − iH t = e f ( p,t ) e ± εωi ( σ +1) pt ρ ω ( ± p ) , ω = 1 , ε = + , ε = − , (3.46)The basic idea to calculate I and I is to use repeatedly the Hausdorff to move theoperators ρ ( − p ) and ρ ( p ) to the right most of the expressions in (3.43) and (3.44), andmove ρ ( p ), ρ ( − p ) to the left most of the above expressions. By formula (2.9) and itsadjoint form we know that these operator annihilate | i and h | , respectively; the survivedterms are those independent of ρ , ( ± p ). Setting ε = 1, we have: I = exp { πL X p p [( e − ip ( σ +1)( t + s ) − φ sinh φ + cosh φ sinh φ )+ ( e ip ( σ +1)( t + s ) −
1) cosh φ sinh φ + e − ipσt ( − cosh φ − sinh φ )+ e ip ( x − z )+ ip ( σ +1) s (cosh φ sinh φ + cosh φ ) − e ip ( x − z )+ ips + e ip ( x − z ) − ip ( σ +1) t ( − sinh φ − sinh φ ) ] }h | ψ x ψ +1 ,z,t,δ | i , (3.47)and I = h | ψ +2 ,z,t,δ ψ x | i exp { πL X p p [( e − ip ( σ +1)( t + s ) −
1) cosh φ sinh φ + ( e ip ( σ +1)( t + s ) − φ sinh φ + 2 cosh φ sinh φ )+ e − ipσt (cosh φ + sinh φ ) − e ip ( x − z ) − ipt + e ip ( x − z )+ ip ( σ +1) s ( − cosh φ sinh φ − sinh φ )+ e ip ( x − z ) − ip ( σ +1) t (sinh φ + cosh φ ) ] } . (3.48)2Combining (3.47) with (3.48) and setting s = t , we get: h | ψ − ,x e iHt ψ +1 ,z ψ − ,z e − iHt ψ +2 ,x | i (3.49)= h | ψ x ψ +1 ,z,t,δ | ih | ψ +2 ,z,t,δ ψ x | i× exp X p p (cid:20) ( e ip ( x − z )+ ip ( σ +1) t − e ip ( x − z )+ ipt )+( e ip ( x − z ) − ip ( σ +1) t − e ip ( x − z ) − ipt )+2 sinh φ cosh φ (sinh φ + cosh φ ) (cos 2 p ( σ + 1) t − (cid:21) . It is useful to derive the asymptotic behavior for the second line in (3.49) and we have:lim δ → lim L →∞ h | ψ x ψ +1 ,zt,δ | ih | ψ +2 ,ztδ ψ x | i = 14 π x − z ) − t (3.50)With the same method we can derive the explicit expression for the other terms in(2.22). Restoring the phase factor e ± ip F ( x − z ) and combine all the terms of (2.22), weobtain the following desired result: < I λ,t | n ( z ) | I λ,t > = 14 π [ 1(( x − z ) − t ) + 1(( x − z ) + t ) ] (3.51)+ 14 π e Z ( t ) ( x − z ) − t (cid:2) e ip F ( x − z ) e Q a ( x,z,t ) + e − ip F ( x − z ) e Q b ( x,z,t ) (cid:3) , where Z ( t ) = X p p sinh φ cosh φ (sinh φ + cosh φ ) (cos 2 p ( σ + 1) t − , (3.52) Q a = X p p [( e ip ( x − z )+ ip ( σ p +1) t − e ip ( x − z )+ ipt )+ ( e ip ( x − z ) − ip ( σ p +1) t − e ip ( x − z ) − ipt )] ,Q b = X p p [( e − ip ( x − z )+ ip ( σ p +1) t − e − ip ( x − z )+ ipt )+ ( e − ip ( x − z ) − ip ( σ p +1) t − e − ip ( x − z ) − ipt )] (cid:9) . (3.53) B. The asymptotic behavior for L → ∞ In this section we shall derive the asymptotic behavior of Formula (3.51) in the limit L → ∞ . Using definitions of the hyper-geometric functions we find thatsech φ ( p ) = 12 (cid:18) v ( p )4 π q v ( p )2 π − (cid:19) , cosh φ = 12 (cid:18) v ( p )4 π q v ( p )2 π + 1 (cid:19) , (3.54)3where v ( p ) is the box potential with strength v (see Formula (2.3)), we have the followingexpression for the critical exponent: γ ( p ) = 2 sinh φ ( p ) cosh φ ( p )(sinh φ ( p ) + cosh φ ( p )) = v ( p )4 π . (3.55)Taking the limit L → ∞ means that we should consider the discrete sum over p asintegral over continuous variables. We have: Z ( t ) = Z ∞ γ ( p ) dpp (cos 2 ω pt − γ Z dpp (cos 2 ω pt − γ Z ω t d (2 ω p )2 ω p (cos 2 ω pt − , (3.56)where γ := v π and ω := q − (cid:0) v π (cid:1) . The second line is true is due to the fact that γ ( p ) = 0 for p ∈ (1 , ∞ ]. Let y = 2 ω pt and w = 2 ω t , Z ( t ) can be written as: γ Z dpp (cos 2 ω pt −
1) = γ Z w dyy (cos y − . (3.57)There are three cases to be considered, depending on the range of t : • when t ≪
1, which corresponds to the short time behavior and implies that y ≪ w ≪ v ( p ) is vanishing for p > Z ( t ) = γ Z w dyy (cos y − ∼ γ Z w dy ( − y O ( y )) ≪ . (3.58)So that Z ( t ) is well defined for y ≪
1. Furthermore, it is vanishing as y → + andwe have e Z ( t ) | t → + → • when t ∈ (0 , Z ( t ) is a bounded function. • when t ∈ [1 , ∞ ]; let p > p and u = 2 ω p t , we have Z ( t ) = γ Z ω tu dyy (cos y − − C − ln u − Z u cos y − y dy − [ln 2 ω t − ln u ]= γ ( − ln 2 ω t − C − Z u cos y − y dy ) , (3.59)4where we have used the integral formula Z ∞ u cos yy dy = − C − ln u − Z u cos y − y dy, (3.60)where C = 0 . · · · is the Euler constant and R u y − y dy is a bounded function.Remark that (3.59) is well defined for u →
0, due to the cancellation of ln u .So we have e Z ( t ) ∼ cst · [ 12 ω t ] γ , f or t ≥ . (3.61)Now we derive the asymptotic formula for Q a and Q b . Replacing the discrete sum over p in (3.53) by integrals and performing the integrations, we can easily find that: Q a = Q b = ln ( x − z ) − t ( x − z ) − ω t . (3.62)Collecting all the above terms we have:lim L →∞ < I λ,t | n ( z ) | I λ,t > = 14 π [ 1(( x − z ) − t ) + 1(( x − z ) + t ) ]+ 12 π cos 2 p F ( x − z ) e Z ( t ) ( x − z ) − ( ω t ) . (3.63)So we proved theorem 2.2.
4. THE BOSONIZATION METHOD
While the Lieb-Mattis method for solving Luttinger model is mathematically rigorous,technically it is very complicated. There exist another very popular method for study-ing the one dimensional interacting Fermions models, called the Bosonizations, whichstates that certain two dimensional models of fermions are equivalent to the correspond-ing Bosonic models: the corresponding Fermionic Hilbert space and the Bosonic one areisomorphic and the the Fermionic operator can be expressed in terms of the Bosonic op-erators. While the Bosonization method can reduce significantly the difficulty for thecalculation, it has the reputation of not mathematically rigorous. A Rigorous proof ofBosonization formulas was given very recently in a paper by Langmann and Moosavi[45]. In this section we shall prove Theorem 2.2 with the exact Bosonization formulas in5[45]. This can be considered as a verification of the use of Bosonization formula in thenon-equilibrium setting.First of all we shall derive Formula (3.51). Following the notations in [45] we have
Proposition 4.1
Let ρ ω be the Bosonic operators introduced before and let R ε ω ω be theKlein factor, then we can express the Fermionic operators ψ − in terms of the Bosonicoperators and the Klein factor as follows: ψ − ω ( x, δ ) = : N δ e iπε ω xQ ω /L R − ε ω ω e iπε ω xQ ω /L × (4.64)exp (cid:8) ε ω X p> πLp [ ρ ω ( p ) e − ipx − δ | p | − ρ ω ( − p ) e ipx − δ | p | (cid:9) , where ω, ω ′ = 1 , , ε = + , ε = − , Q ω = ρ ω (0) and N δ = (cid:20) L (1 − e − πδ/L ) (cid:21) / is thenormalization factor. R ± ω is the Klein factor such that R − ω = ( R + ω ) † . They obey thefollowing commutation relation (see [45] for the detailed derivation): [ ρ ω ( p ) , R ω ′ ] = ε ω δ ω,ω ′ δ p, R ω , [ H , R ω ] = ε ω πL (cid:8) ρ ω (0) , R ω (cid:9) , (4.65) h | R q ω R q ω ′ | i = δ ω,ω ′ δ q , δ q , , R q R q = ( − q q R q R q , [ Q ω , R q R q ] = q ω R q R q , q ω ∈ Z . We shall not repeat the proof here and the interested reader is invited to look at [45] fordetails.Let ˆ Z − ω = e iπε ω xQ ω /L R − ε ω ω e iπε ω xQ ω /L and ˆ Z + be its adjoint, we can write the Fermionicoperators as: ψ ± ω ( x, δ ) = N δ ˆ Z ± ω e ∓ ε ω P p> πLp [ ρ ω ( p ) e − ipx − δ | p | − ρ ω ( − p ) e ipx − δ | p | . (4.66)We calculate first the term h | ψ − ,x e iHt ψ +1 ,z ψ − ,z e − iHt ψ +2 ,x | i in(2.22) forget the phasefactor e ip F ( x − z ) for the moment. Inserting the identity operators I = e iHt e − iHt and I = e iS e − iS we derived Formula (3.28), which is the starting point of our analysis.First of all, it is easily to find that e iS ˆ Z ± ω e − iS = ˆ Z ± ω . (4.67)Using Formula (2.13) we have: e iS ψ +1 ,z e − iS = N δ ˆ Z +1 exp 2 πL X p> p (cid:8) − e − δp e − ipz [cosh φρ ( p ) + sinh φρ ( p )]+ e − δp e ipz [cosh φρ ( − p ) + sinh φρ ( − p )] (cid:9) , e iS ψ − ,z e − iS = N δ ˆ Z − exp 2 πL X p> p (cid:8) − e − δp e − ipz [cosh φρ ( p ) + sinh φρ ( p )]+ e − δp e ipz [cosh φρ ( − p ) + sinh φρ ( − p )] (cid:9) . (4.68)Using the fact that:[ H + D, R ± ω ] = ± π ( σ (0) + 1) L R ± ω (2 ε ω ρ ω (0) + 1) , (4.69)and e i ( H + D ) t R ± ω e − i ( H + D ) t = R ± ω exp [ ± π ( σ (0) + 1) L (2 ε ω ρ ω (0) + 1) t ] , (4.70)we have: e − iS e i ( H + D ) t e iS ψ +1 ,z e − iS e i ( H + D ) t e iS (4.71)= N δ ˆ Z ( t ) exp 2 πL X p> p (cid:8) e − δp [ A ρ ( p ) + A − ρ ( − p ) + A ρ ( p ) + A − ρ ( − p )] (cid:9) , and e − iS e i ( H + D ) t e iS ψ − ,z e − iS e i ( H + D ) t e iS (4.72)= N δ ˆ Z ( t ) exp 2 πL X p> p (cid:8) e − δp [ B ρ ( p ) + B − ρ ( − p ) + B ρ ( p ) + B − ρ ( − p )] (cid:9) , where A ± = ± e ∓ ip [ z +( σ +1) t ] sinh φ ∓ e − ip [ z − ( σ +1) t ] cosh φ ,A ± = ± e ∓ ip [ z − ( σ +1) t ] sinh φ cosh φ ∓ e ∓ ip [ z +( σ +1) t ] cosh φ sinh φ,B ± = ± e ∓ ip [ z +( σ +1) t ] sinh φ cosh φ ∓ e ∓ ip [ z − ( σ +1) t ] cosh φ sinh φ,B ± = ± e ∓ ip [ z − ( σ +1) t ] sinh φ ∓ e ∓ ip [ z +( σ +1) t ] cosh φ ˆ Z ( t ) = e iπxρ (0) /L exp[ − π ( σ (0) + 1) L (2 ρ (0) + 1) t ] R − e iπxρ (0) /L , ˆ Z ( t ) = e iπxρ (0) /L exp { π ( σ (0) + 1) L ( − ρ (0) + 1) t } R e iπxρ (0) /L . (4.73)When p = 0, by using the fact that ρ ω (0) | i = 0 and h | R q ω R q ω ′ | i = δ ω,ω ′ δ q , δ q , , wehave h | ˆ Z ˆ Z +1 ( t ) ˆ Z ( t ) ˆ Z † | i = 1 . (4.74)7So the nontrivial contributions come from the p > ρ ( ± p ) and ρ ( ± p ): h | N δ exp 2 πL X p> p [ e − δp e − ipx ρ ( p ) − e − δp e ipx ρ ( − p )] × N δ exp 2 πL X p> p (cid:8) e − δp [ A +1 ρ ( p ) + A − ρ ( − p ) + A +2 ρ ( p ) + A − ρ ( − p )] (cid:9) × N δ exp 2 πL X p> p (cid:8) e − δp [ B +1 ρ ( p ) + B − ρ ( − p ) + B +2 ρ ( p ) + B − ρ ( − p )] (cid:9) × N δ e πL P p> p [ e − δp e ipx ρ ( p ) − e − δp e − ipx ρ ( − p )] | i =: N δ I I , (4.75)where I = h | e πL P p> p e − δp [ e − ipx ρ ( p ) − e ipx ρ ( − p )] e πL P p> p e − δp [ A +1 ρ ( p )+ A − ρ ( − p )] × e πL P p> p e − δp [ B +1 ρ ( p )+ B − ρ ( − p )] | i , (4.76) I = h | e πL P p> p e − δp [ A +2 ρ ( p )+ A − ρ ( − p )] e πL P p> p e − δp [ B +2 ρ ( p )+ B − ρ ( − p )] × e πL P p> p e − δp [ e − ipx ρ ( p ) − e ipx ρ ( − p )] | i . (4.77)Following exactly the same procedure as section 3 A, namely using repeatedly the Haus-dorff formula and the annihilation formulas we have: I I = exp 2 πL X p> p e − δp [( e ip ( x − z )+ ip ( σ +1) t −
1) + ( e ip ( x − z ) − ip ( σ +1) t − φ cosh φ (sinh φ + cosh φ ) (cos 2 p ( σ + 1) t − . (4.78)In order to reproduce the expressions in (3.51) we need to extract from the aboveformula the noninteracting 2-point correlation function (see [45]), as follows. We writethe terms e ± ip ( x − z ) ± ip ( σ +1) t − e ip ( x − z ) ± ip ( σ +1) t − e ip ( x − z ) ± ip ( σ +1) t ) + ( e ip ( x − z ) ± ip ( σ +1) t − , while the first term gives the factors Q , the second term contributes to the non-interactingcorrelation function: N δ exp 2 πL X p> p e − δp [( e ip ( x − z )+ ipt −
1) + ( e − ip ( x − z )+ ipt − . (4.79)8Now we derive the asymptotic formula for (4.79). Using the Poisson summation for-mula: exp (cid:18) X p> πLp e − δp (cid:19) = exp (cid:18) ∞ X n =1 n e − nπδ/L (cid:19) = LN δ , (4.80)where N δ = 1 L (1 − e − πδ/L ) ∼ πδ f or L → ∞ , (4.81)Formula (4.79) can be written as:lim δ → + lim L →∞ N δ exp 2 πL X p> p e − δp [( e ip ( x − z )+ ipt −
1) + ( e − ip ( x − z )+ ipt − δ → + lim L →∞ N δ L N δ · − e − πL [2 δ + i ( x − z )+ it ] − e − πL [2 δ − i ( x − z )+ it ] = 14 π x − z ) − t . (4.82)Following the same procedure we can calculate all the other terms in (2.22) and deriveFormula (3.51). The asymptotic expressions for the terms in the exponential can bederived with the same procedure as in the last section and we shall not repeat it here. Sowe proved Theorem (2.2) with the exact Bosonization formulas. Acknowledgement
The author is very grateful to V. Mastropietro for many usefuldiscussions and common work on the preliminary version of this paper. Part of this workhas been done during the author’s visit to the Chern Institute of Technology, NankaiUniversity and the Shanghai Institute of Mathematical Science, Fudan University. Theauthor is very grateful for their hospitality. The work is partially supported by theNational Natural Science Foundation of China under the grant No. 11701121.9
5. APPENDIXA. Explicit expressions of the factors in Formulas (3.43) and (3.44)
With some very long but elementary calculation we find that the expressions of theterms in formula (3.43) and (3.44) read: z a = exp 2 πL X p p { ( e − ipσt − } , (5.83) A ± = exp 2 πL X p p ρ ( ± p ) cosh εφ ( ∓ e ∓ ipx ± ipt ± e ∓ ipx ± ipt ( σ +1) ) A ± = exp ± πL X p p ρ ( ± p ) sinh εφ ( e ∓ ipx ± ipt − e ∓ ipx ± ipt ( σ +1) ) .z b = exp 2 πL X p p (1 − e − ipσt ) = z − a , (5.84) B ± = exp ∓ πL X p p ρ ( ± p ) sinh εφ ( ∓ e ∓ ipz ∓ ipt ± e ∓ ipz ∓ ipt ( σ +1) ) ,B ± = exp 2 πL X p p ρ ( ± p ) cosh εφ ( ± e ∓ ipz ∓ ipt ( σ +1) ∓ e ∓ ipz ∓ ipt ) . ˜ W − tε = exp 2 πL X p p [(cosh εφ − e − ipz + ipt ρ ( p ) − (cosh εφ − e ipz − ipt ρ ( − p ) ] , ˜ R − tε = exp − πL X p p [sinh εφe − ipz + ipt ρ ( p ) − sinh εφe ipz − ipt ρ ( − p ) ] . (5.85)¯ W − tε = exp 2 πL X p p [(cosh φ −
1) cosh εφe − ipz + ip ( σ +1) t ρ ( p ) (5.86) − (cosh φ −
1) cosh εφe ipz − ip ( σ +1) t ρ ( − p ) ] , ¯ R − tε = exp − πL X p p [(cosh φ −
1) sinh εφe − ipz + ip ( σ +1) t ρ ( p ) − (cosh φ −
1) sinh εφe ipz − ip ( σ +1) t ρ ( − p ) ]ˆ W − tε = exp − πL X p sinh φp [ ρ ( p ) sinh εφe − ipx − ipt ( σ +1) − ρ ( − p ) sinh εφe ipx + ipt ( σ +1) ] , ˆ R − tε = exp − πL X p sinh φp [ ρ ( − p ) cosh εφe ipx + ipt ( σ +1) − ρ ( p ) cosh εφe − ipx − ipt ( σ +1) ] . W t = exp − πL X p p sinh εφ ( e − ipz − ipt ρ ( p ) − e ipz + ipt ρ ( − p )) , (5.87)˜ R t = exp 2 πL X p p (cosh εφ − e − ipz − ipt ρ ( p ) − e ipz + ipt ρ ( − p )) . ¯ W tε = exp 2 πL X p p sinh φ cosh εφ [ e − ipz + ip ( σ +1) t ρ ( p ) (5.88) − e ipz − ip ( σ +1) t ρ ( − p ) ] , ˆ W tε = exp − πL X p p (cosh φ −
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