Quantum hydrodynamics and nonlinear differential equations for degenerate Fermi gas

Quan tum h ydro dynamics and nonli near differential equations for d egener ate F ermi gas Eldad Bettelheim Racah Institute of Physics, The Hebrew Univ er s it y of Jerusalem, Safra Campus, Giv at Ra m, Jerus alem, Israel 9 1904. E-mail: eldadb @phys .huji.ac.il Alexander G. Abano v Department of Physics and Astronomy , Stony Bro o k Universit y , Stony Bro ok, NY 11794 -3800 . E-mail: alexan dre.a banov@sunysb.edu P aul B. Wiegmann James F r anck Institute of the Universit y of Chica go, 5 640 S.Ellis Avenue, Chicago , IL 6 0637. E-mail: wiegma nn@uc hicago.edu Abstract. W e present n ew nonlinea r differen tial equatio ns for spacetime correlation functions of F ermi gas in one spatial dimension. The correlation functions w e consider describ e non-stationa ry pro cesses out of equilibr ium. The equations we obtain are int egr able equations. They genera liz e kno wn nonlinea r differential equations for correla tion functions a t equilibrium [1, 2, 3, 4] and provide vital too ls to study non- equilibrium dynamics o f electronic sys tems. The metho d we developed is ba sed only on Wick’s theorem and the hydro dy namic description of the F ermi gas. Differential equations app ear directly in bilinea r form. Quantum hydr o dynamics and nonline ar differ ential e quations . . . 2 1. I n tro duction V arious s pacetime correlation functions of free F ermi gas in one spatial dimension (and mo dels related to it – imp enetrable Bo se and F ermi gases, X Y -spin c hain, 2D- Ising mo del, etc.) are F redholm determinants or their minors and, therefore, generally sp eaking, o b ey inte gra ble nonlinear differen tial equations with r espect to spacetime and other pa r a meters lik e temp erature, c hemical p oten tial, etc. In practice, a deriv ation of these equations is a complicated task. F ew equations ar e know n. How eve r, once obtained, they are indisp ensable in studies of off-shell prop erties of F ermi gas. Historically nonlinear differen tial equations for correlation functions first app eared in studies of the 2D Ising model [1]. In Ref. [2] these equations has b een extended, applied to the XY-spin c hain, and ha ve b een deriv ed in explicitly bilinear form with the help of Wic k theorem generalized for thes e purp oses in Ref. [5]. In Refs. [3] it has b een show ed that at zero temp erature the tw o-p oint, equal-time correlatio n function of imp enetrable b osons and of the X Y -spin c hain can be express ed through P ainlev e transcenden ts. These equations ha v e b een extended to time- and temp erature- dep enden t correlation functions at equilibrium for impenetrable b osons and the X Y - c hain in Refs. [4, 6]. The deriv ation of t he know n equations w as essen tially based on equilibrium and translational in v a riance prop erties of correlation functions. Although they ha v e b een recognized as in tegrable, their relations to in tegrable hierarc hies remained unclear. Recen tly , in terest fo cused o n electronic syste ms out of equilibrium [7] and especially on no n-stationary properties of propagating localized states [8 ]. There the ph ysic al prop erties c hange drastically and the dy namics acquires essen tially nonlinear features suc h as h ydro dynamic instabilities [9, 8]. Apart fr o m b eing o f a fundamen tal in terest, researc h in non-equilibrium degenerate F ermi gas is driven by a quest to transmit quan tum information in electronic nano devices and to con trol and manipulate entangled quan tum many - b o dy states. In tegrable equations describing the non-equilibrium – non- stationary states of electronic g as, once av ailable, would b e a v aluable to ol in studies of a complex dynamics of electronic gas. They are deriv ed in this letter. W e will obtain nonlinear differen tial equations with resp ect to spacetime for the correlation functions of vertex op erators e ia L ϕ L · e ia R ϕ R , (1) where ϕ R,L are right/left handed c hiral Bose fields of fermionic curren ts and a L,R are arbitrary (not necessarily real) para meters. The definition of these fields in terms of free fermions is giv en below. Eq uations with resp ect to parameters o f the densit y matrix (e.g., temp erature or c hemic al p oten tial) can b e also obtained, but are not disc ussed here. The correlation functions of v ertex op era t o rs (1) are imp ortant in man y applications. In the case of equilibrium they also satisfy some previously kno wn equations. A few particular cases are w orth me ntioning. The case a L = a R = 1 / 2 describes a one-p oint function of imp enetrable bo sons [3, 4]. The imaginary a L = a R Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 3 and a L = − a R yield a generating function of moments of the distribution function of the dens ity and v elo cit y of ele ctronic gas. The chiral correlation functions a L = 0 or a R = 0 describ e a tunneling in to an electron gas (F ermi Edge Singularit y) [10], etc. In this article we giv e a simple physic al deriv ation of equations on these correlations functions. Our deriv ation is based solely on a general form of the Wic k theorem [11], the represen tation of F ermi-op erator s throug h Bo se fields (see e.g., [12 ]), a nd the hydrodynamic description of F ermi gas [13]. Our equations a pp ear directly in Hirota’s biline ar form [14]. The y are k nown as mo difi e d Kadomtsev - Pe tvia s hvili (or mKP) equations a nd ar e the first a nd the second equation of mKP hierarch y . In a particular case, in free fermion systems these equations hav e previously app eared in [15, 9]. The nonlinear equations, b eing in tegrable ar e prov en to b e an effectiv e to ol in computing asymptotes o f correlation f unctions in differen t regimes. These computations are sp ecific to eac h particular problem and will no t b e discussed here (see, Refs. [9, 8] for some practical applications). In the fo llowing after necess ary preliminaries on the F ermi-Bose corresp ondence, w e describe the correlatio n functions ( 5), presen t the nonlinear differen tial equations they ob ey (16,18,19), and then giv e a short deriv ation of those equations. 2. F ermions and a Bose Field W e consider free fermions on a circle of a circumference L H = X p p 2 2 m ψ † p ψ p , (2) where ψ p is a mo de of a fermion field ψ ( x ) = 1 √ L P p e i ¯ h px ψ p . A cen tral p oint of our approac h is the h ydro dynamic description of the F ermi gas. W e briefly review it. The prop erties of the F e rmi gas are fully de scrib ed in terms of canonical h ydro dynamic v a r iables: densit y and v elo cit y ρ = 1 2 π  J R + J L  , v = ¯ h 2 m  J L − J R  , where the c hiral (righ t, left) curren ts a re J R,L ( x ) = 2 π L X k e i ¯ h k x J R,L k , J R,L k = X ± p> 0 ψ † p ψ p + k . T omonaga’s equal time comm utation relations of curren ts (see, e.g., [12]) [ J R k , J R l ] = [ J L k , J L l ] = Lk 2 π ¯ h δ k + l, 0 , [ J R k , J L l ] = 0 (3) lead to a canonical relatio n b et w een densit y a nd v elo cit y: [ ρ ( x, t ) , v ( y , t )] = − i ¯ h m ∇ δ ( x − y ) (4) and to the canonical Bose field ∇ ϕ R,L ≡ ± J R,L ( x, t ) = m ¯ h v ± π ρ, [ ϕ L , ϕ R ] = 0 . Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 4 W e will b e intereste d in the spacetime dep endence of correlations of t w o v ertex op erators: tr  : e − ib L ϕ L t ( ξ ′ ) − ib R ϕ R ( ξ ′ ) :: e ia L ϕ L ( ξ )+ ia R ϕ R ( ξ ) :   . (5) Here  is the densit y matrix and ξ = ( x, t ). The colo n denotes norma l ordering with resp ect to the v acuum [16]. Belo w w e drop the c hiral subscripts R, L in all formulas and assume that the upp er (lo w er) sign corresp o nds to the r ig h t (left) sectors. 3. C oherent states W e will consider correlatio n functions with respect to c oher ent states . This means t hat the densit y matrix is an elemen t of g l ( ∞ ), i.e., it is an exp onen t bilinear in fermionic mo des:  = exp X p,q A pq ψ † p ψ q ! . This c hoice of the densit y matrix allo ws us to use Wic k’s theorem. The set o f coherent states is rather general. It exhausts most in teresting applications. F or example, the densit y matrix can b e a Bolt zmann distribution, or an y other non-equilibrium distribution as in [7]. Another c hoice w ould b e  = | V ih V | , where | V i = exp ( P k t k J k ) | 0 i is a coheren t state (i.e., an eigenstate of curren t op erators). Such states app ear a s a result o f shak e up of the electronic gas by electromagnetic p otential with harmonics t k . 4. Quantum Hydro dynamics The Ha milto nian o f free fermions (2 ) in the sector of coherent states can b e expressed solely in terms of densit y and v elo cit y [13] H = Z m : ρv 2 : 2 + π 2 ¯ h 2 6 m : ρ 3 : ! dx. (6) Ev olution of densit y and v elo cit y follow s : ˙ ρ : + ∇ : ( ρv ) := 0 Con tin uity equation : ˙ v : + 1 2 ∇ : ( v 2 + π 2 ¯ h 2 m 2 ρ 2 ) := 0 Euler’s equation The h ydro dynamic eq uations are decoupled into tw o independen t c hiral quan tum Riemann equations ˙ ϕ + ¯ h 2 m : ( ∇ ϕ ) 2 := 0 . (7) Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 5 5. V ertex op erator and correlation functions The Heisen berg evolution equations for the c hiral ve rtex op erators f ollo w from the quan tum Riemann equation (7) (see App endix A) ( i∂ t − ¯ h 2 m ∇ 2 ) : e iaϕ : = ¯ h 2 m a ( a + 1) : e iaϕ T : , (8) ( i∂ t + ¯ h 2 m ∇ 2 ) : e i ( a +1) ϕ : = − ¯ h 2 m a ( a + 1) : e i ( a +1) ϕ ¯ T : , (9) where T = : ( ∇ ϕ ) 2 : − i ∇ 2 ϕ and ¯ T = : ( ∇ ϕ ) 2 : + i ∇ 2 ϕ are holomorphic (an tiholomorphic) comp onen ts of the stress-energy tensor of a c hiral Bose field ( with the cen tral charge 1 / 2). A part icular consequence of these equations is tha t the v ertex op erators e ± iϕ R,L ob ey the Sc hro edinger equation. This yields to a f amiliar formula represen ting fermions as expo nents o f a Bose field : e iϕ : ≃ √ Lψ ( x ) , : e − iϕ : ≃ √ Lψ † ( x ) . (10) A useful represen tation f or curren t J = ± : e − iϕ ( − i ∇ ) e iϕ : and the stress energy tensor reads T = − : e − iϕ ∇ 2 e iϕ : , ¯ T = − : e iϕ ∇ 2 e − iϕ : (11) Curren t a nd stress energy tensor can b e further cast in fermionic form using (10) a nd OPE for verte x op erat ors as follows from the curren t algebra (4)[17] : e iaϕ ( x ) e − ibϕ ( x ′ ) := ∓ i 2 π ( x − x ′ ) L ! ab : e iaϕ ( x ) :: e − ibϕ ( x ′ ) : (12) With the help of (12, 11) and (10) w e write T = ∓ 4 π iψ † ∇ ψ , ¯ T = ∓ 4 π iψ ∇ ψ † , J = 2 π ψ † ψ . (13) Finally w e cast the equation of motion of the v ertex op erato r in a suggestiv e bilinear form (Quan tum Hirota equation) iD t − ¯ h 2 m D 2 x ! : e iaϕ · e i ( a +1) ϕ := 0 . (14) Here D is the Hirota deriv ativ e D f · g = ∂ f g − f ∂ g and D 2 f · g = ∂ 2 f g − 2 ∂ f ∂ g + f ∂ 2 g . This equation follo ws from the quan tum Riemann equation (7). 6. N onlinear differential equations The nonlinear equations themselv es do not depend on t he parameters a L,R , b L,R in (5) . F or brevit y w e put L = 2 π in formulas b elo w and consider only the c hiral and neutral ( a = b ) correlation function, whic h w e denote as τ a ( ξ , ξ ′ ) = t r  : e − iaϕ ( ξ ′ ) :: e iaϕ ( ξ ) :   . (15) Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 6 Being presen ted in bilinear form the differen tial equations for correlatio n functions lo ok iden tical to the quan tum equation (14): iD t − ¯ h 2 m D 2 x ! τ a · τ a +1 = 0 . (16) The equations m ust b e view ed as equations for τ a + n ( x, t ; x ′ , t ′ ) in the con tinuous v ariables x, t and the discrete v ariable n , where | a | ≤ 1 / 2. As suc h they are a close d set of equations kno wn as the mKP equations. The correlation function can b e found giv en initial conditions at t = 0, for all in teger n and any x . Certain asymptotes or analytical conditions ma y also b e helpful in solving the problem fo r particular cases. This progr a m is tak en up in practice in [9, 8 ], where in [9] the den sity matrix had the form | A i h 0 | where | 0 i is the gro und state of electron gas a nd | A i is a c hiral cohere nt state of the form | A i =  | 0 i . W e also commen t that τ a where | a | ≤ 1 / 2 is a F redholm determinant, while the τ a + n are its minors. The mKP equations (1 6) has man y different forms discussed in the literature. W e will men tion one of them. Setting Φ = i ¯ h m log τ a τ a +1 ! , ˜ Φ = ¯ h m log( τ a τ a +1 ) the equation ( 16) b ecomes ∂ t Φ = − 1 2 ( ∇ Φ) 2 + ¯ h 2 m ∇ 2 ˜ Φ . (17) In this f o rm the classical equation (17) resem bles the quan tum equation (7). The last term in (17) can b e considered as a quantum correction to the semiclassical Riemann equations. In addition to the ev olution (16) there is another equation whic h do es not in v olv es time evolution. T his equation connects cor r elat io n f unctions with differen t densit y matrices:  and ˜  = P pq a pq ψ † p ψ q , where a pq is an arbitra r y non-degenerate matrix. Then τ a and ˜ τ a = tr  : e − iaϕ ( ξ ′ ) :: e iaϕ ( ξ ) : ˜   are related b y D x D x ′ ˜ τ a · τ a = a 2 ( ˜ τ a +1 τ a − 1 + τ a +1 ˜ τ a − 1 ) . (18) This equation is a minor generalization of familiar 2D T o da lattice equation. An imp o rtan t pa rticular case o ccurs when the mo des p, q in ˜  lie very far fr om the F ermi surface. Then ˜ τ a ∼ τ a and the equation ( 1 8) b ecomes 1D T oda c hain equation D x D x ′ τ a · τ a = 2 a 2 τ a +1 τ a − 1 . (19) This equation, together with (16) b eing written for pairs τ a , τ a +1 and τ a − 1 , τ a giv e a closed set which inv olv es only τ a − 1 , τ a , τ a +1 . 7. T ranslation in v arian t states F urther closure can b e ac hiev ed for sp ecial cases. F or example, if the densit y matrix comm utes with momentum , the correlatio n function depends on the difference x − x ′ (e.g., the density matrix dep ends only on o ccupation n umbers n p = ψ † p ψ p ,  = Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 7 exp P p λ p n p . Then it also comm utes with the Hamiltonian, is a function of t − t ′ , and describes statio nary pro cesses). In this case we denote the tau-function as τ a ( x, t ; x ′ , t ′ ) = F a ( x − x ′ , t, t ′ ) . As follo ws from (16,1 9) t hree functions F a − 1 , F a , F a +1 ob ey three equations with r espect to ( x, t ) iD t − ¯ h 2 m D 2 x ! F a · F a +1 = 0 , iD t + ¯ h 2 m D 2 x ! F a · F a − 1 = 0 , D 2 x F a · F a = − 2 a 2 F a − 1 F a +1 . (20) Here we used D x ′ = − D x . These equations are the bilinear form of the Nonlinear Sc hro edinger equation (without a complex inv olution). In tro ducing Ψ = F a +1 /F a and ¯ Ψ = F a − 1 /F a w e hav e i∂ t + ¯ h 2 m ∇ 2 ! Ψ = ¯ h m a 2 ˜ ΨΨ 2 , − i∂ t + ¯ h 2 m ∇ 2 ! ˜ Ψ = ¯ h m a 2 Ψ ˜ Ψ 2 . (21) These equations were obtained for time and temp era t ure dep enden t tw o- p oint correlation function for imp enetrable b osons at equilibrium (  = e − H/T ) [4]. The imp enetrable b osons are equiv alen t to free fermions. The creation op erator of the b oson is the vertex op erator ( 1 ) with a L = a R = 1 / 2. 8. D er iv ation of Nonlinear equations The fact that the quan tum equation for ve rtex op erator s (14), and the classical equations for the correlation functions, (16), ha v e the same form may not b e an a cciden t but ma y reflect a general prop ert y of integrable equations written in bilinear form. Here w e limit our discussion b y demonstrating this phenomena for the equation (16). W e prov e no w the main form ula (16). First w e adopt the no t a tion: hO 1 ( x ) , O 2 ( x ′ ) i = ( τ a ( x, t ; x ′ , t ′ )) − 1 × tr   e iH t : e iaϕ ( x ) O 1 ( x ) : e − iH ( t − t ′ ) : e − iaϕ ( x ′ ) O 2 ( x ′ ) : e − iH t ′  , where only the spatial dep endence has b een sp ecified in the r.h.s.. Let us m ultiply (8) and (9) by : e − iaϕ ( x ′ ,t ′ ) : and : e − i ( a +1) ϕ ( x ′ ,t ′ ) : resp ectiv ely , tak e the trace, and substitute into Eq. (16). W e find that (16) holds due to the iden tit y h e − iϕ , e iϕ ¯ T i + h 1 , T ih e − iϕ , e iϕ i = 2 h 1 , J ih e − iϕ , e iϕ J i , (22) where 1 is the iden tity op erator, and w e further suppress ed the argumen ts x, x ′ . T o prov e this identit y w e first write : e iϕ ¯ T : in terms of fermions. Using (11) we write : e iϕ ¯ T := − lim y , z → x ∇ 2 y : e iϕ ( x ) e iϕ ( z ) e − iϕ ( y ) :, where w e split spatial p oin ts. W e no w Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 8 use the form ula (12) for vertex op erators and Eq. (11) to write : e iϕ ( x ) ¯ T ( x ) : = ∓ i (2 π ) 3 / 2 lim y , z → x ∇ 2 y ( y − z ) ( y − x ) ( z − x ) ψ ( x ) ψ ( z ) ψ † ( y ) . (23) This form ula allows to write the first term in (22) in terms o f a correlator with fo ur fermions insertion ∓ i (2 π ) 2 lim y , z → x ∇ 2 y ( y − z ) ( y − x ) ( z − x ) h ψ † ( x ′ ) , ψ ( x ) ψ ( z ) ψ † ( y ) i . (24) No w w e use a general form of the Wic k theorem [11] to express the four fermion correlator in terms of correlators containing tw o-fermions h ψ † ( x ′ ) , ψ ( x ) ψ ( z ) ψ † ( y ) i = h ψ ( x ) , ψ † ( x ′ ) ih ψ † ( y ) ψ ( z ) , 1 i − h ψ ( z ) , ψ † ( x ′ ) ih ψ † ( y ) ψ ( x ) , 1 i . (25) After this, we tra ce bac k the pat h whic h lead us to (24). With the help of Eqs. ( 1 0) and (12) w e obtain the form ulas h ψ † ( x ′ ) , ψ ( x ) i = 1 2 π h e − iϕ ( x ′ ) , e iϕ ( x ) i , h 1 , ψ ( z ) ψ † ( y ) i = ± 1 2 π i ( y − z ) h 1 , e iϕ ( z ) e − iϕ ( y ) i . (26) W e use them in order to write the r.h.s. o f (2 5 ) in terms of b osons. After that, one applies the op eration lim y , z → x ∇ 2 y ( y − z )( y − x ) ( z − x ) to the r.h.s of (25), tak es the deriv ative, a nd merges the p oin ts. This leads to (22). The calculations are somewhat cumbersome, but straigh tforward [18]. W e briefly commen t on the pro of of (19 ), the pro of of the more general (18) go es along the same lines. F irst w e write (19) in the f o rm h J, J i − h J, 1 ih 1 , J i = h e iϕ , e − iϕ ih e − iϕ , e iϕ i . (27) The pro o f of this equation is easier: after replacing b osonic exp o nen tials and the curren t op erator b y fermionic op era t ors according to (10,13 ) o ne recognizes Wic k’s theorem. 9. I n tegrable hierarc hy of nonlinear equations As a final remark we men tion that higher equations o f the mKP hierarc h y describe ev olution of F ermi gas with an arbitra r y sp ectrum H = P p ε p ψ † p ψ p . They are generated b y the Hirota equation describ ed in [15]. Equations fo r magnetic c hains equiv alen t to the F ermi gas can b e obtained in a similar manner. 10. A c kno wledgmen t PW thanks V. Korepin for discussion of the results of this pap er. W e thank J. H. H. P erk for p oin ting out to us and discussing t he results of [2, 5]. EB was supp o r ted b y ISF grant n um b er 206/ 07. The w ork of AGA w as supp orted by the NSF under the gran t DMR-0 348358. PW was supp orted b y NSF under the gra n t DMR-054 0811 and MRSEC DMR-02137 45. Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 9 App endix A. Equation of motion for ve rt ex op erators In this App endix w e sk etc h the deriv ation of equations (8,9) fro m the Riemann equation (7). First of all w e sho w ho w to calculate the time deriv ative of the v ertex op erator. A subtlet y here is that ˙ ϕ do es not comm ute with ϕ . W e recall a simple consequence of the Hadamard lemma. If H and A are tw o op erators suc h that [[ H , A ] , A ] commutes with A , then h H , e A i = e A  [ H , A ] + 1 2! [[ H , A ] , A ]  =  [ H , A ] − 1 2! [[ H , A ] , A ]  e A . If H is a Hamiltonian ∂ t A ∼ [ H, A ] and w e obtain f o r the time deriv at ive of an expo nent ∂ t e A = e A  ˙ A + 1 2! h ˙ A, A i  =  ˙ A − 1 2! h ˙ A, A i  e A . (A.1) Let us now compute an ev olution of e iaϕ . W e assume that ϕ is a righ t c hiral field. The left c hiral field ob eys the iden tical equation. W e notice that the comm utator [[ ˙ ϕ + , ϕ + ] , ϕ + ] v anishes , and then apply (A.1). W e obtain ∂ t : e iaϕ : =  ∂ t e iaϕ +  e iaϕ − + e iaϕ +  ∂ t e iaϕ −  = e iaϕ + ia ˙ ϕ + + ( ia ) 2 2! h ˙ ϕ + , ϕ + i ! e iaϕ − + e iaϕ + ia ˙ ϕ − − ( ia ) 2 2! h ˙ ϕ − , ϕ − i ! e iaϕ − . (A.2) Let us now compute the comm utato r [ ˙ ϕ + , ϕ + ]. Using (7 ) and (3) w e obtain for F ourier comp onents [ ˙ ϕ p , ϕ q ] = 1 m ( p + q ) ϕ p + q . (A.3) Then w e pro ceed as h ˙ ϕ + ( x ) , ϕ + ( x ) i = 1 m  2 π L  2 X p,q < 0 e i ¯ h ( p + q ) x ( p + q ) ϕ p + q = 1 m  2 π L  2 X k < 0 e i ¯ h k x k ϕ p X k 0 are placed to the left to the rest of the mo des. F or example : e iaϕ := e iaϕ + e iaϕ − , wher e ϕ − ( x ) = ∓ i 2 π ¯ h L P ± k> 0 1 k J k e i ¯ h kx and ϕ + ( x ) = ∓ i 2 π ¯ h L P ± k< 0 1 k J k e i ¯ h kx . Quantum hydr o dynamics and nonline ar diff er ential e quations . . . 11 [17] In this and further formulas we assume a radial ordering ± I m ( x − x ′ ) > 0 and | x − x ′ | ≪ L . [18] A tec hnical commen t on or dering mixed fermionic and b osonic strings in (24-26) is in o r der. As an example consider h ψ † ( y ) ψ ( x ) , 1 i . A part of this co rrelatio n function is : e aϕ ( x ) ψ † ( y ) ψ ( x ) :. It is understo o d as : e aϕ ( x ) ψ † ( y ) ψ ( x ) := e aϕ + ( x ) ψ † ( y ) ψ ( x ) e aϕ − ( x ) , where fermio nic op era tors are placed in the middle b etw een the p ositive a nd neg a tive bo sonic mo des.