Asymptotic property of current for a conduction model of Fermi particles on finite lattice
AAsymptotic property of current for a conduction model ofFermi particles on finite lattice
Yamaga Kazuki
Abstract
In this paper, we introduce a conduction model of Fermi particles on a finite sample,and investigate the asymptotic behavior of stationary current for large sample size. In ourmodel a sample is described by a one-dimensional finite lattice on which Fermi particlesinjected at both ends move under various potentials and noise from the environment. Weobtain a simple current formula. The formula has broad applicability and is used to studyvarious potentials. When the noise is absent, it provides the asymptotic behavior of thecurrent in terms of a transfer matrix. In particular, for dynamically defined potentialcases, a relation between exponential decay of the current and the Lyapunov exponent ofa relevant transfer matrix is obtained. For example, it is shown that the current decaysexponentially for the Anderson model. On the other hand, when the noise exists but thepotential does not, an explicit form of the current is obtained, which scales as 1 /N forlarge sample size N . Moreover, we provide an extension to higher dimensional systems.For a three-dimensional case, it is shown that the current increases in proportion to crosssection and decreases in inverse proportion to the length of the sample. A unified theory for nonequilibrium systems is still lacking, while statistical mechanics forequilibrium systems well-connects the microscopic and macroscopic world. This occurs mainlyowing to the existence of various states in nonequilibrium systems. Therefore, it is importantto consider a specific, physically interesting subclass of nonequilibrium states. Nonequilibriumstationary state induced by multiple thermal and particle reservoirs should be an importantclass, which has been studied for a long time [1, 2, 3]. For example, [4] considers electricconduction in mesoscopic systems as a problem of nonequilibrium stationary states of manybody Fermi particle systems and derives the Landauer formula. In [5], the problem of how thestructure of a sample between reservoirs determines the property of current is studied, andthe equivalence of a ballistic transport and the existence of absolutely continuous spectrumis confirmed. Thus, if the absolutely continuous spectrum is empty, current goes to 0 inthe limit taking the sample size infinite. There are many physically important models thatdo not have absolutely continuous spectrum such as the Anderson model and the FibonacciHamiltonian, which is considered as the one-dimensional model of a quasi-crystal. While theresult is important, because the real sample size is finite, it is interesting to investigate thescaling of convergence. The scaling behavior depends on the sample structure. This problemhas not been solved yet by the authors of [5, 6]. In their model, a sample is connected to1 a r X i v : . [ m a t h - ph ] S e p nfinitely extended reservoirs at its ends; thus mathematical tools such as operator algebraand scattering theory are used.This study clarifies the problem of ’the scaling of the current’ by introducing a simplefinite dimensional conduction model. We focus on a sample described by a finite lattice onwhich many-body noninteracting Fermi particles are moving under various potentials andcertain noise called dephasing noise from the environment. The exchange of particles betweena sample and reservoirs is performed at the ends of the sample. See figure 1. This effect isdescribed by a Lindblad-type generator. Because our model does not have an infinite part, theentire analysis is performed within linear algebra. The same model has already been studiedin [7, 8]. The difference is that we solve the time evolution using the approach of [9]. Thefollowing simple current formula is obtained, J β ( N ) = 4( α lin α rout − α lout α rin ) (cid:90) ∞ (cid:104) e , T s ( p N ) e (cid:105) ds where ( α lin α rout − α lout α rin ) is a term determined by the strength of interaction at the bothends, and the integral is related with a two-point function which can be evaluated rigorouslyin various models. This formula can be applied to a wide class, which allows various types ofpotentials. Based on this formula, we consider how the scaling of the current is determinedby potentials and noise.Figure 1: conduction model of Fermi particles on finite latticeThis paper is organized as follows. In section 2, we introduce a conduction model ofFermi particles on a one-dimensional lattice. By solving the time evolution of the two-pointfunction, we show that it converges to a constant in the long time limit. In particular, as thecurrent is described by a two-point function, we obtain a simple current formula describedabove. In section 3, we consider the asymptotic behavior of the current for large sample size,using the above formula. We first consider the noiseless case in subsection 3.1 and show thatthe current formula can be evaluated in terms of transfer matrix. This result shows that bothour model and the model in [5] give the same prediction for the asymptotic property of thecurrent. In addition, in case of dynamically defined potential, such as the Anderson modeland the Fibonacci Hamiltonian, the scaling of the asymptotic behavior is shown to be relatedwith the Lyapunov exponent. In subsection 3.2, we introduce the noise called dephasing noise.We obtain an explicit form of the current if the potential is absent. The current decays scales2s 1 /N . This result coincides with that of [8], which takes a different approach from ours.The scaling of the current for general potentials is not obtained yet. But, it is shown thatfor strong noise the main term of current decays as 1 /N , and the current may increase byadding strong noise to random systems. Section 4 is devoted to the generalization to higherdimensional systems. The same formula as the one-dimensional system is obtained. If thenoise exists and potential is absent, it is shown that the current increases in proportion tocross section and decreases in inverse proportion to the length of the sample. In the lastsection, we provide conclusions and discuss the related studies. In this section, we introduce a conduction model of noninteracting Fermi particles on a one-dimensional finite lattice. First we consider the dynamics of a two-point function and itslong time limit (2.1). Then in 2.2, we focus on current and obtain a simple current formula(Theorem 2.2).
Let us consider a many body system of Fermi particles moving under various potentials andnoise on a one-dimensional finite lattice [1 , N ] ∩ Z ( N ∈ N ). The one-particle Hilbert spaceis C N . Denote its standard basis by { e n } Nn =1 . The many body system is described by thecreation and annihilation operators a ∗ ( f ) , a ( f ), where f ∈ C N . Let us write a n for a ( e n )as usual ( a is a ∗ or a ). These operators satisfy the following canonical anti-commutationrelations: { a ∗ i , a j } = δ ij I, { a i , a j } = 0 , where { A, B } = AB + BA , δ ij is the Kronecker delta and I is the identity operator on C N .In the sequel, we write shortly c × I as c ( c ∈ C ). Suppose that the total Hamiltonian is H = N − (cid:88) n =1 (cid:2) − ( a ∗ n a n +1 + a ∗ n +1 a n ) + v ( n ) a ∗ n a n (cid:3) , where v ( · ) is a real-valued function called potential. Since we will consider the limit N → ∞ ,potential v is given as a bounded function on N . Let A be the algebra generated by thecreation and annihilation operators, and θ : A → A be a *-automorphism determined by θ ( a n ) = − a n .For real numbers α lin , α lout , α rin , α rout , β greater than or equal to 0 (at least one of α lin , α lout , α rin , α rout is not 0), define a linear map L : A → A as L ( A ) = i [ H, A ] + α lin (2 a θ ( A ) a ∗ − { a a ∗ , A } ) + α lout (2 a ∗ θ ( A ) a − { a ∗ a , A } )+ α rin (2 a N θ ( A ) a ∗ N − { a N a ∗ N , A } ) + α rout (2 a ∗ N θ ( A ) a N − { a ∗ N a N , A } )+ β N (cid:88) n =1 (cid:18) a ∗ n a n Aa ∗ n a n − { a ∗ n a n , A } (cid:19) . H, A ] = HA − AH .It is obvious from the form of each term that L generates a Quantum Dynamical Semigroup { e tL } t ≥ on A [10]. That is, e tL is a CP (completely positive) map preserving identity (statetransformation) for every t ≥
0. The physical meaning of each term is as follows: i [ H, A ]This term represents the Hamiltonian dynamics of many particles moving independently bya one-particle Hamiltonian( hψ )( n ) = − ψ ( n + 1) − ψ ( n −
1) + v ( n ) ψ ( n )( ψ (0) = ψ ( N + 1) = 0). It operates as i [ H, a ∗ ( f ) a ( g )] = ia ∗ ( hf ) a ( g ) − ia ∗ ( f ) a ( hg ) . The terms with coefficients α lin , α lout , α rin , α rout These terms represent the effects of adding a particle to site 1, removing from site 1, addingto site N and removing from site N , respectively. Put p n = | e n (cid:105)(cid:104) e n | ( n = 1 , , · · · , N ), 1-rankprojections corresponding to the basis { e n } Nn =1 , then these terms operate as2 a θ ( a ( f ) a ∗ ( g )) a ∗ − { a a ∗ , a ( f ) a ∗ ( g ) } = − a ( p f ) a ∗ ( g ) − a ( f ) a ∗ ( p g ) , a ∗ θ ( a ∗ ( f ) a ( g )) a − { a ∗ a , a ∗ ( f ) a ( g ) } = − a ∗ ( p f ) a ( g ) − a ∗ ( f ) a ( p g )(replace p by p N in the case of N ). The dynamics generated by these terms and i [ H, A ] is aspecial case of those in [7, 9].The term with coefficient β (dephasing noise)This term represents noise from the environment called dephasing noise. Dephasing noisepreserves the number of particles and destroys the coherence. Let us check this property.Denote D n ( A ) = 2 a ∗ n a n Aa ∗ n a n − { a ∗ n a n , A } ( n = 1 , , · · · , N ), then since D n commutes with each other, we have e t (cid:80) Nn =1 D n = N (cid:89) n =1 e tD n . Easy calculation shows that D n ( a ∗ i a j ) = (cid:40) i = j = n, i, j (cid:54) = n ) − a ∗ i a j (otherwise) ,e tD n ( a ∗ i a j ) = (cid:40) a ∗ i a j ( i = j = n, i, j (cid:54) = n ) e − t a ∗ i a j (otherwise) . Recall that for every state ω on A , its two-point function is described by a positive operatoron C N : there is an operator R : C N → C N such that 0 ≤ R ≤ I and ω ( a ∗ i a j ) = (cid:104) e j , Re i (cid:105) .
4n the one-particle system, define a linear map d n : M N ( C ) → M N ( C ) as d n ( a ) = 2 p n ap n − { p n , a } , a ∈ M N ( C )( M N ( C ) is the set of N × N complex matrices), then e td n ( a ) = a + (1 − e − t ) d n ( a ) . If i = j = n or i, j (cid:54) = n , (cid:104) e j , e td n ( R ) e i (cid:105) = (cid:104) e j , Re i (cid:105) = ω ( e tD n ( a ∗ i a j )) , and if one of i, j is n , (cid:104) e j , e td n ( R ) e i (cid:105) = (cid:104) e j , Re i (cid:105) + (1 − e − t ) (cid:104) e j , ( − R ) e i (cid:105) = ω ( e tD n ( a ∗ i a j )) . Therefore, the dynamics of the two-point function is described by N (cid:89) n =1 e td n ( R ): ω (cid:16) e t (cid:80) Nn =1 D n ( a ∗ i a j ) (cid:17) = ω (cid:32) N (cid:89) n =1 e tD n ( a ∗ i a j ) (cid:33) = (cid:42) e j , N (cid:89) n =1 e td n ( R ) e i (cid:43) = (cid:68) e j , e t (cid:80) Nn =1 d n ( R ) e i (cid:69) . Set d ( a ) = 12 N (cid:88) n =1 d n ( a ) = (cid:88) n =1 p n ap n − a , then d = − d and e td ( a ) = a + (1 − e − t ) d ( a ) = e − t a + (1 − e − t ) N (cid:88) n =1 p n ap n . The pure state | ψ (cid:105)(cid:104) ψ | is transformed to e − t | ψ (cid:105)(cid:104) ψ | + (1 − e − t ) N (cid:88) n =1 | ψ ( n ) | p n . Thus, this dynamics destroys the coherence and transform a state to a convex combinationof localized states p n . L consists of the above three types of terms. Stationary current induced by the dynamics e tL is the main topic in this paper. From the above discussions it turns out that the dynamicsof the two-point function is described by that of the one-particle system. Suppose that thetwo-point function of the state ω ◦ e tL is expressed as ω ◦ e tL ( a ∗ i a j ) = (cid:104) e j , R ( t ) e i (cid:105) , then by calculating ddt ω ◦ e tL ( a ∗ i a j ) = ω ◦ e tL ( L ( a ∗ i a j )) ,
5e obtain the following differential equation for R ( t ): ddt R ( t ) = − i [ h, R ( t )] − { ( α lin + α lout ) p + ( α rin + α rout ) p N , R ( t ) } + β (cid:32) N (cid:88) n =1 p n R ( t ) p n − R ( t ) (cid:33) + 2 α lin p + 2 α rin p N ,R (0) = R. It is easy to check that R ( t ) = T t ( R ) + (cid:90) t T s (2 α lin p + 2 α rin p N ) ds is a solution of this equation, where T t is an operator semigroup on M N ( C ) generated by l : a (cid:55)→ − i [ h, a ] − { ( α lin + α lout ) p + ( α rin + α rout ) p N , a } + β (cid:32) N (cid:88) n =1 p n ap n − a (cid:33) .T t = e tl is a CP map which does not preserve identity.Let us consider the long time limit t → ∞ . In the case where β = 0, T t ( a ) = e − ith D ae ith ∗ D for h D = h − i ( α lin + α lout ) p − i ( α rin + α rout ) p N . Since the imaginary part of every eigenvalue of h D is less than 0, lim t →∞ e − ith D = 0. Thus, weget lim t →∞ R ( t ) = (cid:90) ∞ T s (2 α lin p + 2 α rin p N ) ds ≡ R ∞ . The integral of the right hand side converges, because0 ≤ (cid:90) t t T s (2 α lin p + 2 α rin p N ) ds ≤ (cid:90) t t T s (2( α lin + α lout ) p + 2( α rin + α rout ) p N ) ds = [ T s ( I )] t t → t , t → ∞ ) . Note that R ∞ does not depend on R . This means that whatever the initial state is, thetwo-point function converges to the same value (cid:104) e j , R ∞ e i (cid:105) . Moreover, it can be shown thatevery state converges to the quasi free state determined by this two-point function [9].In the case where β >
0, we have the same result for the two-point function.
Theorem 2.1. lim t →∞ T t = lim t →∞ e tl = 0 . Proof.
Recall that M N ( C ) is a Hilbert space for the Hilbert-Schmidt inner product, (cid:104) a, b (cid:105) HS =Tr a ∗ b . Let us decompose the generator of l : M N ( C ) → M N ( C ) as l = − iX − Y − βZ for X, Y, Z : M N ( C ) → M N ( C ) defined as Xa = [ h, a ]6 a = { ( α lin + α lout ) p + ( α rin + α rout ) p N , a } Za = a − N (cid:88) n =1 p n ap n .X, Y, Z are self-adjoint and especially Y, Z are positive. Let us check that Z is positive: (cid:104) a, Za (cid:105) HS = Tr (cid:32) a ∗ a − a ∗ N (cid:88) n =1 p n ap n (cid:33) = N (cid:88) n =1 Tr p n a ∗ ( I − p n ) ap n ≥ , ∀ a ∈ M N ( C ) . Let x ∈ C be an eigenvalue of l and a ∈ M N ( C ) be a corresponding unit eigenvector, that is, x, a satisfy la = xa, (cid:104) a, a (cid:105) HS = 1 . By l = − iX − Y − βZ , Re x , the real part of x , satisfies thatRe x = −(cid:104) a, ( Y + βZ ) a (cid:105) ≤ . If Re x = 0, we have (cid:104) a, Y a (cid:105) = 0 , (1) (cid:104) a, Za (cid:105) = 0 , (2)since Y and Z are positive. By equation(2),Tr p n a ∗ ( I − p n ) ap n = 0 , n = 1 , , · · · , N, → ( I − p n ) ap n = 0 . Thus, a is diagonalized for the basis { e n } Nn =1 (we write its entry as a ij ). Assume that α lin + α lout > α rin + α rout > N instead of 1), then by equation(1) a = Tr p ap = 0 . Since a is diagonalized,0 = xa = ( la ) = ia + ia − ia − ( α lin + α lout + β ) a = − ia , xa = ( la ) = ia + ia − ia − ia − ( α lin + α lout ) a = − ia . Repeat these processes until 0 = ( la ) N − N , then we finally get a = a = · · · = a NN = 0.This implies that a = 0. However a = 0 contradicts to the assumption that (cid:104) a, a (cid:105) HS = 1.Thus, Re x < l andlim t →∞ e tl = 0 . By this theorem, in the case where β > t →∞ R ( t ) = (cid:90) ∞ T s (2 α lin p + 2 α rin p N ) ds ≡ R ∞ . .2 current formula In this subsection we will focus on current. Since current is expressed by two-point function,it converges to a constant in the limit t → ∞ . We will consider how the sign of the currentis determined by the relation of constants α lin , α lout , α rin , α rout . The current is shown to beexpressed by a simple formula (Theorem 2.2).At first, recall that the observable of current from site n to n + 1 is j n = − i ( a ∗ n a n +1 − a ∗ n +1 a n ) . As shown in the previous subsection, for any state ω the limit lim t →∞ ω ◦ e tL ( j n ) exists and isindependent of ω . In fact it does not depend on n . Let us check it. By the definition ofgenerator, for any (cid:15) > h > (cid:13)(cid:13)(cid:13)(cid:13) e hL ( a ∗ n a n ) − a ∗ n a n h − L ( a ∗ n a n ) (cid:13)(cid:13)(cid:13)(cid:13) < (cid:15). Thus we have | ω ◦ e tL ( L ( a ∗ n a n )) | < (cid:15) + (cid:12)(cid:12)(cid:12)(cid:12) ω ◦ e tL (cid:18) e hL ( a ∗ n a n ) − a ∗ n a n h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , ∀ t ≥ t →∞ | ω ◦ e tL ( L ( a ∗ n a n )) | ≤ (cid:15) . Since (cid:15) is arbitrary, lim t →∞ ω ◦ e tL ( L ( a ∗ n a n )) = 0. Thisequation and L ( a ∗ n a n ) = − i ( a ∗ n − a n − a ∗ n a n − ) + i ( a ∗ n a n +1 − a ∗ n +1 a n ) = j n − − j n , n = 2 , · · · , N − n . We denote the limit of current by J β ( N ) = lim t →∞ ω ◦ e tL ( j ) (it depends on the sample size N ). Then it is expressed as J β ( N ) = − i (cid:104) e , R ∞ e (cid:105) + i (cid:104) e , R ∞ e (cid:105) = 2Im (cid:104) e , R ∞ e (cid:105) . J β ( N ) has the following simple expression. This is one of our main results in this paper. Theorem 2.2. J β ( N ) = 4( α lin α rout − α lout α rin ) (cid:90) ∞ (cid:104) e , T t ( p N ) e (cid:105) dt = − α lin α rout − α lout α rin ) (cid:104) e , l − ( p N ) e (cid:105) . Proof.
By 2Im | e (cid:105)(cid:104) e | = − i [ h, p ] and the definition of R ∞ , J β ( N ) = − (cid:90) ∞ Tr i [ h, p ] e tl (2 α lin p + 2 α rin p N ) dt = − (cid:90) ∞ Tr p le tl (2 α lin p + 2 α rin p N ) dt − α lin + α lout ) (cid:90) ∞ Tr p e tl (2 α lin p + 2 α rin p N ) dt = 2 α lin Tr p − α lin ( α lin + α lout ) (cid:90) ∞ Tr p e tl ( p ) dt − α rin ( α lin + α lout ) (cid:90) ∞ Tr p e tl ( p N ) dt.
8y the equation I = − (cid:104) e tl ( I ) (cid:105) ∞ = (cid:90) ∞ e tl (2( α lin + α lout ) p + 2( α rin + α rout ) p N ) dt, we have α lin Tr p − α lin ( α lin + α lout ) (cid:90) ∞ Tr p e tl ( p ) dt = 2 α lin ( α rin + α rout ) (cid:90) ∞ Tr p e tl ( p N ) dt. Combining these equations, we get J β ( N ) = 4( α lin α rout − α lout α rin ) (cid:90) ∞ Tr p e tl ( p N ) dt. In order to obtain the latter equation of the theorem, we use the well-known formula foroperator semigroups: for any (cid:15) > (cid:90) ∞ e − (cid:15)t e tl dt = ( (cid:15) − l ) − holds. As discussed before, the real part of every eigenvalue of l is less than 0. This impliesthat ker l = { } and l is invertible. Thus, we get (cid:90) ∞ (cid:104) e , T t ( p N ) e (cid:105) dt = lim (cid:15) ↓ (cid:90) ∞ e − (cid:15)t (cid:104) e , T t ( p N ) e (cid:105) dt = lim (cid:15) ↓ (cid:104) e , ( (cid:15) − l ) − ( p N ) e (cid:105) = −(cid:104) e , l − ( p N ) e (cid:105) . Since (cid:82) ∞ (cid:104) e , T t ( p N ) e (cid:105) dt >
0, the sign of J β ( N ) is completely determined by the coeffi-cient α lin α rout − α lout α rin . Let us check that (cid:82) ∞ (cid:104) e , T t ( p N ) e (cid:105) dt > (cid:104) e , T t ( p N ) e (cid:105) = ∞ (cid:88) n =0 (cid:28) e , ( tl ) n n ! ( p N ) e (cid:29) = (cid:28) e , ( tl ) N (2 N )! ( p N ) e (cid:29) + ∞ (cid:88) n =2 N +1 (cid:28) e , ( tl ) n n ! ( p N ) e (cid:29) = t N (2 N )! N C N (cid:104) e , h N p N h N e (cid:105) + ∞ (cid:88) n =2 N +1 (cid:28) e , ( tl ) n n ! ( p N ) e (cid:29) = t N ( N !) + ∞ (cid:88) n =2 N +1 (cid:28) e , ( tl ) n n ! ( p N ) e (cid:29) . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) n =2 N +1 (cid:28) e , ( tl ) n n ! ( p N ) e (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ t N +1 ∞ (cid:88) n =2 N +1 (cid:28) e , l n n ! ( p N ) e (cid:29) holds for 0 ≤ t <
1, for sufficiently small t > (cid:104) e , T t ( p N ) e (cid:105) > . Asymptotic behavior of current
In the previous section, we obtained a current formula applicable in general settings (Theorem2.2). In this section, using this formula, we investigate how potentials and noise determinethe asymptotic behavior of the current J β ( N ) for large sample size N . Since we would liketo consider the situation that the current J β ( N ) is not 0, let α lin + α lout , α rin + α rout >
0. Atfirst, we deal with the noiseless case ( β = 0). And next, the case where β >
0, mainly v = 0,is considered. β = 0 : noiseless case In this subsection, we first prove the following proposition, which is applicable to arbitrarypotentials.
Proposition 3.1. −(cid:104) e , l − ( p N ) e (cid:105) = 12 π (cid:90) R (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) dE. Using this formula, we relate the current J β ( N ) to transfer matrix. In addition, in caseof dynamically defined potential, such as the Anderson model, the scaling of the asymptoticbehavior is shown to be related with the Lyapunov exponent.Recall that in noiseless case la = − i ( h D a − ah ∗ D ) . As mentioned before, the imaginary part of every eigenvalue of h D is less than 0, and h D isinvertible. Let us prepare a lemma. Lemma 3.2.
For V ∈ M N ( C ) , define a linear map g V : M N ( C ) → M N ( C ) as g V ( x ) = V x − xV ∗ , x ∈ M N . Suppose that V is invertible and the imaginary part of every eigenvalue is less than 0. Then, g V is invertible and g − V ( x ) = i π (cid:90) R ( E − V ) − x ( E − V ∗ ) − dE. Proof.
Since the integrand (operator) in the right hand side is continuous for E and (cid:107) ( V − E ) − (cid:107) ≤ | E | − (cid:107) V (cid:107) for E with large absolute value, the integral converges and defines a linear map on M N ( C ).Let us denote it by h ( x ). Since g V ( x ) = ( V − E ) x − x ( V ∗ − E )for E ∈ R , we have h ◦ g V ( x ) = − i π (cid:90) R x ( E − V ∗ ) − dE + i π (cid:90) R ( E − V ) − xdE. (cid:90) R ( E − V ) − dE. In general, ( i, j )-entry of the inverse matrix of an N × N matrix A = ( a ij ) Ni,j =1 is expressedas det( A ) − ( − i + j det( A ji ) .A ij is an ( N − × ( N −
1) matrix called factor matrix, which is made by removing thei-th row and the j-th column from A . det( E − V ) is a polynomial that has degree of N andthe coefficient of E N is 1. Let us write det( E − V ) = E N + a N ( E ). Set A = E − V , thendet( A ii ) is a polynomial with degree of N − E N − is 1. Let us writedet( A ii ) = E N − + b iN ( E ). If i (cid:54) = j , then det( A ij ) is a polynomial with degree of N − c ijN ( E ). Define C + = { z ∈ C | Im z ≥ } . Since det( E − V ) has no zeros in C + ,( E − V ) − ij is regular in a region containing C + (note that ( E − V ) − ij is the ( i, j )-entry of( E − V ) − , not factor matrix). For R >
0, define a cycle Γ R as { z ∈ C | Im z = 0 , Re z ∈ [ − R, R ] } ∪ { Re iθ | θ ∈ [0 , π ] } , then (cid:73) Γ R ( E − V ) − ij dE = 0 . Set C R = { Re iθ | θ ∈ [0 , π ] } .(i) i = j (cid:90) C R ( z − V ) − ii dz = (cid:90) C R z N − + b iN ( z ) z N + a N ( z ) dz = (cid:90) π R N − e i ( N − θ + b iN ( Re − iθ ) R N e iNθ + a N ( Re iθ ) iRe iθ dθ = i (cid:90) π R N + e − i ( N − θ b iN ( Re − iθ ) R N + e − iNθ a N ( Re iθ ) dθ. This converges to iπ as R → ∞ . By (cid:90) R − R ( E − V ) − ii dE + (cid:90) C R ( z − V ) − ii dz = (cid:73) Γ R ( E − V ) − ij dE = 0 , we have (cid:90) R ( E − V ) − ii dE = lim R →∞ (cid:90) R − R ( E − V ) − ii dE = − iπ. (ii) i (cid:54) = j (cid:90) C R ( z − V ) − ij dz = (cid:90) C R c ijN ( z ) z N + a N ( z ) dz = i (cid:90) π Re − i ( N − θ c ijN ( Re iθ ) R N + e − iNθ a N ( Re iθ ) dθ. R → ∞ . Thus, (cid:90) R ( E − V ) − ij dE = 0 . Summarizing the above calculations, we get (cid:90) R ( E − V ) − dE = − iπI and h ◦ g V ( x ) = − i π iπx + i π ( − iπ ) x = x. This implies that g V is an injection. Since the space that g V operate is finite dimensional, g V is also surjective. Therefore, g V is invertible and g − V ( x ) = h ( x ) = i π (cid:90) R ( E − V ) − x ( E − V ∗ ) − dE. Applying this lemma for V = h D , then we obtain Proposition 3.1: −(cid:104) e , l − ( p N ) e (cid:105) = 12 π (cid:90) R (cid:104) e , ( h D − E ) − p N ( h ∗ D − E ) − e (cid:105) dE = 12 π (cid:90) R (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) dE. By this equation, in order to know the asymptotic behavior of the current J β ( N ), we have toinvestigate that of (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) . As we will see in the following, (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) is related to transfer matrix.Let us recall transfer matrix. Although we are considering a system on finite lattice[1 , N ] ∩ N , potential is given as a function v : N → R in order to take limit N → ∞ . For E ∈ C , if ψ ∈ C N satisfies hψ = Eψ, then the relation (cid:18) ψ ( n + 1) ψ ( n ) (cid:19) = (cid:18) v ( n ) − E −
11 0 (cid:19) (cid:18) ψ ( n ) ψ ( n − (cid:19) , n = 1 , · · · , N holds (here, ψ (0) = ψ ( N + 1) = 0). A 2 × T N ( E ) ≡ (cid:18) v ( N ) − E −
11 0 (cid:19) · · · (cid:18) v (1) − E −
11 0 (cid:19) is called a transfer matrix. It is in SL (2 , C ) and thus (cid:107) T N ( E ) (cid:107) ≥ E ∈ R , define g ij ( E ) = (cid:104) e i , ( h D − E ) − e j (cid:105) . These values are related to transfer matrix as follows.12 emma 3.3. ˜ T N ( E ) (cid:18) g ( E ) g N ( E )1 0 (cid:19) = (cid:18) g N ( E ) g NN ( E ) (cid:19) , where ˜ T N ( E ) is a transfer matrix corresponding to a complex-valued potential ˜ v defined as ˜ v (1) = v (1) − i ( α lin + α lout ) , ˜ v ( N ) = v ( N ) − i ( α rin + α rout ) and ˜ v ( n ) = v ( n ) for n = 2 , · · · , N − . We do not give the proof here, since it is in [11] (Lemma 2.2). By this lemma, we canevaluate (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) using transfer matrix. Lemma 3.4.
There is a constant
M > independent of E ∈ R , N ∈ N such that | g ij ( E ) | ≤ M ( i, j = 1 , N ). There is a constant K such that (cid:107) ˜ T N ( E ) (cid:107) ≤ (cid:12)(cid:12) (cid:104) e , ( h D − E ) − e N (cid:105) (cid:12)(cid:12) ≤ K (cid:107) ˜ T N ( E ) (cid:107) . Proof.
By resolvent formula,( α lin + α lout ) | g ( E ) | + ( α rin + α rout ) | g N ( E ) | = (cid:68) e , ( h ∗ D − E ) − { ( α lin + α lout ) p + ( α rin + α rout ) p N } ( h D − E ) − e (cid:69) = 12 i ( g ( E ) − g ( E )) ≤ | g ( E ) | . From this inequality, we have | g ( E ) | − α lin + α lout | g ( E ) | ≤ , (3)( α rin + α rout ) | g N ( E ) | ≤ − ( α lin + α lout ) | g ( E ) | + | g ( E ) | . (4)By inequality(3), | g ( E ) | ≤ α lin + α lout and by inequality(4), | g N ( E ) | ≤ α lin + α lout )( α rin + α rout ) . Similarly, we get | g N ( E ) | ≤ α lin + α lout )( α rin + α rout ) , | g NN ( E ) | ≤ α rin + α rout . The former inequality of the lemma is obtained.Operating both hand sides of the equation of Lemma 3.3 to a vector (cid:18) (cid:19) , we obtain1 ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) g NN ( E ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:107) ˜ T N ( E ) (cid:107)| g N ( E ) | . g N ( E ) is not 0, (cid:18) g ( E ) g N ( E )1 0 (cid:19) is invertible and by Lemma 3.3 we have g N ( E ) ˜ T N ( E ) = g N ( E ) (cid:18) g N ( E ) g NN ( E ) (cid:19) (cid:18) g ( E ) g N ( E )1 0 (cid:19) − = (cid:18) g N ( E ) g NN ( E ) (cid:19) (cid:18) − g N ( E ) − g ( E ) (cid:19) . Since all the entries of the right hand side are bounded, the norm is also bounded by an
E, N -independent constant K : (cid:107) ˜ T N ( E ) (cid:107) ≤ K | g N ( E ) | . Easy calculation shows that there are
E, N -independent constants a, b > a (cid:107) T N ( E ) (cid:107) ≤ (cid:107) ˜ T N ( E ) (cid:107) ≤ b (cid:107) T N ( E ) (cid:107) . Therefore, the asymptotic behavior of the current is determined by that of (cid:90) R (cid:107) T N ( E ) (cid:107) dE. (5)Denote C = 2 + sup n ∈ N | v ( n ) | , then the spectrum of h , σ ( h ), is contained in the interval[ − C, C ]. Set
R > C + 1. The following facts show that the integral over large energy decaysso rapidly that we do not have to care when considering the asymptotic behavior. This isused when we consider concrete models later.
Theorem 3.5. lim inf N →∞ (cid:18) − N log (cid:90) ∞ R dE (cid:107) T N ( E ) (cid:107) (cid:19) ≥ R − C ) > . It is same for (cid:90) − R −∞ dE (cid:107) T N ( E ) (cid:107) . By this theorem, we immediately obtain the following corollary.
Corollary 3.6.
There is R > such that for all R ≥ R , lim inf N →∞ (cid:18) − N log (cid:90) ∞−∞ dE (cid:107) T N ( E ) (cid:107) (cid:19) = lim inf N →∞ (cid:18) − N log (cid:90) R − R dE (cid:107) T N ( E ) (cid:107) (cid:19) holds. h Z on a doubly infinite lattice Z . Now,potential is given only on N . For n = 0 , − , · · · , we extend it by v ( n ) = 0. Then, h Z is abounded self-adjoint operator on l ( Z ) and σ ( h Z ) ⊂ [ − C, C ] ( C = 2 + sup n ∈ N | v ( n ) | ). Thus, if | E | ≥ R , h Z − E is invertible. Note that there is a solution ψ of the eigenvalue equation h Z ψ = Eψ such that ψ ( n ) = (cid:104) e , ( h − E ) − e n (cid:105) for n = 0 , , , · · · . Such ψ can be constructedas follows: If n ∈ N −(cid:104) e , ( h Z − E ) − e n +1 (cid:105) − (cid:104) e , ( h Z − E ) − e n − (cid:105) + v ( n ) (cid:104) e , ( h Z − E ) − e n (cid:105) = (cid:104) e , ( h Z − E ) − h Z e n (cid:105) = E (cid:104) e , ( h Z − E ) − e n (cid:105) holds. For n = − , − , · · · , determine ψ ( n ) by ψ ( n −
1) = − ψ ( n + 1) + v ( n ) ψ ( n ) − Eψ ( n )inductively.Let us consider the asymptotic behavior of (cid:104) e , ( h Z − E ) − e n (cid:105) . Set q n = −| e n (cid:105)(cid:104) e n − | −| e n − (cid:105)(cid:104) e n | and h n = h Z − q n . By resolvent formula( h Z − E ) − = ( h n − E ) − − ( h Z − E ) − q n ( h n − E ) − , (cid:104) e , ( h Z − E ) − e n (cid:105) = (cid:104) e n , ( h n − E ) − e n (cid:105)(cid:104) e , ( h Z − E ) − e n − (cid:105) . Use this equation for (cid:104) e , ( h Z − E ) − e n − (cid:105) again and repeat this process, then finally we get (cid:104) e , ( h Z − E ) − e n (cid:105) = (cid:104) e , ( h Z − E ) − e (cid:105) n (cid:89) k =1 (cid:104) e k , ( h k − E ) − e k (cid:105) . By spectral decomposition and the condition on E , the absolute value of each factor is boundedby | E |− C . Thus, we have |(cid:104) e , ( h Z − E ) − e n (cid:105)| ≤ (cid:18) | E | − C (cid:19) n +1 . Define α ( n ) = ψ ( n ) / (cid:112) ψ (0) + ψ (1) . Let β ( n ) be the solution of the eigenvalue equationwith the condition β (0) = − α (1) , β (1) = α (0) ( | β (0) | + | β (1) | = 1). By the property oftransfer matrix, (cid:18) α ( n + 1) β ( n + 1) α ( n ) β ( n ) (cid:19) = T n ( E ) (cid:18) α (1) β (1) α (0) β (0) (cid:19) . Since T n ( E ) ∈ SL (2 , C ) and α (1) β (0) − α (0) β (1) = 1, α ( n + 1) β ( n ) − α ( n ) β ( n + 1) = 1 holds.Thus we have1 ≤ | α ( n + 1) β ( n ) | + | α ( n ) β ( n + 1) |≤ (cid:112) | ψ (0) | + | ψ (1) | (cid:34)(cid:18) | E | − C (cid:19) n +1 | β ( n ) | + (cid:18) | E | − C (cid:19) n | β ( n + 1) | (cid:35) ≤ (cid:112) | ψ (0) | + | ψ (1) | (cid:18) | E | − C (cid:19) n ( | β ( n ) | + | β ( n + 1) | ) . | ψ (0) | = |(cid:104) e , ( h − E ) − e (cid:105)| ≥ | E | + C we get | β ( n ) | + | β ( n + 1) | ≥ ( | E | − C ) n | E | + C . By | β ( n ) | + | β ( n + 1) | ≥ ( | β ( n ) | + | β ( n + 1) | ) ≥
12 ( | E | − C ) n ( | E | + C ) and | β (0) | + | β (1) | = 1, we get (cid:107) T n ( E ) (cid:107) ≥ √ | E | − C ) n | E | + C . (cid:90) ∞ R dE (cid:107) T n ( E ) (cid:107) ≤ (cid:90) ∞ R ( | E | + C ) ( | E | − C ) n dE = 22 n − R − C ) n − + 8 C n R − C ) n + 8 C n + 1 1( R − C ) n +1 . Thus, Theorem 3.5 follows (the case of (cid:82) − R −∞ is similarly proven). (cid:3) By this theorem, it turns out that Theorem 1.1 in [5] is also true in our setting. We stateas a theorem here.
Theorem 3.7 ([5]) . Let h N be a discrete Schr¨odinger operator on l ( N ) with a boundedpotential v : N → R . The following statements are equivalent. • h N does not have absolutely continuous spectrum ( σ ac ( h N ) = ∅ ) • lim N →∞ (cid:90) R dE (cid:107) T N ( E ) (cid:107) = 0 . The above results can be applied to arbitrary (bounded) potentials. Next we investigatethe detail for a class of potentials called dynamically defined potentials. This class containsvarious physically important models such as the Anderson model, which is an example ofrandom systems, and the Fibonacci Hamiltonian, which is considered as the one-dimensionalmodel of a quasi-crystal. There are a huge number of studies for the spectrum of Sch¨odingeroperators with dynamically defined potentials [12, 13]. Here, the scaling of the asymptoticbehavior is shown to be related with the Lyapunov exponent.Let us start with the definition of dynamically defined potentials. We deal with the systemon Z , although we are interested in the half of it, N .Let (Ω , F , P, φ ) be an ergodic invertible discrete dynamical system. That is, (Ω , F , P ) isa probability space (in the sequel, we do not write the σ -field F ), φ : Ω → Ω is a measur-able bijection preserving probability P such that the probability of invariant set is 0 or 1(ergodicity). Let f be a bounded real measurable function on Ω. Then, for ω ∈ Ω we have aSchr¨odinger operator h ω with a potential v ω ( n ) = f ( φ n ω ) , n ∈ Z . v ω ( · ) is called a dynamically defined potential and a family of operators { h ω } ω ∈ Ω is calledan ergodic Schr¨odinger operator.Let us denote T N,ω ( E ) the transfer matrix determined by the potential v ω . Then T N,ω ( E )satisfies T N + M,ω ( E ) = T N,φ M ω ( E ) T M,ω ( E )and log (cid:107) T N + M,ω ( E ) (cid:107) ≤ log (cid:107) T N,φ M ω ( E ) (cid:107) + log (cid:107) T M,ω ( E ) (cid:107) . By subadditive ergodic theorem, for a.e. ω lim N →∞ N log (cid:107) T N,ω (cid:107) = L ( E )holds, where L ( E ) ≡ inf N ≥ N (cid:90) Ω log (cid:107) T N,ω ( E ) (cid:107) dP ( ω ) = lim N →∞ N (cid:90) Ω log (cid:107) T N,ω ( E ) (cid:107) dP ( ω ) .L ( E ) is called Lyapunov exponent. Since (cid:107) T N,ω ( E ) (cid:107) ≥ L ( E ) ≥
0. The Lyapunov exponent L ( E ) provides a rate of exponential growth of the norm of the transfer matrix (cid:107) T N,ω ( E ) (cid:107) foreach E ∈ R . What we would like to estimate is the integral I ( N, ω ) ≡ (cid:90) ∞−∞ dE (cid:107) T N,ω ( E ) (cid:107) . Theorem 3.8.
Assume that the Lyapunov exponent L ( E ) is continuous. Then, ≤ lim inf N →∞ (cid:18) − N log I ( N, ω ) (cid:19) ≤ lim sup N →∞ (cid:18) − N log I ( N, ω ) (cid:19) ≤ E ∈ R L ( E ) holds for a.e. ω ∈ Ω . Proof.
Only the last inequality is not trivial. Suppose that ω ∈ Ω satisfieslim N →∞ N log (cid:107) T N,ω ( E ) (cid:107) = L ( E )for a.e. E ∈ R . By Fubini theorem, the probability of the set of such ω is 1. By the discussionof Theorem 3.5 it turns out that inf E ∈ R L ( E ) = min E ∈ R L ( E ). Put γ = min E ∈ R L ( E ) and let E min bethe energy that achieves the minimum (such E min may not be uniquely determined, but thechoice of E min is not important in the following discussion). Since L ( E ) is continuous, forany (cid:15) > δ > E ∈ ( E min − δ, E min + δ ) ≡ R δ ⇒ L ( E ) − γ < (cid:15) . As − logis a monotonically decreasing convex function, we have − N log I ( N, ω ) ≤ − N log (cid:18)(cid:90) R δ (cid:107) T N,ω ( E ) (cid:107) dE (cid:19) = − N log (cid:18) δ (cid:90) R δ (cid:107) T N,ω ( E ) (cid:107) dE (cid:19) − N log 2 δ ≤ δ (cid:90) R δ N log (cid:107) T N,ω ( E ) (cid:107) dE − N log 2 δ.
17y dominated convergence theorem,lim sup N →∞ (cid:18) − N log I ( N, ω ) (cid:19) ≤ δ (cid:90) R δ L ( E ) dE ≤ γ + (cid:15). Since (cid:15) > N →∞ (cid:18) − N log I ( N, ω ) (cid:19) ≤ E ∈ R L ( E ) . By this theorem, if the Lyapunov exponent L ( E ) is continuous and min E ∈ R L ( E ) = 0, thecurrent does not decay exponentially. Examples are given in the last of this section. Althoughthis theorem tells when the decay of the current is slow, it does not tell when the currentdecays exponentially. We do not know whether the equality holds or not in Theorem 3.8. Ifthe following large deviation type estimate and inf E ∈ R L ( E ) > Definition 1 (Large Deviation type estimate) . We say that the property LD (Large Deviationtype estimate) holds, if the following condition is satisfied: For any (cid:15) > and any finite closedinterval [ a, b ] , there are constants C, η > such that P (cid:18)(cid:26) ω ∈ Ω | (cid:12)(cid:12)(cid:12)(cid:12) N log (cid:107) T N,ω ( E ) (cid:107) − L ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:27)(cid:19) ≤ Ce − ηN , ∀ N ∈ N , ∀ E ∈ [ a, b ] . Theorem 3.9.
Suppose that the property LD holds and inf E ∈ R L ( E ) > , then lim inf N →∞ (cid:18) − N log I ( N, ω ) (cid:19) > , a.e. ω ∈ Ω . Although the proof is obvious from the discussion in the proof of Lemma 3.2 in [14], werepeat it here.
Proof.
Set γ = inf E ∈ R L ( E ) > (cid:15), R that satisfy 0 < (cid:15) < γ and R > (cid:107) f (cid:107) ( (cid:107) f (cid:107) is thenorm in L ∞ (Ω , P )). By the property LD, there are η, C > P (cid:18)(cid:26) ω ∈ Ω | (cid:12)(cid:12)(cid:12)(cid:12) N log (cid:107) T N,ω ( E ) (cid:107) − L ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:27)(cid:19) ≤ Ce − ηN , ∀ N ∈ N , ∀ E ∈ [ − R, R ] . Let us denote m Lebesgue measure on R . DenoteΩ N(cid:15) = (cid:26) ( E, ω ) ∈ [ − R, R ] × Ω | (cid:12)(cid:12)(cid:12)(cid:12) N log (cid:107) T N,ω ( E ) (cid:107) − L ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:27) , Ω N(cid:15) ( ω ) = { E ∈ [ − R, R ] | ( E, ω ) ∈ Ω N(cid:15) } , then we have m × P (Ω N(cid:15) ) ≤ RCe − ηN . δ such that 0 < δ < η and set X Nδ = { ω ∈ Ω | m (Ω N(cid:15) ( ω )) ≤ e − δN } , then we get P ( X N,Cδ ) ≤ e δN (cid:90) X N,Cδ m (Ω N(cid:15) ( ω )) P ( dω ) ≤ e δN m × P (Ω N(cid:15) ) ≤ RCe − ( η − δ ) N . ∞ (cid:88) N =1 P ( X N,Cδ ) < ∞ holds and by Borel-Cantelli lemma, P (cid:18) lim inf N →∞ X Nδ (cid:19) = 1 . This means that for a.e. ω there is N ( ω ) ∈ N such that if N ≥ N ( ω ) then m (Ω N(cid:15) ( ω )) ≤ e − δN holds. Obviously such ω satisfies (cid:90) R − R dE (cid:107) T N,ω ( E ) (cid:107) ≤ (cid:90) Ω N(cid:15) ( ω ) dE (cid:107) T N,ω ( E ) (cid:107) + (cid:90) Ω N(cid:15) ( ω ) C dE (cid:107) T N,ω ( E ) (cid:107) ≤ e − δN + (cid:90) R − R dEe L ( E ) − (cid:15) ) N ≤ e − δN + 2 Re − γ − (cid:15) ) N for N ≥ N ( ω ). By this estimate and Theorem 3.5, we obtainlim inf N →∞ (cid:18) − N log (cid:18)(cid:90) R dE (cid:107) T N,ω ( E ) (cid:107) (cid:19)(cid:19) ≥ min { δ, γ − (cid:15) ) , R − C ) } > . The continuity and the Large deviation type estimate of the Lyapunov exponent are alreadywell investigated in the context of ergodic Schr¨odinger operators [15]. Here we show somephysically important examples. See [16] for well-organized results for the continuity and thelarge deviation type estimate of the Lyapunov exponent. Here we would like to show someexamples.The Anderson modelLet K ⊂ R be a compact subset, ρ be a probability measure on K such that ρ ≥ K Z and P = ρ Z . Let φ be a shift on Ω, thatis, ( φω ) n = ω n +1 . f ( ω ) = ω . This is a model such that the value of the potential at each siteis the i.i.d. random variable. As is well known, this model exhibits Anderson localization.The following theorem is a statement called spectral localization [17].19 heorem 3.10. For a.e. ω ∈ Ω , the following statements hold: • h ω has pure point spectrum. • Every eigenvector decays exponentially.
By Theorem 3.7, the current converges to 0 as N → ∞ for a.e. ω (we can apply Theorem3.7 for the system on Z , since absolutely continuous spectrum is stable under trace classperturbations). Moreover, since the Lyapunov exponent satisfies the large deviation typeestimate and inf E ∈ R L ( E ) > T , P : Lebesgue measure. φω = ω + α , where α = √ − . f ( ω ) = − λχ [1 − α, ( ω ).The spectrum is independent of ω ∈ T (we denote it by Σ λ ) and singular continuous. Itis known that the Lyapunov exponent L ( E ) is continuous and is 0 on Σ λ . Thus by Theorem3.7, 3.8, although the current converges to 0 as N → ∞ , it does not decay exponentially.The more can be said for this model. In the case where ω = 0, it is shown that the normof the transfer matrix is bounded by the power of the sample size N on the spectrum [22] :There is an E -independent constant θ > E ∈ Σ λ then (cid:107) T N ( E ) (cid:107) ≤ N θ . Notethat this fact does not imply the power law decay of the current immediately, because theLebesgue measure of the spectrum Σ λ is 0. However, by combining the results in [23, 24], wecan conclude the power law decay of the current. Theorem 3.11.
Let dim H Σ λ be the Hausdorff dimension of Σ λ ( dim H Σ λ ∈ (0 , by [21]).For any ξ ∈ (0 , dim H Σ λ ) , there is a constant C ξ > such that I ( N ) ≥ C ξ N ( ξ − θ . Almost Mathieu operatorThis model is the representative example of quasi-periodic potential. Ω = T , P : Lebesuguemeasure. φω = ω + α for fixed α ∈ T . f ( ω ) = − λ cos(2 πω ). This model has two parameters α ∈ T , λ >
0, and the properties vary according to them. Since if α is rational, the porential isperiodic, we assume that α is irrational. If λ <
1, then for every ω ∈ T the spectrum of h ω ispurely absolutely continuous. If λ ≥
1, then for every ω ∈ T , absolutely continuous spectrumis empty, σ ac ( h ω ) = ∅ . So our interest is in the case where λ ≥
1. The Lyapunov exponent L ( E ) is continuous and its minimum is max { log λ, } , which is the value on the spectrum [15].Thus, the current does not show the exponential decay for λ = 1. If λ >
1, it is shown thatthe property LD holds for appropriate α , and the current decays exponentially [25]. β > : with noise In this subsection we consider the current under dephasing noise. We obtain an explicit formof the current, which scales as 1 /N for large N , in the case where the potential is absent(3.2.1). 3.2.2 deals with the general potential case. Unfortunately, the scaling of the currentfor general potentials is not obtained yet. But we can say a little about the current for strongnoise regime. 20 .2.1 v = 0Let us start with the case where v = 0. In this case we can obtain an explicit form of thecurrent J β ( N ), using the equation J β ( N ) = − α lin α rout − α lout α rin ) (cid:104) e , l − ( p N ) e (cid:105) . Set X = l − ( p N ). X is a self-adjoint operator on C N . Let us denote X ij = (cid:104) e i , Xe j (cid:105) .Since X is self-adjoint, X ji = X ij . Denote α lin + α lout = ζ l > , α rin + α rout = ζ r >
0. By l ( X ) = p N , we have0 = (cid:104) e , p N e (cid:105) = (cid:104) e , l ( X ) e (cid:105) = − ζ l X + iX − iX → Im X = ζ l X . And for n = 2 , · · · , N − (cid:104) e n , p N e n (cid:105) = iX n − n − iX nn − + iX n +1 n − iX nn +1 → Im X n − n = Im X nn +1 = ζ l X . By (cid:90) ∞ T ∗ t (2 ζ l p + 2 ζ r p N ) dt = I, where T ∗ t is the dual action of T t (Tr aT t ( b ) = Tr T ∗ t ( a ) b ), we get2 ζ l X + 2 ζ r X NN = − → X NN = (cid:18) − ζ r − ζ l ζ r X (cid:19) . We have 0 = (cid:104) e , p N e (cid:105) = − ζ l X − βX + iX − iX − iX , (cid:104) e N − , p N e N (cid:105) = − ζ r X N − N − βX N − N + iX NN + iX N − N − iX N − N − , and for n = 2 , · · · , N −
20 = (cid:104) e n , p N e n +1 (cid:105) = − βX nn +1 + iX n +1 n +1 + iX n − n +1 − iX nn − iX nn +2 . Adding the imaginary part of the above three equations, we finally obtain0 = X NN − X − ζ l · ζ l X − ζ r · ζ l X − β ( N − · ζ l X = (cid:18) − ζ r − ζ l ζ r X (cid:19) − X − ζ l X − ζ l ζ r X − βζ l ( N − X . → X = −
12 1 ζ l + ζ r + ζ l ζ r ( ζ l + ζ r + β ( N − . Thus the current J β ( N ) is expressed as follows:21 heorem 3.12. When v = 0 , then J β ( N ) = 2( α lin α rout − α lout α rin ) α lin + α lout + α rin + α rout + ( α lin + α lout )( α rin + α rout )( α lin + α lout + α rin + α rout + β ( N − . The current J β ( N ) decays as 1 /N for large N and its coefficient is2( α lin α rout − α lout α rin ) β ( α lin + α lout )( α rin + α rout ) . For α lin = Γ − µ , α lout = Γ µ , α rin = Γ µ , α rout = Γ − µ , β = 2 γ , we have J γ ( N ) = − µ Γ + 1 / Γ + γ ( N − . This corresponds to the result of [8] (note that the Hamiltonian in [8] corresponds to 2 H inour setting). v : general potentials In the case of general potentials, the scaling of J β ( N ) is not obtained. But for large β , we canknow a little about the current. First, we consider the strong noise limit β → ∞ . And then,large but finite noise β = (cid:15)N is discussed and it is shown that the current may be increasedby adding large noise in the case of random potentials.The same calculation as the case where v = 0 shows that[ ζ l + ζ r + ζ l ζ r ( ζ l + ζ r + β ( N − X = −
12 + ζ r N − (cid:88) n =1 ( v ( n + 1) − v ( n ))Re X nn +1 . Since X is bounded: 0 ≤ − X = − l − ( p N ) = (cid:90) ∞ e tl ( p N ) dt ≤ ζ r I, we have | X nn +1 | ≤ ζ r β (3 + max { ζ l , ζ r , } ) → β → ∞ ) . Thus we obtain lim β →∞ β J β ( N ) = 2( α lin α rout − α lout α rin )( α lin + α lout )( α rin + α rout )( N − . This means that when one expands J β ( N ) in terms of 1 /β for large β , the dominant term is2( α lin α rout − α lout α rin )( α lin + α lout )( α rin + α rout )( N −
1) 1 β , which is independent of potentials and scales 1 /N for large N . But there is a gap betweenthis fact and the claim that J β ( N ) scales as 1 /N .22ext, we consider large β not taking limit β → ∞ . Denote C = 3 + max { ζ l , ζ r , } . Fix (cid:15) > (cid:107) v (cid:107) C and put β = (cid:15)N , then we have[ ζ l + ζ r + ζ l ζ r ( ζ l + ζ r + β ( N − X = −
12 + ζ r N − (cid:88) n =1 ( v ( n + 1) − v ( n ))Re X nn +1 ≤ − (cid:18) − (cid:107) v (cid:107) C(cid:15) (cid:19) . Therefore, the current J β ( N ) is bounded below as J (cid:15)N ( N ) ≥ α lin α rout − α lout α rin ) ζ l + ζ r + ζ l ζ r ( ζ l + ζ r + (cid:15)N ( N − (cid:18) − (cid:107) v (cid:107) C(cid:15) (cid:19) > . Let us consider the Anderson model as an example. Recall that if β = 0, the current showsthe exponential decay for a.e. ω . It turns out that by the above inequality, for such ω , J (cid:15)N ( N, ω ) ≥ J ( N, ω )holds for sufficiently large N . Thus, strong noise increases the current in this example. Itis remarkable that although the noise is symmetric and does not have the effect to flow theparticles to a specific direction, it could increase the current. Note that the noise does notalways increase the current (consider the case where v = 0). d -dimensional systems In the previous sections we focused on one-dimensional systems. In this section we consider anextension to general d -dimensional systems. As in the one-dimensional case, we assume thatparticles go in and out in a specific direction. Although the case where d = 2 , d -dimensional systems here. Since the analysis is almost thesame as one-dimensional systems, we do not discuss the detail here.For N , N , · · · , N d ∈ N , let us consider a finite d -dimensional lattice L = { , , · · · , N } × { , , · · · , N } × { , , · · · , N d } . An element of this lattice is written as ν = ( ν , ν , · · · , ν d ) ∈ L . We assume that particles go in and out in the direction’1’. For i = 1 , , · · · , N , define M i = { ν ∈ L | ν = i } . This is a plane vertical to the direction’1’. Suppose that particles go in and out at the surfaces M , M N . For ν ∈ L \ M N , define ν + = ( ν + 1 , ν , · · · , ν d ) ∈ L . N N ( ν ) be the set of nearest-neighbors of ν in L .one-particle Hilbert space that describes Fermi particles moving on this lattice is C | L | ,where | L | = d (cid:89) n =1 N n . We denote its standard basis by { e ν } ν ∈ L . one-particle Hamiltonian h isgiven as ( hψ )( ν ) = − (cid:88) µ ∈ NN ( ν ) ψ ( µ ) + v ( ν ) ψ ( v ) , ψ ∈ C | L | . Let H be the total Hamiltonian constructed by this one-particle Hamiltonian h . Let usconsider the following generator L in many body system: L ( A ) = i [ H, A ]+ α lin (cid:88) ν ∈ M (2 a ν θ ( A ) a ∗ ν − { a ν a ∗ ν , A } ) + α lout (cid:88) ν ∈ M (2 a ∗ ν θ ( A ) a ν − { a ∗ ν a ν , A } )+ α rin (cid:88) ν ∈ M N (2 a ν θ ( A ) a ∗ ν − { a ν a ∗ ν , A } ) + α rout (cid:88) ν ∈ M N (2 a ∗ ν θ ( A ) a ν − { a ∗ ν a ν , A } )+ β (cid:88) ν ∈ L (cid:18) a ∗ ν a ν Aa ∗ ν a ν − { a ∗ ν a ν , A } (cid:19) . Here, we denote a ( e ν ) = a ν as usual. α lin , α lout , α rin , α rout , β are real numbers that are greaterthan or equal to 0, and we assume that α lin + α lout > , α rin + α rout >
0. By the samecalculation as one-dimensional case, it turns out that the dynamics of the two point functionis described in terms of that of one-particle system. For e ν ∈ C | L | , denote a 1-rank projectionby p ν = | e ν (cid:105)(cid:104) e ν | . If ω ( a ∗ ( f ) a ( g )) = (cid:104) g, Rf (cid:105) , then R ( t ) defined by the relation ω ◦ e tL ( a ∗ ( f ) a ( g )) = (cid:104) g, R ( t ) f (cid:105) , is expressed as R ( t ) = e tl ( R ) + (cid:90) t e sl α lin (cid:88) ν ∈ M p ν + 2 α rin (cid:88) ν ∈ M N p ν ds, where l is a linear map on M | L | ( C ) defined as l ( a ) = − i [ h, a ] − ( α lin + α lout ) (cid:88) ν ∈ M p ν , a − ( α rin + α rout ) (cid:88) ν ∈ M N p ν , a + β (cid:32)(cid:88) ν ∈ L p ν ap ν − a (cid:33) . It generates a semigroup of CP maps e tl . By the same discussion as the one-dimensionalsystem, we obtain lim t →∞ e tl = 0. Thus R ( t ) converges to (cid:90) ∞ e tl (2 α lin P + 2 α rin P N ) dt t → ∞ , where P = (cid:88) ν ∈ M p ν and P N = (cid:88) ν ∈ M N p ν . In the long time limit, the number ofparticles which move from M n to M n +1 per time (current) becomes (cid:88) ν ∈ M n Im (cid:90) ∞ (cid:68) e ν + , e tl (2 α lin P + 2 α rin P N ) e ν (cid:69) dt. It is independent of n (we denote it by J ( N , · · · , N d )). The same calculation as one-dimensional system shows that J ( N , · · · , N d ) = 4( α lin α rout − α lout α rin ) (cid:90) ∞ Tr P e tl ( P N ) dt = − α lin α rout − α lout α rin )Tr P l − ( P N ) . In the case where v = 0, we obtain the explicit form of the current: Theorem 4.1. J ( N , · · · , N d ) = 2( α lin α rout − α lout α rin ) (cid:81) dn =2 N n ( β ( N −
1) + α lin + α lout + α rin + α rout )( α lin + α lout )( α rin + α rout ) + α lin + α lout + α rin + α rout . Especially in the case where d = 3, the current decreases in inverse proportion to thelength of the sample N and increases in proportion to the cross section N × N . In this paper, we investigated the current for a conduction model of Fermi particles on afinite lattice. When the dephasing noise is absent ( β = 0), this model is a special case ofthose in [7, 9]. First, we obtained the dynamics of two point function and proved that itconverges to a constant independent of initial state. Next, we investigated the current, whichis an important quantity in nonequilibrium systems and described by two point function andobtained a simple current formula (Theorem 2.2). Based on this formula, we considered theasymptotic behavior of the current. The results are as follows: noiseless ( β = 0 ) One can evaluate the current using transfer matrix. For dynamically de-fined potentials, the asymptotic behavior is related to the property of the Lyapunovexponent. For example, the Anderson model shows the exponential decay of current. with noise ( β > ) For the case where v = 0, the current is explicitly obtained and decaysas 1 /N . The same analysis can be applied to higher dimensional systems. In three-dimensional case, the current increases in proportion to cross section and decreases ininverse proportion to the length of the sample for large sample size.Apart from the case where v = 0, we gave only inequalities for the asymptotic property inthis paper. To obtain the exact scaling of the current for various models is our future work.Finally we would like to discuss some related studies. As previously mentioned, the noise-less case is also studied in more general settings in [7, 9]. But we believe that it is our original25ork to obtain the current formula (Theorem 2.2) and investigate the asymptotic propertybased on it. In [7], Prosen discussed the conduction model as an example and said that thecurrent would decay exponentially for random potentials. But he did not give an exact prooffor it. The model that noise exists and potential v = 0 is studied in [8], and the same currentformula as ours (subsection 3.2.1) is obtained for special values of α lin , α lout , α rin , α rout . How-ever, the approach is different from ours. We solved the time evolution of the current andshowed that the current converges to a stable value independent of initial states. On the otherhand, in [8] ˇZnidariˇc tried to obtain a nonequilibrium stationary state directly as a state ρ which satisfies L ( ρ ) = 0. Since he obtained a stationary state based on an ansatz, it is notobvious if this state is the unique stationary state and the system converges to it (and if ’thestationary state’ he obtained satisfies the condition of state, ρ ≥ (cid:90) µ L µ R dE (cid:107) T N ( E ) (cid:107) [5, 6], where µ L , µ R ( µ L > µ R ) are chemical potentials of the reservoirs. The differencebetween our model and this model is only the region of integral, one is R and the other is[ µ R , µ L ]. But by Theorem 3.4, if [ µ R , µ L ] is sufficiently large, this difference does not matterand both model give the same prediction for the asymptotic behavior. References [1] Herbert Spohn and Joel L Lebowitz. Irreversible thermodynamics for Quantum Sys-tems Weakly Coupled to Thermal Reservoirs.
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