A translation invariant bipolaron in the Holstein model and superconductivity
aa r X i v : . [ c ond - m a t . o t h e r] F e b Noname manuscript No. (will be inserted by the editor)
A translation invariant bipolaron in the Holsteinmodel and superconductivity.
Victor LakhnoAbstract
Large-radius translation invariant (TI) bipolarons are considered in aone-dimensional Holstein molecular chain. Criteria of their stability are obtained.The energy of a translation invariant bipolaron is shown to be lower than that of abipolaron with broken symmetry. The results obtained are applied to the problemof superconductivity in 1D-systems. It is shown that TI-bipolaron mechanism ofBose-Einstein condensation can support superconductivity even for infinite chain.
Keywords
Delocalized · broken symmetry · strong coupling · canonicaltransformation · Hubbard Hamiltonian · Bose condensate
The problem of possible existence of superconductivity in low-dimensional molec-ular systems has long been of interest to researchers [1]-[6]. Presently, it is believedthat this phenomenon may occur via a bipolaron mechanism. In three-dimensionalsystems a bipolaron gas is thought to form a Bose condensate possessing supercon-ducting properties. It is well known that in one-and two-dimensional systems theconditions for bipolarons formation are more favorable than in three-dimensionalones. The main problem in this regard is the fact that in one- and two-dimensionalsystems Bose-condensation is impossible [7].In papers [8]-[12] a concept of translation invariant polarons and bipolaronswas introduced. Under certain conditions these quasiparticles can possess super-conducting properties even if they do not form a Bose condensate. Papers [8]-[12]dealt with three-dimensional translation-invariant polarons and bipolarons. In thecontext of the aforesaid it would be interesting to consider the conditions underwhich translation invariant bipolarons arise in low-dimensional systems. Here theresults of [8]-[12] are applied to the quasione- dimensional case corresponding tothe Holstein model of a large-radius polaron.
Victor LakhnoInstitute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics,Russian Academy of Sciences, Pushchino, Moscow Region, 142290, Russia.E-mail: [email protected] Victor Lakhno
In recent years increased interest in physics of 1D polarons and 1D bipolaronshas been considerably provoked by the development of a lot of new materials,such as metal-oxyde ceramics with layered ( La ( Sr, Br ) CuO and ( Bi, T l ) ( Sr, Ba ) CaCuO ) or layered-chain ( Y Ba Cu O ) structure, demonstratinghigh-temperature superconductivity [13]-[16], chain organic (polyacetylene) andinorganic (( SN ) x ) polymers, quasi-one-dimensional conducting compounds wherecharge transfer takes place (TTF TCNQ), etc. [1]-[19]. Much the same as 1Dsystems can be materials with huge anisotropy where polarons or bipolaronscan emerge [17]-[19]. Development of DNA-based nanobioelectronics [20], [21]is also closely related with calculation of polaron and bipolaron properties inone-dimensional molecular chains [22]-[25]. Despite great theoretical efforts, manyproblems of polaron physics have not been solved yet.One of the central problems of polaron physics is that of spontaneous breakingof symmetry of the ”electron + lattice” system. In most papers on polaron physics,following initial Landau hypothesis [26] valid for classical lattice, (see books andreviews [27]-[34]) it is thought that at rather a large coupling an electron deforms alattice so heavily that it becomes self-trapped in the deformed region. In this casethe initial symmetry of the Hamiltonian is broken: an electron passes on from thedelocalized state having the Hamiltonian symmetry to the localized self-trappedstate with broken symmetry. This problem is still more actual for bipolarons sincea bipolaron state can arise only in the case of large values of the coupling constant.As showed in Ref. [35] for 1D Holstein polaron in a continuum limit for all thevalues of the coupling constant, the minimum of its energy in quantum lattice isreached in the class of delocalized wave functions. So in Ref. [35] it is shown that inthe case of a strong-coupling polaron, symmetry is not broken and a self-trappedstate is not formed.In this paper the results of paper [35] are generalized to the case of 1D bipo-laron.In § § § § translation invariant bipolaron in the Holstein model and superconductivity. 3 a quasicontinuous spectrum. The concept of an ideal gas of TI-bipolarons is sub-stantiated.With the use of the spectrum obtained, in § § § According to Ref. [35]–[37] Holstein Hamiltonian in a one-dimensional chain in acontinuum limit has the form: H = − m ∆ x − m ∆ x + X k h V k (cid:16) e ikx + e ikx (cid:17) a k + h.c. i + (1) X k ~ ω k a + k a k + U ( x − x ) , V k = g √ N , ω k = ω , where a + k , a k are operators of the phonon field, m is the electron effective mass, ω k is the frequency of optical phonons, g is the constant of electron-phonon interac-tion, N is the number of atoms in the chain, U ( x − x ) is the Coulomb repulsionbetween electrons depending on the difference of electron coordinates which willbe taken to be: U ( x − x ) = Γ δ ( x − x ) (2)where Γ is a certain constant, δ ( x ) is a delta function. In the case of brokentranslation invariance the bipolaron state is described by localized wave functions Ψ = Ψ ( x , x ) and in the strong coupling limit the functional of the total energy¯ H = h Ψ | H | Ψ i is written as [38]:¯ H = − m X i =1 , h Ψ | ∆ x i | Ψ i − X V k ~ ω D Ψ (cid:12)(cid:12)(cid:12) e ikx + e ikx (cid:12)(cid:12)(cid:12) Ψ E + (3) h Ψ | U ( x − x ) | Ψ i The exact solution of problem (3) is a complicated computational problem [39]-[40]. For the purposes of this section, however, it will suffice to illustrate theproperties of the ground state of a bipolaron with broken symmetry with the useof a direct variational method. Towards this end let us choose the probe function Ψ = Ψ ( x , x ) in the form Ψ ( x , x ) = ϕ ( x ) ϕ ( x ). Notice that this choice of theprobe function corresponds to the exact solution of problem (3) for U = 0, i.e. inthe absence of the Coulomb interaction between electrons.As a result, from (3) we get the functional of the ground state energy: Victor Lakhno ¯ H = 1 m Z |∇ x ϕ ( x ) | dx − (cid:18) g a ~ ω − Γ (cid:19) Z | ϕ ( x ) | dx, (4)where a is the lattice constant. Variation of (4) with respect to ϕ ( x ), the normal-ization requirement being met, leads to Schroedinger equation: ~ m ∆ x ϕ + 2 (cid:18) g a ~ ω − Γ (cid:19) | ϕ | ϕ + W ϕ = 0 , (5)whose solution has the form: ϕ ( x ) = ± (cid:16) √ rch x − x r (cid:17) − , r = 2 ~ m g a ) / ( ~ ω ) − Γ ) , (6) W = − (cid:18) g a ~ ω − Γ (cid:19) m ~ , E bp = − (cid:18) g a ~ ω − Γ (cid:19) m ~ , where x is an arbitrary constant, E bp = min ¯ H is the energy of the bipolaronground state. Notice, that the polaron state energy E p in the case under consid-eration is [35]: E P = − (cid:18) g a ~ ω (cid:19) m ~ . (7)Let us introduce the notation: γ = Γ ~ ω /a g . (8)From (6) it follows that for: γ > < γ < γ < x , the bipo-laron state discussed has an infinite degeneracy and can move along the chain. Anyarbitrarily small violation of the chain leads to elimination of the degeneration andlocalization of the bipolaron state on defects with attracting potential. A quali-tatively different situation arises in the case of a translation invariant bipolaronconsidered below. translation invariant bipolaron in the Holstein model and superconductivity. 5 To construct a translation invariant bipolaron theory in the Holstein model, inHamiltonian (1) we pass on to coordinates of the center of mass. In this systemHamiltonian (1) takes the form: H = − ~ M ∆ R − ~ µ ∆ r + X k V k cos kr (cid:16) e ikR a k + h.c. (cid:17) + (12) X k ~ ω k a + k a k + U ( r ) ,R = ( x + x ) / , r = x − x , M = 2 m, µ = m/ . In what follows we will use units, putting ~ = 1, ω = 1, M = 1 (accordingly µ = 1 / R in Hamiltonian (2) can be eliminatedvia Heisenberg canonical transformation [41]:ˆ S = exp ( − i X k ka + k a k R ) . (13)As a result, the transformed Hamiltonian: ˜ H = ˆ S − H ˆ S is written as:˜ H = − ∆ r + X k V k cos kr a + k + a k ) + X k a + k a k + U ( r ) + (14)12 X k a + k a k ! From (14) it follows that the exact solution of the bipolaron problem is deter-mined by the wave function Ψ ( r ) which depends only on the relative coordinates r and, therefore, is automatically translation invariant. It corresponds to the statedelocalized over the coordinates of the center of mass of two electrons.Averaging Hamiltonian (14) over Ψ ( r ), we will write the averaged Hamiltonianas: ¯˜ H (15) ¯˜ H = ¯ T + X k ¯ V k ( a + k + a k ) + X k a + k a k + 12 X k a + k a k ! + ¯ U , (15)¯ V k = 2 V k (cid:28) Ψ (cid:12)(cid:12)(cid:12)(cid:12) cos kr (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:29) , ¯ U = h Ψ | U ( r ) | Ψ i , ¯ T = − h Ψ | ∆ r | Ψ i Subjecting Hamiltonian (15) to Lee-Low-Pines transformation [42]:ˆ S = exp (X k f k ( a k − a + k ) ) , (16) Victor Lakhno we get: ˜˜ H = ˆ S − ¯˜ H ˆ S ˜˜ H = H + H (17)where: H = ¯ T + 2 X k ¯ V k f k + X k f k + 12 X k kf k ! + ¯ U + H (18) H = X k ω k a + k a k + 12 X k,k ′ kk ′ f k f k ′ (cid:16) a k a k ′ + a + k a + k ′ + a + k a k ′ + a + k ′ a k (cid:17) , (19) H = X k ( V k + f k ω k ) (cid:16) a k + a + k (cid:17) + X k,k ′ kk ′ f k ′ (cid:16) a + k a k a k ′ + a + k a + k ′ a k (cid:17) + (20)12 X k,k ′ kk ′ a + k a + k ′ a k a k ′ ,ω k = ω + k k X k ′ k ′ f k ′ . (21)According to Ref. [8], contribution of H into the energy vanishes if the eigenfunction of Hamiltonian H transforming the quadratic form H to the diagonalone, is chosen properly. Diagonalisation of H leads to the total energy of theaddition ∆E : ∆E = 12 X k ( ν k − ω k ) = − πi Z c ds √ s lnD ( s ) , (22)where ν k are phonon frequencies renormalized by the interaction with the electron.The contour of integration c involved in (22) is the same as in Ref. [8], [35]. In theone-dimensional case under consideration: D ( s ) = 1 − π Z ∞−∞ k f k ω k s − ω k dk (23)Repeating calculations carried out in Ref. [8], [35] in the strong coupling limit, weexpress ∆E as: ∆E = 14 π Z ∞−∞ k f k dk Q ) + (24)14 π ZZ ∞−∞ k f k p f p ω p ( ω k ω p + ω k ( ω k + ω p )) + 1( ω k + ω p ) ( ω p − (cid:12)(cid:12) D + ( ω p ) (cid:12)(cid:12) dpdk,D + ( ω p ) = 1 + 1 π Z ∞−∞ f k k ω k dkω k − ω p − iǫ ,Q = 1 π Z ∞−∞ k f k ω k dkω k − E bp is written as: E bp = ∆E + 2 X k ¯ V k f k + X k f k + ¯ T + ¯ U (25) translation invariant bipolaron in the Holstein model and superconductivity. 7 Fig. 1
The dependence of E min (29) on γ . We could have derived an exact equation for determining the bipolaron energyby varying (25) with respect to Ψ and f k . The quantities Ψ and f k obtained assolutions of this equation, being substituted into (25) determine the bipolarontotal energy E bp . Since finding a solution of the equation obtained by variation of E bp is rather a complicated procedure, we will use the variational approach. Tothis end let us choose the probe functions Ψ and f k in the form: Ψ ( r ) = (cid:18) π (cid:19) / √ l e − r /l , (26) f k = − N ge − k / a , (27)where N , l , a are variational parameters. As a result, after minimization of (25)on N , the bipolaron energy will be: E bp = ma ~ g ~ ω min ( x,y ) E ( x, y ; γ ) , (28) E ( x, y ; γ ) ≈ . x + 2 y − x √ π (1 + x y /
16) + r π γy ! (29)The expression for the bipolaron energy is given in dimension units. The resultsof minimization of function E ( x, y ; γ ) with respect to dimensionless parameters x, y are presented in Fig. 1 for various values of the parameter γ . Fig. 1 suggests thatas distinct from a bipolaron with broken symmetry (inequality (9)), a translationinvariant bipolaron exists for all the values of the parameter γ . In the region: γ > .
02 (30)
Victor Lakhno
Fig. 2
The dependence of x min , y min on γ . a translation invariant bipolaron is unstable relative to its decay into both in-dividual polarons with spontaneously broken symmetry, i. e. Holstein polaronswith the energy 2 E p = − (1 / ma g / ~ ω (upper horizontal line in Fig. 1 inenergy units ma g / ~ ω ) and translation invariant polarons with the energy2 E p = − . ma g / ~ ω [35] (lower horizontal line in Fig. 1). For:2 . < γ < .
02 (31)a translation invariant bipolaron becomes stable relative to its decay into individ-ual Holstein polarons, but remains unstable relative to decomposition into indi-vidual translation invariant polarons. For: γ < γ c = 2 .
775 (32)a translation invariant bipolaron becomes stable relative to its decay into twoindividual polarons. Notice that for γ = 0, the energy of a translation invariantbipolaron is equal to: E bp = − . ma g / ~ ω , i.e. lies much lower than theexact value of the energy of a bipolaron with broken symmetry, which, accordingto (6) is equal to E bp = − (4 / ma g / ~ ω . The energy of a translation invariantbipolaron also lies below the variational estimate of the energy of a bipolaron withspontaneously broken symmetry (6) for all the values of γ [39].The dimensionless parameters x, y involved in (29) are related to the variationalparameters a and l (26), (27) as: a = (2 ma g / ~ ω ) x , l = ( ~ ω / ma g ) y . Theparameter l determine the characteristic size of the electron pair, i.e. the correlationlength L ( γ ), whose dependence on γ is given by the expression: L ( γ ) = ~ ma ~ ω g y min ( γ ) . (33)The dependencies of y min and x min on γ are presented in Fig. 2. translation invariant bipolaron in the Holstein model and superconductivity. 9 Fig. 2 suggests that the correlation length L ( γ ) in the region of a bipolaronstability 0 < γ < γ c does not change greatly and for its critical value γ c = 2 . L ( γ ) approximately three times exceeds the value of L (0), i.e. the cor-relation length in the absence of the Coulomb repulsion. This qualitatively differsfrom the case of a bipolaron with broken symmetry for which the correspondingvalue, according to (6), for γ = γ c turns to infinity. According to the results obtained in [8], [43], the spectrum of excited states ofHamiltonian (18), (19) is determined by the expression:˜˜ H = E bp + X k ν k α + k α k (34)where α + k , α k are operators in which quadric form H (19) is diagonal. Op-erators α + k , α k can be considered as operators of birth and annihilation of TI-bipolarons in excited states obeying Bose commutation relations: h α n , α + n ′ i = α n α + n ′ − α + n α n = δ n,n ′ (35)Renormalized frequencies involved in (34), according to [8], [43], are determinedby the equation for s : 1 = 2 X k k f k ω k s − ω k (36)solutions of which give the spectrum of s = (cid:8) ν k (cid:9) solutions.It is convenient to present Hamiltonian (34) in the form:˜˜ H = X n =0 , , E n α + n α n (37) E n = ( E bp , n = 0; ν n = E bp + ω + k n , n = 0 . (38)where k n for a discrete chain of atoms is equal to: k n = ± π ( n − N a , n = 1 , ..., N a / N a is the number of atoms in the chain.Let us prove the validity of (38). The energy spectrum of TI-bipolarons, ac-cording to (36), reads: F ( s ) = 1 (39) Fig. 3
Graphical solution of Eqs. (39), (40) F ( s ) = 2 X n k n f k n ω k n s − ω k n (40)It is convenient to solve equation (39) graphically (Fig.3)Fig.3 suggests that frequencies ν k n occur between the frequencies ω k n and ω k n +1 . Hence, the spectrum of ν k n as well as the spectrum of ω k n is quasi contin-uous in the continuum limit: ν k n − ω k n = 0( N − a ), which proves the validity of(37), (38).Therefore the spectrum of a TI-bipolaron has a gap between the ground stateof E bp and the quasi continuum spectrum, which is equal to ω .Below we will consider the case of low concentration of TI-bipolarons in thechain. In this case they can be adequately considered as Bose-gas, whose propertiesare determined by Hamiltonian (37). Let us consider the rare (the pair correlation length is much smaller then theaverage distance between pairs) one-dimensional ideal Bose-gas of TI-bipolaronswhich is a system of N particles, occurring in a one-dimensional chain of length L . Let us write N for the number of particles in the lower one-particle state, and N for the number of particles in higher states. Then: N = X n =0 , , ,... ¯ m n = X n e ( E n − µ ) /T − translation invariant bipolaron in the Holstein model and superconductivity. 11 φ e ω φ e ω φ e ω φ e ω φ e ω φ e ω e T φ e ω .
82 14 . . . . . . Fig. 4
Solutions of equation (43) C D = 34 .
69 and ˜ ω i = { .
2; 1; 2; 10; 15; 20 } , which corre-spond to ˜ T c i = ˜ T c = 5 . T c = 14 .
1; ˜ T c = 20 .
87; ˜ T c = 53 .
47; ˜ T c = 68 .
33; ˜ T c = 81 . N = N + N ′ , N = 1 e ( E − µ ) /T − , N ′ = X n =0 e ( E n − µ ) /T − N ′ (42) we will replace summation by integration over quasicontinuous spectrum (37), (38) and take µ = E bp . As a result we will get from(41), (42) an expression for the temperature of Bose condensation T c : C D = Φ ˜ ω ( T c ) (43) Φ ˜ ω = ˜ T / c F / (cid:18) ˜ ω ˜ T c (cid:19) , F / ( α ) = Z ∞ dx √ x ( e x + α − ,C D = 2 √ π n ~ M / ω ∗ / , ω ∗ = ω ω , ˜ T c = Tω ∗ , where n = N/L . Fig.4 shows a graphical solution of equation (43) for theparameter values: M e = 2 m = 2 m , where m is the mass of a free electron invacuum, ω ∗ = 5 meV, ( ≈ K ), n = 10 cm − and the values: ˜ ω = 0 .
2; ˜ ω = 1;˜ ω = 2; ˜ ω = 10; ˜ ω = 15; ˜ ω = 20 ( C D = 34 . ω = 0. The equality T c = 0 for ω = 0 corresponds e ω e ω e ω e ω e ω e ω e T − N ′ / N .
82 14 . . . . . Fig. 5
Temperature dependencies of the relative number of supracondensate particles N ′ /N and condensate particles N /N for the values of ˜ ω i , given in Fig. 4. to the known result, that Bose-condensation is impossible in ideal gas in a one-dimensional case.Fig. 4 also suggests that it is just the increase in the concentration of TI-bipolarons which will lead to an increase in the critical temperature, while theincrease in the electron mass m to its decrease.It follows from (41), (42) that: N ′ (˜ ω ) N = ˜ T / C D F / (cid:18) ˜ ω ˜ T (cid:19) (44) N (˜ ω ) N = 1 − N ′ (˜ ω ) N (45)Fig. 5 illustrates temperature dependencies of the supracondensate particles N ′ and the particles in the condensate N for the above-cited values of ˜ ω c parameters.From Fig. 5 it follows that, as we might expect, the number of particles in thecondensate grows as the gap ω i increases.The energy of TI-bipolaron gas E reads: E = X n =0 , , ... ¯ m n E n = E bp N + X n =0 ¯ m n E n (46) translation invariant bipolaron in the Holstein model and superconductivity. 13 With the use of (37), (38) for the specific energy (i.e. energy per one TI-bipolaron) ˜ E ( ˜ T ) = E/N ω ∗ , ˜ E bp = E bp /ω ∗ (46) transforms into:˜ E = ˜ E bp + ∆ ˜ E (47) ∆ ˜ E = ˜ T / C D F / (cid:18) ˜ ω − ˜ µ ˜ T (cid:19) ˜ ω ˜ T + F / (cid:16) ˜ ω − ˜ µ ˜ T (cid:17) F / (cid:16) ˜ ω − ˜ µ ˜ T (cid:17) , (48) F / ( α ) = Z ∞ √ xdxe x + α − µ is determined by the equation: C D = ˜ T / c F / (cid:18) ˜ ω − ˜ µ ( ˜ T )˜ T (cid:19) (50)˜ µ = ( , ˜ T < ˜ T c ;˜ µ ( ˜ T ) , ˜ T > ˜ T c . Relation between ˜ µ and the chemical potential of the system µ is given by theexpression ˜ µ = ( µ − E bp ) /ω ∗ . Formulae (49), (50) also yield expressions for Ω -potential: Ω = − E and entropy S = − ∂Ω/∂T ( F = − E , S = − ∂F/∂T ).Fig. 6 demonstrates temperature dependencies of ∆ ˜ E = ˜ E − ˜ E bp for the above-cited values of ˜ ω i . Salient points of ∆ ˜ E i ( ˜ T ) curves correspond to the values ofcritical temperatures T c i .These dependencies enable us to find the heat capacity of TI-bipolaron gas: C V ( ˜ T ) = d ˜ E/d ˜ T .Fig. 7 shows temperature dependencies of the heat capacity C V ( ˜ T ) for theabove-cited values of ˜ ω i . Table 1 lists the heat capacity jumps for the values ofparameters ˜ ω i : ∆ ∂C V ( ˜ T ) ∂ ˜ T = ∂C V ( ˜ T ) ∂ ˜ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ T = ˜ T c +0 − ∂C V ( ˜ T ) ∂ ˜ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ T = ˜ T c − (51)at the transition points.The dependencies obtained enable us to find the latent heat of the transition q = T S , where S is the entropy of supracondensate particles. At the transitionpoint this value is q = 2 T c C V ( T c − C V = d ˜ E/d ˜ T and ˜ E is determined byformulae (47), (48). The values of the heat of transition q i for the above-citedvalues of ˜ ω i are given in Table 1. e T ∆ e E .
82 14 . . . . . Fig. 6
The temperature dependencies ∆ ˜ E = ˜ E ( ˜ T ) − ˜ E bp for various values of ˜ ω i (see Table 1). Table 1
Dependence of critical temperatures ˜ T c i , heat capacities C V ( ˜ T c i ± ∆ on the values of ˜ ω i . i ω i T c i C V ( ˜ T c i −
0) 0.24 0.37 0.45 0.71 0.79 0.86 C V ( ˜ T c i + 0) 0.17 0.23 0.25 0.32 0.33 0.34 ∂C V ∂ ˜ T ( ˜ T c i −
0) 20 . × − . × − . × − . × − . × − . × − ∂C V ∂ ˜ T ( ˜ T c i + 0) 10 . × − . × − . × − . × − . × − . × − ∆ − . × − − . × − − . × − − . × − − . × − − . × − Earlier we considered the problem of symmetry breakdown for one electron inter-acting with oscillations of a one-dimensional quantum chain [35]. According to Ref.[35], a rigorous quantum-mechanical treatment leads to delocalized translation-invariant electron states, or to a lack of soliton-type solutions, breaking the initialsymmetry of the Hamiltonian.In this paper we have shown that when the chain contains two electrons whichinteract with its oscillations and suffer Coulomb repulsion determined by the inter-action, a stable state can be formed which does not violate translation invarianceand has a lower energy than the localized solution which breaks TI symmetry does. translation invariant bipolaron in the Holstein model and superconductivity. 15 e T c V .
82 14 . . . . . . . . . . . . . . Fig. 7
The temperature dependencies of the heat capacity for various values of ˜ ω i (see Ta-ble 1). Presently most papers describing electron states in discrete molecular chainsare based on Holstein-Hubbard Hamiltonian [36], [44], [45], [46]: H = η X i,j,σ,σ ′ c + iσ c jσ ′ + X i ~ ω (cid:16) a + j a j + 1 / (cid:17) + X j g ˆ n j (cid:16) a + j + a j (cid:17) + (52) X j,σ,σ ′ U ˆ n jσ ˆ n jσ ′ , where ˆ n jσ = c + jσ c jσ ′ , ˆ n j = P σ ˆ n jσ , c + jσ , c jσ - are operators of the birth andannihilation of an electron with spin σ at the j-th site; η - is the matrix elementof the transition between nearest sites ( i, j ).Numerical investigations of Hamiltonian (52) are based on the use of ansatzfor the wave functions of the ground state: | Ψ i = X i,j,σ,σ ′ Ψ ij c + iσ c + jσ ′ | i , (53)where | i - is a vacuum wave function which is a product of electron and latticevacuum functions.Hamiltonian (1) considered in this work is a continuum analog of Hamiltonian(52) if in (52) we put: m = ~ / ηa , Γ = U a . As is shown in Ref. [43], presentationof the wave function as a product of the electron wave function by the lattice Table 2
Coupling energies ∆ for U = 0 for a discrete model ∆ d , for continuum Holsteinmodel ∆ H , and translation invariant bipolaron ∆ TI . ∆ κ ∆ d ∆ H ∆ TI one (Pekar ansatz) does not give an exact solution of Hamiltonian (1). A similarconclusion is valid for Hamiltonian (52). In this context it would be interesting todiscuss the limits of applicability of ansatz (53) in a discrete case using a particularexample.By way of example of a discrete model let us take the results of calculationof bipolaron states in a Poly G/Poly C nucleotide chain given in paper [47]. TheTable 2 lists the values of the coupling energy ∆ = | E bp − E p | in the case of U = 0, η = 0 .
084 eV for a discrete model (52) with using ansatz (53): ∆ = ∆ d ;for a continuum Holstein bipolaron with broken symmetry: ∆ = ∆ H (6)-(7); for acontinuum TI-bipolaron: ∆ = ∆ T I (28). These results suggest that ∆ H virtuallycoincides with ∆ d and becomes less than ∆ d as κ ≤ . ∆ T I for κ ≤ . ∆ d and become lessthan ∆ d as gets larger. In the general case we can say that discreteness violatescontinual translation invariance of the chain only when some threshold value ofthe coupling constant is exceeded. In particular, for the discrete model of a PolyG/Poly C chain [47] with parameters U ≈ κ = 4 g / ~ ω = 0 . γ ≈ .
6, the TI-bipolaron states considered in thepaper are probably unstable, since they do not fall on the stability interval γ < γ c (32). In this case the states should be calculated based on a discrete model. ForDNA, such a calculation, as applied to the possibility of superconductivity in DNAwas carried out in papers [25], [47]. It should also be noted that apart from thecondition γ < γ c , for continuum TI-bipolarons to exist, the condition of continuityshould also be met. According to Ref. [43] it implies that the characteristic phononvectors making the main contribution into the energy of TI-bipolarons shouldsatisfy the inequality ka <
1. From (27) it follows that the main contribution intothe energy is given by the values of k ≤ a . For U = 0, this yields k ≤ g / ~ ωνa .Accordingly, the condition of continuity takes on the form: g / ~ ων ≤ U = 0, this condition is equivalent to the requirement r/a ≥ r is determined by (6). From (6) it also follows that for U = 0 Hol-stein polaron becomes lengthier, since its characteristic size becomes equal to r = r (1 − γ/ − , where r - is the characteristic size for U = 0. For a TI-bipolaron, the same conclusion follows from expression (33) for the correlationlength and Fig. 2. Physically this is explained by the fact that Coulomb repulsionleads to an increase of the characteristic distance between the electrons in thebipolaron state. Earlier this result was also obtained in Ref. [40]. Hence, thoughTI-bipolarons are delocalized, the requirements of continuity for TI-bipolaron andHolstein bipolaron turn out to be similar. translation invariant bipolaron in the Holstein model and superconductivity. 17 The Table 2 lists the values of ∆ for which the continuum model is morepreferable than the ’exact’ discrete one.The results obtained suggest that for parameter values when the continuummodel is valid and conditions of strong coupling are met, TI-bipolarons are ener-getically more advantageous. Therewith the question of the character of a transi-tion from the continuum description to the discrete one remains open. One wouldexpect that such a transition will occur with a sharp increase in the bipolaroneffective mass as a result of which the molecular chain will change from highlyconducting state to low conducting one. The estimate of the value of the coupling constant g c = g/ ~ ω sufficient for theformation of translation-invariant bipolaron states in the region where the criterionof their existence is met 0 < γ < γ c can be obtained by comparing the totalenergy of a strong coupling bipolaron with twice energy of individual weak couplingpolarons. Weak coupling polarons, by their treatment per se (perturbation theory)are translation invariant with the energy [37]: E p = − g q ma / ~ ω In particular, for γ = 0 we get: g c ≈ . ~ /ma ω ) / . Hence, for the over-whelming majority of various systems g c ≤ H shifts by − g L µ B H/
2, where g L is Lande factor, µ B = | e | ~ / mc isa Bohr magneton. Being singlet, bipolarons do not experience such a shift. Hence,the region of a bipolaron stability is determined by the inequality H < H c , where: H c = 1 √ πy min ( γ ) (cid:18) γ c − γγ (cid:19) ma ~ g ~ ω This estimate is valid for the case of non-quantizing magnetic fields.As is known, the main mechanism leading to finite resistance in solid bodies isdissipation of charge carriers on phonons [48]. In the case of translation invariantbipolarons the separation of the system into bipolarons and optical phonons ispointless. For a translation invariant bipolaron in the strong coupling limit, thewave function of the system cannot be divided into electron and phonon parts.The total momentum of a translation invariant bipolaron is a conserving value,the relevant wave function is delocalized over the space and a translation invariantbipolaron occurring in a system consisting only of electrons and phonons, willbe superconducting. Inclusion of acoustical phonons into consideration leads to alimitation on the possible value of the velocity v of a translation invariant polaronor bipolaron at which they have superconducting properties, namely, according tothe laws of energy and momentum conservation, this velocity should be less thanthat of sound s . For v > s , a translation invariant polaron and bipolaron becomedissipative. In a real system containing defects or structural imperfections with attractivepotential, these defects and imperfections will always trap polarons and bipolaronswith spontaneously broken symmetry. On the contrary translation invariant bipo-larons will form a bound state only if the potential well is deep enough. Otherwise,even in an imperfect system, translation invariant bipolarons will be delocalized.Notwithstanding the lack of bound states in the presence of defects, the total mo-mentum of a bipolaron no longer commutates with the Hamiltonian and thereforeis not an integral of the system’s motion. In this case a bipolaron will scatterelastically on a defect as a result of which only its momentum will change. Thisscattering does not lead to an energy loss. In the absence of dissipation the motionof bipolarons will occur without friction and superconductivity in the system willbe retained. In the presence of large defects or imperfections possessing a greattrapping (scattering) potential, the system under discussion cannot be consideredas infinite any longer.
Conclusions
In this paper we demonstrate that TI-bipolaron mechanism of Bose condensationcan support superconductivity even for infinite chain. According to Fig. 6 thecondensation in 1D systems is the phase transition of second kind.The theory resolves the problem of the great value of the bipolaron effectivemass. As a consequence, formal limitations on the value of the critical temperatureof the transition are eliminated too. The theory quantitatively explains such ther-modynamic properties of HTSC-conductors as availability and value of the jumpin the heat capacity lacking in the theory of Bose condensation of an ideal gas.The theory also gives an insight into the occurrence of a great ratio between thewidth of the pseudogap and T c . It accounts for the small value of the correlationlength and explains the availability of a gap and a pseudogap in HTSC materials.Accordingly, isotopic effect automatically follows from expression (43), wherethe phonon frequency ω acts as a gap.Earlier the 3D TI-bipolaron theory was developed by author in [9], [10], [12],[43]. Consideration of 1D case carried out in the paper can be used to explain 3Dhigh-temperature superconductors (3D TI-bipolaron theory of superconductivitywas developed in Ref. [49]) where 1D stripes play a great role. As the consid-eration suggests, artificially created nanostripes with enhanced concentration ofcharge carriers can be used to increase the critical temperature of superconduc-tors. Theoretical description of the nanostripes can also be based on the approachdeveloped. Declarations
AcknowledgementsThe work was supported by projects RFBR N 16-07-00305 and RSF N 16-11-10163. translation invariant bipolaron in the Holstein model and superconductivity. 19