Strong short-range interactions in one-dimensional proton conductor

The behaviour of one-dimensional proton conductor is investigated on the basis of orientational-tunnelling model. The previously proposed technique which permits the proton interactions to be taken into account in the zeroorder Hamiltonian, and is based on Green’s function’s expansion in terms of irreducible Green’s function parts, is generalized. The Larkin equation for the one-particle proton Green’s function is solved, while irreducible Larkin part is calculated within the framework of the first order approximation. The proton energy spectrum and chemical potential behaviour are investigated, and the results are compared with the ones previously obtained in the Hubbard-1 type approximation.


Introduction
The systems with proton conductivity (superionic conductors with hydrogen bonds, quasi-one-dimensional proton conductors and others) have lately become the objects of great interest in experimental and theoretical investigations.The conductivity in these systems arises due to the proton transport, thus the problem of studying its microscopic mechanisms is very important.However, the behaviour of these systems is also affected by other factors, including the interactions of protons with phonons, proton-proton interactions, etc.
There were many attempts to describe the proton conductors using the fermion lattice models.One of the first utilizations of the lattice model for the description of such systems [1,2] made it possible to obtain the superionic phase transition in Rb 3 H(SeO 4 ) 2 when considering the proton potential on hydrogen bond in simple one-minimum approximation.More complete and correct approach, which took into account the system symmetry changes and was in better agreement with the experiment, was made in [3] for a similar crystal (NH 4 ) 3 H(SeO 4 ) 2 .
Some attention was also paid to theoretical aspects of proton-phonon interactions [4][5][6], while the effect of the short-range proton-proton interactions is still not sufficiently studied.However, these interactions are strong and can considerably affect the system's behaviour.Several steps in this direction were made in [7] when calculating thermodynamic functions, but the method which would permit to take into account the short-range proton interactions as well as the proton transfer has not been developed.In this paper we will try to generalize the previously proposed approach [8][9][10], which permits the proton interactions to be taken into account exactly.
Our investigations are based on the orientational-tunnelling model for the onedimensional proton conductor [11].This model takes into account the two-stage proton transfer (according to Grotthuss mechanisms) as well as the short-range proton interactions and treats the protons as fermions.
It is worth pointing out that the protons should be rather described by the Pauli statistics than by the Fermi one.However, theoretical investigations of proton conductivity systems have shown a good agreement with experimental results when treating the protons as fermions.
The orientational-tunnelling model can be simplified to the well-known spinless fermion model, which apart from the nearest neighbours interactions takes into account the transfer of fermions as well.This model has been studied a lot, for example in [12][13][14] using the method of high density expansions.However, those results cannot be used for the case of strong fermion interactions.
In this paper we extend the technique previously proposed by us, which is based on generalized Wick's theorem.We use Green's function method to calculate the proton energy spectrum.The series for the one-particle proton Green's function is built in terms of the irreducible parts according to Larkin.We calculate the irreducible part, which is needed to solve the Larkin equation for Green's function, in the approximation which permits to take into account the simplest proton scattering.To do this, one needs to obtain the two-particle irreducible Green's function parts, which are calculated within the framework of cluster approach.By solving the Larkin equation we obtain the desired one-particle proton Green's function.The proton spectrum and chemical potential behaviour with the change of the proton concentration is investigated.The results are compared with the ones obtained in Hubbard-1 type approximation in [8][9][10].

The model
The investigations of the energy spectrum and thermodynamics of the proton subsystem of the one-dimensional proton conductor are based on the orientationaltunnelling model.For the first time this model was used in [11] (see also [15]) to describe the molecular complexes with hydrogen bonds.Later it was used to describe proton transport and to calculate the conductivity coefficients in superionic crystals of Me 3 H(XO 4 ) 3 (Me = Cs, Rb, NH 4 ; X = S, Se) group [16,17].This model introduces two types of proton transport according to Grotthuss mechanisms: tunnelling on the bond (Ω 0 ) and reorientational transfer (Ω R ).It also takes into account the strong short-range interactions, produced by the differences between the energies of proton configurations near the ionic groups, as well as the energies of D-and L-defects.
We consider the one-dimensional proton conductor, which consists of the ionic groups, connected by hydrogen bonds (figure 1).Two equilibrium positions (i, a) and (i, b) on each bond are taken into account.The Hamiltonian of the system in the second quantization form is as follows: where Here, n iα (α = a, b) is the occupation number in position α; U and V are the energies of formation of D-(two protons on the bond) and L-(broken bond) defects respectively; w, ε, w -configuration energies with one, two or none protons near the ionic group (figure 2).
The model ( 1) is one of the fermionic lattice models.Its spectrum has a band character and its structure is defined by the ratio between the effective transfer pa- rameter Ω eff (with the value of the order of max(Ω 0 , Ω R )) and the effective energy of short-range interaction U eff (with the value of the order of min(U, V, w, w )).If Ω eff U eff the interaction between protons can be taken into account in the framework of the mean field approximation (MFA), that just causes the shift of the proton energy band (such an approach was used in [17] to calculate the dispersion curves E( k) in proton bands, which describe the motion of protons in quasi-one-dimensional layers of hydrogen bonds in (NH 4 ) 3 H(SeO 4 ) 2 ).In the opposite case, (Ω eff U eff ) correlation leads to an additional splitting in the spectrum.These results are wellknown in the Hubbard model and in similar ones [18].Corresponding theoretical investigations cannot be carried out in the MFA, since short-range interactions have to be taken into account in the zero-order Hamiltonian.
Thus, the zero-order Hamiltonian has the following form: It includes the short-range proton interactions, while the expansion is made in terms of the transfer Hamiltonian Ĥt .

Perturbation theory and diagram technique
We will use Green's function method to calculate the proton energy spectrum.Let us consider the proton Green's function which is constructed with the creation and the annihilation operators, defined in Heisenberg representation (0 τ, τ β = 1/kT ).We use the thermodynamic perturbation theory [19] to calculate this function, considering the term H t as a perturbation.In the interaction representation where the mean values are calculated using the statistical Gibbs distribution with Hamiltonian H 0 .In the middle part of the expression (5), numerator, as well as denominator, can be expanded into a series in terms of H t , but some of the terms are cancelled.The remaining terms in the diagrammatic representation are denoted by connected diagrams.Expansion in terms of H t leads to the following series: where we introduced the matrix of the proton transfer the elements of which are and the others are equal to zero.Since the Hamiltonian Ĥ0 includes the terms, which describe the interaction between protons, Wick's theorem in its usual form cannot be used to calculate the mean values of the products of Fermi operators.To avoid additional expansions in terms of interaction constants, we use the method similar to that introduced in [20] and later developed in [21,22] for Hubbard model, employing the irreducible many-particle parts.Let us define the one-particle Green's function as or in shorthand notation − T c 1 c + 2 0 = g 0 12 .Two and three-particle Green's functions are presented in the following form: 12,34 , − T τ c 1 c + 2 c 3 c + 4 c 5 c + 6 c 0 = g 0 12 g 0 34 g 0 56 − g 0 12 g 0 36 g 0 54 − g 0 14 g 0 32 g 0 56 − g 0 14 g 0 52 g 0 36 + g 0 16 g 0 32 g 0 54 − g 0 16 g 0 34 g 0 52 + g 0 12 Γ (2) 34,56 − g 0 14 Γ (2) 32,56 + g 0 16 Γ (2) 32,54 + Γ 12,34,56 , where are irreducible Green's function parts.Many-particle Green's function can be written in a similar manner.Expressions (9,10) are a generalization of Wick's theorem for the case, when Hamiltonian H 0 includes the terms, which describe the interaction.In this case, in addition to the sum of usual terms having the form of products of one-particle Green's function, which appear due to the all possible pairings, the expression for Green's function also includes the irreducible parts (10).
We will use the diagram technique to operate with perturbation theory expansions.To do so we introduce the following notations iα,jβ,mµ,nν,kγ,lδ (τ which makes it possible to present a series for the one-particle Green function in the following form: This diagram representation is the schematic one.We leave out the site and time indices and show only the topologically nonequivalent diagrams, which give the nonzero contribution to Green's function.Furthermore, the coefficients at the diagrams in (11) are omitted and have to be calculated via the combinatoric method.
The wavy lines in the diagrams represent the proton transfer.These lines are joined to Green's function vertices or to the vertices of irreducible parts Γ (n) (2nvertex polygons).The proton transfer always connects the incoming vertex of Green's function or polygon (this vertex corresponds to the fermionic annihilation operator c iα ) with the outgoing vertex (it corresponds to the creation operator c + iα ).Time arguments of the vertices, connected with transfer, are equal and have to be integrated, whereas the corresponding site indices are subject to summation.Expression (11) contains the terms up to the second order of perturbation theory (in terms of transfer) as well as the several terms of the third order.
The obtained series for function G can be formally presented as the infinite sum which corresponds to the Larkin equation where Σ is the full irreducible part according to Larkin.

Contributions to Σ can be presented as the expansion in terms of irreducible Green's function parts Γ
iα,jβ,mµ,nν (τ, τ , τ 1 , τ 1 ) where we introduced the generalized transfer The simplest approximation for Σ (Σ = g 0 , the diagram 14.1) which is similar to the well-known Hubbard-1 approximation for the strongly correlated electron systems [23], was used by us in [8][9][10].In the present work the irreducible Larkin part will be calculated in the first order approximation with respect to transfer, which besides 14.1 takes into account the diagram 14.2 (but enclosed with the simple, not generalized transfer).Such an approximation permits the simplest proton scattering processes to be included in our consideration.
In addition to the one-particle proton Green's function (4) we can also consider the two-particle one which can be expanded into a series, similar to (6) Using generalized Wick's theorem (9,10) and the diagram technique, proposed above, this series can be presented in the following form: Similarly to (11) this series includes only the topologically nonequivalent diagrams.The diagrams of the first and the second order as well as the several ones of the third order are shown.
The series (18) can be rewritten using the lines of generalized transfer ( 15) and the full one-particle Green's function ( 11) It is worth pointing out that the diagrams (19.9, 19.10) are the representatives of the diagram class, which is taken into account in the Generalized Random Phase Approximation (GRPA), first proposed in [24] for (t − J) model.Later this approximation was used in the Hubbard and pseudospin-electron model investigations.For instance, in [25] the authors have shown that GRPA results are in good agreement with the results of dynamical mean-field approximation (which is exact in the limit of infinite dimension) in the case of strong one-site electron interactions.This makes it possible to use GRPA when taking into account the strong (in our case proton) interactions to calculate Green's function ( 16).This will be the aim of our future investigation.
Analyzing the diagram series ( 11) and ( 18), one can see that the common structural elements of all the diagrams are zero-order Green's function and irreducible parts Γ (n) .The zero-order Green's function was calculated by us in [8,9].Thus, we will focus on the evaluation of the most simple irreducible part -Γ (2) (four-vertex polygon), which will enable us to obtain the first-order contribution to the irreducible Larkin part.
According to ( 9) Thus, we have to calculate the function This mean value is not equal to zero only if all site indices are equal or when iα = jβ, mµ = nν or iα = nν, jβ = mµ.At first we will consider the case when all site indices are equal, i.e. i = j = m = n; α In this case we need to consider the auxiliary problem for the cluster, which consists of a single site in the nearest surroundings (figure 3) and calculate the function (21).The corresponding mean value of this kind is calculated on the basis of the zero-order Hamiltonian (without transfer), but this Hamiltonian also includes an interaction with the site neighborhood.
Let us consider the cluster i, a (figure 3a) as an example.Its neighborhood consists of the sites i − 1, b and i, b, so Hamiltonian of this cluster has the following form: Hamiltonian (22) reduces to Using Wick's theorem and definition ( 8) we obtain Let us perform the Fourier transformation to the frequency representation G 0 iα,jβ,mµ,nν (ω n 1 , ω n 2 , ω n 3 , ω n 4 )e iωn 1 τ 1 e −iωn 2 τ 2 e iωn 3 τ 3 e −iωn 4 τ 4 (25) and write (24) in the following form: The function g 0 ia,ia can be calculated, for example, using the method of equations of motion Hamiltonian (22) in this case will be as follows: It differs from (23) and this leads to the change of the poles of the function g 0 ia,ia : Expression ( 26) is valid in this case as well.
Similarly we can consider the other possible occupations of the neighbouring sites: (c) Each of the cases mentioned above will give the contribution to the final Green's function with the corresponding statistical weight C α .Finally, where C α are the following correlation functions: and E α are the energies which are the poles of one-particle Green's function at different occupations of neighbouring sites The evaluation of Γ iα,jβ,mµ,nν (τ 1 , τ 2 , τ 3 , τ 4 ) in the cases when site indices are pairwise by two is based on the procedure, similar to the one described above, which involves the calculation of two-particle Green's function localized on a two-site cluster.Two sites of the corresponding cluster can be both the nearest neighbours and the ones separated with some number of sites.The details of calculation of the fourvertex irreducible part on the two-site cluster of the nearest neighbours are shown in appendix A. Final expression for the four-vertex irreducible part in this case will be as follows: This expression consists of 16 terms (only the first four of them are shown for brevity).Each term includes its own energy poles and statistical weight but has the structure similar to that of the shown ones.The correlation functions which appeared during the calculation of Γ ib;ib;i+1,a;i+1,a are taken in mean filed approximation in (33) (for details see appendix A).

Green's function and energy spectrum
In the first order approximation the irreducible Larkin part of Green's function is equal to including the first order correction for Σ, which takes into account the simplest contribution of the proton scattering.
After applying the Fourier-transformation to the momentum and frequency representation we have to calculate all possible sums and integrals to obtain the expression for irreducible Larkin part.Due to the fact that the transfer line in (34) connects the sites, which are the nearest neighbours, only the contributions with Γ (2) ib;ib;i+1,a;i+1,a and Γ (2) ia;ia;ib;ib should be left among the others.Thus, we can present the irreducible Larkin part in the form of a matrix which elements are given by the following expressions F (ω n ) consists of 16 terms which arise from the respective ones of (33).Here, n + (E α ) and n − (E α ) are Fermi and Bose distributions with the energy E α , respectively, and is the first derivative of Fermi distribution.These functions arise as a result of the following summations in the expression for Σ 1 Here, it is taken into account, that the frequencies ω n , ω n are the Fermi-frequencies and their sum is the Bose-frequency.
From the Larkin equation for Green's function we obtain In general, the poles of this function cannot be found analytically.Therefore, we use an independent subband approximation.We assume that each energy level E α gives an independent contribution to G aa .It is valid when the distance between these levels is larger than the effective value of the transfer parameter responsible for the width of the bands.The distance between the levels is determined by proton correlation energies which, in our problem, are considerably larger than the transfer parameters t ab = Ω 0 , Ω R .In this approximation, Green's function will have the following form: The A α coefficients can be presented as follows: but only in the case when correlation functions ( 31) and (A.16) are calculated in the mean field approximation (A.18).The first correction to the irreducible Larkin part contains A α .Thus, taking A α = 0 in (40) we obtain G aa in Hubbard-1 type approximation, i.e. the same approximation that was used in [9,10].
Using the spectral theorem in (40) we can obtain the equation, which links up the chemical potential with the average proton concentration By solving this equation, one can get the dependence of the chemical potential on the average proton concentration.Such a dependance obtained numerically is given in figure 4 (together with the dependence of the proton band edges on the proton concentration).The following parameter values were used in calculations: [29], Ω 0 = 40 . . .250 cm −1 , Ω R = 50 . . .2500 cm −1 [30].
As one can see in figure 4 and from (40), the energy spectrum of the proton subsystem consists of 16 bands, which are separated into four groups.Every group is connected with the corresponding energy level E α .As was mentioned above taking A α = 0, we restrict ourselves to the Hubbard-1 type approximation.In this case the proton energy spectrum consists of 8 bands.As it was shown in [8][9][10], at low temperatures there exist the regions of proton concentration, where the chemical potential decreases with the increase of concentration (∂µ/∂n < 0).It points out to the possible separation of the system on the areas with different average concentration of protons.This effect was considered as a possible explanation of the so-called precursor effect, observed in some superionic conductors, particularly in CsDSO 4 [31][32][33].
By taking into account the first-order contribution to the irreducible Larkin part (A α = 0) one gets a more complicated spectrum, but the areas of instability 0,00 ,5 .
-1 4 0,00 ,5 (∂µ/∂n < 0) still exist.Furthermore, their number increases (figure 4).At the given values of the energy constants and of the temperature, the negative slope of the chemical potential exists at n ≈ 0.54 . . .0.62, n ≈ 0.69 . . .0.76, n ≈ 1.24 . . .1.31 and n ≈ 1.38 . . .1.46.With the increase of temperature these intervals reduce and disappear.The important point to note here is the fact that these areas exist only at Ω 0 /Ω R 1 or Ω R /Ω 0 1, and the larger is the difference between the transfer parameter values, the wider are the intervals of ∂µ/∂n < 0. We also have to note that the contribution of A α is small at low temperatures and it affects the spectrum only at considerably high temperatures (figure 5).
Thus, the first-order approximation for the irreducible Larkin part confirms the fact of the existence of the proton concentration regions, with the separation instabil-ity (∂µ/∂n < 0), obtained in the Hubbard-1 type approximation in [8][9][10].However, for a more complete analysis one needs to perform more accurate calculations of the correlation functions.

Conclusions
The present paper extends and generalizes the method based on generalized Wick's theorem and expansions in terms of irreducible Green's function parts which permits to take into account the short range proton interactions in an exact way.The corresponding diagram technique is proposed which enables us to sum Green's function series.The cluster technique of the calculation of zero-order Green's functions is extended to the case of the two-particle irreducible Green's function parts, which made it possible to obtain the irreducible Larkin part in the first order four-vertex approximation.The proton energy spectrum in the corresponding approximation is calculated.The investigations of chemical potential behaviour have confirmed the fact of the existence of the regions with ∂µ/∂n < 0, previously obtained in [8][9][10].In this case the proton subsystem is unstable with respect to the phase separation.

Acknowledgement
This work was partially supported by the Fundamental Research Fund of the Ministry of Ukraine for Science and Education (Project No. 02.07/266).

A. Evaluation of four-vertex irreducible part on a two-site cluster
Here we will consider the evaluation of four-vertex irreducible part Γ (2) iα,jβ,mµ,nν (τ, τ , τ 1 , τ 1 ) in the case when the site indices are pairwise equal and correspond to the nearest neighbouring sites, separated by ionic group (i = j = m+1 = n+1, α = β = b, µ = ν = a) or placed on the same bond (i = j = m = n, α = β = a, µ = ν = b).Despite the fact that the general idea of calculations remains the same as described in section 4, this case is more complicated since the short range proton interactions act inside the cluster.
We will take the cluster shown in figure 6a as an example.Calculations for the cluster 6b can be done in the same manner and will give the same result.
The neighbourhood of the chosen cluster consists of two sites (i, a and i+1, b).An important problem is that we cannot use usual Wick's theorem for Fermi operators to calculate Green's function (21), because the cluster Hamiltonian (which is used for averaging) includes interaction.Therefore, we will pass from the Fermi creation and annihilation operators to the Hubbard ones.The quantum state basis of the given cluster consists of four states |n ib , n i+1,a : The Hubbard operators defined on this basis, are connected with the Fermi operators by the following relations Now, we have to consider all possible cases of occupation of the cluster neighbourhood.
The Hamiltonian will be as follows: or, in the X-operator representation where where we leave only the terms, which give non-zero contributions.
For every term in (A.6) we use Wick's theorem for Hubbard operators [26].We use the priority rule: X 12 > X 34 > X 24 > X 23 > X 14 at the pairing of operators.
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R 0 Figure 1 .
Figure 1.Proton conductor model.The large circles correspond to the ionic groups, the small ones correspond to the possible proton positions.

Figure 3 .
Figure 3.The simplest (one-site) cluster which consists of the given site (in figure (a) -(i,a), in (b) -(i,b)) and its neighbourhood.

Figure 4 .
Figure 4. Dependence of the proton bands' edges and the chemical potential on the proton concentration.T = 10 K, Ω 0 = 250 cm −1 , Ω R = 2000 cm −1 .Dotted lines limit the areas, where ∂µ/∂n < 0; (a) is the whole spectrum, (b) is the lowest group of the bands.

Figure 6 .
Figure 6.The cluster which consists of two neighbouring sites on different bonds (a), or on the same bond (b).