We present studies of the atomic limit of the extended Hubbard model with
pair hopping for arbitrary electron density and arbitrary chemical potential.
The Hamiltonian consists of (i) the effective on-site interaction $U$ and (ii)
the intersite charge exchange term $I$, determining the hopping of electron
pairs between nearest-neighbour sites. In the analysis of the phase diagrams
and thermodynamic properties of this model we treat the intersite interactions
within the mean-field approximation. In this report we focus on metastable
phases and determine their ranges of occurrence. Our investigations in the
absence of the external magnetic field show that the system analysed exhibits
tricritical behaviour. Two metastable phases (superconducting and nonordered)
can exist inside the regions of the phase separated state stability and a
first-order transition occurs between these metastable phases.Comment: 5 pages, 3 figures; pdf-ReVTeX; submitted to: Journal of
  Superconductivity and Novel Magnetis

Kapcia, Konrad

Journal of Superconductivity and Novel Magnetism

English

arXiv

Konrad Kapcia

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Metastability and Phase Separation in a Simple Model of a Superconductor with Extremely Short Coherence Length

This is an author-created, un-copyedited version of an article accepted for publication in Journal of Superconductivity and Novel Magnetism. The Version of Record is available online at http://dx.doi.org/10.1007/s10948-013-2409-8We present studies of the atomic limit of the extended Hubbard model with pair hopping for arbitrary electron density and arbitrary chemical potential. The Hamiltonian consists of (i) the effective on-site interaction U and (ii) the intersite charge exchange term I, determining the hopping of electron pairs between nearest-neighbour sites. In the analysis of the phase diagrams and thermodynamic properties of this model we treat the intersite interactions within the mean-field approximation. In this report we focus on metastable phases and determine their ranges of occurrence. Our investigations in the absence of the external magnetic field show that the system analysed exhibits tricritical behaviour. Two metastable phases (superconducting and nonordered) can exist inside the regions of the phase separated state stability and a first-order transition occurs between these metastable phases.National Science Center (NCN) as a research project in years 2011-2013, under grant No. DEC-2011/01/N/ST3/00413; 
National Science Center (NCN) as a doctoral scholarship in years 2013-2014  No. DEC- 2013/08/T/ST3/00012; 
European Commission and Ministry of Science and Higher Education (Poland) - partial financial  support from European Social Fund – Operational Programme "Human Capital" – POKL.04.01.01-00-133/09-00 – "Proinnowacyjne kształcenie, kompetentna kadra, absolwenci przyszłości";
The Fundation of Adam Mickiewicz University in Pozna

Name not available

Metastability and phase separation in a simple model of a superconductor with extremely short coherence length

AMUR Adam Mickiewicz University Repository

Repozytorium Uniwersytetu im. Adama Mickiewicza (AMUR)

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J Supercond Nov Magn (2014) 27:913–917DOI 10.1007/s10948-013-2409-8O R I G I NA L PA P E RMetastability and Phase Separation in a Simple Modelof a Superconductor with Extremely Short Coherence LengthKonrad KapciaReceived: 19 September 2013 / Accepted: 28 September 2013 / Published online: 14 November 2013© The Author(s) 2013. This article is published with open access at Springerlink.comAbstract We present studies of the atomic limit of the ex-tended Hubbard model with pair hopping for arbitrary elec-tron density and arbitrary chemical potential. The Hamilto-nian consists of (i) the effective on-site interaction U and(ii) the intersite charge exchange term I , determining thehopping of electron pairs between nearest-neighbour sites.In the analysis of the phase diagrams and thermodynamicproperties of this model we treat the intersite interactionswithin the mean-field approximation. In this report we fo-cus on metastable phases and determine their ranges of oc-currence. Our investigations in the absence of the externalmagnetic field show that the system analysed exhibits tri-critical behaviour. Two metastable phases (superconductingand nonordered) can exist inside the regions of the phaseseparated state stability and a first-order transition occursbetween these metastable phases.Keywords Extended Hubbard model · Phase separation ·Superconductivity · Metastability · Pair hopping · Phasediagrams1 IntroductionThe superconductivity (SS) with very short coherence lengthand the phase separation (PS) phenomenon involving SSstates are very current topics (for a review see [1–5] and ref-erences therein). It is worthwhile to mention that metastableK. Kapcia (B)Electron States of Solids Division, Faculty of Physics,Adam Mickiewicz University in Poznan´, Umultowska 85,61-614 Poznan´, Polande-mail: konrad.kapcia@amu.edu.pland unstable states have been found in many physical sys-tems experimentally and theoretically.In our work we will study a model which directly pertainsto that problem. The effective Hamiltonian considered hasthe following form:Hˆ = U∑inˆi↑nˆi↓ − 2I∑〈i,j〉ρˆ+i ρˆ−j− μ∑inˆi − B∑isˆzi , (1)where nˆi = ∑σ nˆiσ , nˆiσ = cˆ+iσ cˆiσ , ρˆ+i = (ρˆ−i )† = cˆ+i↑cˆ+i↓.B = gμBHz is external magnetic field andsˆzi = (1/2)(nˆi↑ − nˆi↓) is z-component of the total spin ati site.∑〈i,j〉 indicates the sum over nearest-neighbour sitesi and j independently; cˆ+iσ (cˆiσ ) denotes the creation (an-nihilation) operator of an electron with spin σ = ↑,↓ at thesite i, which satisfies canonical anticommutation relations:{cˆiσ , cˆ+jσ ′ } = δij δσσ ′ , {cˆiσ , cˆjσ ′ } = {cˆ+iσ , cˆ+jσ ′ } = 0, where δijis the Kronecker delta; μ is the chemical potential, con-nected with the concentration of electrons by the formulan = (1/N)∑i 〈nˆi〉, with 0 ≤ n ≤ 2, and N is the total num-ber of lattice sites. I0 = zI , where z is a number of thenearest-neighbour sites and 〈Aˆ〉 indicates the average valueof the operator Aˆ in the grand canonical ensemble.Model (1) exhibits (in the absence of the field con-jugated with the superconducting (SS) order parameterΔ = 1N∑i〈ρˆ−〉) a symmetry between I > 0 (s-pairing) andI < 0 (η-pairing, ηS, ΔηS = 1N∑i exp (iQ · Ri )〈ρˆ−i 〉, Q be-ing half of the smallest reciprocal lattice vector) cases. Thus,we restrict ourselves to the I > 0 case only. In the presenceof finite single electron hopping tij = 0 the symmetry is bro-ken in the general case [6–11].Model (1) has been intensively analysed for B = 0[4, 12–16] as well as for B = 0 [5, 15] (in particular, in the914 J Supercond Nov Magn (2014) 27:913–917context of the phase separation [4, 5]). In the analysis wehave adopted a variational approach (VA), which treats theon-site interaction term (U ) exactly and the intersite inter-action (I ) within the mean-field approximation (MFA). Oneobtains two equations for n and Δ, which are solved self-consistently. Explicit forms of equations for the energy andother thermodynamical properties are derived in Refs. [4, 5,15]. Condition Δ = 0 is in the superconducting (SS) phase,whereas in the nonordered (NO) phase Δ = 0. For fixed n,the model can exhibit also the phase separation (PS) whichis a state with two coexisting domains (SS and NO) withdifferent electron concentrations, n− and n+. The free en-ergy of the PS state can be derived in standard way, usingMaxwell’s construction (e.g. [4, 5, 17–19]). It is importantto find all homogeneous solutions at which grand canoni-cal potential ω (free energy f ) has the local minimum withrespect to Δ if system is considered for fixed μ (or n).We say that the solution (of the set of two self-consistentequations for n and Δ) corresponds to a metastable phase ifit gives a (local) minimum of ω (or f ) with respect to Δ andthe stability condition ∂μ/∂n > 0 (system with fixed n) isfulfilled. Otherwise, we say that the phase is unstable. A sta-ble (homogeneous) phase is a metastable phase with thelowest free energy (among all metastable phases and phaseseparated states).In the paper we have used the following convention.A second- (first-)order transition is a transition between ho-mogeneous phases with a (dis-)continuous change of theorder parameter at the transition temperature. A transitionbetween homogeneous phase and PS state is symbolicallynamed as a “third-order” transition [4, 5]. At this transitiona size of one domain in the PS state decreases continuouslyto zero at the transition temperature. We have also distin-guish a first-order transition between metastable phases.The phase diagrams obtained are symmetric with respectto half-filling because of the particle–hole symmetry of theHamiltonian (1) [1, 4, 5], so the diagrams will be presentedonly in the range μ¯ = μ − U/2 ≤ 0 and 0 ≤ n ≤ 1.In present report we will focus on the possibility ofthe metastable phases occurrence on the phase diagrams ofmodel (1) in the absence of magnetic field (B = 0). The ef-fects of B = 0 are rather similar to those of U > 0 [4, 5, 15]and we leave deeper analysis of the B = 0 case to futurepublications.2 Numerical Results and Discussion (B = 0)The overall behaviour of the system has been shown in [4, 5,15]. The model considered exhibits interesting multicriticalbehaviour including tricritical points.In the range 2 < U/I0 < +∞, only the NO phase is sta-ble at any T ≥ 0. For on-site attraction U/I0 < 0 (“localFig. 1 kBT /I0 vs. U/I0 phase diagram for n = 1 (I0 = zI ). Dottedand solid lines denote first- and second-order transitions between sta-ble phases. Dashed-dotted lines denote the boundaries of metastablephase occurrence (names of metastable phases in brackets). T denotestricritical pointpair” limit), only the second-order SS–NO transitions be-tween homogeneous phases occur with increasing tempera-ture. The transition temperature is maximal for U → −∞,μ¯ = 0 (n = 1) and decreases monotonically with increasingU/I0 and |μ¯|/I0 = |n − 1|.The most interesting is the range 0 < U/I0 < 2. In thisrange there is smooth crossover into the “pair breaking”limit and the SS–NO transition can also be of a first or-der (for fixed μ¯) and the system exhibits phase separation(for fixed n). The metastable phases exist in several defi-nite ranges of model parameters as it will be discussed be-low.One should stress that metastable phases can occuronly at T > 0. At T = 0 one phase (state) can be stableonly. For T = 0 the discontinuous SS–NO transition oc-curs at U/I0 = (μ¯/I0)2 + 1 (for fixed |μ¯|/I0 < 1) whereasthe continuous SS–NO transition occurs at |μ¯|/I0 = 1 andU/I0 < 2. The PS state stability region is determined byconditions U/I0 ≤ 2 and |n − 1|2 ≤ U/I0 − 1 (n = 1). Atn = 1 (μ¯ = 0) the discontinuous SS–NO transition occurfor U/I0 = 1. The extension (to the ground state) of theend of the first-order transition line between metastablephases (SS and NO) is located at U/I0 = 1 + |1 − n| (forfixed n). The boundaries for the regions of the metasta-bility of homogeneous phases at T > 0 near the groundstate are: for the NO phase, U/I0 = 2|μ¯|/I0 and |μ¯|/I0 < 1(U/I0 = 2|n − 1|, any n); for the SS phase, U/I0 = 2 and|μ¯|/I0 < 1 (any n). Note that for both homogeneous phasesthe condition ∂μ/∂n ≥ 0 is fulfilled at T = 0 (in particularin the ranges of the PS state occurrence)—cf. Sect. 3 of [4].Let us point out that for T = 0 the discontinuous transitionbetween two NO phases with |n − 1| = 1 and n = 1 (Mottstate) occurs at U/I0 = 2|μ¯|/I0 and |μ¯|/I0 > 1, but it doesnot exist for any T > 0.J Supercond Nov Magn (2014) 27:913–917 915Fig. 2 kBT /I0 vs. μ¯/I0 phase diagrams (upper row) and correspond-ing kBT /I0 vs. n diagrams (lower row) for U/I0 = 0.4, 0.9, 1.25 (aslabelled). Dotted, solid and dashed lines indicate first-order, second-order and “third-order” boundaries, respectively. Dashed-dotted linesindicate the boundaries of metastable phase occurrence (names ofmetastable phases in brackets). T denotes tricritical point2.1 The Half-Filling (μ¯ = 0, n = 1)In Fig. 1 we present the phase diagram involving metastablephases for half-filling (μ¯ = 0, n = 1). One can distinguishfour ranges of on-site repulsion, in which a different be-haviour can occur: (i) 0 < U/I0 < 23 ln 2, the second-orderSS–NO transition is present and at low temperatures theNO phase is metastable; (ii) 23 ln 2 < U/I0 < 0.557, the first-order SS–NO transition occurs (it takes place in the wholerange 23 ln 2 < U/I0 < 1). Above this transition tempera-ture the SS phase is metastable, whereas the NO phaseis metastable (close) below the transition temperature andat low temperatures there is another region where the NOphase is metastable; (iii) 0.557 < U/I0 < 1, there is one re-gion of metastability of the NO phase, which extends fromT = 0; (iv) 1 < U/I0 < 2, there is no transitions with in-creasing temperature and only the NO phase is stable. Atsufficiently low temperatures the SS phase is metastable.Note that at n = 1 the VA results for model (1) can besimply mapped onto these of the U–W model with W > 0[17, 18, 20–24]. In such a case the SS phase correspondsto the charge-ordered phase on the phase diagram [24]. Theresults from Fig. 1 can be also transformed into the U–Jmodel [25–27] for n = 1 by generalized U ↔ −U Shiba’stransformation [1, 28, 29]. In such a case the SS phase cor-responds to the magnetic phase with simultaneous changeU → −U in the diagram in Fig. 1.2.2 Arbitrary Electron ConcentrationsIn this section we present results for arbitrary concen-tration n (and arbitrary chemical potential μ¯ = μ − U/2).A few particular phase diagrams are shown in Fig. 2. Let usdiscuss them in the order which corresponds to the ranges ofU/I0 mentioned in Sect. 2.1.(i): 0<U/I0 < 23 ln 2. The phase diagrams for U/I0 =0.4are shown in Fig. 2(a, b). The SS–NO transition between(stable) homogeneous phases is a continuous one and itstemperature decreases monotonically with increasing U/I0and |μ¯|/I0 = |n − 1|. Moreover, at sufficiently low temper-atures, there is a region (extending from half-filling) of theNO phase metastability.(ii)/(iii): 23 ln 2 < U/I0 < 1. With the increasing of U/I0in the vicinity of n = 1 (μ¯ = 0) the SS–NO transitionchanges its order from second order to the first order andthe tricritical point T appears on the phase diagram (cf.Fig. 2(c, d) for U/I0 = 0.9 and Fig. 3). It is quite obviousthat in the neighbourhood of the first-order SS–NO transi-tion (for fixed μ¯) the regions of the metastable phases oc-currence are present (above the transition temperature theSS phase is metastable, whereas below the transition tem-perature the NO phase is metastable). The first-order SS–NO transition line (on the diagram for fixed μ¯) splits intotwo “third-order” lines (on the diagram for fixed n) and thePS state is stable at T > 0 in definite range of parameters(between the “third-order” lines, for n = 1). In the regionof the PS state occurrence (in which the PS state has the916 J Supercond Nov Magn (2014) 27:913–917Fig. 3 kBT /I0 vs. μ¯/I0 phase diagrams for U/I0 = 0.55, 0.56 (as la-belled). Above the first-order SS–NO boundary a narrow region of theSS phase metastability is present (not indicated explicitly). Denotationsas in Fig. 1lowest energy fPS) the first-order transition between twometastable (homogeneous) phases (SS, NO) exists at T > 0.Above this line the SS phase has the highest energy (i.e.fSS > fNO > fPS), whereas below the line the energy ofthe NO phase is higher than the energy of the SS phase(fNO > fSS > fPS). The line of SS–NO first-order transitionbetween metastable phases ends at n = 1 and T > 0. Onemetastable phase (SS or NO) can also exist in the regions ofhomogeneous phases (NO or SS, respectively) stability forfixed n (where the PS state does not exist), cf. Fig. 2(d).The only difference between cases (ii) and (iii) is thatfor U/I0 < 0.557 the separated region of the NO phasemetastability exists also at sufficiently low temperatures. ForU/I0 ≈ 0.557 that region connects with the NO phase re-gion of metastability at higher temperatures (at half-filling,cf. Figs. 1 and 3).(iv) 1 < U/I0 < 2. The exemplary phase diagrams forU/I0 = 1.25 are shown in Fig. 2(e, f). The line of SS–NOfirst-order transition between stable phases (for fixed μ¯) andmetastable phases (for fixed n) ends at T = 0 and μ¯ < 0(n < 1). The region of the PS state stability extends fromthe ground state. The rest of the discussion is similar to thecase (ii)/(iii).The thermodynamic properties of the model have beenanalysed in [4, 5], therefore, we refer the reader to these pub-lications. In particular, the behaviour of thermodynamic pa-rameters in the PS state (as well as in the metastable phases)has been widely discussed in Sect. 5 of [4].3 Concluding RemarksThe results obtained are important for physics of phasetransitions as they involve the investigation of metastablephases. They show that the SS phase metastable boundaryis not dependent on n and μ¯ for |μ¯| (|1 − n|) smaller thanthose of T -point and that the (meta-)stable solutions for theSS phase can exist only for temperatures lower than those ofT -point. The SS solution can be stable or metastable and ex-ists only in regions indicated on phase diagrams. On the con-trary, the NO phase solutions exist at any model parametersand temperature. Outside the regions where the NO phaseis (meta-)stable, it is unstable. The first-order boundariesfound in [15] correspond to transitions between metastablephases.Note that the behaviour of metastable phases in model (1),where two metastable phases (SS, NO) can exist in theranges of the PS state stability, is different than that in modelU–W1–W2 [17, 18, 23, 24] (with W1 > 0 and W2 < 0),where the metastable phases cannot exist in the PS occur-rence regions at sufficiently low temperatures (at T = 0 forW2 < 0 ∂μ/∂n < 0 for fixed n in all homogeneous phases)[17, 18, 24].The on-site U term is the main factor determining thepair binding energy and the on-site density-density fluctua-tions in the model [4, 7, 15, 30]. Due to rigorous treatmentof this term within VA our major conclusions of the paperconcerning the behaviour of the model are reliable for arbi-trary U . Moreover, the MFA treatment of the I term is exactin the limit of infinite dimensions and for Iij of infinite range(Iij = 1N I for any i, j , I > 0) [4, 13, 15] (e.g. effective long-range Iij interaction derived from the coupling between thewide band electrons and local pairs [1]).The interesting problem is the competition and inter-play between superconductivity and charge orderings (gen-erated by density-density interaction) [17, 18, 20–24] ormagnetism [25–27, 31]. Some preliminary results of suchinvestigations have been presented in [4, 32–36].Acknowledgements The author is indebted to Professor StanisławRobaszkiewicz for very fruitful discussions during this work and care-ful reading of the manuscript. The work has been financed by Na-tional Science Centre (NCN) as a research project in the years 2011–2013, under Grant No. DEC-2011/01/N/ST3/00413 and as a doctoralscholarship in the years 2013–2014 No. DEC-2013/08/T/ST3/00012.We thank the European Commission and the Ministry of Science andHigher Education (Poland) for the partial financial support from theEuropean Social Fund—Operational Programme “Human Capital”—POKL.04.01.01-00-133/09-00—“Proinnowacyjne kształcenie, kompe-tentna kadra, absolwenci przyszłos´ci” as well as the Foundation ofAdam Mickiewicz University in Poznan´ for the support from its schol-arship programme.Open Access This article is distributed under the terms of the Cre-ative Commons Attribution License which permits any use, distribu-tion, and reproduction in any medium, provided the original author(s)and the source are credited.J Supercond Nov Magn (2014) 27:913–917 917References1. Micnas, R., Ranninger, J., Robaszkiewicz, S.: Rev. Mod. Phys. 62,113 (1990)2. 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Metastability and phase separation in a simple model of a superconductor with extremely short coherence length

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