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Opening of a pseudogap in a quasi-two dimensional superconductor due to critical thermal fluctuations

We examine the role of the anisotropy of superconducting critical thermal fluctuations in the opening of a pseudogap in a quasi-two dimensional superconductor such as a cuprate-oxide high-temperature superconductor. When the anisotropy between planes and their perpendicular axis is large enough and its superconducting critical temperature T_c is high enough, the fluctuations are much developed in its critical region so that lifetime widths of quasiparticles are large and the energy dependence of the selfenergy deviates from that of Landau's normal Fermi liquids. A pseudogap opens in such a critical region because quasiparticle spectra around the chemical potential are swept away due to the large lifetime widths. The pseudogap never smoothly evolves into a superconducting gap; it starts to open at a temperature higher than T_c while the superconducting gap starts to open just at T_c. When T_c is rather low but the ratio of varepsilon_G(0)/k_BT_c, with varepsilon_G(0) the superconducting gap at T=0K and k_B the Boltzmann constant, is much larger than a value about 4 according to the mean-field theory, the pseudogap must be closing as temperature T approaches to the low T_c because thermal fluctuations become less developed as T decreases. Critical thermal fluctuations cannot cause the opening of a prominent pseudogap in an almost isotropic three dimensional superconductor, even if its T_c is high.


I. INTRODUCTION
The elucidation of the mechanism of high critical-temperature (high-T c ) superconductivity occurring in cuprate-oxide superconductors is an important and long standing issue since its discovery in 1986. 1 On the other hand, many unconventional normal-state properties are observed: the so called spin-gap behavior or the reduction of the nuclear magnetic relaxation (NMR) rate with decreasing temperatures T , 2 the opening of a pseudogap in quasiparticle spectra, 3,4,5,6,7,8,9,10 and so on. The issue on the mechanism of high-T c superconductivity cannot be settled unless not only high-T c superconductivity but also such unconventional properties are explained within a theoretical framework. It is widely believed that the reduction of the NMR rate above T c is due to the opening of the pseudogap. It is a key issue to clarify the relation between the pseudogap above T c and a superconducting (SC) gap below T c or whether or not the pseudogap smoothly evolves into the SC gap.
One may argue that the opening of a pseudogap must be a precursor effect of a possible low-temperature instability, antiferromagnetism, superconductivity, or an exotic one. Another may argue that a pseudogap must open due to fluctuations corresponding to one or some of antiferromagnetism, superconductivity, and exotic ones.
In actual, many possible mechanisms of pseudogaps have already been proposed along these scenarios. 11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 Since cuprate-oxide superconductors are highly anisotropic quasi-two dimensional ones, it is also a reasonable argument that, even if either of these scenarios is relevant, low dimensionality must play a crucial role. If a second-order phase transition occurred at a non-zero critical temperature T c in one or two dimensions, eventual effects of critical thermal fluctuations or their integrated effects over the wave-number space would diverge at the nonzero T c . This leads to a conclusion that no order is possible at non-zero temperatures in one and two dimensions. 28 It also leads to a speculation that T c of highly anisotropic quasi-low dimensional superconductors must be substantially reduced by SC critical thermal fluctuations and the reduction of T c must be accompanied by some normal-state anomalies. Since pseudogap structures are certainly substantial in SC critical regions of cuprate-oxide superconductors, it must be clarified first of all how a crucial role SC critical fluctuations can play in the reduction of T c and the opening of pseudogaps, even if any other type of fluctuations or any other mechanism plays a role or a major role.
Critical temperatures T c are reduced by SC critical thermal fluctuations. On the other hand, SC gaps at T = 0 K, ε G (0), can never be reduced by them because they vanish at T = 0 K. Therefore, large ratios of ε G (0)/k B T c 8,7,8,9 with k B the Boltzmann constant, are pieces of evidence that T c are actually substantially reduced even if observed T c are high. It is plausible that the opening of a pseudogap must be one of normal-state anomalies accompanying the reduction of T c .
It has been shown in a previous paper 29 When transfer integrals t ij between nearest and next nearest neighbors on a plane, −t and −t ′ , are considered, the dispersion relation of electrons is given by with a the lattice constant of square lattices; the bandwidth is 8|t|. We denote attractive interactions U ij between onsite and nearest-neighbor pairs on a plane by U 0 and U 1 . We consider two models: (i) U 0 /|t| ≃ −4 and U 1 = 0, and (ii) U 0 = 0 and U 1 /|t| ≃ −4. In model (i), SC fluctuations corresponding to isotropic s-wave superconductivity are developed. In model (ii), SC fluctuations corresponding to anisotropic s-wave, p-wave, or d x 2 −y 2 -or dγwave superconductivity can be developed. We consider only two cases: the isotropic s-wave case in model (i) and the dγ-wave case in model (ii). We consider quasi-two dimensional features phenomenologically by introducing an anisotropy factor for correlation lengths of SC fluctuations, as is discussed below.
We define a SC susceptibility for singlet superconductivity by 30 where ω l = 2lπk B T , with l an integer, is a bosonic energy. Here, η Γ (k) is a form factor of Γ-wave Cooper pairs: for the anisotropic s and dγ waves of model (ii). We assume that the conventional condensation of Cooper pairs with zero total momenta occurs below a critical temperature T c .
When superconductivity of the isotropic Γ = s or Γ = dγ wave occurs, the homogeneous and static part of the SC susceptibility shows a divergence at T c . Superconducting T c can be determined from the condition of The divergence implies that critical fluctuations can play a role, at least, in SC critical regions of highly anisotropic quasi-two dimensions.
We divide the selfenergy correction into two terms: where ε n = (2n+1)k B T , with n an integer, is a fermionic energy. The first term Σ is due to SC fluctuations of the isotropic s or dγ wave and is of linear order in χ ΓΓ (iω l , q), and the second term Σ ′ σ (iε n , k) is due to other fluctuations such as SC fluctuations of other waves, charge fluctuations, higher-order terms in SC and charge fluctuations, and so on.
When fluctuations of a single wave, the s or dγ wave, are considered, 30 it follows that 8) with N the number of unit cells, an effective attractive interaction for the s or dγ wave, and the renormalized Green function, with µ the chemical potential. Since we are interested in SC critical fluctuations of Γ-wave superconductivity, we assume that critical points of other instabilities are a little far way from the critical point or region of Γ-wave superconductivity.
Then, the energy and wave-number dependences of Σ ′ σ (iε n , k) can play no significant role in the SC critical region, so that we simply assume Here, γ is the lifetime width of quasiparticles due to other fluctuations except for those of the considered s-wave or dγ-wave SC fluctuations. Although it is desirable to calculate selfconsistently the total selfenergy Σ σ (iε n , k) to satisfy Eq. (2.8), 31 we approximately replace as a first-order approximation G σ (iε n , k) in the right-hand side of Eq. (2.8) by an unperturbed one given by . (2.12) Since critical fluctuations are restricted to a narrow region around q = 0, Eq. (2.8) is approximately given by where the summation over q = (q x , q y ) is restricted to |q | ≤ q c . The density of states is given by the retarded Green function in such a way that , (2.15) being the spectral weight of quasiparticles with energy ε and wave number k.
When an anisotropic three dimensional model is considered, it is straightforward to carry out selfconsistently a numerical calculation according to the above formulation. Since the anisotropy of SC thermal critical fluctuations plays the most crucial role in the opening of a pseudogap, it is much more convenient to use a phenomenological SC susceptibility, which can explicitly include the anisotropy factor for SC correlation lengths, within the twodimensional model (2.1) than to use a microscopically derived one for the anisotropic three dimensional model. The SC retarded susceptibility can be approximately but well described by a phenomenological one: (2. 16) This is similar to the wellknown one for the spin susceptibility except for the existence of the so called ω-linear real term, αω, with α being real. According to a microscopic calculation, such as is carried out in the previous paper, 29 we can show that |α| ≪ 1; the ω-linear term is ignored because it plays no significant role. We can also show that Eq. (2.16) can be used not only for |q a| ≪ 1 and |ω/t| ≪ 1 but also for a little larger region than that, that is, at least for |q a| 2 and |ω/t| 2. In Eq. (2.16), χ Γ (0) is the static homogeneous one or χ Γ (0) = χ ΓΓ (0, |q| → 0), and Γ SC |t| is an energy scale of SC fluctuations. We introduce no cutoff in the ω or energy integration of Eq. (2.13) because Eq. (2.13) is never sensitive of the cutoff energy ω c as long as ω c /|t| 2. In Eq. (2.16), c is introduced for the lattice constant along the z axis, and a factor κ and an anisotropy factor δ are introduced in such a way that SC correlation lengths parallel to the x-y planes and along the z axis are a/κ and δc/κ, respectively. We introduce a cut-off wave number q c = π/3a in the q integration of Eq. (2.13); the q integration is carried out over a region of |q |a ≤ π/3 and |q z |c ≤ π.
According to the definition of the anisotropy factor δ by Eq. (2.16), the absolute magnitudes of a and c, the anisotropy of the lattice constants, or the difference between a and c plays no role in the framework of this paper.
In general, in both the two cases. We also assume t ′ = −0.3t < 0 for transfer integrals and µ/|t| = −0.5 for the chemical potential.

III. OPENING OF A PSEUDOGAP AT A CRITICAL TEMPERATURE
In this section, we restrict our examination to the SC critical temperature, T = T c , so that κ = 0. Although T c , which is determined from Eq. (2.6), depends on other parameters, we treat it as an independent one. Then, free parameters are T c , δ, Γ SC , and γ. Qualitative features are the same among results for different γ, unless γ are extremely large such as γ/|t| ≫ 1; if a fine structure appears in a physical property it is sharper for a smaller γ. 33 We present here only results for γ/|t| = 0.5, that is, results for γ/|t| = 0.5, and various sets of T c , δ, and Γ SC . Figure 1 shows the imaginary part of the selfenergy, as a function of ε for three cases of E(k) − µ; the k dependence of Eq. (3.1) comes through E(k). When fluctuations are isotropic (δ = 1), the selfenergy is small and its ε dependence is consistent with that of Landau's normal Fermi liquids, as is shown in Fig. 1

(d). As long as
T c or T is low enough, lifetime widths of quasiparticles are also small and the ε dependence of the selfenergy is also consistent with that of Landau's normal Fermi liquids, even when the anisotropy is large. When fluctuations are anisotropic (δ ≪ 1) and T c or T is high enough, on the other hand, the imaginary part of the selfenergy is large and it has a peak around the chemical potential, as is shown in Figs  quasiparticles is about 8|t|, η 2 s (k) = 1 for s wave, η 2 dγ (±π/a, 0) = η 2 dγ (0, ±π/a) = 4 for dγ wave, and we assume g Γ = 4 for both the waves. Therefore, quasiparticles are not well defined or incoherent on the whole Fermi surface in case of the s wave provided that They are incoherent around (±π/a, 0) and (0, ±π/a) in case of the dγ wave provided that −Im [Σ σ (ε + i0, k)] /g dγ η 2 dγ (k)|t| 1/4. For example, Fig. 2 shows the spectral weight ρ k (ε = 0) for dγ-wave case, which is defined by Eq. (2.15).
implies that the reduction of T c is large in a highly anisotropic quasi-two dimensional superconductor even when its observed T c is high. The reduction of T c must be small in an almost isotropic three dimensional one. Large anisotropy of critical fluctuations or a small δ such as δ < 0.1-0.3 is indispensable for the opening of a prominent pseudogap at T c . Smaller energy scales Γ SC or Γ SC |t| are favorable for the opening of pseudogaps. A pseudogap is more prominent for a higher T c .
Since higher T c are mainly caused by larger g Γ , this tendency must be larger when the dependence of T c on g Γ is considered than it is in Figs. 3 and 4. In the isotropic case (δ = 1), on the other hand, a pseudogap structure is absent or subtle at T c .
Since a pseudogap opens because quasiparticle spectra around the chemical potential are swept away due to large lifetime widths, the size of the pseudogap is mainly determined from the peak width of the imaginary part of the selfenergy. The peak width is about 2|t| for parameters considered in this paper so that the size is also about 2|t|; the peak width and the size are larger for a higher T c and a smaller δ.
Although spectra of ρ(ε) are slightly different between the s-wave case shown in Fig. 3 and the dγ-wave case shown in Fig. 4, there is no essential difference between them as long  as U s ≃ U dγ or g s ≃ g dγ . The anisotropy of critical fluctuations within planes, the s or d wave, plays a minor role in the opening of pseudogaps.

IV. APPLICATION TO CUPRATE-OXIDE SUPERCONDUCTORS
When cuprate oxides are considered, the formulation presented in Sec. II should be extended to treat a repulsive strong-coupling regime, U 0 /|t| 8 and |U 1 /t| ≪ 1; we should use the so called d-p or the t-J model. 34 The most serious issues are what are single-particle elementary excitations or quasiparticles, which are bound into Cooper pairs, and what is an effective attractive interaction, which works between the quasiparticles. These two issues can be solved by a Kondo-lattice theory, as is argued in Appendix. We present an alternative in this paper is to demonstrate the essence of the proposed mechanism on pseudogaps due to thermal SC critical fluctuations, which does not depend on microscopic models.
According to the mean-field theory for dγ-wave superconductivity, 42 with ε G (0) the superconducting gap at T = 0 K. Since SC thermal fluctuations vanish at T = 0 K, the reduction of ε G (0) must be very small. As is discussed in Introduction, therefore, observed large ratios 7,8,9 of if other mechanisms work there. It is interesting to confirm that pseudogaps above T c never smoothly evolve into superconducting gaps below T c in cuprate-oxide superconductors.
We consider one of effective Hamiltonians where the on-site U repulsion plays a crucial role, such as the Hubbard model, the periodic Anderson model, the d-p model, the t-J or t-J-infinite U model, 51 and so on; the formulation and argument of this Appendix can be extended almost in parallel to such various on-site U models. We assume the repulsive strong-coupling regime for electron correlations or we assume that the on-site U is as large as the bandwidth of unrenormalized electrons or is larger than it.
According to Hubbard's theory, 52,53 a band splits into two subbands called the lower Hubbard band (LHB) and the upper Hubbard band (UHB). According to Gutzwiller's theory, 39,40 with the use of the Fermi-liquid theory, 37,38 a narrow band of quasiparticles appears around the chemical potential; we call them Gutzwiller's band and Gutzwiller's quasiparticles. The combination of the two theories implies that the density of states must be of a three-peak structure, Gutzwiller's band between LHB and UHB. Actually the formation of Gutzwiller's band at the top of LHB in less-than-half filling cases is demonstrated in a previous paper. 54 The Mott-Hubbard splitting occurs in both metallic and insulating phases, and Gutzwiller's band or quasiparticles are responsible for metallic behaviors.
The third term −4Λ(iω l , q) includes mode-mode coupling terms among intersite spin fluctuations, which correspond to those considered in the selfconsistent renormalization (SCR) it can treat a crossover between them. 50 The selfenergy in the SSA is expanded at T T K in such a way that with g being the g factor, µ B the Bohr magneton, and H an infinitesimally small magnetic field. According to the Ward relation, 67 the irreducible single-site three-point vertex function in spin channels,λ s (iε n , iε n + iω l ; iω l ), is given by 68 for |ε n | → +0 and |ω l | → +0. When we approximately use Eq. (A.4), with higher-order terms in 1/Uχ s (iω l ) being ignored, for small enough |ε n | and |ω l |, it follows that .
The left-hand side is the mutual interaction due to spin fluctuations; the single-site term is subtracted because it is considered in the SSA. The exchange interaction I * s (iω l , q) is nothing but an exchange interaction enhanced by spin fluctuations. The right-hand side is that due to the enhanced exchange interaction;φ s appear as effective single-site vertex functions.
The two mechanisms of attractive interactions to form Cooper pairs, the so called spinfluctuation mechanism and the exchange-interaction mechanism, are essentially the same as each other. 69 A starting or unperturbed state is constructed in the SSA; it is definitely a normal Fermi liquid. Then, intersite effects can be perturbatively considered in terms of I s (iω l , q) or I * s (iω l , q). Since this formulation, which is a perturbation theory starting from the unperturbed state constructed in the non-perturbative SSA theory, is consistent with the physical picture for Kondo lattices, we should call it a Kondo-lattice theory. Since the SSA is rigorous for paramagnetic phases with no order parameter in infinite dimensions, 70 the perturbative theory can also be formulated as a 1/d-expansion theory, with d the spatial dimensionality. The SSA is also called the dynamical mean-field theory (DMFT) 71 or the dynamical coherent potential approximation (DCPA). 72 Leading order effects in 1/d are not only local correlations considered in the SSA, which correspond to dynamical mean fields considered in DMFT, the dynamical coherent potential considered in DCPA, and single-site terms or local effects related with them, but also conventional Weiss's mean fields of certain instabilities. 73 All the other effects or terms are of higher order in 1/d. Not only the Weiss's mean fields but also higher order effects in 1/d can be perturbatively considered.
Taking the Kondo-lattice theory, we can develop a theory of superconductivity occurring in the vicinity of the Mott-Hubbard transition almost in parallel to that of this paper. 29,42,50,58 Effectively or eventually, t and t ′ of this paper are replaced by those of Gutzwiller's quasiparticles. The on-site part of the eventual mutual interaction is definitely strongly repulsive.
Therefore, T c of s-wave or BCS superconductivity cannot be high. On the other hand, it plays no role in the effective coupling constant of dγ-wave superconductivity. The attractive interaction given by Eq. (A.5) includes not only the superexchange interaction and the novel exchange interaction themselves but also their enhanced ones, which include effects due to the nesting of the Fermi surface, the so called inter-nodal scatterings in the dγ-wave case, and so on. Although it works between not only nearest-neighbor sites but also neighboring sites, its nearest-neighbor part play a major role, at least, provided that the system is a little far away from the critical point of antiferromagnetism. The attractive interaction of this paper is replaced by the nearest-neighbor component of I * s (iω l , q); we simply denote an averaged one over its low-energy part by I * s . When the unperturbed state is constructed in the SSA, U dγ for the dγ wave is replaced by where the factor 3 is due to the three spin channels. However, Gutzwiller's quasiparticles constructed in the SSA are further renormalized by SC and antiferromagnetic spin fluctuations. We should use the mass renormalization factor φ γ (k), which includes such an intersite renormalization in addition to the single-site renormalization, instead ofφ γ . Then, we should take with φ γ (k) an average over the Fermi surface.
When we consider cuprate-oxide superconductors, we should use the d-p or t-J model 34 rather than the Hubbard model. 74 In the SSA, it follows thatφ s /φ γ ≃ 2 for almost half waves are different from each other. 31 If the selfconsistency is completed for a repulsive model that is relevant for cuprate-oxide superconductors, the renormalization of the effective attractive interaction, which is discussed in Appendix, has also to be included in the selfconsistent procedure. 32 In the random-phase approximation, where the reduction of T c due to SC critical fluctuations is not considered, it follows that λ = 1 for the critical exponent.
constant around the chemical potential and the Fermi surface shows no nesting. The mode-mode coupling term is responsible for the CW law of such itinerant-electron magnets. 62 On the other hand, the novel exchange interaction is responsible for the CW law of ferromagnets 64 where the density of state has a sharp peak around the chemical potential and antiferromagnets 63 where the Fermi surface shows a sharp nesting. 66 In the periodic Anderson model, strongly hybridized admixtures of almost localized electrons and conduction electrons are formed into Gutzwiller's quasiparticles or the so called heavy electrons at T T K . On the other hand, they are independent compositions at T ≫ T K .
A relevant physical picture for T ≫ T K is the coexistence of localized spins and conduction electrons; the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction arises from the virtual exchange of pair excitations of conduction electrons, and it works between localized spins. Then, we can argue that a crossover occurs as a function of T in such a way that the novel exchange interaction at T T K turns into the RKKY exchange interaction at T ≫ T K .
The RKKY exchange interaction is never relevant at low temperatures such as T T K