Continuous unitary transformation approach to pairing interactions in statistical physics

We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter. For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization g...

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Published in:	Condensed Matter Physics
Date:	2008
Main Author:	Domański, T.
Format:	Article
Language:	English
Published:	Інститут фізики конденсованих систем НАН України 2008
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/119045
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Cite this:	Continuous unitary transformation approach to pairing interactions in statistical physics / T. Domański // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 195-207. — Бібліогр.: 25 назв. — англ.

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author	Domański, T.
author_facet	Domański, T.
citation_txt	Continuous unitary transformation approach to pairing interactions in statistical physics / T. Domański // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 195-207. — Бібліогр.: 25 назв. — англ.
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container_title	Condensed Matter Physics
description	We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter. For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently. We further generalize this procedure to the case of fermions interacting with the discrete boson mode. Andreev-type interactions are responsible for developing a gap in the excitation spectrum. However, the long-range coherence is destroyed due to strong quantum fluctuations. Ми застосовуємо метод потокового рiвняння до дослiдження фермiонних систем з взаємодiями парування, що ведуть до БКШ-нестiйкостi, яка виявляється у появi недiагонального параметра порядку. З цiєю метою ми переформульовуємо неперервне перетворення Боголюбова в ренормгрупових термiнах, де низько- i високоенергетичнi сектори розглядаються послiдовно. Ми узагальнюємо цю процедуру на випадок, коли фермiони взаємодiють з дискретною бозонною модою. Взаємодiї типу Андреєва спричинюють появу щiлини у спектрi збуджень, однак далекосяжна кореляцiя руйнується завдяки сильним квантовим флуктуацiям.
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fulltext	Condensed Matter Physics 2008, Vol. 11, No 2(54), pp. 195–207 Continuous unitary transformation approach to pairing interactions in statistical physics T.Domański Institute of Physics, M. Curie Skłodowska University, 20–031 Lublin, Poland Received January 31, 2008 We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter. For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently. We further generalize this procedure to the case of fermions interacting with the discrete boson mode. Andreev-type interactions are responsible for developing a gap in the excitation spectrum. However, the long-range coherence is destroyed due to strong quantum fluctuations. Key words: continuous unitary transformation, pairing interactions, superconductivity PACS: 05.10.Cc, 74.20.Fg, 71.10.Li 1. Introduction The flow equation method introduced by Wegner [1] and independently proposed by Wilson and G lazek [2] turned out to be a very useful theoretical tool in the condensed matter and high energy physics (for a recent survey of applications see for instance [3]). The main idea behind this technique is to transform the Hamiltonians to diagonal or block-diagonal structure through the continuous unitary transformation upon flexibly adjusting the generating operator. Such continuous diagonalization can be thought of as a process of renormalizing the coupling constants in a way similar to the Renormalization Group (RG) approach [4]. However, the differ- ence, in comparison to the RG scheme, is that instead of integrating out the high energy excitations (fast modes) one renormalizes them in the beginning of the transformation and afterwards the low energy excitations (slow modes) are transformed. We should emphasize that in the quantum me- chanics these low energy excitations determine most of the measurable physical properties, the proper description of which is extremely important [5]. In the first parts of this paper we show the way of constructing the continuous unitary trans- formation for a reduced BCS model. Its exact solution can be obtained using various methods, including the standard single step Bogoliubov transformation [6]. Here we rederive this rigorous solution using the continuous unitary method and illustrate how the pairing instability gradually builds up as an asymptotic fixed point. Recently there appeared in the literature several concepts [7,8] where the authors addressed the Cooper pairing by means of the functional RG. Since the continuous unitary transformation belongs to the same family of RG treatments, it would be in- structive to adopt Wegner’s procedure [9] in order to deal with the symmetry broken states induced by the pairing interactions. In the second part we generalize this transformation to the case where fermions interact with a single boson mode. Such a situation can be encountered for example when the conduction band electrons are scattered by the Andreev type interaction on the localized electron pairs [10]. On some more general grounds it can also refer to the boson type modes coupled with the correlated electrons [11]. Another realization of this scenario is possible in ultracold gases where fermion atoms interact with the weakly bound molecules through the Feshbach resonance which finally leads to the atomic superfluidity [12]. c© T.Domański 195 T.Domański 2. The BCS pairing To have a simple example illustrating how the fast and slow energy modes are scaled during the continuous unitary transformation we first consider the exactly solvable bilinear Hamiltonian Ĥ = ∑ k,σ ξkĉ†kσ ĉkσ − ∑ k ( ∆kĉ†k↑ĉ † −k↓ + h.c. ) (1) which describes the fermions coupled to the pairing field ∆k. In (1) we use the standard notation for the creation (annihilation) operator ĉ†kσ (ĉkσ) and energy ξk = εk−µ is measured from the chemical potential µ. Pairing field can be thought of, for instance, as a mean field approximation to the weak pairing potential between electrons ∆k = −∑ q Vk,q〈ĉ−q↓ĉq↑〉, where Vk,q < 0. In general, one can treat ∆k as the boson operator, e.g., arising from the Hubbard Stratonovich transformation for the interacting fermion system. It can also correspond to a boson whose exchange mediates the pairing between electrons. Coupling to the boson operator will be discussed in the last section. Hamiltonians of the fermion and/or boson systems having a bilinear structure (1) have been already considered using the flow equation method [1,9,13,14]. However, the authors have focused so far on the determination of quasiparticle energies. Here we supplement their analysis by cal- culating the correlation functions that contain information about the order parameter as well as characteristics of the excitation spectrum. It is well known that the bilinear Hamiltonian (1) can be diagonalized by the (single step) Bogoliubov transformation [6]. All physical quantities, static and dynamic, can be exactly deter- mined. In this work we get the same rigorous solution in a process of continuous diagonalization of the coupled states \|k, ↑〉 and \| − k, ↓〉. How fast this can be achieved depends on the distance from the Fermi surface \|k−kF\| (or on the energy ξk). Simultaneously with diagonalization there emerge the coherence factors uk, vk. Their final values depend on the relative distance of the momentum k from the Fermi surface. We construct the continuous unitary transformation Û(l), where l stands for a formal flow parameter varying between the initial l = 0 value and the value l > 0 such that the transformed Hamiltonian Ĥ(l) = Û(l)ĤÛ−1(l) is simplified to the needed structure (diagonal, block-diagonal, tridiagonal or any other). With an increase of the flow parameter l, the Hamiltonian evolves according to the general flow equation [1] dĤ(l) dl = [η̂(l), Ĥ(l)], (2) where η̂(l) ≡ dÛ(l) dl Û−1(l). We now require the transformed Hamiltonian Ĥ(l) to preserve its bilinear structure (1) and let only the coupling constants become l-dependent (renormalized) Ĥ(l) = ∑ k,σ ξk(l)ĉ†kσ ĉkσ − ∑ k ( ∆k(l)ĉ†k↑ĉ † −k↓ + h.c. ) + const(l) (3) with the initial conditions ξk(0) = ξk, ∆k(0) = ∆k and const(0) = 0. This strategy reminds us the scheme of RG approach and it has been shown [1] that the flow parameter l roughly corresponds to 1/ √ Λ [3]. There are several ways of choosing the generating operator η̂(l). In order to reduce the Hamil- tonian to a diagonal structure, we follow the Wegner’s proposal [1] η̂(l) = [Ĥ0(l), Ĥ(l)], where Ĥ0(l) = ∑ k,σ ξk(l)ĉ†kσ ĉkσ denotes a diagonal part. With this choice, the Hamiltonian becomes diagonal in the limit l → ∞. An explicit form of the generating operator for (3) is given by η̂(l) = −2 ∑ k ξk(l) ( ∆k(l)ĉ†k↑ĉ † −k↓ − h.c. ) . (4) Substituted into the equation (2) it gives dĤ(l) dl = 4 ∑ k,σ ξk(l)\|∆k(l)\|2 ( ĉ†kσ ĉkσ − 1 ) − 4 ∑ k (ξk(l))2 ( ∆∗ k(l)ĉ†k↑ĉ † −k↓ + h.c. ) (5) 196 Continuous unitary transformation approach to pairing interactions in statistical physics which is identical to the following set of flow equations dξk(l) dl = 4ξk(l)\|∆k(l)\|2, (6) d∆k(l) dl = −4(ξk(l))2∆∗ k(l) (7) and dconst(l)/dl = −4 ∑ k,σ ξk(l)\|∆k(l)\|2. Formally, the solution of the equation (7) can be given as \|∆k(l)\| = \|∆k\|e−4 � l 0 dl′[ξk(l′)]2 (8) which yields that liml→∞ ∆k(l) = 0 for all k 6= kF. When we multiply the equation (6) by ξk(l) and the equation (7) by ∆k(l)∗, their sum gives the following invariance d dl { (ξk(l))2 + \|∆k(l)\|2 } = 0. (9) This constraint, together with ∆k(l) vanishing in the limit l → ∞, implies the known Bogoliubov spectrum ξk(∞) = sgn(ξk) √ (ξk)2 + \|∆k\|2. (10) −10 −5 0 5 10 −1 0 1 −10 −5 0 5 10 0 1 l=0.01 l=0.05 l=0.1 l=0.5 l=1 l=10 ξ k ∆ k (l ) ξ k (l )− ξ k ξ k Figure 1. Renormalization of ∆k(l) and ξk(l) during the flow for several values of l as denoted in the legend. At the starting point l = 0, we used the isotropic coupling ∆k(0) = ∆. Energies are expressed in units of ∆ while l is expressed in units of ∆−2. In figure 1 we plot the pairing field ∆k(l) and the change (renormalization) of fermion energies ξk(l) − ξk at several stages of the continuous transformation. We notice that fast modes (the states distant from the Fermi energy) are transformed in the first part of the process and the change of their energies is rather small. The slow modes (i.e. the low energy excitations) have to be worked on much longer. Asymptotically, at l → ∞, the entire spectrum reduces to the Bogoliubov structure (10). Now we turn our attention to the dynamical quantities which can be expressed via the normal 〈〈ĉkσ; ĉ†kσ〉〉ω and the anomalous single particle Green’s function 〈〈ĉk↑; ĉ−k↓〉〉ω. We introduced 197 T.Domański the standard Fourier transforms for the retarded fermion Green’s function ∫ dωeiωt〈〈Â; B̂〉〉ω ≡ iΘ(t)〈Â(t)B̂ + B̂Â(t)〉 and time evolution is given by Â(t) = eitĤÂe−itĤ . The statistical averaging for an arbitrary observable Ô is defined as 〈Ô〉 = Tr { e−βĤÔ } /Tr { e−βĤ } , where β−1 = kBT . Since the trace is invariant under the unitary transformations we can write down Tr { e−βĤÔ } = Tr { Û(l)e−βĤÔÛ−1(l) } = Tr { e−βĤ(l)Ô(l) } , (11) where Ô(l) = Û(l)ÔÛ−1(l). It is convenient to compute the trace in the limit l = ∞ when Hamil- tonian becomes diagonal. However, the price which we pay for having the simplified Hamiltonian at l = ∞ is the extra necessity to transform the observables Ô → Ô(l) → Ô(∞). This is to be done similarly to the Hamiltonian (2) using dÔ(l) dl = [η̂(l), Ô(l)]. (12) With the choice of the generating operator η(l) given by (4) we find from the commutator [η̂(l), ĉk↑] that for l > 0 the annihilation operator ĉk↑ gets convoluted with ĉ†−k↓. Thus, it is natural to propose the following Ansatz for the l-dependent operator ĉk↑(l) = uk(l)ĉk↑ + vk(l)ĉ†−k↓. (13) Similar analysis for ĉ†−k↓ operator leads to ĉ−k↓(l)† = −vk(l)ĉk↑ + uk(l)ĉ†−k↓ (14) with the initial conditions uk(0) = 1, vk(0) = 0. These parametrized equations (13,14) substituted into (12) with η̂(l) given in (4) lead to the coupled flow equations duk(l) dl = 2ξk(l)∆k(l)vk(l), (15) dvk(l) dl = −2ξk(l)∆k(l)uk(l). (16) After straightforward algebra for (15,16) we obtain the following invariance \|uk(l)\|2 + \|vk(l)\|2 = const = 1 which assures that the l-dependent fermion operators (13,14) preserve the anticommu- tation relations { ĉkσ(l), ĉ†k′σ′(l) } = δk,k′δσ,σ′ . (17) Without a loss of generality we assume ∆k(l) to be real so that the unknown coefficients uk(l), vk(l) become real too. To determine their asymptotic l = ∞ values we can rewrite (15) as duk(l) vk(l) = 2ξk(l)∆k(l)dl (18) and we integrate both sides in the limits ∫ l=∞ l=0 . Using vk(l) = √ 1 − (uk(l))2 we get for the l.h.s. ∫ l=∞ l=0 duk(l) √ 1 − (uk(l))2 = arccos uk(∞) (19) 198 Continuous unitary transformation approach to pairing interactions in statistical physics due to uk(0) = 1. Using equation (7) we can replace 2ξk(l)∆k(l)dl by −d∆k(l)/2ξk(l) and from the invariance (9) we obtain that ξk(l) = √ ξ2 k(∞) − \|∆k(l)\|2. The r.h.s. of equation (18) gives after integration ∫ l=∞ l=0 2ξk(l)∆k(l)dl = − ∫ l=∞ l=0 d∆k(l) 2 √ ξ2 k(∞) − \|∆k(l)\|2 = −1 2 ( arccos \|∆k\| \|ξk(∞)\| − π 2 ) (20) because ∆k(∞) = 0. Combining the results (19,20) we obtain 2 arccos uk(∞) = π 2 − arccos \|∆k\| \|ξk(∞)\| (21) and finally we determine the l=∞ factors u2 k(∞) = 1 2 [ 1 + ξk ξk(∞) ] = 1 − v2 k(∞). (22) In the same way one can show that 2uk(∞)vk(∞) = ∆k/\|ξk(∞)\|. 0 1 2 3 4 5 0.5 0.6 0.7 0.8 0.9 1 0 5 10 0.5 1 l u k 2 (l) ξ k u k 2 ( )81 0.1 0.01 Figure 2. Flow of the coherence factor u2 k(l) for three fermion states of the initial energy ξk = 0.01, 0.1 and 1 (in units ∆). Inset shows the effective value of the coherence factor u2 k(∞) versus ξk. In figure 2 we present the coherence factor u2 k(l) as a function of the flow parameter l. Again we notice that for the momenta distant from the Fermi surface the coherence factors evolve rather quickly from the initial value u2 k(0) = 1 to the asymptotic values (see the inset). The coherence factors of the slow modes establish later on, similarly to renormalization of the corresponding excitation energies shown in figure 1. Saturation occurs around l ∝ 1/ξk(∞)2. Since the transformed Hamiltonian Ĥ(∞) is diagonal, we can easily derive the single particle Green’s functions (and the higher order Green’s functions as well). With the parameterizations (13,14) we obtain 〈〈ĉk↑; ĉ†k↑〉〉ω = u2 k(∞) ω − ξk(∞) + v2 k(∞) ω + ξk(∞) , (23) 〈〈ĉk↑; ĉ−k↓〉〉ω = uk(∞)vk(∞) ω − ξk(∞) − uk(∞)vk(∞) ω + ξk(∞) . (24) The equal time expectation values defined by 〈B̂Â〉 = − 1 π ∫ ∞ −∞ dωf(ω, T )Imag〈〈Â; B̂〉〉ω+i0+ (where f(ω, T ) = 1/ [1 + exp{ω/kBT}] is the Fermi distribution function) yield the following equations for the average occupancy 〈ĉ†k↑ĉk↑〉 = 1 2 [ 1 − ξk ξk(∞) tanh ξk(∞) 2kBT ] , (25) 199 T.Domański and for the off-diagonal order parameter 〈ĉ−k↓ĉk↑〉 = ∆k 2ξk(∞) tanh ξk(∞) 2kBT . (26) These equations (25,26) exactly reproduce the rigorous solution of the reduced BCS Hamiltonian. In the next section we generalize this treatment to account for the scattering of finite momentum fermion pairs. 3. Finite momentum fermion pairs The continuous Bogoliubov transformation discussed in the preceding section can be extended to the more general bilinear Hamiltonian Ĥ = ∑ k,σ ξkĉ†kσ ĉkσ − ∑ k,q ( Gk,qĉ†k↑ĉ † q−k↓ + h.c. ) . (27) (27) eventually reduces to (1) when Gk,q = δq,0∆k. For q 6= 0, this model allows for scattering of the finite momentum fermion pairs. In what follows below we analyze the effect of such finite momentum scattering. The generator from Wegner’s proposal η̂0(l) = − ∑ k,q [ξk(l) + ξq−k(l)] ( Gk,q(l)ĉ†k↑ĉ † q−k↓ ) − h.c. (28) applied to the flow equation (2) induces the off-diagonal terms c†kσcp 6=kσ. In order to eliminate them from the transformed Hamiltonian H(l) we replace (28) by η̂(l) = η̂0(l) + ∑ k,q,σ γk,q,σ(l) ( ĉ†kσ ĉqσ − ĉ†qσ ĉkσ ) . (29) After simple algebraic calculations we find that the off-diagonal terms are exactly cancelled if γk,q,↑(l) = ∑ k′ ξk(l) + ξk′(l) ξk(l) − ξq(l) Gk′,k′+q−k(l)Gk,q(l), (30) γk,q,↓(l) = ∑ k′ ξk(l) + ξk′(l) ξk(l) − ξq(l) Gk′,k+k′(l)Gk′,k′+q(l). (31) The modified generating operator (29) has a virtue to preserve the initial structure of the Hamilto- nian Ĥ(l) = ∑ k,σ ξk(l)ĉ†kσ ĉkσ − ∑ k,q ( Gk,q(l)ĉ†k↑ĉ † q−k↓ + h.c. ) + const(l). The corresponding set of flow equations is dξk(l) dl = 2 ∑ q (ξk(l) + ξq−k(l)) \|Gk,q(l)\|2, (32) dGk,q(l) dl = − (ξk(l) + ξq−k(l)) 2 G∗ k,q(l) + 2 ∑ k′,σ (γk,k′,σ(l) − γk′,k,σ(l)) Gk′,q−k+k′(l), (33) d const(l) dl = −2 ∑ k,q,σ (ξk(l) + ξq−k(l))\|Gk,q(l)\|2. (34) We were not able to analytically solve the equations (32)–(34). Therefore, we explored them nu- merically assuming the tight binding dispersion ξk = −2t cos(kxa) and using the pairing potential Gk,q in a Lorentian form Gk,q = N (n) ∆ n [qxa] 2 + 1 . (35) 200 Continuous unitary transformation approach to pairing interactions in statistical physics The normalization factor N (n) was taken such that ∑ q Gk,q = ∆ and ∆/4t = 0.01. The flow of the operators c (†) kσ(l) takes the form ĉk↑(l) = ∑ q uk,q(l)ĉq+k↑ + ∑ q vk,q(l)ĉ†q−k↓ , (36) ĉ†−k↓(l) = − ∑ q vk,q(l)ĉq+k↑ + ∑ q uk,q(l)ĉ†q−k↓ (37) which generalizes the previous equations (13, 14) due to finite momentum scattering. Substituting (36,37) into the flow equation (12) we obtain duk,q(l) dl = [ξk(l) + ξq−k(l)] Gk,q(l)vk,0(l) + ∑ p 6=k [ξk−q(l) + ξq+p(l)] Gq−k,2q−k+p(l)vk,q(l), (38) dvk,q(l) dl = − [ξk(l) + ξq−k(l)] Gk,q(l)uk,0(l) − ∑ p 6=k [ξk−q(l) + ξq+p(l)] Gq−k,2q−k+p(l)uk,q(l). (39) From these equations (38)–(39) we derive the invariance ∑ q \|uk,q(l)\|2 + ∑ q \|vk,q(l)\|2 = 1 (40) which again leads to the anticommutation relations (17). Using the parameterization (36,37) we can determine the single particle Green’s functions. Let us first consider the diagonal part (in the Nambu representation). The spectral function A(k, ω) = −π−1Imag〈〈ĉk↑; ĉ†k↑〉〉ω+i0+ turns out to consist of two contributions A(k, ω) = Acoh(k, ω) + Ainc(k, ω). (41) The usual structure with two delta peaks Acoh(k, ω) = \|uk,0(∞)\|2 δ [ω − ξk(∞)] + \|vk,0(∞)\|2 δ [ω + ξ−k(∞)] (42) describes the long-lived quasiparticle modes of energies ±ξk(∞) similar to the Bogoliubov modes (10) discussed previously for Gk,q = ∆kδq,0. The remaining part Ainc(k, ω) = ∑ q6=0 \|uk,q(∞)\|2 δ [ω − ξq+k(∞)] + ∑ q6=0 \|vk,q(∞)\|2 δ [ω + ξq−k(∞)] . (43) 0−2 −1 1 2 ω/∆ ρ(ω) n=100 50 10 0.0 0.5 1.0 Figure 3. The local density of states ρ(ω) = � k A(k, ω) of fermions coupled to the potential Gk,q given in (35) with n = 10, 50 and 100. The density is normalized to ρ0(ω) of the nointeracting system. 201 T.Domański corresponds to the background spectrum which is continuously spread over a certain energy regime. Due to the invariance (40), the total spectral weight properly obeys the sum rule ∫ ∞ −∞ dωA(k, ω) = 1. In figure 3 we plot the local density of states ρ(ω) = ∑ k A(k, ω). We notice that the excitation spectrum is no longer truly gaped. There appears a partial suppression of the fermion states around the Fermi energy (corresponding to ω = 0). This property can be assigned to the effect of finite momentum fermion pairs scattered on the potential Gk,q introduced ad hoc in (27). Physically this means that only a certain fraction of the fermion states is expelled from the vicinity of the Fermi surface. k/kF ω/∆0.9 1.0 1.1 -2 -1 0 1 2 0 0.5 1 A(k,ω) Figure 4. The single particle spectral function A(k, ω) obtained for the potential (35) with n = 100. The thick vertical lines represent the spectral weights \|uk,0(∞)\|2 and \|vk,0(∞)\|2 of the long-lived quasiparticles ±ξk(∞). The thin solid lines show ω dependence of the incoherent background Ainc(k, ω). Figure 4 shows both parts of the spectral function for the pairing potential (35) with n=100, which yields the strongest suppression of the low lying fermion states plotted in figure 3. Two modes of the coherent part have similar spectral weights as the BCS coherence factors. However, in distinction from the standard BCS solution, the corresponding quasiparticle dispersion ±ξk(∞) in this case is gapless. This is evidently caused by coupling to the finite momentum fermion pairs. As regards the background part, it builds up mainly near the Fermi surface kF. Such fermion states are located around the quasiparticle energies ±ξk(∞). We estimated that for k = kF, these states contribute nearly 30 percent of the total spectral weight. The off-diagonal single particle Green’s function 〈〈ĉk↑; ĉ−p↓〉〉ω has a structure similar to (43). Skipping unnecessary technical details we show the resulting expression for the expectation value of the order parameter 〈ĉ−p↓ĉk↑〉 = −1 2 ∑ q ( uk,q+p−k(∞) vp,q(∞) exp [βξq+p(∞)] + 1 − up,q(∞) vk,q−p+k(∞) exp [−βξq−p(∞)] + 1 ) . (44) We numerically investigated the momentum dependence of the order parameter 〈ĉq−k↓ĉk↑〉 and we noticed its rather fast decrease against \|q\|. For n = 10, 50 and 100, we obtained that the magnitude of the order parameter at \|q\| = 0.2π/a, which is nearly a hundred times smaller than 〈ĉ−k↓ĉk↑〉. This indicates that finite momentum fermion pairs are less favored in the system. 202 Continuous unitary transformation approach to pairing interactions in statistical physics 4. Coupling to the boson mode In this section we apply the same method to a non-trivial problem describing the itinerant fermions coupled to the dispersionless boson mode Ĥ = ∑ k,σ ξkĉ†kσ ĉkσ + ∑ k ( gkb̂ ĉ†k↑ĉ † −k↓ + h.c. ) + Ω b̂†b̂. (45) The boson mode can be regarded, for instance, as a pairing field derived from the Hubbard Stratonovich transformation in the system of correlated fermions. The same model (45) can also describe the Andreev tunneling between c-fermions and some pair reservoir denoted by b-particles. Another possibility is to think of the case where itinerant fermions (e.g. conduction band electrons) coexist and interact with some localized fermion pairs [10]. The model (45) is also frequently ap- plied to the description of the ultracold fermion atoms coupled to the weakly bound molecules effectively leading to the resonant Feshbach scattering [15]. To keep a conserved total number of particles N̂tot = ∑ k,σ ĉ†kσ ĉkσ + 2b̂†b̂ we apply the grand canonical ensemble. Symbol Ω stands for the boson energy measured from 2µ and gk denotes the coupling constant. It can be shown that physical properties of this model depend only on the magnitude of gk. In other words, all \|gk\|eiφk lead to identical results independently of the phase φk. In the mean field treatment of the Hamiltonian (45), one usually introduces the linearization b̂ĉ†k↑ĉ † −k↓ ' 〈b̂〉ĉ†k↑ĉ † −k↓ + b̂〈ĉ†k↑ĉ † −k↓〉 [10]. This idea relies on the assumption that there exists a finite amount of the Bose Einstein (BE) condensation of b particles. Hamiltonian (45) is simplified to the ordinary BCS problem, where ∆ ≡ −g〈b̂〉. However, let us notice that the BE condensation of infinitely heavy boson (a discrete energy level Ω means that the effective mass m =∞) is not allowed. Our present analysis based on the continuous unitary transformation shows a route to go beyond such a mean field approximation. To construct the continuous transformation decoupling c from b particles we choose the following generating operator η̂(l) = ∑ k [2ξk(l) − Ω(l)] gk(l)b̂ĉ†k↑ĉ † −k↓ − h.c. (46) Applying (46) to the flow equation (2) we obtain the following l-dependent Hamiltonian [16] Ĥ(l) ' ∑ k,σ ξk(l)ĉ†kσ ĉkσ + ∑ k ( gk(l)b̂ĉ†k↑ĉ † −k↓ + h.c. ) + Ω(l)b̂†b̂ + ∑ k,k′ Uk,k′(l)ĉ†k↑ĉ † −k↓ĉ−k′↓ĉk′↑ (47) with Uk,k′(0) = 0 and the higher order terms involving density-density interaction between c and b objects are linearized through the normal ordering [1]. In a standard way we derive the following set of flow equations dξk(l) dl = 2\|gk(l)\|2 [2ξk(l) − Ω(l)] nB , (48) dΩ(l) dl = 2 ∑ k \|gk(l)\|2 [2ξk(l)−Ω(l)] ( 2nF kσ−1 ) , (49) dgk(l) dl = −gk(l) [2ξk(l) − Ω(l)] 2 , (50) dUk,k′(l) dl = −gkg∗k′(l) [2ξk(l) − Ω(l)] − gk′g∗k(l) (2ξk′(l) − Ω(l)) , (51) where nF kσ denotes the fermion occupancy and nB = 〈b̂†b̂〉 refers to the statistically averaged boson number. From the formal equation (50) gk(l) = gk exp{− ∫ l 0 dl′[2ξk(l′)−Ω(l′)]2} we conclude that gk(l→∞) = 0. 203 T.Domański To have a complete set of the flow equations (48)–(48) we need the distribution function nF kσ. From the flow equation (12) we determine the following parametrizations [18] ĉk↑(l) = uk(l) ĉk↑ + vk(l) b̂ ĉ†−k↓, (52) ĉ†−k↓(l) = −vk(l) b̂† ĉk↑ + uk(l) ĉ†−k↓, (53) where l-dependent coefficients satisfy uk(l) dl = vk(l)gk(l) [2εk(l) − Ω(l)] (nB + nF −k↓), (54) vk(l) dl = −uk(l)gk(l) [2εk(l) − Ω(l)] . (55) with the boundary condition uk(0)=1 and vk(0)=0. In equations (54,55) we neglected the terms proportional to gk(l)Uk1,k2 (l) which can eventually contribute some higher order corrections, of the order ∼ g3 k. Combining (54) and (55) we obtain the invariance \|uk(l)\|2 + (nB + nF −k↓)\|vk(l)\|2 = const = 1 (56) which guarantees that operators ĉ (†) kσ(l) defined in (52,53) obey the anticommutation relations. Various expectation values 〈Ô〉 are easy to carry out in the limit l → ∞ because fermions are decoupled from the boson field. Some delicate problem causes the two-body interaction Uk,k′(∞). As will be shown below this potential is small, so in the lowest order perturbation theory we incorporate its effect via the Hartree shift Uk,k(∞)nF kσ to fermion energies. However, this weak point of the present analysis can be systematically improved. Figure 5 confirms that the two-body potential is indeed weak. −0.5 −0.4 −0.3 −0.2 −0.1 −0.1 0.0 0.1 ε k /D U k,k /D µ Figure 5. The resonant-like scattering potential Uk,k(∞) of the interactions induced between fermions by eliminating the boson mode. Using the parameterization (52,53) we find the normal single particle Green’s function 〈〈ĉk↑; ĉ†k↑〉〉ω = \|uk(∞)\|2 ω − Ek,1 + (nB + nF −k↓) \|vk(∞)\|2 ω − Ek,2 (57) with quasiparticle energies Ek,1 = ξk(∞)+Uk,k(∞)nF kσ and Ek,2 = Ω(∞)−Ek,1. The corresponding spectral function A(k, ω) = − 1 π Imag〈〈ĉk↑; ĉ†k↑〉〉ω+i0 satisfies the sum rule ∫ ∞ −∞ dωA(k, ω) = 1 due to the invariance (56). Finally, the momentum distribution function is given by nF kσ = \|uk(∞)\|2 eβEk,1 + 1 + (nB + nF −k↓) \|vk(∞)\|2 eβEk,2 + 1 . (58) We numerically explored the set of coupled flow equations (48)–(51,54,55) and (58) using the Runge Kutta algorithm. In our calculations we considered the finite fermion band −D/2 6 εk 6 D/2 and assumed the flat density of states. Total particle concentration was fixed for 〈N̂tot〉 = 1 which corresponds to nearly equal populations of fermions and bosons ∑ k nF kσ ∼ nB . 204 Continuous unitary transformation approach to pairing interactions in statistical physics −0.5 −0.4 −0.3 −0.2 −0.1 −0.2 −0.1 0.0 0.1 0.2 ε k /D E k,1E k,2 µ e x c it a ti o n e n e rg ie s Figure 6. The single particle excitation energies. The thin dashed line indicates that the excita- tions Ek,ν are gapped around the renormalized boson energy 1 2 Ω(∞). We used for computations gk = 0.05D, Ω/2 + µ = −0.3D, kBT = 0.015D. In figure (6) we show the single particle fermion spectrum which turns out to be gapped. Two branches of the excitations Ek,ν are discontinuous around the energy Ω(∞)/2 (the dashed line). We have checked that the value of the gap is rather independent of temperature. The corresponding spectral weights are shown in figure 7. They have a behavior similar to the BCS formfactors but we notice only a negligible dependence on temperature. −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.5 1.0 ε k /D µs p e c tr a l w e ig h ts \|u k \| 2 1−\|u k \| 2 Figure 7. The spectral weights corresponding to the energy branches shown in figure 6. Let us emphasize that the gapped spectrum shown in figure 6 is not related to any off-diagonal order parameter. According to the parameterization (52, 53) we determine the anomalous Green’s function 〈〈ĉ†−k↓; c†k↑〉〉 and we find it to be identically zero. The above mentioned structure should be referred to as a pseudogap. The broken symmetry can arise if and only if a certain fraction of b particles becomes BE condensed [19]. There are two possibilities for this to occur: a) either we assume bosons to be mobile right from the outset of the problem, or b) we allow for a finite momentum exchange q 6= 0 in the interaction term ĉ†k↑ĉ † q−k↓b̂q. The second option has been explored in the literature by several groups using various many- body techniques (for a representative list of the references see the review paper [20]). The purpose of our present study was to show that the temporal quantum fluctuations alone can induce the gapped excitation structure without involving any spatial fluctuations of the order parameter. In the real systems there would always exist both temporal and spatial fluctuations and the latter cause that upon lowering the temperature, the pseudogap smoothly evolves into the true gap of superconducting state [21]. 205 T.Domański 5. Conclusions By means of the flow equation method [1] we analyzed the bilinear Hamiltonians describing fermion systems having pairing interactions. This new technique becomes more and more popular [3] owing to its conceptual simplicity which allows us to go beyond the framework of various per- turbative approximations [22]. This method is based on the continuous canonical transformations in the course of which all the parameters are gradually renormalized. In some way this can be compared to the renormalization of the coupling constants within the RG procedure [4]. The high energy excitations are renormalized here mainly in the beginning, while the low energy excitations should be dealt with until the end of transformation. The first part of our paper illustrates the way of reproducing the rigorous solution of the BCS model using a continuous version of the Bogoliubov transformation [6]. In particular, we show how one can approach the symmetry broken problems which usually is not an easy task for the RG methods [7,8]. This scheme can be extended to the analysis of the two-body interactions where various kinds of instabilities can arise. Some results for the 2-dimensional Hubbard model using the flow equation method have been already reported [23]. In the second part we focused on the case where fermion pairs are coupled to some single level boson field. Such a situation can take place when the correlated fermion system is affected by bosonic modes originating from the quantum fluctuations in the systems with a reduced dimen- sionality like the high temperature superconductors [24]. This sort of physics has recently been intensively studied for the ultracold fermion atoms where the interaction with the weakly bound molecules leads to the Feshbach resonance and turns out to be a driving mechanism for the atomic superfluidity [25]. We selfconsistently solved the corresponding set of flow equations using the numerical Runge Kutta algorithm. We showed that the interactions are responsible for the appearance of a gap in the single particle fermion spectrum. This gap is centered around the boson energy 1 2Ω(∞) and does not signify any symmetry breaking because no order parameter is present in the system. The off- diagonal order parameter can eventually appear if the bosons have a finite mobility (characterized by some dispersion Ωq different from the discrete energy Ω considered here) when those bosons undergo the BE condensation [19,16]. Here, we emphasize that a single particle gap can appear in a normal state solely due to the temporal quantum fluctuations. In the realistic systems with additional spatial fluctuations, such a normal state pseudogap can be expected to smoothly evolve into the true gap of the superconducting state. 6. Acknowledgement The author kindly acknowledges F. Wegner, J. Ranninger and K.I. Wysokiński for valuable discussions. This work is partly supported by the Polish Ministry of Science and Education under the grant NN202187833. References 1. Wegner F., Ann. Physik, Leipzig, 1994, 3, 77. 2. G lazek S.D., Wilson K.G., Phys. Rev. D, 1994, 49, 4214; Phys. Rev. D, 1993, 48, 5863. 3. Kehrein S., The flow equation approach to many-particle systems, Springer Tracts in Mod. Phys., 2006, 217, Springer-Verlag, Berlin. 4. Wilson K.G., Rev. Mod. Phys., 1975, 47, 773; Shankar R., Rev. Mod. Phys., 66, 129, 1994. 5. Nozieres P., Theory of interacting fermi systems, Benjamin W.A. Inc., New York, 1964. 6. Bogoliubov N.N., Sov. Phys. JETP, 7, 1948, 41; J.G. Valatin, Nuovo Cimento, 1958, 7, 843. 7. Salmhofer M., Honercamp C., Metzner W., Lauscher O., Prog. Theor. Phys., 2004, 112, 943; Katanin A., Phys. Rev. B, 70, 2004, 115109; Honercamp C., Salmhofer M., Prog. Theor. Phys., 2005, 113, 1145. 8. Schütz F., Bartosch L., . Kopietz P., Phys. Rev. B, 2005, 72, 035107; Schütz F., Kopietz P., J. Phys. A, 2006, 39, 8205. 206 Continuous unitary transformation approach to pairing interactions in statistical physics 9. Stelter G., Zur Lösung von Flussgleichungen für einfache Systeme, Diploma thesis no., 3291, Heidel- berg, 1996 unpublished. 10. Ranninger J., Robaszkiewicz S., Physica B, 1985, 135, 468; Robaszkiewicz S., Micnas R., Ranninger J., Phys. Rev. B, 1987, 36, 180; Friedberg R., Lee T.D., Phys. Rev. B, 1989, 40, 423. 11. Eremin I., Morr D.K., Chubukov A.V., Bennemann K., Norman M.R., Phys. Rev. Lett., 2005, 94, 147001. 12. Duine R.A., Stoof H.T.C., Phys. Rep., 2004, 396, 115. 13. Stein J., Eur. Phys. J. B, 1998, 5, 193; Europhys. Lett., 2000, 50, 68; J. Phys. G, 2000, 26, 377. 14. Safonov V.L., quant-ph/0202095. 15. Timmermans E., Tommasini P., Hussein M., Kerman A., Phys. Rep., 1999, 315, 199. 16. Domański T., Ranninger J., Phys. Rev. B, 2001, 63, 134505; Domański T., Phys. Rev. A, 2003, 68, 013603. 17. Domański T., Ranninger J., Phys. Rev. B, 2004, 70, 184503. 18. The most general form for parametrization of the fermion operators is given by ĉk↑(l) = uk(l)ĉk↑ + vk(l)b̂ĉ†−k↓ + ṽk(l)ĉ−k↓ + ũk(l)b̂†ĉk↑. However, it has been shown in the section IV.B of our previous paper [17] that the coefficients ṽk(l) and ṽk(l) become identically zero if 〈b̂〉 = 0 which is the case here. 19. Kostyrko T., Ranninger J., Phys. Rev. B, 1996, 54, 13105. 20. Chen Q., Stajic J., Tan S., Levin K., Phys. Rep., 2005, 412, 1. 21. Domański T., Ranninger J., Phys. Rev. Lett., 2003, 91, 255301. 22. Kriel J.N., cond-mat/0602022. 23. Grote I., Körding E., Wegner F., Low J., Temp. Phys., 2002, 126, 1385; Hankevych V., Grote I., Wegner F., Phys. Rev. B, 2002, 66, 094516. 24. Lee P.A., Nagaosa N., Wen X.-G., Rev. Mod. Phys., 2006, 78, 17. 25. Holland M., Kokkelmans S.J.J.M.F., Chiofalo M.L., Walser R., Phys. Rev. Lett., 2001, 87, 120406. Пiдхiд з використанням неперервних унiтарних перетворень до взаємодiй спарювання в статистичнiй фiзицi Т.Доманьскi Iнститут фiзики, Унiверситет М. Кюрi, Польща Отримано 31 сiчня 2008 р. Ми застосовуємо метод потокового рiвняння до дослiдження фермiонних систем з взаємодiями па- рування, що ведуть до БКШ-нестiйкостi, яка виявляється у появi недiагонального параметра поряд- ку. З цiєю метою ми переформульовуємо неперервне перетворення Боголюбова в ренормгрупових термiнах, де низько- i високоенергетичнi сектори розглядаються послiдовно. Ми узагальнюємо цю процедуру на випадок, коли фермiони взаємодiють з дискретною бозонною модою. Взаємодiї типу Андреєва спричинюють появу щiлини у спектрi збуджень, однак далекосяжна кореляцiя руйнується завдяки сильним квантовим флуктуацiям. Ключовi слова: неперервне унiтарне перетворення, взаємодiї спарювання, надпровiднiсть PACS: 05.10.Cc, 74.20.Fg, 71.10.Li 207 208
id	nasplib_isofts_kiev_ua-123456789-119045
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1607-324X
language	English
last_indexed	2025-11-29T05:03:39Z
publishDate	2008
publisher	Інститут фізики конденсованих систем НАН України
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spelling	Domański, T. 2017-06-03T04:33:09Z 2017-06-03T04:33:09Z 2008 Continuous unitary transformation approach to pairing interactions in statistical physics / T. Domański // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 195-207. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 05.10.Cc, 74.20.Fg, 71.10.Li DOI:10.5488/CMP.11.2.195 https://nasplib.isofts.kiev.ua/handle/123456789/119045 We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter. For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently. We further generalize this procedure to the case of fermions interacting with the discrete boson mode. Andreev-type interactions are responsible for developing a gap in the excitation spectrum. However, the long-range coherence is destroyed due to strong quantum fluctuations. Ми застосовуємо метод потокового рiвняння до дослiдження фермiонних систем з взаємодiями парування, що ведуть до БКШ-нестiйкостi, яка виявляється у появi недiагонального параметра порядку. З цiєю метою ми переформульовуємо неперервне перетворення Боголюбова в ренормгрупових термiнах, де низько- i високоенергетичнi сектори розглядаються послiдовно. Ми узагальнюємо цю процедуру на випадок, коли фермiони взаємодiють з дискретною бозонною модою. Взаємодiї типу Андреєва спричинюють появу щiлини у спектрi збуджень, однак далекосяжна кореляцiя руйнується завдяки сильним квантовим флуктуацiям. The author kindly acknowledges F. Wegner, J. Ranninger and K.I. Wysoki´nski for valuable discussions. This work is partly supported by the Polish Ministry of Science and Education under the grant NN202187833. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Continuous unitary transformation approach to pairing interactions in statistical physics Пiдхiд з використанням неперервних унiтарних перетворень до взаємодiй спарювання в статистичнiй фiзицi Article published earlier
spellingShingle	Continuous unitary transformation approach to pairing interactions in statistical physics Domański, T.
title	Continuous unitary transformation approach to pairing interactions in statistical physics
title_alt	Пiдхiд з використанням неперервних унiтарних перетворень до взаємодiй спарювання в статистичнiй фiзицi
title_full	Continuous unitary transformation approach to pairing interactions in statistical physics
title_fullStr	Continuous unitary transformation approach to pairing interactions in statistical physics
title_full_unstemmed	Continuous unitary transformation approach to pairing interactions in statistical physics
title_short	Continuous unitary transformation approach to pairing interactions in statistical physics
title_sort	continuous unitary transformation approach to pairing interactions in statistical physics
url	https://nasplib.isofts.kiev.ua/handle/123456789/119045
work_keys_str_mv	AT domanskit continuousunitarytransformationapproachtopairinginteractionsinstatisticalphysics AT domanskit pidhidzvikoristannâmneperervnihunitarnihperetvorenʹdovzaêmodiisparûvannâvstatističniifizici

Continuous unitary transformation approach to pairing interactions in statistical physics

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