Interplay between condensation energy, pseudogap and the specific heat of a Hubbard model in a n-pole approximation
A. C. Lausmann, E. J. Calegari, S. G. Magalhaes, C. M. Chaves, A. Troper
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov A. C. Lausmann · E. J. Calegari · S. G.Magalhaes · C. M. Chaves · A. Troper Interplay between condensation energy,pseudogap and the specific heat of aHubbard model in a n-pole approximation
XX.XX.2014
Keywords superconductivity, pseudogap, specific heat, condensation energy,Hubbard model
Abstract
The condensation energy and the specific heat jump of a two-dimensionalHubbard model, suitable to discuss high- T c superconductors, is studied. In thiswork, the Hubbard model is investigated by the Green’s function method withina n -pole approximation, which allows to consider superconductivity with d x − y -wave pairing. In the present scenario, the pseudogap regime emerges when theantiferromagnetic (AF) correlations become sufficiently strong to move to lowerenergies the region around of the nodal point ( p , p ) on the renormalized bands. Itis observed that above a given total occupation n T , the specific heat jump D C andalso the condensation energy U ( ) decrease signaling the presence of the pseudo-gap. It is believed that the two-dimensional Hubbard model is able to capture theessential physics of the high temperature superconductivity (HTSC) in copper-oxides . In such systems, understanding the interplay between the superconduc-tivity and the pseudogap regime could be the key to clarify the mechanisms behindthe unconventional superconductivity. Experimental results for some cuprates in-dicate a close relation among specific heat, condensation energy and the pseudo-gap . More precisely, due to the presence of a pseudogap on the normal statedensity of states, the jump in the specific heat and the superconducting conden-sation energy decrease below a given doping. Besides, according to references ,
1: Laborat´orio de Teoria da Mat´eria Condensada, Universidade Federal de Santa Maria,97105-900, Santa Maria, RS, BrazilE-mail: [email protected]: Universidade Federal Fluminense, Av. Litorˆanea s/n, 24210-346 Niter´oi, RJ, Brazil3: Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro,RJ, Brazil the HTSC phase diagram can be separated in two regimes: a weak coupling anda strong coupling regime . The weak coupling regime could be described, ap-proximately, in terms of the conventional BCS superconductivity while the strongcoupling regime would be governed by unconventional superconductivity. In thisscenario, the pseudogap is a property of the strong coupling regime . In this con-text, the investigation of the specific heat and the condensation energy of two-dimensional Hubbard model may give us important insights about the physics ofthe HTSC.In the present work, the normal-state pseudogap and the superconducting regimeof a two-dimensional Hubbard model is investigated within the Green’s functionstechnique . The pseudogap emerges on the strongly correlated regime in whichthe antiferromagnetic correlations associated with the spin-spin correlation func-tion h S i · S j i becomes sufficiently strong to open a pseudogap in the region ( p , ) on the Fermi surface. Such normal-state pseudogap is also observed in the ( p , ) point of the renormalized band. The repulsive ( U >
0) one band two-dimensional Hubbard model studied here is H = (cid:229) hh i j ii s t i j c † i s c j s + U (cid:229) i s n i , s n i , − s − m (cid:229) i s n i s (1)which takes into account hopping to first and second nearest neighbors. The quan-tity m represents the chemical potential, n i , s = c † i s c i s is the number operatorand c † i s ( c i s ) is the fermionic creation (annihilation) operator at site i with spin s = {↑ , ↓} . We use the Green’s function technique in the Zubarev’s formalism .The equation of motion of the Green’s functions are treated within the n -pole ap-proximation introduced by L. Roth . In this procedure, a set of operators { ˆ A n } is introduced in order to describe the most important excitations of the system. The n -pole approximation assumes that the commutator [ ˆ A n , ˆ H ] , which appears in theequation of motion of the Green’s functions, can be written as [ ˆ A n , ˆ H ] = (cid:229) m K nm ˆ A m where the elements K nm are determined by anti-commuting both sides of this rela-tion with the operator set { ˆ A n } and taking the thermal average. We get K = EN − with E nm = h [[ ˆ A n , ˆ H ] , ˆ A † m ] + i and N nm = h [ ˆ A n , ˆ A † m ] + i . (2)In terms of E and N , the Green’s function matrix is G ( w ) = N ( w N − E ) − N . Both E and N can be determined through equations (2) if the set of operators { ˆ A n } isknown. As we are interested in investigating both the normal and the supercon-ducting regimes, we use the operator set { ˆ A n } = { ˆ c i s , ˆ c i s ˆ n i − s , ˆ c † i − s , ˆ n i s ˆ c † i − s } .The energy per particle can be obtained from the Green’s function followingthe procedure described by Kishore and Joshi . For the Hubbard model intro-duced in equation (1), the internal energy per particle in the superconducting stateis: E = L (cid:229) k , s (cid:229) i = Z i , k s ( e k + m + E i , k s ) f ( E i , k s ) − m n T (3) where n T = n − s + n s is the total occupation, Z i , k s are the spectral weights ofthe Green’s function G ( ) k , s = hh c k , s ; c † k , s ii and f ( w ) is the Fermi function. In thesuperconducting state, the renormalized bands are: E i , k s = ( − ) ( i + ) vuut w j , k s + ( − ) ( j + ) | g k | [( e k + Un − s − m ) − w j , k s ] n − s ( − n − s ) ( w , k s − w , k s ) , (4)with j = i = j = i = g k = t g ( cos ( k x a ) − cos ( k y a )) isthe gap function and g is the superconducting order parameter with d x − y -wavesymmetry . In the normal state, the renormalized bands are: w j , s k = U + e k + W k , s − m − ( − ) ( j + ) X k , s X k = p ( U − e k + W k s ) + h n − s i U ( e k − W k s ) and e k is the unperturbedband energy e k = t [ cos ( k x a ) + cos ( k y a )] + t cos ( k x a ) cos ( k y a ) where t is thefirst-neighbor and t is the second-neighbor hopping amplitudes. W k , s , is a bandshift that depends on the correlation function h S i · S j i .The specific heat jump is D C = [ ( C S − C N ) C N ] T = Tc with C S , N = ¶ E S , N ¶ T , E S and E N being the energy per particle in the superconducting and in the normal sate, re-spectively. E N is obtained from equation (3) keeping the superconducting orderparameter equal to zero ( g = ) . Now, let’s define U ( T ) = F N − F S as the differ-ence between the normal ( F N ) and the superconducting ( F S ) states Helmholtz freeenergy. The superconducting condensation energy is defined as: U ( ) = E N − E S . (6) The main focus of the present work is the strong coupling regime in which un-conventional superconductivity may occurs. For this purpose, we analyzed therenormalized bands, the superconducting condensation energy and the specificheat jump as a function of the total occupation n T and of the interaction U .Figure 1 shows the renormalized band w , s k . In the left panel, w , s k is shownfor different values of total the occupation n T . The inset displays the region nearthe point ( p , ) in which a pseudogap develops when the occupation is increased.For instance, when n T = .
81, the band intersects the Fermi energy e F , but, for n T = .
85 the band does not reaches the Fermi energy giving rise to a pseudogapbetween the band and e F . The right panel shows w , s k for n T = .
85 and differ-ent interactions U . The inset highlights a pseudogap on ( p , ) for U = . | t | and U = . | t | and the absence of pseudogap for U = . | t | . In the n -pole approxi-mation used in this work the Green’s functions naturally present a pole struc-ture which contains the spin-spin correlation function h S i · S j i . In the presentscenario the pseudogap emerges when the correlation function h S i · S j i becomessufficiently strong to move to lower energies the region of the nodal point ( p , p ) of the renormalized band w , s k . This occurs because the renormalized band w , s k -5-4-3-2-1 0 1 2 w , k k U=8.0|t|t= -1.0 eVt =0.2|t|k B T=0.0(0,0) ( p , p ) ( p ,0) (0,0) n T =0.81n T =0.83n T =0.85n T =0.87 -0.2-0.1 0 k n T =0.85t= -1.0 eVt =0.2|t|k B T=0.0 ( p , p ) ( p ,0) (0,0) U=6.0|t|U=8.0|t|U=10.0|t| -0.2-0.1 0
Fig. 1 (Color online) The left panel shows the renormalized band w , s k for different occupa-tions. In the inset, the region of the ( p , ) point shows the pseudogap for n T & .
83. In the rightpanel, the bands for n T = .
85 and different interactions U . In the inset, the region of the ( p , ) point shows the pseudogap for U & . | t | . D O S w U=8.0|t|t=-1.0 eVt =0.2|t|k B T=0.0 n T =0.81n T =0.83n T =0.85n T =0.87 Fig. 2 (Color online) The density of states (DOS) for different occupations n T . The vertical linein w = e F . The inset shows the region near the Fermienergy e F . For n T & .
83 the DOS at e F is decreased indicating the presence of a pseudogap. is deeply influenced by the momentum structure of the spin-spin correlation func-tion h S i · S j i . Due to the antiferromagnetic character of h S i · S j i , the regionof the nodal point ( p , p ) of w , s k is strongly affected (see figure 1). As a conse-quence a pseudogap arises at the anti-nodal point ( p , ) . Moreover, the h S i · S j i isvery sensitive to n T and U , indeed, |h S i · S j i| increases with n T and U . Therefore,when |h S i · S j i| reaches a critical value |h S i · S j i| c the pseudogap emerges.The density of states (DOS) for the renormalized band w , s k is shown in figure2 for different occupations n T . The vertical line in w = e F and the model parameters are shown in the figure. When n T increases, the correlations become stronger, resulting in a narrowing of thedensity of states. However, the most important feature observed in the DOS isthe reduction of the DOS on e F for n T & .
83. Such reduction is an effect of thepresence of a pseudogap in the strong correlated regime of the system. U ( ) (a)t=-1.0U=8|t|t =0.2|t|-0.17-0.16-0.15-0.14-0.13-0.12 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 Æ S i • S j æ n T (b) t=-1.0t =0.2|t|(c) n T =0.83n T =0.85n T =0.87 5 6 7 8 9 10 11 12U/|t|(d) n T =0.83n T =0.85n T =0.87 Fig. 3 (Color online) In (a), the condensation energy as a function of the total occupation n T . (b)The behavior of the spin-spin correlation function for the same model parameters considered in(a). In (c) the condensation energy as a function of the interaction U for different occupations. (d)The spin-spin correlation function as a function of U for the same model parameters consideredin (c). Details about these results are given in the text. The condensation energy U ( ) as a function of the total occupation n T isshown in figure 3(a). Notice that U ( ) increases with n T , reaches a maximum andthen start to decrease. The depletion of U ( ) for a n T greater than a given value isrelated to the development of a pseudogap near the anti-nodal points on the Fermisurface. This result is in qualitative agreement with experimental data obtainedfor some cuprate systems . Figure 3(b) shows h S i · S j i as a function of n T . Thehorizontal dotted line indicates approximately the value of h S i · S j i , from whichthe system enters in the underdoped strong coupling regime. Figure 3(c) displaysthe behavior of the condensation energy U ( ) as a function of the interaction U for several occupations. It is interesting to note that there is an optimal value of U which produces a maximum U ( ) . However, such optimum value changes withthe occupation n T . This feature is associated to the opening of the pseudogapwhich occurs in the strong coupling regime. Therefore, if n T decreases, a highervalue of U is necessary for the system to access the strong coupling regime. Thecorrelation function h S i · S j i shown in figure 3(d) may serve as a parameter to indi-cate that the system is reaching the strong coupling regime. As in figure 3(b), thehorizontal dotted line indicates approximately the value of h S i · S j i , from whichthe system enters in the underdoped strong coupling regime. The results for con-densation energy U ( ) versus n T , figure 3(a), are in qualitatively agreement witha method bansed on the resonanting valence bond (RVB) spin liquid and alsowith results from the fluctuation-exchange (FLEX) approximation . In the FLEXapproximation, U ( ) increases with U but does not present a maximum like the D C n T U=8.0|t|t=-1.0 eVt =0.2|t| Fig. 4 (Color online) The jump in the specific heat as a function of the total occupation (see thetext). one observed in figure 3(c). There are no available results for U ( ) versus U inthe RVB method .The figure 4 shows the specific heat jump D C as a function of n T . Noticethat initially the D C increases slightly with n T but, above n T ≈ . D C startsto decrease. The decreasing in D C is an evidence of the presence of a pseudogapin the underdoped regime and is close related to the development of a pseudogapon the density of states (DOS) (see figure 2). The decreasing of the DOS on e F when n T increases, is directly related to the pseudogap on e F and agrees witha high-resolution photoemission study of La − x Sr x CuO which suggests thata pseudogap is the main responsible for the similar behavior between the specificheat jump and the DOS( e F ) observed in the underdoped regime. This result for D C agrees at least qualitatively, with a method based on the resonating valence bond(RVB) spin liquid in which the specific heat jump and the condensation energydecrease due to the opening of a pseudogap in the underdoped regime. Also, theresult for D C shown in figure 4 is in qualitative agreement with experimental datafor some cuprates . In this work we have investigated the superconducting condensation energy U ( ) and the specific heat jump D C of a two-dimensional Hubbard model. The resultsshow that both U ( ) and D C decreases in the strong coupling underdoped regime.It has been verified that this behavior is related to the opening of a pseudogapat the anti-nodal point ( p , ) on the renormalized band w , s k . In the strong cou-pling regime, the correlation function h S i · S j i presents in the band shift becomessufficiently strong to move to lower energies the renormalized band w , s k in theregion of the nodal point ( p , p ) and as a consequence, a pseudogap opens in the ( p , ) point. The results obtained here corroborate the scenario that attributes thepseudogap to the strong correlations present in the underdoped regime . Acknowledgements
This work was partially supported by the Brazilian agencies CNPq, CAPESand FAPERGS.
References
1. Hubbard J.,
J. Proc. Roy. Soc. London A , 238 (1963).2. D. Scalapino, in Handbook of High-Temperature Superconductivity, editedby J. Schrieffer and J. Brooks (Springer, New York, 2007), pp. 495526.3. E. Gull, A. J. Millis,
Phys. Rev. B , 241106(R) (2012).4. J. W. Loram, K. A. Mirza, J. R. Cooper, W. Y. Liang, J. M. Wade, J. Super-cond. , 243 (1994).5. J. W. Loram, J. Luo, J. R. Cooper, W. Y. Liang, J. L. Tallon, J. Phys. Chem.Solids , 56 (2001).6. J. W. Loram, J. L. Luo, J. R. Cooper, W. Y. Liang, J. L. Tallon, Physica C ,831 (2000).7. T. Matsuzaki, N. Momono, M. Oda, M. Ido,
J. Phys. Soc. Japan , 2232(2004).8. E. Gull, O. Parcollet, A. J. Millis, Phys. Rev. Lett. , 216405 (2013).9. L. M. Roth,
Phys. Rev. , 451 (1969).10. J. Beenen, D. M. Edwards,
Phys. Rev. B , 13636 (1995).11. D. N. Zubarev, Sov. Phys. Usp. , 320 (1960).12. R. Kishore, S. K. Joshi, J. Phys. C: Solid St. Phys. , 2475 (1971).13. E. J. Calegari, S. G. Magalhaes, C. M. Chaves, A. Troper, Solid State Com-mun. , 20 (2013).14. E. J. Calegari, S. G. Magalhaes, Int. J. Mod. Phys.
B 25 , 41 (2011).15. T. Herrmann, W. Nolting,
J. Magn. Magn. Mater. , 253 (1997).16. J. P. F LeBlanc, E. J. Nicol and J. P. Carbotte,
Phys. Rev. B J. Phys. Soc. Japan , 1534 (2005).18. A. Ino, T Mizokawa, K. Kobayashi and A. Fujimori Phys. Rev. Lett.81