Unidirectional Charge Instability of the d-wave RVB Superconductor
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Vol. (2009)
ACTA PHYSICA POLONICA A
No. 1(2)
Proceedings of the European Conference “Physics of Magnetism 08”, Pozna´n 2008
Unidirectional Charge Instability ofthe d -wave RVB Superconductor Marcin Raczkowski a , Manuela Capello b and Didier Poilblanc b a Marian Smoluchowski Institute of Physics, Jagellonian University,Reymonta 4, PL-30059 Krak´ow, Poland b Laboratoire de Physique Th´eorique UMR5152, CNRS, F-31062 Toulouse, France (Version date: October 31, 2018)
Starting from a uniform d -wave superconducting phase we study theenergy cost due to imposed unidirectional defects with a vanishing pair-ing amplitude. Both renormalized mean-field theory and variational MonteCarlo calculations within the t - J model yield that the energies of inhomo-geneous and uniform phases are very close to each other. This suggeststhat small perturbations in the microscopic Hamiltonian, might lead to in-homogeneous superconducting phases in real materials as observed in recentscanning tunneling microscopy on Ca − x Na x CuO Cl .PACS numbers: 74.72.-h, 74.20.Mn, 74.81.-g, 75.40.Mg
1. Introduction
Recent progress in spectroscopic techniques has provided a wide varietyof interesting data concerning electronic states of the high- T c superconductors.For example, scanning tunneling microscopy (STM) on different cuprate fami-lies Ca − x Na x CuO Cl and Bi Sr Dy . Ca . Cu O δ , has revealed short-range unidirectional charge domains coexisting with inhomogeneous d -wave supercon-ductivity [1]. In particular, it has been found, that the doped holes primarilyenter oxygen sites leading to a bond-centered charge pattern with a period of fourlattice spacings. Motivated by this result, we have recently shown [2], that sucha charge order might be naturally interpreted in terms of a valence bond crystal[3], i.e., paramagnetic phase with both spatially varying bond charge hoppingand short-range antiferromagnetic (AF) correlations. In this case, an inhomoge-neous antiphase domain resonating valence bond ( π DRVB) phase was obtainedby assuming a π -phase shift in the superconducting (SC) order parameter acrossdomain walls (DWs). While the antiphase solution is particularly intriguing since,in contrast to its inphase counterpart, offers a simple explanation of the suppres-sion of the effective interlayer Josephson coupling observed in some stripe-orderedhigh- T c compounds [4, 5], both types of the modulation of the SC order parameterare the subject of intense ongoing studies [6, 7, 8, 9]. Therefore, in this paper(1)we shall study the energy cost due to imposed defects with a vanishing pairingamplitude (no a π -shift is assumed across the DWs) and compare the resultingcharge modulation with the corresponding one found in the π DRVB phase.
2. Model and the approach
We investigate a t - J model Hamiltonian, H = − t X h ij i ,σ (˜ c † iσ ˜ c jσ + h.c. ) + J X h ij i S i · S j , (1)where ˜ c † iσ = (1 − n i, − σ ) c † iσ is the Gutzwiller projected electron operator anduse a renormalized mean field theory (RMFT) in which the local constraints ofno doubly occupied sites are replaced by statistical Gutzwiller weights g tij ( g Jij )for hopping (superexchange) processes, respectively [10]. Hence the mean-fieldHamiltonian reads, H MF = − t X h ij i ,σ g tij ( c † i,σ c j,σ + h.c. ) − µ X i,σ n i,σ − J X h ij i ,σ g Jij [( χ ji c † i,σ c j,σ + ∆ ji c † i,σ c † j, − σ + h.c. ) − | χ ij | − | ∆ ij | ] , (2)with the Bogoliubov-de Gennes self-consistency conditions for the bond- χ ji = h c † j,σ c i,σ i and pair-order ∆ ji = h c j, − σ c i,σ i = h c i, − σ c j,σ i parameters in the unpro-jected state. We consider here the so-called modified Gutzwiller factors, g Jij = 4(1 − n hi )(1 − n hj ) α ij + 8 n hi n hj β − ij (2) + 16 β + ij (4) , (3) g tij = s n hi n hj (1 − n hi )(1 − n hj ) α ij + 8(1 − n hi n hj ) | χ ij | + 16 | χ ij | , (4)where α ij = (1 − n hi )(1 − n hj ), β ± ij ( n ) = | ∆ ij | n ± | χ ij | n while n hi are local holedensities. By including the effects of the nearest-neighbor correlations χ ij and∆ ij they are known to give a better agreement with a more accurate VariationalMonte Carlo (VMC) technique [2]. Hereafter, we shall assume a typical value t/J = 3 and fix the doping level x = 1 /
8. Finally, using unit cell translationsymmetry [11], RMFT calculations were carried out on large 256 ×
256 clustersat a low temperature βJ = 500 approaching thermodynamic limit.
3. Results and discussion
In Fig. 1 we show the hole profiles as well as the values of the bond- andpair-order parameters across the unit cell found in the π DRVB (top) and inphaseDRVB (bottom) state. The obtained modulations clearly reflect the competi-tion between the superexchange energy E J and kinetic energy E t of doped holes.However, a detailed charge profile depends on the assumed type of the SC orderparameter. On the one hand, suppression of the pair-order amplitude ∆ ij along n h χ x ( y ) ∆ x ( y ) i n h i χ x ( y ) i -0.20.00.2 ∆ x ( y ) (a) (c) (e)(b) (d) (f) Fig. 1. (a,b) Hole density n hi and variational parameters: (c,d) ∆ i,i + α as well as (e,f) χ i,i + α found in the π DRVB (top) and DRVB (bottom) phase. Solid (open) circles inpanels (c-f) correspond to the x ( y ) direction, respectively. the DWs automatically involves a deviation of the bond-order parameter χ ij fromthe value found in the areas with finite ∆ ij . Remarkably, the deviation is par-ticularly strong in the case of the antiphase SC order parameter. On the otherhand, the absence of the π shift across the stripe boundary in the DRVB phaseallows the system (as confirmed by the VMC method [8]) to avoid a reduction of∆ ij on the adjacent vertical bonds which remains almost intact. Therefore, thecharge redistributes from the hole rich areas with enhanced ∆ ij in the π DRVBphase [2], towards DWs with vanishing ∆ ij in the DRVB state (see Fig. 1).In order to appreciate better the reason of a different charge profile in bothphases we show in Fig. 2(a-d) the corresponding short-range AF correlations, S αi = − g Ji,i + α ( | χ i,i + α | + | ∆ i,i + α | ) , (5)with α = { x, y } , as well as bond charge hopping, T αi = 2 g ti,i + α Re { χ i,i + α } , (6)across the unit cell. Here one finds that a local reduction of the SC order param-eter (and the concomitant strong suppression of the superexchange energy on therelated bonds) enables, in the π DRVB phase, a large bond charge hopping alongthe DWs as in the usual stripe scenario [2]. In fact, it also determines the actualhole profile arranged in the way which minimizes the loss of the superexchangeenergy at the DWs. This can be easily accomplished by expelling the holes andstrengthening locally the corresponding g Jij factors. In contrast, small modulationof χ ij in the DRVB phase results in a much weaker, with respect to the π DRVB T x ( y ) -0.32-0.160.00 S x ( y ) ∆ s c x ( y ) i -0.32-0.160.00 S x ( y ) i T x ( y ) i ∆ s c x ( y ) (a) (c) (e)(b) (d) (f) Fig. 2. (a,b) Spin correlation S αi , (c,d) bond charge T αi , and (e,f) SC order parameter∆ SC iα found in the π DRVB (top) and DRVB (bottom) phase. Solid (open) circles corre-spond to the x ( y ) direction, respectively; solid line in panels (e,f) depicts the SC orderparameter in the uniform d -wave RVB phase. one, modulation of both the spin correlations and bond-charge hopping. Conse-quently, the system does not have to further improve the superexchange energyat the defect lines but it rather tries to regain some kinetic energy released on thebroken RVB bonds. This is reached by adjusting the hole profile and attractingthe holes to the DWs which enlarges locally renormalization factors g tij . As aresult, the DRVB phase has a very good kinetic energy being even slightly betterthan that of the uniform d -wave RVB phase (see Table I). Let us point out,however, that even though both the RMFT and VMC methods predict exactlythe same hole profiles in the DRVB phase (as well as its remarkably good energy),a discrepancy appears concerning kinetic energy gain at the DWs, strongly en-hanced in the VMC method [8]. The difference simply follows from the fact thatin the RMFT both the short-range AF correlations and bond-charge hopping are ∝ χ ij . Hence its suppression involves a reduction of both the energy contributionsunless the system is disposed towards a strong phase separation so that they canbe further modified by the Gutzwiller factors [12]. TABLE IRMFT kinetic energy E t , magnetic energy E J , and free energy F as well as VMC energy E VMC of the locally stable phases: π DRVB, DRVB, and d -wave RVB one at x = 1 / E t /J E J /J F/J E VMC /Jπ
DRVB − − − − − − − − − − − − Finally in order to discuss the SC properties of our inhomogeneous phaseswe plot in Fig. 2(e,f) the modulus of SC order parameter,∆ SC iα = g ti,i + α | ∆ i,i + α | , (7)across the unit cell. One of the key qualitative differences between the π DRVBand its inphase counterpart is evident in this figure. Namely, while the SC orderparameter deviates, in the regions between defect lines, only slightly in bothstates from the value found in the uniform d -wave RVB phase, the absence of the π shift across the stripe boundary in the DRVB phase decouples the horizontaland vertical bonds constituting DWs. Therefore, in contrast to the π DRVB phase,the latter retain the value of the SC order parameter of the uniform state.
4. Summary and conclusions
In this paper we have studied two possible modulations of the SC orderparameter across the DWs: inphase and antiphase. Remarkably, we have foundthat the energy of the unidirectional modulated phases (especially of the inphaseconfiguration) approaches the energy of the uniform d -wave RVB superconductor.In fact, the energy difference might be further reduced by the tetragonal latticedistortion that often appears in the high- T c compounds [8]. We conclude thereforethat the d -wave RVB phase is capable of efficient minimizing the energy cost dueto unidirectional defects with broken RVB bonds which in turn might induce thecharge modulation similar to that observed in the STM experiments [1]. Acknowledgments
M.R. acknowledges support from the Foundation for Polish Science (FNP)and from Polish Ministry of Science and Education under Project No. N202068 32/1481. M.C. and D.P. acknowledge the Agence Nationale de la Recherche(France) for support.
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