d-wave Superconductivity, Orbital Magnetism, and Unidirectional Charge Order in the t-J Model
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y d -wave Superconductivity, Orbital Magnetism, andUnidirectional Charge Order in the t - J Model ∗ Marcin Raczkowski
Marian Smoluchowski Institute of Physics, Jagellonian University,Reymonta 4, PL-30059 Krak´ow, PolandRecent scanning tunneling microscopy in the superconducting regimeof two different cuprate families has revealed unidirectional bond-centeredmodulation in the local electronic density of states. Motivated by thisresult we investigate the emergence of modulated d -wave superconductivitycoexisting with charge domains that form along one of the crystal axes.While detailed stripe profiles depend on the used form of the Gutzwillerfactors, the tendency towards a valence bond crystal remain robust. Wealso find closely related stripe phase originating from the staggered fluxphase, a candidate for the pseudogap phase of lightly doped cuprates.PACS numbers: 74.72.-h, 74.20.Mn, 74.81.-g, 75.40.Mg
1. Introduction
It is now well established that the simplest model proposed to describethe physics of the high- T c superconductors, the so-called t - J model [1], 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 and n iσ is the particle number operator, yields apart from the true long-rangemagnetic order, characteristic of the undoped parent Mott insulators, anarray of quantum SU(2)-invariant ground states [2]. In fact, it is quite nat-ural to expect that strong quantum fluctuations arising from both low spin S = 1 / planes, should lead to quantum disordered states with only short-range an-tiferromagnetic (AF) spin correlations. The most famous example of such ∗ Presented at the XIII National School ”Superconductivity, . . . ”, L¸adek Zdr´oj 2007 (1)
Marcin Raczkowski states is a resonating valence bond (RVB) phase [3]. Remarkably, Ander-son’s RVB theory based on a Gutzwiller projected BCS trial wave function,which parameters are usually determined either by using renormalized meanfield theory (RMFT) [4] or by Variational Monte Carlo (VMC) method [5],not only predicted correctly the d -wave symmetry of the superconducting(SC) order parameter [6], but in addition, it reproduced experimental dop-ing dependence of a variety of physical observables in the SC regime [7].Moreover, the tendency towards valence bond amplitude maximizationmight enhance charge and spin stripe correlations in the d -wave RVB state.The presence of charge and long-range spin stripe order has been detectedin neutron scaterring experiments and confirmed in resonance x -ray scatter-ing in a few special cuprate compounds, namely La . − x Nd . Sr x CuO andLa − x Ba x CuO [8]. However, such stripe order competes with supercon-ductivity and thus strongly reduces T c [9]. In contrast, recent scanning tun-neling microscopy (STM) on different cuprate families Ca − x Na x CuO Cl and Bi Sr Dy . Ca . Cu O δ , has revealed intense spatial variations inasymmetry of electron tunneling currents with bias voltage that forms uni-directional domains coexisting with inhomogeneous d -wave superconduc-tivity [10]. In particular, it has been found, that the asymmetry occursprimarily at the oxygen sites being indicative of a short-range bond-centered charge pattern with a period of four lattice spacings. In this paper weshow that the bond-centered modulation observed in the STM experimentsmight be naturally interpreted in terms of a valence bond crystal, i.e., spin-rotationally invariant phase with spatially varying bond charge hopping anda concomitant modulation of short-range AF correlations [11].
2. Renormalized mean-field theory
We begin by discussing RMFT of the t - J model applied to the casewith homogeneous charge distribution. In this approach, the Gutzwillerprojection removing double occupancy is handled with statistical weightfactors g t = 2 x/ (1 + x ) and g J = 4 / (1 + x ) which account for differentprobabilities of hopping and superexchange processes in the projected andunprojected wave functions. Hence the mean-field Hamiltonian reads, H = − 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 -wave Superconductivity, Orbital Magnetism, and Unidirectional . . . Γ S M X S Y Γ -6-4-20246 Γ S M X S Y Γ -6-4-20246 Γ S M X S Y Γ -6-4-20246 E k / J (b) (c)(a) Fig. 1. Electronic structure along the main directions of the Brillouin zone of: (a) d -wave RVB/SF ( x = 0); (b) d -wave RVB and (c) SF phase (both for x = 1 / the unprojected state. Hereafter, we shall assume a typical value t/J = 3.Even though the original proposal for the high- T c superconductivity wasthe s -wave BCS wave function, it immediately turned out that it is the d -wave BCS state with ∆ ij = ± ∆ for the nearest-neighbor pairs along the x ( y ) axis, respectively, which gives the lowest energy [6]. At half-filling,such a phase is equivalent to the staggered flux (SF) state with complex χ ij = | χ | exp[( − i x + j y iφ ] yielding circulating currents whose chirality al-ternates from plaquette to plaquette [12]. In this limit, the t - J model re-duces to the Heisenberg Hamiltonian with the local SU(2) gauge symmetrycorresponding to the following particle-hole transformation, (cid:18) c † i ↑ c i ↓ (cid:19) = (cid:18) α i β i − β ∗ i α ∗ i (cid:19) (cid:18) c † i ↑ c i ↓ (cid:19) , (3)with α i α ∗ i + β i β ∗ i = 1. It mixes an ↑ -spin particle with a ↓ -spin hole andhence decouplings in terms of ∆ or χ become indeed equivalent. In order toappreciate this better let us consider the related Hamiltonian matrices usingthe Bogoliubov-Nambu formalism with η k = ( c k ↑ , c †− k ↓ ) for the d -wave RVBphase and η k = ( c k ,σ , c k + Q ,σ ) with Q = ( π, π ) for the SF phase: M RVB k = (cid:18) − ε k − µ ∆ k ∆ k ε k + µ (cid:19) , M SF k = (cid:18) − ε k − µ iχ k − iχ k ε k − µ (cid:19) , (4)where ε k = ( tg t + J g J Reχ ) γ + , ∆ k = J g J ∆ γ − , χ k = J g J Imχγ − with γ ± = 2(cos k x ± cos k y ). Therefore, the corresponding spectra are given by: E RV B k = ± q ( ε k − µ ) + ∆ k , and E SF k = − µ ± q ε k + χ k . (5)At half-filling ( µ = 0), one finds ∆ = χ = 0 .
169 ( | χ | = 0 .
239 and φ = π/ d -wave RVB (SF) phase, respectively. Hence, in the latter case Marcin Raczkowski x ∆ , ∆ s c x Φ , - S (a) (b) Fig. 2. Doping dependence of: (a) pairing amplitude ∆ (open squares) and SC orderparameter ∆ SC (solid squares) in the d -wave RVB phase, as well as (b) plaquetteflux Φ (open circles) and spin correlations S (solid circles) in the SF phase. Reχ = Imχ = 0 .
169 and, since ∆ k = χ k , both spectra become degener-ate. As shown in Fig. 1(a), the key feature of the obtained spectrum isthat the energy gap vanishes linearly along the S = ( π/ , π/
2) point form-ing a cone-like dispersion. While this cone remains pinned to the Fermisurface in the d -wave RVB phase, finite doping takes the node of the SForder away from the Fermi surface and opens hole pockets around the S point [see Fig. 1(b,c)]. Nevertheless, both excitation spectra remain similar,which makes the SF phase an excellent candidate for the normal pseudogapphase that emerges below a characteristic temperature T ∗ . Moreover, short-range staggered orbital current-current correlations have been found in theGutzwiller-projected d -wave RVB phase [13], in the exact ground state ofthe t - J model with a negative two-hole binding energy [14], as well as byanalyzing motion of a hole pair in the AF background [15].Next, as shown in Fig. 2(a), variational parameter ∆ is the largest at x = 0 and vanishes linearly with doping. In contrast, g t is an increasingfunction of x so that the resulting SC order parameter ∆ SC = g t ∆ re-produces qualitatively the SC dome. Finally, Fig. 2(b) depicts the dopingdependence of the fictitious flux (in unit of the flux quantum) defined by asum over the four bonds of the plaquette Φ ✷ = π P h ij i∈ ✷ Θ ij as well as theAF spin correlations S = − g J | χ | . Here, the appearance of a finite flux at x ≃ .
15 clearly strengthens S with respect to the Fermi liquid state where χ is entirely real.
3. Unidirectional charge order
We turn now to the discussion of bond-centered (with a maximum of thehole density spread over two-leg ladders) inhomogeneous RVB (chiral) statesderived from the parent d -wave RVB (SF) phases, respectively. Hereafter we -wave Superconductivity, Orbital Magnetism, and Unidirectional . . . T x ( y ) n h -0.32-0.160.00 S x ( y ) ∆ s c x ( y ) i n h i -0.32-0.160.00 S x ( y ) i T x ( y ) i -0.030.000.03 ∆ s c x ( y ) (a) (c) (e) (g)(b) (d) (f) (h) Fig. 3. (a,b) Hole density n hi , (c,d) SC order parameter ∆ SC iα , (e,f) spin correlation S xi , and (g,h) bond charge T yi , found in the π DRVB phase. Top (bottom) panelsdepict the results obtained using original (modified) Gutzwiller factors; solid (open)circles in panels (c-h) correspond to the x ( y ) direction, respectively. refer to the former as π -phase domain RVB phase ( π DRVB), as it involvestwo out-of-phase SC domains (see also Ref. [16]), separated by horizontalbonds with vanishing pairing amplitudes, named as “domain wall” (DW),where ∆ ij gains a phase shift of π . Similarly, due to the existence of DWswhich act as nodes for the staggered current and introduce into the SF orderparameter a phase shift of π , we refer to the latter as π DSF state.We consider both original ( q = 0) and modified ( q = 1) Gutzwillerfactors depending on local hole densities n hi , g Jij = 4(1 − n hi )(1 − n hj ) α ij + q [8 n hi n hj β − ij (2) + 16 β + ij (4)] , (6) g tij = s n hi n hj (1 − n hi )(1 − n hj ) α ij + q [8(1 − n hi n hj ) | χ ij | + 16 | χ ij | ] , (7)with α ij = (1 − n hi )(1 − n hj ) and β ± ij ( n ) = | ∆ ij | n ±| χ ij | n . Note, however, thatthe Gutzwiller renormalization scheme becomes substantially more compli-cated in the case of inhomogeneous charge distribution as the local densitymay change before/after projection [17]. As a consequence, Eqs. (6) and (7)may provide only an approximate way of the Gutzwiller projection. Finally,using the unit cell translation symmetry [18], calculations were carried outon a large 256 ×
256 cluster at a low temperature βJ = 500.The corresponding stripe profiles in both phases shown in Figs. 3 and 4are clearly a compromise between the superexchange energy E J and kinetic Marcin Raczkowski T x ( y ) n h -0.32-0.160.00 S x ( y ) Φ π i n h i -0.32-0.160.00 S x ( y ) i T x ( y ) i -0.40.00.4 Φ π (a) (c) (e) (g)(b) (d) (f) (h) Fig. 4. (a,b) Hole density n hi , (c,d) modulated flux Φ πi , (e,f) spin correlation S xi ,and (g,h) bond charge T yi , found in the π DSF phase. Top (bottom) panels depictthe results obtained using original (modified) Gutzwiller factors; solid (open) circlesin panels (e-h) correspond to the x ( y ) direction, respectively. energy E t of doped holes. On the one hand, a reduction of the SC or fluxorder parameters (the latter known to frustrate coherent hole motion [19])enables a large bond charge hopping T yi = 2 g ti,i + y Re { χ i,i + y } along the DWsas in the usual stripe scenario [18]. On the other hand, it simultaneouslyresults in the suppression of the AF correlations S xi = − g Ji,i + x ( | χ i,i + x | + | ∆ i,i + x | ) along the transverse bonds.However, a closer inspection of Figs. 3 and 4 as well as Table I indicatesthat this competition is especially subtle in the π DRVB state. Indeed, in-stead of increasing hole level in the SC areas in order to reinforce the SCorder parameter, the system prefers a more spread out charge distribution,which suggests that the d -wave RVB state is less disposed to phase separa-tion than the SF one where in fact also other charge instabilities have beenfound [20, 21]. Moreover, as listed in Table I, both E t and E J are reducedwith respect to the uniform d -wave RVB superconductor. In contrast, π DSFphase fully optimizes both energy contributions simply by expelling holesfrom the regions between the stripes and accommodating them at the DWs.Indeed, a low local doping level strengthens plaquette flux which reachesthe value Φ ✷ ≃ .
35 expected for x ≃ . g tij andallow the phase to retain a favorable E t . Taken together, these two effectsare responsible for a much stronger charge modulation of the π DSF phaseas compared to its π DRVB counterpart. -wave Superconductivity, Orbital Magnetism, and Unidirectional . . . E t , magnetic energy E J , and free energy F of the locallystable phases: π DSF, SF, π DRVB, and d -wave RVB one at x = 1 / g ij modified g ij phase E t /J E J /J F/J E t /J E J /J F/Jπ DSF − − − − − − − − − − − − π DRVB − − − − − − − − − − − − Unfortunately, the total RMFT energy in both phases differs substan-tially from the one obtained within the VMC scheme: E DRVB /J ≃ − . E DSF /J ≃ − .
33) for the π DRVB ( π DSF) phase, respectively [11]. We con-sider therefore the so-called modified Gutzwiller factors where the effects ofthe nearest-neighbor correlations χ ij and ∆ ij are also included [22]. Firstof all, one observes that the inclusion of the intersite correlations weakens(strengthens) stripe order in the π DRVB ( π DSF) phase, respectively. In-deed, in both cases the holes are ejected from the regions in between stripesinto the DWs defined as nodes of the SC/flux order parameter. Similartendency towards accommodating the holes at the DWs has also been es-tablished in the VMC calculations [23]. Furthermore, while the short-rangeAF correlations remain either unaltered or they are changed in such a waythat E J remains almost constant, modified Gutzwiller factors mainly renor-malize bond charge hopping. As a consequence, the total energy in bothphases approaches the one found in the VMC scheme (see Table I).In summary, we believe that our results provide insights into the for-mation of the recently found bond-centered charge order that coexists withmodulated d -wave superconductivity. Moreover, we expect a further en-hancement of the proposed valence bond crystal near impurities that breakthe space group symmetry of the t - J Hamiltonian by producing a modu-lation in the magnitude of the superexchange coupling [24]. In fact, it hasrecently been argued that the dopant-induced spatial variation of the atomiclevels indeed strengthens locally the AF superexchange interaction [25].
Acknowledgments
The author acknowledges support from the Foundation for Polish Sci-ence (FNP), Minist`ere Fran¸cais des Affaires Etrang`eres under Bourse deRecherche, as well as from Polish Ministry of Science and Education underProject No. N202 068 32/1481.
Marcin Raczkowski
REFERENCES [1] K. A. Chao, J. Spa lek, A. M. Ole´s,
J. Phys. C , L271 (1977); Phys. Rev. B , 3453 (1978); F. C. Zhang, T. M. Rice, ibid. , 3759 (1988).[2] S. Sachdev, Rev. Mod. Phys. , 913 (2003).[3] P. W. Anderson, Science , 1196 (1987).[4] F. C. Zhang, C. Gros, T. M. Rice, H. Shiba,
Supercond. Sci. Technol. , 36(1988).[5] S. Sorella, G. B. Martins, F. Becca, C. Gazza, L. Capriotti, A. Parola, E.Dagotto, Phys. Rev. Lett. , 117002 (2002).[6] G. Kotliar, J. Liu, Phys. Rev. B , 5142 (1988).[7] K.-Y. Yang, C. T. Shih, C. P. Chou, S. M. Huang, T. K. Lee, T. Xiang, F. C.Zhang, Phys. Rev. B , 224513 (2006).[8] N. B. Christensen, H. M. Rønnow, J. Mesot, R. A. Ewings, N. Momono, M.Oda, M. Ido, M. Enderle, D. F. McMorrow, A. T. Boothroyd, Phys. Rev. Lett. , 197003 (2007); P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A.Sawatzky, D. L. Feng, Nature Physics , 155 (2005).[9] T. Valla, A. V. Fedorov, J. Lee, J. C. Davis, G. D. Gu, Science , 1914(2006).[10] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M.Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, J. C. Davis,
Science ,1380 (2007); see also J. Zaanen, ibid. , 1372 (2007).[11] M. Raczkowski, M. Capello, D. Poilblanc, R. Fr´esard, A. M. Ole´s,
Phys. Rev.B , 140505(R) (2007); M. Vojta, O. R¨osch, Phys. Rev. B , 094504 (2008).[12] I. Affleck, J. B. Marston, Phys. Rev. B , R3774 (1988).[13] D. A. Ivanov, P. A. Lee, X.-G. Wen, Phys. Rev. Lett. , 3958 (2000).[14] P. W. Leung, Phys. Rev. B , R6112 (2000).[15] P. Wr´obel, R. Eder, Phys. Rev. B , 184504 (2001).[16] A. Himeda, T. Kato, M. Ogata, Phys. Rev. Lett. , 117001 (2002).[17] B. Edegger, V. N. Muthukumar, C. Gros, Adv. Phys. ,
927 (2007).[18] M. Raczkowski, R. Fr´esard, A. M. Ole´s,
Phys. Rev. B , 174525 (2006); Europhys. Lett. , 128 (2006).[19] D. Poilblanc, Y. Hasegawa, Phys. Rev. B , 6989 (1990).[20] D. Poilblanc, Phys. Rev. B , 060508(R) (2005).[21] M. Raczkowski, D. Poilblanc, R. Fr´esard, A. M. Ole´s, Phys. Rev. B , 094505(2007).[22] M. Sigrist, T. M. Rice, F. C. Zhang, Phys. Rev. B , 12058 (1994).[23] M. Capello, M. Raczkowski, D. Poilblanc, in press, Phys. Rev. B ,arXiv:0801.2722.[24] M. A. Metlitski, S. Sachdev, Phys. Rev. B , 054411 (2008).[25] M. M. Ma´ska, ˙Z. ´Sled´z, K. Czajka, M. Mierzejewski, Phys. Rev. Lett.99