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Superconductivity and a Mott transition in a Hubbard model on an anisotropic triangular lattice
Physica C 445–448 (2006) 166–170 www.elsevier.com/locate/physc
Superconductivity and a Mott transition in a Hubbard model on an anisotropic triangular lattice Tsutomu Watanabe b
a,b,*
, Hisatoshi Yokoyama c, Yukio Tanaka
a,b
, Jun-ichiro Inoue
a
a Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan CREST Japan Science and Technology Corporation (JST), Nagoya 464-8603, Japan c Department of Physics, Tohoku University, Sendai 980-8578, Japan
Available online 4 May 2006
Abstract With a mind to a Mott transition and superconductivity (SC) arising in j-(BEDT-TTF)2X, we study a half-filled-band Hubbard model on an anisotropic triangular lattice (t in two bond directions and t 0 in the other). Using an optimization variational Monte Carlo method, we find that with doublon–holon binding factors in the trial functions, a first-order Mott transition takes place at U = Uc approximately of the band width. For a moderate frustration (t 0 /t = 0.4), the d-wave superconducting correlation Pd is sizably enhanced just under Uc/t, whereas for a strong frustration (t 0 /t = 0.8), Pd is hardly enhanced for any value of U/t. Since the behavior of Pd is closely connected to that of the spin structure factor S(p, p), the SC in this model must be induced by the antiferromagnetic correlation. 2006 Elsevier B.V. All rights reserved. PACS: 74.70.b; 74.20.z Keywords: Hubbard model; Anisotropic triangular lattice; Variational Monte Carlo method; Mott transition
1. Introduction Organic layered compounds, j-BEDT-TTF (ET) salts, abbreviated as j-(ET)2X [1], are very intriguing in some points. (i) The j-type compounds have quasi-two-dimensional conducting planes whose structure is magnetically frustrated and quite similar to that of high-Tc cuprates. (ii) Superconductor (SC)-to-insulator (antiferromagnetic (AF) or nonmagnetic) transitions take place in low temperatures, by applying (chemical) pressure. According to extended Hu¨ckel calculations [2], the conduction band in j-(ET)2X is composed of antibonding orbitals of dimerized ET molecules, and a hole exists on each unit dimer, forming a half-filled band. The connectivity of the ET dimers is properly modeled by an anisotropic triangular lattice with * Corresponding author. Address: Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan. Tel.: +81 52 789 3700; fax: +81 52 789 3298. E-mail address: [email protected] (T. Watanabe).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.03.107
a hopping integral t in two directions and t 0 in the third direction. Within the extended Hu¨ckel approximation, the anisotropy (or frustration) jt 0 /tj is estimated at 0.5– 1.1, which is fairly larger than the values of cuprates 0.1– 0.3 in a model with t 0 in both diagonal bond directions. Thus, it is an important issue to clarify whether a new superconducting (SC) mechanism is realized due to the strong frustration or the AF spin correlation (or fluctuation) causes superconductivity (SC) as in the cuprates [3,4]. Another vital issue is to consider the mechanism of a SC-insulator (so-called Mott) transition. In mean-field [5,6], Gutzwiller [7] and fluctuation exchange (FLEX) [8] approximations, the term ‘Mott transition’ is used as a transition between a conductive state and an AF insulator. However, the mechanism of a Mott transition can be basically independent of magnetic properties, because a nonmagnetic insulating state seems to be realized in j-(ET)2Cu2(CN)3 [9]. The purpose of this work is to illuminate the above two issues, by studying a Hubbard model on an anisotropic
T. Watanabe et al. / Physica C 445–448 (2006) 166–170
2. Formulation As a model of the conducting plane in j-(ET)2X, we consider a Hubbard model on an anisotropic triangular lattice (or an extended square lattice with next-nearest-neighbor hopping t 0 only in one diagonal direction [1, 1]) [5]: X y X H¼ ek ckr ckr þ U ni" ni# ð1Þ i
kr
with ek ¼ 2tðcos k x þ cos k y Þ 2t0 cosðk x þ k y Þ;
ð2Þ
and t, t 0 , U > 0. Throughout this paper, we fix the electron density at half filling. To this model, we apply an optimization VMC method [10], which can treat the whole parameter space spanned by U/t and t 0 /t. As a variational wave function, we use a two-body Jastrow type: W = PU, where U is a one-body (Hartree–Fock) part expressed as a Slater determinant, and P is a correlation factor. As forQP, in addition to the onsite (Gutzwiller) factor, P G ¼ i ½1 ð1 gÞni" ni# , we introduce a crucial intersite factor PQ (P = PQPG), which binds a doublon (doubly occupied site) to a holon (empty site) within short distances [11]: Y 0 ð1 lQsi Þð1 l0 Qsi Þ; ð3Þ PQ ¼ i sðs0 Þ Qi
¼
Y
d i ð1 eiþsðs0 Þ Þ þ ei ð1 d iþsðs0 Þ Þ ;
3. Results We start with a comparison of stability between the dwave and AF states. We know that the AF state is much stabler for t 0 /t = 0 (normal square lattice) as far as U is positive, due to the complete nesting condition. As t 0 /t increases, however, the AF state becomes less stable. In Fig. 1, we show condensation energies per site of the AF and d-wave states, EdðAFÞ ¼ Ed ðEAF Þ EF , as functions of c U/t. For t 0 /t = 0.4, the stable range of the AF state is reduced to 5 < U/t < 6.5 for L = 10, and EAF is always c smaller than Edc for L = 12, as observed in the inset. It follows that the ground state switches to the d-wave state from the AF state for t 0 /t J 0.4. Actually, for t 0 /t = 0.8, the AF state is never stabilized with respect to the normal state. Next, let us look at Edc . For small values of U/t, Edc is very small, whereas it abruptly increases at U/t 6.5 (8.0) for t 0 /t = 0.4 (0.8), regardless of system size. This is nothing but a sign of a Mott transition, as we will see in the following. In Fig. 2, we plot the doublon density, D = hni"ni#i, which is considered as the order parameter of Mott transitions. For t 0 /t = 0.4 (0.8), D abruptly decreases at U = Uc 6.55t ± 0.05t (8.05t ± 0.05t). These critical values agree with those of Edc discussed in the preceding paragraph. For a large value of t 0 /t(=0.8), the behavior of D at
0.15
ð4Þ Ec / t
sðs0 Þ
0.1
0.05
t'/t=0.8 L=10 L=12
AF
t'/t=0.4
0.015
t'/t=0.4 L=10 L=12
0.01
0.005 0
0 4
6
8
5
10 U/t
6
U /t
12
7
14
Fig. 1. Condensation energies of the d-wave and AF states for t 0 /t = 0.4 and 0.8 as functions of U/t. The systems are of 10 · 10 and 12 · 12. The inset shows the magnification of the region of U Uc for t 0 /t = 0.4.
0.15 d-wave
0.1
t'/t =0.4, L=10 t'/t =0.4, L=12 t'/t =0.8, L=10 t'/t =0.8, L=12
D
where di = ni"ni#, ei = (1 ni")(1 ni#), and s(s 0 ) runs over all nearest-neighbor sites in the directions of t(t 0 ). For l ! 1 in Eq. (3), a doublon (negative-charged) and a holon (positive-charged) are bound at adjacent sites, so that a current cannot flow. Without PQ, a Mott transition cannot be described by W [12]. Regarding U, we study three states: (i) the Fermi sea: UF, (ii) a BCS function with a dx2 y 2 -wave gap Dk = Dd(cos kx cos ky): Ud(Dd, f), and (iii) a Hartree– Fock-type AF state: UAF(DAF). Here, Dd, f, and DAF are, in VMC, variational parameters to be optimized, which correspond to the d-wave gap, chemical potential, and AF gap, respectively, in the Hartree–Fock theory. Furthermore, we take account of a renormalization effect of the quasi-Fermi surface in the d-wave state due to the electron correlation [13]. This effect is introduced by optimizing the value of t 0 /t in ek in the wave function Ud as a variational parameter, independently of t 0 /t given in the Hamiltonian Eq. (2). The renormalization is conspicuous in an insulating phase, as we will see later. Since the trial wave functions we treat have up to six parameters to be optimized, we adopt an optimization VMC scheme [14]. In this study, optimization is carried out with 2–5 · 105 samples. We use the systems of Ns = L · L (L = 10 and 12) sites with periodic-antiperiodic boundary conditions, and impose the closed shell condition.
d-wave t'/t=0.4 L=10 L=12
Ec / t
triangular lattice with a variational Monte Carlo (VMC) method, which is known to be an effective scheme for cuprates [10].
167
0.05
0 5
6
7
8 U/ t
9
10
11
Fig. 2. Density of doublon (doubly occupied site) of the d-wave state for t 0 /t = 0.4 and 0.8 as a function of U/t. Data for two system sizes (L = 10 and 12) are shown for each value of t 0 /t.
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T. Watanabe et al. / Physica C 445–448 (2006) 166–170
U = Uc has a clear discontinuity even for a small system size (L = 10), whereas for a small value of t 0 /t(=0.4), this critical behavior gradually becomes clear as L increases. This is directly related to the existence of a SC phase for U < Uc for t 0 /t = 0.4. Although we do not mention details here, all the variational parameters also exhibit discontinuities at U = Uc. In particular, the binding factor l in Eq. (3) jumps to a value near 1. Furthermore, the behaviors of charge structure factor N(q) and momentum distribution function n(k) indicate that a gap opens in the charge degree of freedom for U > Uc [15]. Thus, we conclude that a Mott transition of first order takes place at U = Uc. Similar transitions have been also found using a dynamical cluster approximation [16] and a path integral renormalization group calculation [17]. Now, we discuss the properties of SC. In Fig. 3, we depict a long-range average of a d-wave nearest-neighbor pair correlation function, 0 1 X X P d ðrÞ ¼ ð1Þ1dðs;s Þ Dys ðjÞDs0 ðj þ rÞ ; ð5Þ 4N s j s;s0 ¼x;y pffiffiffi where Dys ðjÞ ¼ ðcyj" cyjþs# þ cyjþs" cyj# Þ= 2. For a moderately frustrated case (t 0 /t = 0.4), Pd(r) is very small for small
U/t, whereas it is remarkably enhanced as U/t approaches the Mott critical point Uc/t (= 6.55) with negligible system size dependence. This indicates that there exists a region of robust SC in the close vicinity of the Mott critical point. On entering the insulating regime (U > Uc), Pd(r) abruptly drops to almost zero, as naturally expected. In this case, the system size dependence is appreciable, and Pd(r) will completely vanishes in the thermodynamic limit. In contrast with this case, for a strongly frustrated system (t 0 /t = 0.8), we cannot find an enhancement of Pd(r) below Uc/t (= 8.05) (Fig. 3(b)). The observed values of Pd(r) for U/t = 4–8 are comparable to the statistical errors. In the insulating regime, Pd(r) vanishes as L becomes large as in the case of t 0 /t = 0.4. Thus, robust SC does not appear for a strongly frustrated (or nearly isotropic) case. To trace the above SC to its origin, we study a spin correlation function. In the insets of Fig. 4, the spin structure factor, 1 X iqðijÞ D z z E SðqÞ ¼ e Si Sj ð6Þ N s ij is plotted for values somewhat below the critical value Uc, where the states are conductive. For t 0 /t = 0.4, S(q) has a
0.002 0.006 L=10 L=12
0.004
Pdave
Pdave
t'/ t =0.4
0.001
t'/ t =0.8 L=10 L=12
0.002
0 4 (a)
0 5
6
7 Uc / t
U/t
8
9
10
4 (b)
5
6
7
U/t
8
9
10
Uc / t
Fig. 3. A pair correlation function of the d wave for (a) t 0 /t = 0.4 and (b) 0.8 as a function of U/t. Data of two different system sizes are compared. Notice that the scales of vertical axes are different. Since Pd(r) rapidly decreases with increasing jrj, and broadly constant for jrj > 3 as far as U/t is not small, we show here the average of Pd(r) for jrj > 3 as a long-distance value.
Fig. 4. AF [q = (p, p)] component of the spin structure factor S(q) of the d-wave state are depicted for (a) t 0 /t = 0.4 and (b) 0.8 as a function of U/t. Data of two system sizes are shown. In the insets, S(q) in a quarter of the Brillouin zone is shown for the values of U/t somewhat below the Mott critical points (in the conductive phases): (a) U/t = 6 and (b) U/t = 7.5. Here, the system size is 12 · 12.
T. Watanabe et al. / Physica C 445–448 (2006) 166–170
sharp peak at the AF wave number (p, p), and we find in the main graph of Fig. 4(a) that this component S(p, p) gradually increases as U/t increases even for U < Uc. On the other hand, S(q) for t 0 /t = 0.8 exhibits no special enhancement at (p, p) (inset of Fig. 4(b)), and the tendency of gradual increase of S(p, p) for U < Uc can be hardly observed in this case (Fig. 4(b)). This indicates that a strong frustration destroys not only a long-range AF order but short-range AF correlation (or fluctuation). Note that this feature of S(p, p) is faithfully reflected in the behavior of Pd(r) in the conductive phases, namely, when P ave increases, S(p, p) necessarily increases. We cond firm this feature for various values of t 0 /t and L without exception. Consequently, the SC in this model is considered to be induced by the AF correlation as is the highTc cuprates. Lastly, we discuss the insulating states (U > Uc). In these states, the renormalization of t 0 /t is considerable, and the quasi-Fermi surface becomes almost the same as that of the simple square lattice. Thus, the conspicuous peaks at q = (p, p) in S(q) are restored for both t 0 /t = 0.4 and 0.8, as seen in Fig. 4. Although the value of S(p, p) somewhat increases as the system size increases, the AF long-range order in the Pd-wave function is always absent, which is confirmed by i hð1Þi S zi i ¼ 0. The state for U > Uc is a spinliquid state, which has basically the same character as Anderson’s RVB state for t–J-type models [3]. Since the d-wave function used here does not have seeds of the AF order, we should include them in W to grasp the competition between AF and nonmagnetic insulators more reliably, for example, by employing a function in which Dd and DAF coexist [18]. 4. Summary and discussions With j-(ET)2X in mind, we have studied a Hubbard model on an anisotropic triangular lattice at half filling, using an optimization VMC method. Our main results are: (I) A Mott transition takes place for any values of t 0 /t we have studied. The critical value Uc/t is almost constant at 6.5–6.6 for t 0 /t [ 0.5, and gradually increases as the frustration further increases. (II) As the frustration t 0 /t increases, an AF long-range order vanishes and a robust d-wave SC appears at t 0 /t 0.4 in the U region just below the Mott critical value Uc. As the t 0 /t further increases, the SC disappears simultaneously with AF short-range correlation (t 0 /t 0.7), which supports the idea that the SC in j-(ET)2X is induced by the AF spin correlation. These results agree with main aspects of experiments in j-(ET)2X: (1) The pairing symmetry is likely dx2 y 2 wave [19]. (2) A compound with a smaller value of t 0 /t exhibits higher Tc. (3) SC-insulator transitions are caused by (chemical) pressure. (4) A nonmagnetic insulating phase appears for a compound with a strong frustration [X = Cu2(CN)3] [9]. Although this compound is considered to have a nearly isotropic value of t 0 /t(=1) [20], it exhibits SC with low pressure at relatively low Tc. According to the present work,
169
however, robust SC is unlikely to occur for t 0 /t J 0.8. One can interpret this problem as follows: Since organic compounds are soft, the effect of pressure is not only to enhance the band width (reduce U/t) but to substantially reduce the frustration t 0 /t [21]. Finally, we briefly refer to a couple of related theoretical studies. Although a Mott transition cannot be treated within FLEX approximations [8], their results of SC are basically consistent with ours, (II) above. A recent VMC study [22] tackled the same problem with a very similar trial function, and found a similar enhancement of Pd for t 0 /t = 0.7. However, their data were not precise enough and the analysis was not careful enough to lead the correct answers we have shown in this proceedings [23]. Detailed description of this work will be given elsewhere [25]. Acknowledgements This work is partly supported by Grant-in-Aid from the Ministry of Education, etc. Japan, from the Supercomputer Center, ISSP, Univ. of Tokyo, from NAREGI Nanoscience Project and for the 21st Century COE ‘‘Frontiers of Computational Science’’. References [1] R.H. McKenzie, Science 278 (1997) 820; K. Kanoda, Physica C 282–287 (1997) 299. [2] T. Komatsu, N. Matsukawa, T. Inoue, G. Saito, J. Phys. Soc. Jpn. 65 (1996) 1340; R.H. McKenzie, Comments Cond. Mat. Phys. 18 (1998) 309. [3] P.W. Anderson, Science 235 (1987) 1196. [4] For instance, D.J. Scalapino, Phys. Rep. 250 (1995) 329. [5] H. Kino, H. Fukuyama, J. Phys. Soc. Jpn. 65 (1996) 2158. [6] H. Kondo, T. Moriya, J. Phys. Soc. Jpn. 65 (1996) 2559. [7] J.Y. Gan, Y. Chen, Z.B. Su, F.C. Zhang, Phys. Rev. Lett. 94 (2005) 067005. [8] H. Kino, H. Kontani, J. Phys. Soc. Jpn. 67 (1998) 3691; H. Kondo, T. Moriya, J. Phys. Soc. Jpn. 67 (1998) 3695; J. Schmalian, Phys. Rev. Lett. 81 (1998) 4232. [9] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, G. Saito, Phys. Rev. Lett. 91 (2003) 107001. [10] H. Yokoyama, Y. Tanaka, M. Ogata, H. Tsuchiura, J. Phys. Soc. Jpn. 73 (2004) 1119; H. Yokoyama, in preparation. [11] T.A. Kaplan, P. Horsch, P. Fulde, Phys. Rev. Lett. 49 (1982) 889; H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 59 (1990) 3669. [12] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 56 (1987) 1490; H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. [13] A. Himeda, M. Ogata, Phys. Rev. Lett. 85 (2000) 4345. [14] C.J. Umrigar, K.G. Wilson, J.W. Wilkins, Phys. Rev. Lett. 60 (1988) 1719. [15] As U/t increases, the behaviors of N(q) and n(k) suddenly change at U = Uc. For U < Uc, N(q) / q and n(k) has a discontinuity at k = kF in the node-of-gap direction, whereas for U > Uc, N(q) / q2 and n(k) becomes smooth with a finite slope even at k = kF. [16] O. Parcollet, G. Biroli, G. Kotliar, Phys. Rev. Lett. 92 (2004) 226402. [17] H. Morita, S. Watanabe, M. Imada, J. Phys. Soc. Jpn. 71 (2002) 2109. [18] T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943; A. Himeda, M. Ogata, Phys. Rev. B 60 (1999) R9935. [19] H. Mayaffre, P. Wzietek, D. Jerome, C. Lenoir, P. Batail, Phys. Rev. Lett. 75 (1995) 4122;
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S.M. De Soto, C.P. Slichter, A.M. Kini, H.H. Wang, U. Geiser, J.M. Williams, Phys. Rev. B 52 (1995) 10364; K. Kanoda, K. Miyagawa, A. Kawamoto, Y. Nakazawa, Phys. Rev. B 54 (1996) 76. [20] W. Zheng, R.R.P. Singh, R.H. McKenzie, R. Coldea, Phys. Rev. B 71 (2005) 134422. [21] The behavior of o(t 0 /t)/op has not yet been established [1]. This is an urgent problem to study. [22] J. Liu, J. Schmalian, N. Trivedi, Phys. Rev. Lett. 94 (2005) 127003.
[23] The authors of Ref. [22] believe the mistaken conclusion of Millis and Coppersmith [24] that a Mott transition cannot be described within a doublon–holon binding factor like Eqs. (3) and (4), and consider that a SC region continues to large value of U beyond Uc. Thus, they have not checked the system-size dependence of Pd, notwithstanding it is decisively important near the phase transition. [24] A.J. Millis, S.N. Coppersmith, Phys. Rev. B 43 (1991) 13770. [25] T. Watanabe, Y. Yokoyama, Y. Tanaka, J. Inoue, cond-mat/ 0602098.
Superconductivity and a Mott transition in a Hubbard model on an anisotropic triangular lattice Tsutomu Watanabe b
a,b,*
, Hisatoshi Yokoyama c, Yukio Tanaka
a,b
, Jun-ichiro Inoue
a
a Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan CREST Japan Science and Technology Corporation (JST), Nagoya 464-8603, Japan c Department of Physics, Tohoku University, Sendai 980-8578, Japan
Available online 4 May 2006
Abstract With a mind to a Mott transition and superconductivity (SC) arising in j-(BEDT-TTF)2X, we study a half-filled-band Hubbard model on an anisotropic triangular lattice (t in two bond directions and t 0 in the other). Using an optimization variational Monte Carlo method, we find that with doublon–holon binding factors in the trial functions, a first-order Mott transition takes place at U = Uc approximately of the band width. For a moderate frustration (t 0 /t = 0.4), the d-wave superconducting correlation Pd is sizably enhanced just under Uc/t, whereas for a strong frustration (t 0 /t = 0.8), Pd is hardly enhanced for any value of U/t. Since the behavior of Pd is closely connected to that of the spin structure factor S(p, p), the SC in this model must be induced by the antiferromagnetic correlation. 2006 Elsevier B.V. All rights reserved. PACS: 74.70.b; 74.20.z Keywords: Hubbard model; Anisotropic triangular lattice; Variational Monte Carlo method; Mott transition
1. Introduction Organic layered compounds, j-BEDT-TTF (ET) salts, abbreviated as j-(ET)2X [1], are very intriguing in some points. (i) The j-type compounds have quasi-two-dimensional conducting planes whose structure is magnetically frustrated and quite similar to that of high-Tc cuprates. (ii) Superconductor (SC)-to-insulator (antiferromagnetic (AF) or nonmagnetic) transitions take place in low temperatures, by applying (chemical) pressure. According to extended Hu¨ckel calculations [2], the conduction band in j-(ET)2X is composed of antibonding orbitals of dimerized ET molecules, and a hole exists on each unit dimer, forming a half-filled band. The connectivity of the ET dimers is properly modeled by an anisotropic triangular lattice with * Corresponding author. Address: Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan. Tel.: +81 52 789 3700; fax: +81 52 789 3298. E-mail address: [email protected] (T. Watanabe).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.03.107
a hopping integral t in two directions and t 0 in the third direction. Within the extended Hu¨ckel approximation, the anisotropy (or frustration) jt 0 /tj is estimated at 0.5– 1.1, which is fairly larger than the values of cuprates 0.1– 0.3 in a model with t 0 in both diagonal bond directions. Thus, it is an important issue to clarify whether a new superconducting (SC) mechanism is realized due to the strong frustration or the AF spin correlation (or fluctuation) causes superconductivity (SC) as in the cuprates [3,4]. Another vital issue is to consider the mechanism of a SC-insulator (so-called Mott) transition. In mean-field [5,6], Gutzwiller [7] and fluctuation exchange (FLEX) [8] approximations, the term ‘Mott transition’ is used as a transition between a conductive state and an AF insulator. However, the mechanism of a Mott transition can be basically independent of magnetic properties, because a nonmagnetic insulating state seems to be realized in j-(ET)2Cu2(CN)3 [9]. The purpose of this work is to illuminate the above two issues, by studying a Hubbard model on an anisotropic
T. Watanabe et al. / Physica C 445–448 (2006) 166–170
2. Formulation As a model of the conducting plane in j-(ET)2X, we consider a Hubbard model on an anisotropic triangular lattice (or an extended square lattice with next-nearest-neighbor hopping t 0 only in one diagonal direction [1, 1]) [5]: X y X H¼ ek ckr ckr þ U ni" ni# ð1Þ i
kr
with ek ¼ 2tðcos k x þ cos k y Þ 2t0 cosðk x þ k y Þ;
ð2Þ
and t, t 0 , U > 0. Throughout this paper, we fix the electron density at half filling. To this model, we apply an optimization VMC method [10], which can treat the whole parameter space spanned by U/t and t 0 /t. As a variational wave function, we use a two-body Jastrow type: W = PU, where U is a one-body (Hartree–Fock) part expressed as a Slater determinant, and P is a correlation factor. As forQP, in addition to the onsite (Gutzwiller) factor, P G ¼ i ½1 ð1 gÞni" ni# , we introduce a crucial intersite factor PQ (P = PQPG), which binds a doublon (doubly occupied site) to a holon (empty site) within short distances [11]: Y 0 ð1 lQsi Þð1 l0 Qsi Þ; ð3Þ PQ ¼ i sðs0 Þ Qi
¼
Y
d i ð1 eiþsðs0 Þ Þ þ ei ð1 d iþsðs0 Þ Þ ;
3. Results We start with a comparison of stability between the dwave and AF states. We know that the AF state is much stabler for t 0 /t = 0 (normal square lattice) as far as U is positive, due to the complete nesting condition. As t 0 /t increases, however, the AF state becomes less stable. In Fig. 1, we show condensation energies per site of the AF and d-wave states, EdðAFÞ ¼ Ed ðEAF Þ EF , as functions of c U/t. For t 0 /t = 0.4, the stable range of the AF state is reduced to 5 < U/t < 6.5 for L = 10, and EAF is always c smaller than Edc for L = 12, as observed in the inset. It follows that the ground state switches to the d-wave state from the AF state for t 0 /t J 0.4. Actually, for t 0 /t = 0.8, the AF state is never stabilized with respect to the normal state. Next, let us look at Edc . For small values of U/t, Edc is very small, whereas it abruptly increases at U/t 6.5 (8.0) for t 0 /t = 0.4 (0.8), regardless of system size. This is nothing but a sign of a Mott transition, as we will see in the following. In Fig. 2, we plot the doublon density, D = hni"ni#i, which is considered as the order parameter of Mott transitions. For t 0 /t = 0.4 (0.8), D abruptly decreases at U = Uc 6.55t ± 0.05t (8.05t ± 0.05t). These critical values agree with those of Edc discussed in the preceding paragraph. For a large value of t 0 /t(=0.8), the behavior of D at
0.15
ð4Þ Ec / t
sðs0 Þ
0.1
0.05
t'/t=0.8 L=10 L=12
AF
t'/t=0.4
0.015
t'/t=0.4 L=10 L=12
0.01
0.005 0
0 4
6
8
5
10 U/t
6
U /t
12
7
14
Fig. 1. Condensation energies of the d-wave and AF states for t 0 /t = 0.4 and 0.8 as functions of U/t. The systems are of 10 · 10 and 12 · 12. The inset shows the magnification of the region of U Uc for t 0 /t = 0.4.
0.15 d-wave
0.1
t'/t =0.4, L=10 t'/t =0.4, L=12 t'/t =0.8, L=10 t'/t =0.8, L=12
D
where di = ni"ni#, ei = (1 ni")(1 ni#), and s(s 0 ) runs over all nearest-neighbor sites in the directions of t(t 0 ). For l ! 1 in Eq. (3), a doublon (negative-charged) and a holon (positive-charged) are bound at adjacent sites, so that a current cannot flow. Without PQ, a Mott transition cannot be described by W [12]. Regarding U, we study three states: (i) the Fermi sea: UF, (ii) a BCS function with a dx2 y 2 -wave gap Dk = Dd(cos kx cos ky): Ud(Dd, f), and (iii) a Hartree– Fock-type AF state: UAF(DAF). Here, Dd, f, and DAF are, in VMC, variational parameters to be optimized, which correspond to the d-wave gap, chemical potential, and AF gap, respectively, in the Hartree–Fock theory. Furthermore, we take account of a renormalization effect of the quasi-Fermi surface in the d-wave state due to the electron correlation [13]. This effect is introduced by optimizing the value of t 0 /t in ek in the wave function Ud as a variational parameter, independently of t 0 /t given in the Hamiltonian Eq. (2). The renormalization is conspicuous in an insulating phase, as we will see later. Since the trial wave functions we treat have up to six parameters to be optimized, we adopt an optimization VMC scheme [14]. In this study, optimization is carried out with 2–5 · 105 samples. We use the systems of Ns = L · L (L = 10 and 12) sites with periodic-antiperiodic boundary conditions, and impose the closed shell condition.
d-wave t'/t=0.4 L=10 L=12
Ec / t
triangular lattice with a variational Monte Carlo (VMC) method, which is known to be an effective scheme for cuprates [10].
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0.05
0 5
6
7
8 U/ t
9
10
11
Fig. 2. Density of doublon (doubly occupied site) of the d-wave state for t 0 /t = 0.4 and 0.8 as a function of U/t. Data for two system sizes (L = 10 and 12) are shown for each value of t 0 /t.
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U = Uc has a clear discontinuity even for a small system size (L = 10), whereas for a small value of t 0 /t(=0.4), this critical behavior gradually becomes clear as L increases. This is directly related to the existence of a SC phase for U < Uc for t 0 /t = 0.4. Although we do not mention details here, all the variational parameters also exhibit discontinuities at U = Uc. In particular, the binding factor l in Eq. (3) jumps to a value near 1. Furthermore, the behaviors of charge structure factor N(q) and momentum distribution function n(k) indicate that a gap opens in the charge degree of freedom for U > Uc [15]. Thus, we conclude that a Mott transition of first order takes place at U = Uc. Similar transitions have been also found using a dynamical cluster approximation [16] and a path integral renormalization group calculation [17]. Now, we discuss the properties of SC. In Fig. 3, we depict a long-range average of a d-wave nearest-neighbor pair correlation function, 0 1 X X P d ðrÞ ¼ ð1Þ1dðs;s Þ Dys ðjÞDs0 ðj þ rÞ ; ð5Þ 4N s j s;s0 ¼x;y pffiffiffi where Dys ðjÞ ¼ ðcyj" cyjþs# þ cyjþs" cyj# Þ= 2. For a moderately frustrated case (t 0 /t = 0.4), Pd(r) is very small for small
U/t, whereas it is remarkably enhanced as U/t approaches the Mott critical point Uc/t (= 6.55) with negligible system size dependence. This indicates that there exists a region of robust SC in the close vicinity of the Mott critical point. On entering the insulating regime (U > Uc), Pd(r) abruptly drops to almost zero, as naturally expected. In this case, the system size dependence is appreciable, and Pd(r) will completely vanishes in the thermodynamic limit. In contrast with this case, for a strongly frustrated system (t 0 /t = 0.8), we cannot find an enhancement of Pd(r) below Uc/t (= 8.05) (Fig. 3(b)). The observed values of Pd(r) for U/t = 4–8 are comparable to the statistical errors. In the insulating regime, Pd(r) vanishes as L becomes large as in the case of t 0 /t = 0.4. Thus, robust SC does not appear for a strongly frustrated (or nearly isotropic) case. To trace the above SC to its origin, we study a spin correlation function. In the insets of Fig. 4, the spin structure factor, 1 X iqðijÞ D z z E SðqÞ ¼ e Si Sj ð6Þ N s ij is plotted for values somewhat below the critical value Uc, where the states are conductive. For t 0 /t = 0.4, S(q) has a
0.002 0.006 L=10 L=12
0.004
Pdave
Pdave
t'/ t =0.4
0.001
t'/ t =0.8 L=10 L=12
0.002
0 4 (a)
0 5
6
7 Uc / t
U/t
8
9
10
4 (b)
5
6
7
U/t
8
9
10
Uc / t
Fig. 3. A pair correlation function of the d wave for (a) t 0 /t = 0.4 and (b) 0.8 as a function of U/t. Data of two different system sizes are compared. Notice that the scales of vertical axes are different. Since Pd(r) rapidly decreases with increasing jrj, and broadly constant for jrj > 3 as far as U/t is not small, we show here the average of Pd(r) for jrj > 3 as a long-distance value.
Fig. 4. AF [q = (p, p)] component of the spin structure factor S(q) of the d-wave state are depicted for (a) t 0 /t = 0.4 and (b) 0.8 as a function of U/t. Data of two system sizes are shown. In the insets, S(q) in a quarter of the Brillouin zone is shown for the values of U/t somewhat below the Mott critical points (in the conductive phases): (a) U/t = 6 and (b) U/t = 7.5. Here, the system size is 12 · 12.
T. Watanabe et al. / Physica C 445–448 (2006) 166–170
sharp peak at the AF wave number (p, p), and we find in the main graph of Fig. 4(a) that this component S(p, p) gradually increases as U/t increases even for U < Uc. On the other hand, S(q) for t 0 /t = 0.8 exhibits no special enhancement at (p, p) (inset of Fig. 4(b)), and the tendency of gradual increase of S(p, p) for U < Uc can be hardly observed in this case (Fig. 4(b)). This indicates that a strong frustration destroys not only a long-range AF order but short-range AF correlation (or fluctuation). Note that this feature of S(p, p) is faithfully reflected in the behavior of Pd(r) in the conductive phases, namely, when P ave increases, S(p, p) necessarily increases. We cond firm this feature for various values of t 0 /t and L without exception. Consequently, the SC in this model is considered to be induced by the AF correlation as is the highTc cuprates. Lastly, we discuss the insulating states (U > Uc). In these states, the renormalization of t 0 /t is considerable, and the quasi-Fermi surface becomes almost the same as that of the simple square lattice. Thus, the conspicuous peaks at q = (p, p) in S(q) are restored for both t 0 /t = 0.4 and 0.8, as seen in Fig. 4. Although the value of S(p, p) somewhat increases as the system size increases, the AF long-range order in the Pd-wave function is always absent, which is confirmed by i hð1Þi S zi i ¼ 0. The state for U > Uc is a spinliquid state, which has basically the same character as Anderson’s RVB state for t–J-type models [3]. Since the d-wave function used here does not have seeds of the AF order, we should include them in W to grasp the competition between AF and nonmagnetic insulators more reliably, for example, by employing a function in which Dd and DAF coexist [18]. 4. Summary and discussions With j-(ET)2X in mind, we have studied a Hubbard model on an anisotropic triangular lattice at half filling, using an optimization VMC method. Our main results are: (I) A Mott transition takes place for any values of t 0 /t we have studied. The critical value Uc/t is almost constant at 6.5–6.6 for t 0 /t [ 0.5, and gradually increases as the frustration further increases. (II) As the frustration t 0 /t increases, an AF long-range order vanishes and a robust d-wave SC appears at t 0 /t 0.4 in the U region just below the Mott critical value Uc. As the t 0 /t further increases, the SC disappears simultaneously with AF short-range correlation (t 0 /t 0.7), which supports the idea that the SC in j-(ET)2X is induced by the AF spin correlation. These results agree with main aspects of experiments in j-(ET)2X: (1) The pairing symmetry is likely dx2 y 2 wave [19]. (2) A compound with a smaller value of t 0 /t exhibits higher Tc. (3) SC-insulator transitions are caused by (chemical) pressure. (4) A nonmagnetic insulating phase appears for a compound with a strong frustration [X = Cu2(CN)3] [9]. Although this compound is considered to have a nearly isotropic value of t 0 /t(=1) [20], it exhibits SC with low pressure at relatively low Tc. According to the present work,
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however, robust SC is unlikely to occur for t 0 /t J 0.8. One can interpret this problem as follows: Since organic compounds are soft, the effect of pressure is not only to enhance the band width (reduce U/t) but to substantially reduce the frustration t 0 /t [21]. Finally, we briefly refer to a couple of related theoretical studies. Although a Mott transition cannot be treated within FLEX approximations [8], their results of SC are basically consistent with ours, (II) above. A recent VMC study [22] tackled the same problem with a very similar trial function, and found a similar enhancement of Pd for t 0 /t = 0.7. However, their data were not precise enough and the analysis was not careful enough to lead the correct answers we have shown in this proceedings [23]. Detailed description of this work will be given elsewhere [25]. Acknowledgements This work is partly supported by Grant-in-Aid from the Ministry of Education, etc. Japan, from the Supercomputer Center, ISSP, Univ. of Tokyo, from NAREGI Nanoscience Project and for the 21st Century COE ‘‘Frontiers of Computational Science’’. References [1] R.H. McKenzie, Science 278 (1997) 820; K. Kanoda, Physica C 282–287 (1997) 299. [2] T. Komatsu, N. Matsukawa, T. Inoue, G. Saito, J. Phys. Soc. Jpn. 65 (1996) 1340; R.H. McKenzie, Comments Cond. Mat. Phys. 18 (1998) 309. [3] P.W. Anderson, Science 235 (1987) 1196. [4] For instance, D.J. Scalapino, Phys. Rep. 250 (1995) 329. [5] H. Kino, H. Fukuyama, J. Phys. Soc. Jpn. 65 (1996) 2158. [6] H. Kondo, T. Moriya, J. Phys. Soc. Jpn. 65 (1996) 2559. [7] J.Y. Gan, Y. Chen, Z.B. Su, F.C. Zhang, Phys. Rev. Lett. 94 (2005) 067005. [8] H. Kino, H. Kontani, J. Phys. Soc. Jpn. 67 (1998) 3691; H. Kondo, T. Moriya, J. Phys. Soc. Jpn. 67 (1998) 3695; J. Schmalian, Phys. Rev. Lett. 81 (1998) 4232. [9] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, G. Saito, Phys. Rev. Lett. 91 (2003) 107001. [10] H. Yokoyama, Y. Tanaka, M. Ogata, H. Tsuchiura, J. Phys. Soc. Jpn. 73 (2004) 1119; H. Yokoyama, in preparation. [11] T.A. Kaplan, P. Horsch, P. Fulde, Phys. Rev. Lett. 49 (1982) 889; H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 59 (1990) 3669. [12] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 56 (1987) 1490; H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. [13] A. Himeda, M. Ogata, Phys. Rev. Lett. 85 (2000) 4345. [14] C.J. Umrigar, K.G. Wilson, J.W. Wilkins, Phys. Rev. Lett. 60 (1988) 1719. [15] As U/t increases, the behaviors of N(q) and n(k) suddenly change at U = Uc. For U < Uc, N(q) / q and n(k) has a discontinuity at k = kF in the node-of-gap direction, whereas for U > Uc, N(q) / q2 and n(k) becomes smooth with a finite slope even at k = kF. [16] O. Parcollet, G. Biroli, G. Kotliar, Phys. Rev. Lett. 92 (2004) 226402. [17] H. Morita, S. Watanabe, M. Imada, J. Phys. Soc. Jpn. 71 (2002) 2109. [18] T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943; A. Himeda, M. Ogata, Phys. Rev. B 60 (1999) R9935. [19] H. Mayaffre, P. Wzietek, D. Jerome, C. Lenoir, P. Batail, Phys. Rev. Lett. 75 (1995) 4122;
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[23] The authors of Ref. [22] believe the mistaken conclusion of Millis and Coppersmith [24] that a Mott transition cannot be described within a doublon–holon binding factor like Eqs. (3) and (4), and consider that a SC region continues to large value of U beyond Uc. Thus, they have not checked the system-size dependence of Pd, notwithstanding it is decisively important near the phase transition. [24] A.J. Millis, S.N. Coppersmith, Phys. Rev. B 43 (1991) 13770. [25] T. Watanabe, Y. Yokoyama, Y. Tanaka, J. Inoue, cond-mat/ 0602098.