- Email: [email protected]
Variational Monte Carlo analysis of singlet-pairing state in a system with inhomogeneous potential
ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 69 (2008) 3388– 3391
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Variational Monte Carlo analysis of singlet-pairing state in a system with inhomogeneous potential Yusuke Fujihara , Akihisa Koga, Norio Kawakami Department of Physics, Kyoto University, Kyoto 606-8502, Japan
a r t i c l e in f o
Keywords: D. Superconductivity
a b s t r a c t We investigate the singlet-pairing superfluid state of two-component correlated fermions with repulsive interactions in a one-dimensional harmonic potential. By introducing a BCS-type wave function with spatially modulated order parameter, we show how the spin-singlet pairs are formed in the inhomogeneous potential. We then perform variational calculations with particular emphasis on the effect of particle correlations in order to examine the stability of the superfluid state. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Superfluidity in strongly correlated particle systems with spatially modulated potentials has attracted much interest. A typical example is the well-known cuprate, where an observed stripe structure is suggested to play an important role in realizing the d-wave superconductivity with a high critical temperature [1]. More recently, a possibility of the superconductivity in heterostructure of transition-metal oxides such as LaTiO3/SrTiO3 [2] has been discussed, which stimulates intensive investigations on the superconductivity in correlated systems with inhomogeneous structures. Another remarkable example in a slightly different field is the optical lattice system, where the Bose–Einstein condensation of trapped bosonic atom gas has been observed [3]. Owing to its high controllability, a variety of optical lattice systems have been realized such as one-dimensional (1D), twodimensional (2D) and three-dimensional (3D) lattices with bosons [3–7], fermions [8–11] and their mixture [12,13]. Among them, for fermionic systems, a number of interesting properties such as the superfluidity, the Mott transition, etc. have been observed [11] and also theoretically predicted [14–19]. It is particularly interesting and challenging to realize superfluidity of correlated fermions with repulsive interactions in optical lattices, which may share common properties with the high-T c cuprates. Although such inhomogeneous systems provide us with rich physics, it is not easy to investigate their properties theoretically because of strong correlations under inhomogeneous potentials. Systematic studies of such strong correlations in confined systems are thus highly desirable.
Corresponding author. Tel.: +81757533674.
E-mail address: [email protected] (Y. Fujihara). 0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.092
In this paper, we consider two-component correlated fermions trapped in an optical lattice to discuss how the superfluid state emerges under the confining potential. Although we focus our discussions on the optical lattice to make the point clear, we think that the discussions given here may also be applied to some related systems, such as transition-metal oxides with heterostructure, for which site-dependent correlations play a central role. To be more specific, we consider a 1D optical lattice with a confining potential to clarify how the spatially modulated pairing state is realized in mean-field theory. To this end, we make use of the Bogoliubov–de Gennes (BdG) equations. Furthermore, by using the variational Monte Carlo (VMC) [20] simulations, we examine the stability of the pairing state obtained in mean-field theory. Since we study the 1D system in mean-field theory in this paper, the results obtained may be applied to quasi1D systems.
2. Spatially modulated pairing state We investigate two-component correlated fermions with repulsive interactions confined in a 1D harmonic well. Although a proper model system for our purpose may be the Hubbard model with on-site repulsions, we instead consider the t2J model having the nearest-neighbor antiferromagnetic exchange coupling, which is more convenient to observe how the superfluidity emerges in inhomogeneous potentials. Discussions on the Hubbard model will be given later in the paper. The Hamiltonian we consider here is X 1 H ¼ H0 þ J Si Sj ni nj , 4 hi;ji X y X H0 ¼ t cis cjs þ ðV i mÞnis , (1) hi;jis
is
ARTICLE IN PRESS Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
j
j
1.2 1 0.8 〈ni〉
where cyis (cis ) is a creation (annihilation) Fermi operator with spin sð¼"; #Þ at ith site. Here we specify two distinct components of fermions in terms of "; # spins. nis ¼ cyis cis , ni ¼ P y s nis and Si ¼ cia rab cib , where s is the Pauli matrix. t and J ð40Þ are the hopping matrix and the antiferromagnetic exchange between nearest-neighbor sites and m the chemical potential. Interacting fermions are assumed to be trapped by the harmonic potential V i . We introduce a site-dependent BCS-type order parameter Dij ¼ hci# cj" i. Here, we focus on the singlet pairing expected naively for our Hamiltonian with antiferromagnetic coupling, where Dij ¼ Dji . Although doubly occupied states should be excluded in the t2J Hamiltonian, we ignore this constraint for a while to make the problem more tractable. In terms of the P Bogoliubov transformation, cis ¼ l fuli als svil ayls¯ g, we obtain the BdG equation [21], !0 l 1 ! uj F ij X Hij uli @ A ¼ El , (2) F ji Hji vl vl
3389
0.6
J/t = 0.0 J/t = 2.0 J/t = 3.0
0.4 0.2 0
0
10
20
30
40
50 60 i (site)
70
80
90
100
i
with Hij ¼ t dj;iþt þ ðV i mÞdij ,
(3)
F ij ¼ J Dij dj;iþt ,
(4)
which are supplemented by the self-consistent equation, X Dij ¼ uli vjl ,
(5)
0.2
0.1 0.05
-0.05 0
10 20 30 40 50 60 70 80 90 100 i (site)
1 0.5 0
Fig. 1. Spatial distributions of (a) the particle density hni i and (b) the amplitude of the order parameter Di;iþ1.
i = 10 i = 16 i = 30 i = 49 ρ (i,ε)
where t ¼ 1 specifies the nearest-neighbor sites. In our calculation, we deal with a 1D system with L ¼ 100 sites and N" ¼ N# ¼ 30, where Ns is the total number of particles with spin s. To moderate the effects of the edges, the harmonic potential V i is chosen as V L=21 ¼ V L=2 ¼ 0 at the center of the system and V 0 ¼ V L1 ¼ 4:0t at the edges. We also set m as the Fermi level for the non-interacting case ðJ ¼ 0Þ. In Fig. 1, we show the spatial distribution of the particle density P hni i ¼ l jvli j2 and the nearest-neighbor pairing amplitude Di;iþ1 . When J=t ¼ 0, the system is reduced to the tight-binding model, where the spatially modulated particle density appears owing to the confining potential, as shown in Fig. 1(a). It is found that the introduction of J has little effect on both quantities for small J ðoJ c ¼ 0:8tÞ. Further increase of J induces the singlet-pairing state. We note that the amplitude of Di;iþ1 has a maximum around the 16th (equivalently 83th) site, while the profile of the particle density hardly changes. This seems somewhat puzzling at a glance since the particle density hni i0:4 at i ¼ 16 is far from half filling hni i ¼ 1. These results are indeed contrasted to those for the Hubbard (or t2J) model on the square lattice, where the gap function is enhanced near half filling. However, close observations of the local density of states (LDOS) give natural understanding of the above results. Recall that the DOS around the Fermi level mainly contributes to the singlet-pairing formation. In our case, the filling factor alters spatially, so that we look at the LDOS rði; Þ ¼ hijdð H0 Þjii. It is seen in Fig. 2 that a peak structure in the LDOS is located around Fermi level at i ¼ 16, which has a tendency to stabilize the pairing state in that region. Note that this type of peak structure in the LDOS is inherent in (quasi-) 1D systems, and therefore the above results on the superfluiddominant region are, we think, characteristic of (quasi-) 1D systems. Another point we wish to remark in Fig. 1(b) is that there exists a critical value Jc =t0:8 at which the pairing state develops. We have indeed checked that for small interactions JoJ c , the formation of the spin-singlet pairs yields no energy gain. Only after J4Jc, the pairing state starts to appear in the region
3 2.5 2 1.5
0
l
J/t
Δi,i+1
0.15
-8
-6
-4
-2
0 ε
2
4
6
8
Fig. 2. Local density of states rði; Þ of the 1D confined system H0 for i ¼ 10; 16; 30; 49.
mentioned above. We think that this also characterizes the present system. Namely, in our system with confinement, the spatial region where the paring state is stabilized is rather narrow, so that the energy gain to stabilize the superfluidity may not be sufficient enough for small J values. We will address this problem later again for the Hubbard model.
ARTICLE IN PRESS 3390
Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
3. Impact of particle correlations The above mean-field treatment captures some essential features of the paring state in our confined system. However, we are still left with serious problems to be resolved: (a) we have so far neglected the impact of on-site repulsions, which has tendency to suppress the superfluidity; (b) we have employed the t2J model, in which the exchange coupling is artificially introduced by hand, instead of the Hubbard model. In this section, we try to address these problems, and examine how stable the singletpairing state is in the presence of particle correlations due to onsite repulsions. We start with the first problem (a). For this purpose, we write down a superfluid trial state with a spatially modulated order parameter. Following the BCS theory, we consider a trial state, Y jFBCS i ¼ ðun þ vn cyn" cyn# Þj0i, (6) n
2
3
2
16 xn 16 xn 7 7 (7) 41 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; v2n ¼ 41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5, 2 2 2 2 2 2 xn þ Dn xn þ Dn P n where xn and cns ¼ i fi cis are eigenvalues and eigenstates of P the non-interacting Hamiltonian, H0 ¼ ns xn cyns cns . In order to determine the gap function, X n n Dn ¼ hcn# cn" i ¼ fi fj Dij , (8) i;j
we make use of the solution of the BdG equation in Eq. (2), where the effect of the confining potential is already incorporated. However, in this treatment, the prohibition of the doubly occupied states has not been taken into account, and therefore the corresponding mean-field Dij has finite on-site components Dii , which should be zero due to the above-mentioned constraint. To amend this, we introduce a modified pairing state which incorporates the constraint due to the on-site repulsions. As in the case of dx2 y2 -wave pairing in cuprates in 2D, we only consider the nearest-neighbor pairing that should be most relevant and discard the on-site pairing: we set Dij ¼ jhci# cj" ij for j ¼ i 1 and Dij ¼ 0 otherwise in Eq. (8). We carry out the calculation with the above improved trial function for the t2J model. As a result, we confirm that the singlet-pairing state is still stabilized even if the impact of on-site repulsions is considered. In Fig. 3, we show the resulting one-
7 J/t = 0.0 J/t = 1.0 J/t = 2.0 J/t = 3.0
0.4 0.35
5
0.25
4
0.15
3
0.05
0.3 0.2
Δ
En
0.1 0 26
2
28
30
32
34
1 0
0
10
20
30
40
50 n
60
70
defined as the lowest value of En is very small, D=t101 even for large J=t. This implies that the condensation energy is not so large although the superfluid state is realized. As mentioned above, the pairing state has its dominant weight in the narrow regions with hni i0:4, where the attractive interaction is considered to be small in contrast to the regions with hni i1:0. The small energy gain is thus related to the nature of the LDOS in 1D systems. If we treat 2D systems, the situation becomes much better, since the high LDOS is realized near half filling. Finally, we address the problem (b) by starting from the original Hubbard model. We investigate whether the superfluid state can be stabilized the Hubbard model with on-site repulsion U, X H ¼ H0 þ U ni" ni# . (9) i
3
u2n ¼
6
particle excitation spectra obtained for the superfluid state, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 En ¼ xn þ D2n . As seen in Fig. 3, the excitation gap D that is
80
90
Fig. 3. One-particle excitation spectra in the superfluid state. D in the inset shows the excitation energy gap.
In comparison with the t2J model, it is not easy to treat the superfluid state in the Hubbard model because we should properly consider not only on-site correlations but also nearestneighbor correlations under the confining potential. To this end, it is legitimate to use (6) as a trial state describing the superfluidity at mean-field level. To incorporate particle correlations due to U explicitly, we make use of the VMC simulations with the following trial state [19]: (10)
jCi ¼ PQ PG jFBCS i, "
PG ¼ exp
X
#
ai ni" nj# ,
(11)
# b , Q i
(12)
b iþt Þ, ð1 H
(13)
i
"
PQ ¼ exp a0
X i
b ¼D bi Q i
Y t
b i ¼ ð1 ni" Þð1 ni# Þ. PG is a site-dependent b i ¼ ni" ni# , H where D Gutzwiller projection operator with variational parameters ai , which can reduce the weight of the doubly occupied states at each site. In addition, we introduce the projection operator PQ called doublon-holon binding factor, which can incorporate the higher order effect of U [20,22,23]. We consider the nearest-neighbor pairing amplitude Di;iþ1 in jFBCS i as additional variational parameters. Thus we end up with quite a few of variational parameters f. . . ; ai ; . . . ; a0 ; . . . ; Di;iþ1 ; . . .g to be optimized. Unfortunately the minimization procedure is practically beyond the standard VMC method. To overcome this difficulty, we make use of a stochastic reconfiguration with Hessian acceleration (SRH) scheme to minimize the ground state energy in a given parameter space [19,24]. Since the calculation with the SRH scheme is rather tuff and now in progress, we report here the preliminary results we have obtained so far. The calculated energies for U ¼ 6:0; 8:0; 10:0 and 12.0 are E=t ¼ 4:55 0:02; 0:08 0:04; 2:80 0:05 and 4:85 0:05, respectively. We find that the energy difference among the different sets of Di;iþ1 obtained for J=t ¼ 0:0; 1:0; 2:0 and 3.0 in the BdG equations is of the same order as the convergence error that arises from the multivariable optimization process within the SRH scheme. Therefore we could not yet figure out whether the superfluid state is more stable than the normal state. This subtle situation partly comes from the fact that Dij has large amplitudes only in the narrow spatial regions in the system, making the energy gain of the superfluid state very small, as mentioned before. To draw a definite conclusion for the 1D
ARTICLE IN PRESS Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
system, we are now trying to improve the variational calculations. Our next target in the future study is 2D correlated fermions. In that case, it is assumed that the superfluid state becomes stabilized much easier, since the large LDOS appears near half filling and also the superfluid-dominant region should be extended.
4. Summary We have investigated the singlet-pairing state in 1D fermionic optical lattice systems with repulsive interactions. By taking account of the spatially modulated singlet pairing, we have discussed how the amplitude of the pairing state distributes under a harmonic potential. It has been shown that the superfluid order parameter has large values in the region where the LDOS has a maximum structure around the Fermi level. Furthermore, we have incorporated the correlation effects, and have confirmed that the superfluid state is still stabilized for the t2J model. Concerning the Hubbard model, we have reported the preliminary results obtained by the VMC simulations with the SRH scheme. The calculation done so far is unfortunately not precise enough to figure out whether the superfluid state can be stabilized. We are now in progress to improve the accuracy of calculations and also to introduce other trial states more suitable for inhomogeneous systems. In this paper, we have restricted our discussions to the superfluid state. There may possibly appear other ordered states, such as the Mott insulating state and magnetically ordered states, etc. [14,19,25]. It is particularly worth studying whether or how the superfluid state and the antiferromagnetically ordered state can coexist in the Hubbard model with confining potentials, especially in 2D systems. This issue is now under consideration.
3391
Acknowledgments The numerical computations were carried out at the Supercomputer Center, the Institute for Solid State Physics, University of Tokyo. We would like to thank M. Yamashita for valuable discussions. This work was supported by Grant-in-Aids for Scientific Research [Grant nos. 17740226 (AK) and 18043017 (NK)] from The Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25]
J.M. Tranquada, et al., Nature 375 (1995) 561. A. Ohtomo, et al., Nature 419 (2002) 378. M. Greiner, et al., Nature 415 (2002) 39. O. Mandel, et al., Nature 425 (2003) 937. B. Paredes, et al., Nature 429 (2004) 277. F. Gerbier, et al., Phys. Rev. Lett. 95 (2005) 050404. G.K. Campbell, et al., Science 313 (2006) 649. G. Roati, et al., Phys. Rev. Lett. 92 (2004) 230402. M. Ko¨hl, et al., Phys. Rev. Lett. 94 (2005) 080403. T. Sto¨ferle, et al., Phys. Rev. Lett. 96 (2006) 030401. J.K. Chin, et al., Nature 443 (2006) 961. K. Gu¨nter, et al., Phys. Rev. Lett. 96 (2006) 180402. S. Ospelkaus, et al., Phys. Rev. Lett. 96 (2006) 180403. M. Rigol, et al., Phys. Rev. Lett. 91 (2003) 130403. S. Trebst, et al., Phys. Rev. Lett. 96 (2006) 250402. T. Yamashita, et al., Phys. Rev. A 74 (2006) 063624. C. Kollath, et al., Phys. Rev. A 74 (2006) 041604. F.K. Pour, et al., Phys. Rev. B 75 (2007) 161104. Y. Fujihara, et al., J. Phys. Soc. Jpn. 76 (2007) 034716; Y. Fujihara, et al., J. Magn. Magn. Mater. 310 (2007) 882. H. Yokoyama, et al., J. Phys. Soc. Jpn. 56 (1987) 1490; H. Yokoyama, et al., J. Phys. Soc. Jpn. 56 (1987) 3582; H. Yokoyama, et al., J. Phys. Soc. Jpn. 59 (1990) 3669. M. Ogata, J. Phys. Soc. Jpn 66 (1997) 3375. T.A. Kaplan, et al., Phys. Rev. Lett. 49 (1982) 889. P. Fazekas, K. Penc, Int. J. Mod. Phys. B 2 (1988) 485. S. Sorella, Phys. Rev. B 71 (2005) 241103. B.M. Andersen, G.M. Bruun, cond-mat 0706.3611.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Variational Monte Carlo analysis of singlet-pairing state in a system with inhomogeneous potential Yusuke Fujihara , Akihisa Koga, Norio Kawakami Department of Physics, Kyoto University, Kyoto 606-8502, Japan
a r t i c l e in f o
Keywords: D. Superconductivity
a b s t r a c t We investigate the singlet-pairing superfluid state of two-component correlated fermions with repulsive interactions in a one-dimensional harmonic potential. By introducing a BCS-type wave function with spatially modulated order parameter, we show how the spin-singlet pairs are formed in the inhomogeneous potential. We then perform variational calculations with particular emphasis on the effect of particle correlations in order to examine the stability of the superfluid state. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Superfluidity in strongly correlated particle systems with spatially modulated potentials has attracted much interest. A typical example is the well-known cuprate, where an observed stripe structure is suggested to play an important role in realizing the d-wave superconductivity with a high critical temperature [1]. More recently, a possibility of the superconductivity in heterostructure of transition-metal oxides such as LaTiO3/SrTiO3 [2] has been discussed, which stimulates intensive investigations on the superconductivity in correlated systems with inhomogeneous structures. Another remarkable example in a slightly different field is the optical lattice system, where the Bose–Einstein condensation of trapped bosonic atom gas has been observed [3]. Owing to its high controllability, a variety of optical lattice systems have been realized such as one-dimensional (1D), twodimensional (2D) and three-dimensional (3D) lattices with bosons [3–7], fermions [8–11] and their mixture [12,13]. Among them, for fermionic systems, a number of interesting properties such as the superfluidity, the Mott transition, etc. have been observed [11] and also theoretically predicted [14–19]. It is particularly interesting and challenging to realize superfluidity of correlated fermions with repulsive interactions in optical lattices, which may share common properties with the high-T c cuprates. Although such inhomogeneous systems provide us with rich physics, it is not easy to investigate their properties theoretically because of strong correlations under inhomogeneous potentials. Systematic studies of such strong correlations in confined systems are thus highly desirable.
Corresponding author. Tel.: +81757533674.
E-mail address: [email protected] (Y. Fujihara). 0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.092
In this paper, we consider two-component correlated fermions trapped in an optical lattice to discuss how the superfluid state emerges under the confining potential. Although we focus our discussions on the optical lattice to make the point clear, we think that the discussions given here may also be applied to some related systems, such as transition-metal oxides with heterostructure, for which site-dependent correlations play a central role. To be more specific, we consider a 1D optical lattice with a confining potential to clarify how the spatially modulated pairing state is realized in mean-field theory. To this end, we make use of the Bogoliubov–de Gennes (BdG) equations. Furthermore, by using the variational Monte Carlo (VMC) [20] simulations, we examine the stability of the pairing state obtained in mean-field theory. Since we study the 1D system in mean-field theory in this paper, the results obtained may be applied to quasi1D systems.
2. Spatially modulated pairing state We investigate two-component correlated fermions with repulsive interactions confined in a 1D harmonic well. Although a proper model system for our purpose may be the Hubbard model with on-site repulsions, we instead consider the t2J model having the nearest-neighbor antiferromagnetic exchange coupling, which is more convenient to observe how the superfluidity emerges in inhomogeneous potentials. Discussions on the Hubbard model will be given later in the paper. The Hamiltonian we consider here is X 1 H ¼ H0 þ J Si Sj ni nj , 4 hi;ji X y X H0 ¼ t cis cjs þ ðV i mÞnis , (1) hi;jis
is
ARTICLE IN PRESS Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
j
j
1.2 1 0.8 〈ni〉
where cyis (cis ) is a creation (annihilation) Fermi operator with spin sð¼"; #Þ at ith site. Here we specify two distinct components of fermions in terms of "; # spins. nis ¼ cyis cis , ni ¼ P y s nis and Si ¼ cia rab cib , where s is the Pauli matrix. t and J ð40Þ are the hopping matrix and the antiferromagnetic exchange between nearest-neighbor sites and m the chemical potential. Interacting fermions are assumed to be trapped by the harmonic potential V i . We introduce a site-dependent BCS-type order parameter Dij ¼ hci# cj" i. Here, we focus on the singlet pairing expected naively for our Hamiltonian with antiferromagnetic coupling, where Dij ¼ Dji . Although doubly occupied states should be excluded in the t2J Hamiltonian, we ignore this constraint for a while to make the problem more tractable. In terms of the P Bogoliubov transformation, cis ¼ l fuli als svil ayls¯ g, we obtain the BdG equation [21], !0 l 1 ! uj F ij X Hij uli @ A ¼ El , (2) F ji Hji vl vl
3389
0.6
J/t = 0.0 J/t = 2.0 J/t = 3.0
0.4 0.2 0
0
10
20
30
40
50 60 i (site)
70
80
90
100
i
with Hij ¼ t dj;iþt þ ðV i mÞdij ,
(3)
F ij ¼ J Dij dj;iþt ,
(4)
which are supplemented by the self-consistent equation, X Dij ¼ uli vjl ,
(5)
0.2
0.1 0.05
-0.05 0
10 20 30 40 50 60 70 80 90 100 i (site)
1 0.5 0
Fig. 1. Spatial distributions of (a) the particle density hni i and (b) the amplitude of the order parameter Di;iþ1.
i = 10 i = 16 i = 30 i = 49 ρ (i,ε)
where t ¼ 1 specifies the nearest-neighbor sites. In our calculation, we deal with a 1D system with L ¼ 100 sites and N" ¼ N# ¼ 30, where Ns is the total number of particles with spin s. To moderate the effects of the edges, the harmonic potential V i is chosen as V L=21 ¼ V L=2 ¼ 0 at the center of the system and V 0 ¼ V L1 ¼ 4:0t at the edges. We also set m as the Fermi level for the non-interacting case ðJ ¼ 0Þ. In Fig. 1, we show the spatial distribution of the particle density P hni i ¼ l jvli j2 and the nearest-neighbor pairing amplitude Di;iþ1 . When J=t ¼ 0, the system is reduced to the tight-binding model, where the spatially modulated particle density appears owing to the confining potential, as shown in Fig. 1(a). It is found that the introduction of J has little effect on both quantities for small J ðoJ c ¼ 0:8tÞ. Further increase of J induces the singlet-pairing state. We note that the amplitude of Di;iþ1 has a maximum around the 16th (equivalently 83th) site, while the profile of the particle density hardly changes. This seems somewhat puzzling at a glance since the particle density hni i0:4 at i ¼ 16 is far from half filling hni i ¼ 1. These results are indeed contrasted to those for the Hubbard (or t2J) model on the square lattice, where the gap function is enhanced near half filling. However, close observations of the local density of states (LDOS) give natural understanding of the above results. Recall that the DOS around the Fermi level mainly contributes to the singlet-pairing formation. In our case, the filling factor alters spatially, so that we look at the LDOS rði; Þ ¼ hijdð H0 Þjii. It is seen in Fig. 2 that a peak structure in the LDOS is located around Fermi level at i ¼ 16, which has a tendency to stabilize the pairing state in that region. Note that this type of peak structure in the LDOS is inherent in (quasi-) 1D systems, and therefore the above results on the superfluiddominant region are, we think, characteristic of (quasi-) 1D systems. Another point we wish to remark in Fig. 1(b) is that there exists a critical value Jc =t0:8 at which the pairing state develops. We have indeed checked that for small interactions JoJ c , the formation of the spin-singlet pairs yields no energy gain. Only after J4Jc, the pairing state starts to appear in the region
3 2.5 2 1.5
0
l
J/t
Δi,i+1
0.15
-8
-6
-4
-2
0 ε
2
4
6
8
Fig. 2. Local density of states rði; Þ of the 1D confined system H0 for i ¼ 10; 16; 30; 49.
mentioned above. We think that this also characterizes the present system. Namely, in our system with confinement, the spatial region where the paring state is stabilized is rather narrow, so that the energy gain to stabilize the superfluidity may not be sufficient enough for small J values. We will address this problem later again for the Hubbard model.
ARTICLE IN PRESS 3390
Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
3. Impact of particle correlations The above mean-field treatment captures some essential features of the paring state in our confined system. However, we are still left with serious problems to be resolved: (a) we have so far neglected the impact of on-site repulsions, which has tendency to suppress the superfluidity; (b) we have employed the t2J model, in which the exchange coupling is artificially introduced by hand, instead of the Hubbard model. In this section, we try to address these problems, and examine how stable the singletpairing state is in the presence of particle correlations due to onsite repulsions. We start with the first problem (a). For this purpose, we write down a superfluid trial state with a spatially modulated order parameter. Following the BCS theory, we consider a trial state, Y jFBCS i ¼ ðun þ vn cyn" cyn# Þj0i, (6) n
2
3
2
16 xn 16 xn 7 7 (7) 41 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; v2n ¼ 41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5, 2 2 2 2 2 2 xn þ Dn xn þ Dn P n where xn and cns ¼ i fi cis are eigenvalues and eigenstates of P the non-interacting Hamiltonian, H0 ¼ ns xn cyns cns . In order to determine the gap function, X n n Dn ¼ hcn# cn" i ¼ fi fj Dij , (8) i;j
we make use of the solution of the BdG equation in Eq. (2), where the effect of the confining potential is already incorporated. However, in this treatment, the prohibition of the doubly occupied states has not been taken into account, and therefore the corresponding mean-field Dij has finite on-site components Dii , which should be zero due to the above-mentioned constraint. To amend this, we introduce a modified pairing state which incorporates the constraint due to the on-site repulsions. As in the case of dx2 y2 -wave pairing in cuprates in 2D, we only consider the nearest-neighbor pairing that should be most relevant and discard the on-site pairing: we set Dij ¼ jhci# cj" ij for j ¼ i 1 and Dij ¼ 0 otherwise in Eq. (8). We carry out the calculation with the above improved trial function for the t2J model. As a result, we confirm that the singlet-pairing state is still stabilized even if the impact of on-site repulsions is considered. In Fig. 3, we show the resulting one-
7 J/t = 0.0 J/t = 1.0 J/t = 2.0 J/t = 3.0
0.4 0.35
5
0.25
4
0.15
3
0.05
0.3 0.2
Δ
En
0.1 0 26
2
28
30
32
34
1 0
0
10
20
30
40
50 n
60
70
defined as the lowest value of En is very small, D=t101 even for large J=t. This implies that the condensation energy is not so large although the superfluid state is realized. As mentioned above, the pairing state has its dominant weight in the narrow regions with hni i0:4, where the attractive interaction is considered to be small in contrast to the regions with hni i1:0. The small energy gain is thus related to the nature of the LDOS in 1D systems. If we treat 2D systems, the situation becomes much better, since the high LDOS is realized near half filling. Finally, we address the problem (b) by starting from the original Hubbard model. We investigate whether the superfluid state can be stabilized the Hubbard model with on-site repulsion U, X H ¼ H0 þ U ni" ni# . (9) i
3
u2n ¼
6
particle excitation spectra obtained for the superfluid state, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 En ¼ xn þ D2n . As seen in Fig. 3, the excitation gap D that is
80
90
Fig. 3. One-particle excitation spectra in the superfluid state. D in the inset shows the excitation energy gap.
In comparison with the t2J model, it is not easy to treat the superfluid state in the Hubbard model because we should properly consider not only on-site correlations but also nearestneighbor correlations under the confining potential. To this end, it is legitimate to use (6) as a trial state describing the superfluidity at mean-field level. To incorporate particle correlations due to U explicitly, we make use of the VMC simulations with the following trial state [19]: (10)
jCi ¼ PQ PG jFBCS i, "
PG ¼ exp
X
#
ai ni" nj# ,
(11)
# b , Q i
(12)
b iþt Þ, ð1 H
(13)
i
"
PQ ¼ exp a0
X i
b ¼D bi Q i
Y t
b i ¼ ð1 ni" Þð1 ni# Þ. PG is a site-dependent b i ¼ ni" ni# , H where D Gutzwiller projection operator with variational parameters ai , which can reduce the weight of the doubly occupied states at each site. In addition, we introduce the projection operator PQ called doublon-holon binding factor, which can incorporate the higher order effect of U [20,22,23]. We consider the nearest-neighbor pairing amplitude Di;iþ1 in jFBCS i as additional variational parameters. Thus we end up with quite a few of variational parameters f. . . ; ai ; . . . ; a0 ; . . . ; Di;iþ1 ; . . .g to be optimized. Unfortunately the minimization procedure is practically beyond the standard VMC method. To overcome this difficulty, we make use of a stochastic reconfiguration with Hessian acceleration (SRH) scheme to minimize the ground state energy in a given parameter space [19,24]. Since the calculation with the SRH scheme is rather tuff and now in progress, we report here the preliminary results we have obtained so far. The calculated energies for U ¼ 6:0; 8:0; 10:0 and 12.0 are E=t ¼ 4:55 0:02; 0:08 0:04; 2:80 0:05 and 4:85 0:05, respectively. We find that the energy difference among the different sets of Di;iþ1 obtained for J=t ¼ 0:0; 1:0; 2:0 and 3.0 in the BdG equations is of the same order as the convergence error that arises from the multivariable optimization process within the SRH scheme. Therefore we could not yet figure out whether the superfluid state is more stable than the normal state. This subtle situation partly comes from the fact that Dij has large amplitudes only in the narrow spatial regions in the system, making the energy gain of the superfluid state very small, as mentioned before. To draw a definite conclusion for the 1D
ARTICLE IN PRESS Y. Fujihara et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3388–3391
system, we are now trying to improve the variational calculations. Our next target in the future study is 2D correlated fermions. In that case, it is assumed that the superfluid state becomes stabilized much easier, since the large LDOS appears near half filling and also the superfluid-dominant region should be extended.
4. Summary We have investigated the singlet-pairing state in 1D fermionic optical lattice systems with repulsive interactions. By taking account of the spatially modulated singlet pairing, we have discussed how the amplitude of the pairing state distributes under a harmonic potential. It has been shown that the superfluid order parameter has large values in the region where the LDOS has a maximum structure around the Fermi level. Furthermore, we have incorporated the correlation effects, and have confirmed that the superfluid state is still stabilized for the t2J model. Concerning the Hubbard model, we have reported the preliminary results obtained by the VMC simulations with the SRH scheme. The calculation done so far is unfortunately not precise enough to figure out whether the superfluid state can be stabilized. We are now in progress to improve the accuracy of calculations and also to introduce other trial states more suitable for inhomogeneous systems. In this paper, we have restricted our discussions to the superfluid state. There may possibly appear other ordered states, such as the Mott insulating state and magnetically ordered states, etc. [14,19,25]. It is particularly worth studying whether or how the superfluid state and the antiferromagnetically ordered state can coexist in the Hubbard model with confining potentials, especially in 2D systems. This issue is now under consideration.
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Acknowledgments The numerical computations were carried out at the Supercomputer Center, the Institute for Solid State Physics, University of Tokyo. We would like to thank M. Yamashita for valuable discussions. This work was supported by Grant-in-Aids for Scientific Research [Grant nos. 17740226 (AK) and 18043017 (NK)] from The Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25]
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