- Email: [email protected]
Analysis of superfluid state of ultracold fermions with attractive interactions in two-dimensional optical lattices
Physica C 470 (2010) S991–S992
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Analysis of superfluid state of ultracold fermions with attractive interactions in two-dimensional optical lattices Yusuke Fujihara a, Akihisa Koga b,*, Norio Kawakami a a b
Department of Physics, Kyoto University, Kyoto, Japan Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 2 December 2009 Available online 11 December 2009
We discuss the ground state properties of a Fermi gas with attractive interactions trapped in a twodimensional optical lattice. By making use of the variational Monte Carlo method, we show that the system has an instability toward a density-wave state as well as a superfluid state. We also discuss how the additional staggered potential and inter-site Coulomb repulsion stabilize the supersolid region where a superfluid state coexists with a density-wave state. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Optical lattice Ultracold fermion Hubbard model Superfluid Variational Monte Carlo method
Optical lattice systems have attracted recent interest in condensed matter physics. In such systems, it is expected that the inhomogeneity due to a trap potential provides not only spatially-modulated ordered states, but also other intriguing states which are hardly realized in uniform lattice systems. One of the examples is a supersolid state [1,2], where a superfluid state coexists with a density-wave state. Since most of the numerical investigations for the supersolid state in fermionic systems are based on local approximations, it is not clear whether inter-site correlations are also important to realize the supersolid state in the low dimensional optical lattice system. To clarify this, we investigate the ground state properties of attractive fermions by means of the variational Monte Carlo (VMC) method, which allows us to treat onsite and inter-site particle correlations systematically. In particular, we discuss how a supersolid state is realized in the system. Here, we consider two-component (pseudo-spin r ¼"; #) fermionic atoms on a two-dimensional square lattice with L L ðL ¼ 20Þ sites, which is described by the following Hubbard Hamiltonian [3],
H ¼ t
X hi;jir
cyir cjr þ
X X ðV i lÞnir U ni" ni# ; ir
ð1Þ
i
where cyir ðcir Þ is a creation (annihilation) operator of a fermion P P with pseudo-spin r at site i; ni ¼ r nir ¼ r cyir cir . We note that * Corresponding author. E-mail address: [email protected] (A. Koga). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.12.001
the hopping matrix t, the onsite attractive interaction U (>0) and the harmonic potential V i are experimentally controllable. To discuss the ground state properties in the framework of the VMC method, we introduce the following trial state:
jWi ¼ exp
" X i
0
ci ni" ni# þ c
X
# Q i jUSF fDi gi;
ð2Þ
i
where fci g is a set of the site-dependent Gutzwiller variational parameters [4]. c0 is the so-called spinon–spinon binding parameter [5] which can incorporate inter-site spin correlations. jUSF fDi gi denotes the superfluid state with spatially-modulated pair potentials fDi ð¼ hci# ci" iÞg, which are obtained by Pong’s method [6]. In the following: we fix the total number of atoms as N ¼ 160 and set a two dimensional harmonic potential such that V i ¼ 0 at the center of the system and V i ¼ 12t at the corners. By optimizing the variational parameters fci g; c0 ; fDi g to minimize the ground state energy, we obtain the results, as shown in Fig. 1a. In the weak coupling case ðU < 2tÞ, the particles are smoothly distributed, and the pair potential Di is almost zero at all sites (not shown). Therefore, the ground state is metallic. On the other hand, the strong attractive interaction induces Di mainly at the regions with hni i 1:0, where a superfluid state is realized. Although in this parameter space, we could not find a spatial modulation in the density profile indicating a density-wave state, its fluctuations are indeed enhanced. To discuss an instability toward a density-wave state, we P P focus on the quantity DDi ¼ 14 s hni niþs i s0 hni niþs0 i , where 0 s ðs Þ runs over all the nearest (the next nearest) sites. The computed results are shown in Fig. 1b. It is found that DDi has large negative values around i ¼ 4 and 15. This implies that the checkerboard-type
S992
Y. Fujihara et al. / Physica C 470 (2010) S991–S992
1
U/t=7 U/t=4 U/t=1
0.5
0 0.15
ΔDi
(a)Density
(a)
site
Density
2 1.5
(b)
0
-0.15
0
2
4
6
8
10
12
14
16
Fig. 1. Profiles of: (a) the atom density hni i and (b) DDi (see text) along y ¼ L=2 line of the system for U=t ¼ 7:0 (circle), 4:0 (square) and 1.0 (diamond).
Density difference
0.2
U/t=7 U/t=4 U/t=1
0.1 0 -0.1 -0.2
0
2
4
6
8
10
12
14
16
site
18
site
18
site Fig. 2. Profiles of the density differences between those for DV i – 0 and for DV i ¼ 0 along y ¼ L=2 line of the system.
density correlations develop since the density correlations between the next nearest neighbors are larger than the next nearest ones. In order to discuss the instability toward a coexisting state, we consider small perturbations which can stabilize a density-wave state. First, we introduce a staggered potential DV i ¼ 0:02ð1Þi t, which is fairly smaller than other energy scales. We note again that this potential is experimentally realized by a double-well superlattice potential. By making use of the BdG equations, we calculate the density profile hni i self-consistently. The result is shown in Fig. 2. We find that a notable change is induced in the density profiles around i ¼ 4 and 15, where hni i 1. This may be consistent with the fact that in the attractive Hubbard model without an inhomogeneous potential, the density-wave state is stabilized only at half filling [7]. In our case, the density-wave state is induced by the tiny staggered potential, and at the same time, the superfluid state is realized in the region. Therefore, we can say that the supersolid state is indeed realized in the optical lattice system.
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
(b)Pair potential
site
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Fig. 3. Density plots of: (a) the atom density hni i and (b) the pair potential Di for U=t ¼ 6:0; U 0 =t ¼ 0:06.
We also consider a repulsive interaction between the nearest P neighbor sites U 0 hi;ji ni nj with U 0 =U ¼ 0:01. When U ¼ 6t, we can find that the density-wave state develops around the region with hni i 1, as shown in Fig. 3a. Although the density-wave state suppresses the pair potential Di in the area, it still remains finite. This means that both the density-wave state and the superfluid state coexist, that is, the supersolid state is realized in the region. In this paper, we have demonstrated that the optical lattice systems with perturbations have an instability toward the supersolid state around the regions with hni i 1. Further detailed discussions on the stability of the supersolid state are required, which are now under consideration. Acknowledgments The numerical computations were carried out at the Supercomputer Center, ISSP, University of Tokyo. This work is supported by Grant-in-Aids for Scientific Research [Grant Nos. 20740194 (A.K.), and 19014013, 20029013, 20102008 (N.K.)], and the Global COE Programs ‘‘The Next Generation of Physics, Spun from Universality and Emergence” and ‘‘Nanoscience and Quantum Physics” from the MEXT of Japan. Y.F. is supported by JSPS Research Fellowships for Young Scientists. References [1] F.K. Pour et al., Phys. Rev. B 75 (2007) 161104 (R). [2] A. Koga et al., J. Phys. Soc. Jpn. 77 (2008) 073602; A. Koga et al., Phys. Rev. A 79 (2009) 013607. [3] D. Jaksch et al., Phys. Rev. Lett. 81 (1998) 3108. [4] Y. Fujihara et al., J. Phys. Soc. Jpn. 76 (2007) 034716. [5] H. Yokoyama, Prog. Theor. Rhys. 108 (2002) 59. [6] Y.H. Pong, C.K. Law, Phys. Rev. A 74 (2006) 013618. [7] R.T. Scalettar et al., Phys. Rev. Lett. 62 (1989) 1407.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Analysis of superfluid state of ultracold fermions with attractive interactions in two-dimensional optical lattices Yusuke Fujihara a, Akihisa Koga b,*, Norio Kawakami a a b
Department of Physics, Kyoto University, Kyoto, Japan Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 2 December 2009 Available online 11 December 2009
We discuss the ground state properties of a Fermi gas with attractive interactions trapped in a twodimensional optical lattice. By making use of the variational Monte Carlo method, we show that the system has an instability toward a density-wave state as well as a superfluid state. We also discuss how the additional staggered potential and inter-site Coulomb repulsion stabilize the supersolid region where a superfluid state coexists with a density-wave state. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Optical lattice Ultracold fermion Hubbard model Superfluid Variational Monte Carlo method
Optical lattice systems have attracted recent interest in condensed matter physics. In such systems, it is expected that the inhomogeneity due to a trap potential provides not only spatially-modulated ordered states, but also other intriguing states which are hardly realized in uniform lattice systems. One of the examples is a supersolid state [1,2], where a superfluid state coexists with a density-wave state. Since most of the numerical investigations for the supersolid state in fermionic systems are based on local approximations, it is not clear whether inter-site correlations are also important to realize the supersolid state in the low dimensional optical lattice system. To clarify this, we investigate the ground state properties of attractive fermions by means of the variational Monte Carlo (VMC) method, which allows us to treat onsite and inter-site particle correlations systematically. In particular, we discuss how a supersolid state is realized in the system. Here, we consider two-component (pseudo-spin r ¼"; #) fermionic atoms on a two-dimensional square lattice with L L ðL ¼ 20Þ sites, which is described by the following Hubbard Hamiltonian [3],
H ¼ t
X hi;jir
cyir cjr þ
X X ðV i lÞnir U ni" ni# ; ir
ð1Þ
i
where cyir ðcir Þ is a creation (annihilation) operator of a fermion P P with pseudo-spin r at site i; ni ¼ r nir ¼ r cyir cir . We note that * Corresponding author. E-mail address: [email protected] (A. Koga). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.12.001
the hopping matrix t, the onsite attractive interaction U (>0) and the harmonic potential V i are experimentally controllable. To discuss the ground state properties in the framework of the VMC method, we introduce the following trial state:
jWi ¼ exp
" X i
0
ci ni" ni# þ c
X
# Q i jUSF fDi gi;
ð2Þ
i
where fci g is a set of the site-dependent Gutzwiller variational parameters [4]. c0 is the so-called spinon–spinon binding parameter [5] which can incorporate inter-site spin correlations. jUSF fDi gi denotes the superfluid state with spatially-modulated pair potentials fDi ð¼ hci# ci" iÞg, which are obtained by Pong’s method [6]. In the following: we fix the total number of atoms as N ¼ 160 and set a two dimensional harmonic potential such that V i ¼ 0 at the center of the system and V i ¼ 12t at the corners. By optimizing the variational parameters fci g; c0 ; fDi g to minimize the ground state energy, we obtain the results, as shown in Fig. 1a. In the weak coupling case ðU < 2tÞ, the particles are smoothly distributed, and the pair potential Di is almost zero at all sites (not shown). Therefore, the ground state is metallic. On the other hand, the strong attractive interaction induces Di mainly at the regions with hni i 1:0, where a superfluid state is realized. Although in this parameter space, we could not find a spatial modulation in the density profile indicating a density-wave state, its fluctuations are indeed enhanced. To discuss an instability toward a density-wave state, we P P focus on the quantity DDi ¼ 14 s hni niþs i s0 hni niþs0 i , where 0 s ðs Þ runs over all the nearest (the next nearest) sites. The computed results are shown in Fig. 1b. It is found that DDi has large negative values around i ¼ 4 and 15. This implies that the checkerboard-type
S992
Y. Fujihara et al. / Physica C 470 (2010) S991–S992
1
U/t=7 U/t=4 U/t=1
0.5
0 0.15
ΔDi
(a)Density
(a)
site
Density
2 1.5
(b)
0
-0.15
0
2
4
6
8
10
12
14
16
Fig. 1. Profiles of: (a) the atom density hni i and (b) DDi (see text) along y ¼ L=2 line of the system for U=t ¼ 7:0 (circle), 4:0 (square) and 1.0 (diamond).
Density difference
0.2
U/t=7 U/t=4 U/t=1
0.1 0 -0.1 -0.2
0
2
4
6
8
10
12
14
16
site
18
site
18
site Fig. 2. Profiles of the density differences between those for DV i – 0 and for DV i ¼ 0 along y ¼ L=2 line of the system.
density correlations develop since the density correlations between the next nearest neighbors are larger than the next nearest ones. In order to discuss the instability toward a coexisting state, we consider small perturbations which can stabilize a density-wave state. First, we introduce a staggered potential DV i ¼ 0:02ð1Þi t, which is fairly smaller than other energy scales. We note again that this potential is experimentally realized by a double-well superlattice potential. By making use of the BdG equations, we calculate the density profile hni i self-consistently. The result is shown in Fig. 2. We find that a notable change is induced in the density profiles around i ¼ 4 and 15, where hni i 1. This may be consistent with the fact that in the attractive Hubbard model without an inhomogeneous potential, the density-wave state is stabilized only at half filling [7]. In our case, the density-wave state is induced by the tiny staggered potential, and at the same time, the superfluid state is realized in the region. Therefore, we can say that the supersolid state is indeed realized in the optical lattice system.
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
(b)Pair potential
site
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Fig. 3. Density plots of: (a) the atom density hni i and (b) the pair potential Di for U=t ¼ 6:0; U 0 =t ¼ 0:06.
We also consider a repulsive interaction between the nearest P neighbor sites U 0 hi;ji ni nj with U 0 =U ¼ 0:01. When U ¼ 6t, we can find that the density-wave state develops around the region with hni i 1, as shown in Fig. 3a. Although the density-wave state suppresses the pair potential Di in the area, it still remains finite. This means that both the density-wave state and the superfluid state coexist, that is, the supersolid state is realized in the region. In this paper, we have demonstrated that the optical lattice systems with perturbations have an instability toward the supersolid state around the regions with hni i 1. Further detailed discussions on the stability of the supersolid state are required, which are now under consideration. Acknowledgments The numerical computations were carried out at the Supercomputer Center, ISSP, University of Tokyo. This work is supported by Grant-in-Aids for Scientific Research [Grant Nos. 20740194 (A.K.), and 19014013, 20029013, 20102008 (N.K.)], and the Global COE Programs ‘‘The Next Generation of Physics, Spun from Universality and Emergence” and ‘‘Nanoscience and Quantum Physics” from the MEXT of Japan. Y.F. is supported by JSPS Research Fellowships for Young Scientists. References [1] F.K. Pour et al., Phys. Rev. B 75 (2007) 161104 (R). [2] A. Koga et al., J. Phys. Soc. Jpn. 77 (2008) 073602; A. Koga et al., Phys. Rev. A 79 (2009) 013607. [3] D. Jaksch et al., Phys. Rev. Lett. 81 (1998) 3108. [4] Y. Fujihara et al., J. Phys. Soc. Jpn. 76 (2007) 034716. [5] H. Yokoyama, Prog. Theor. Rhys. 108 (2002) 59. [6] Y.H. Pong, C.K. Law, Phys. Rev. A 74 (2006) 013618. [7] R.T. Scalettar et al., Phys. Rev. Lett. 62 (1989) 1407.