- Email: [email protected]
Ground-state properties of 1D Hubbard model in optical superlattice
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 910–912 www.elsevier.com/locate/jmmm
Ground-state properties of 1D Hubbard model in optical superlattice A. Yamamotoa,, T. Yamashitaa, M. Yamashitab, N. Kawakamia a
Department of Applied Physics, Osaka University, 2-1 Yamada, Suita, Osaka, 565-0871, Japan NTT Basic Research Laboratories, NTT Corporation, 3-1, Morinosato-Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan
b
Available online 30 October 2006
Abstract We study the ground-state properties of a cold Fermi gas trapped in a one-dimensional optical superlattice with periodically modulated lattice potentials. By applying the density matrix renormalization group method to the Hubbard model with harmonic confinement, we compute the density profile and the spin correlation function for several different choices of superlattice potential. The ground state is characterized by the coexistence phase with the metallic and several types of Mott-insulating domains. It is shown that the spin correlation can be enhanced between the two spatially separated Mott-insulating domains. r 2006 Elsevier B.V. All rights reserved. PACS: 32.80.P; 05.30.Fk; 71.30.+h Keywords: Optical lattice; DMRG; Spin correlation
Recent rapid advances in the laser cooling technology have made it possible to realize optical lattice systems. In particular, physics of cold atoms confined in an optical lattice has opened a new research area closely related to condensed matter physics. This includes, for example, a current hot topic of correlated fermions trapped in an optical lattice [1–3], which covers a wide variety of phenomena such as the BCS-BEC crossover, the Mott transition, etc. Since we focus on the repulsive Hubbard model in this paper, we do not go into the detail of superconducting correlations. We will address the problems related to superconducting correlations such as BCSBEC crossover by properly generalizing the model to the attractive cases in future studies. The behavior of a Fermi gas trapped in an optical lattice can be described by the Hubbard model with the confining potential. It is known that the metallic and insulating domains coexist in such confined systems [4]. Furthermore, a superlattice extension of the optical lattice has been demonstrated in the recent report [5]. It is expected that systematic studies on fermions in an optical superlattice will be done experimentally in the near future. Corresponding author. Tel.: +81 6 6879 7873.
E-mail address: [email protected] (A. Yamamoto). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.137
Motivated by these hot topics, we study the zerotemperature properties of the Hubbard model [5] in a onedimensional optical superlattice confined by harmonic potential, which may be a simplified model for multispecies cold fermions. By computing the density profile and the spin correlation function by means of the density matrix renormalization group (DMRG) [6], we study how the metallic state and the insulating states coexist and exhibit characteristic density/spin correlations. We consider spin-12 fermionic atoms confined in a onedimensional optical superlattice, which is modeled by the Hubbard Hamiltonian with harmonic confinement, X y X X H¼ t ðci;s ciþ1;s þ h:c:Þ þ U ni# ni" þ V i ni;s i;s
þ Vc
8 L=2
i L X
i;s
9 =
ði L=2Þ2 ni;s þ ði L=2 1Þ2 ni;s , :i¼1;s ; i¼L=2þ1;s
where cyi;s ; ci;s and ni;s ¼ cyi;s ci;s are, respectively, the fermion creation, annihilation and number operators relevant to the site labeled i with spin sð¼"; #Þ. Here t is the transfer integral, U the on-site repulsive interaction, V i the periodic superlattice potential, V c the curvature of confinement potential and L the number of sites. The potential V i is shown schematically in Fig. 1, where we
ARTICLE IN PRESS A. Yamamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 910–912
introduce the difference DV to specify the superlattice structure. We make use of DMRG [6], which allows us to deal with a large system with high numerical accuracy as required for the analysis of optical lattices. The essence of the idea is that we first diagonalize a given small system, and then enlarge the system size step by step via a renormaliza-
V
V unit cell
(a)
(c)
unit cell
V (b)
unit cell
Fig. 1. Schematic description of the optical superlattice: (a) 2-site periodic case; (b),(c) 3-site periodic case. These figures do not include a confining potential for simplicity.
911
tion procedure by retaining relevant low-energy states which have large eigenvalues of the density matrix. This procedure gives precise results in one-dimensional quantum systems. In Fig. 2(a)–(c), we show the results for the density profile hni i, which, respectively, correspond to the superlattices in Fig. 1(a)–(c). We can see characteristic sawtoothed behavior reflecting the multiple periodicities. Let us observe the results of the 2-site periodic superlattice in Fig. 2(a). It is seen that the local density alternates its amplitude between even and odd sites reflecting the 2-site periodic potential. Looking at the profile from left to right, we first encounter a small metallic domain around i ¼ 30, which is immediately followed by a kind of Mott insulating domain around i ¼ 35 (ni 0; 1 at every two sites). Then we come to another Mott insulating domain around i ¼ 45 (ni 1 at every site). In the center of the system, we have another metallic-like domain with localized atom at every two sites (i.e. ni 1 or ni 41 alternately), which is closely related to a heavy-fermion model for rare-earth compounds in condensed matter physics. Similar domain structures appear for the 3-site periodic superlattice cases shown in Figs. 2(b) and (c). In these cases, the local density oscillates periodically like 1,0,0 or 1,1,0. In this way, several
2 (d)
0.2 〈siz sjz〉
ni
(a)
1
j=35
0.1 0 -0.1
0 0
50
100
0
50
100
2 (e)
0.2 〈siz sjz〉
ni
(b)
1
j=45
0.1 0 -0.1
0 0
50
100
0
2
50
100
0.1 (c)
(f)
j=55
〈siz sjz〉
ni
0.05 1
0 -0.05
0 0
50
100 i
0
50
100 i
Fig. 2. The density profile hni i and the spin correlation function hS zi Szj i: (a), (b) and (c) show hni i, which respectively correspond to (a), (b) and (c) in Fig.1. We choose U=t ¼ 7:0, DV =t ¼ 3:0, L ¼ 120, and V c =t ¼ 0:01. The number of fermions is N f ¼ 60. In (d), (e) and (f), we represent the spin correlation function hSzi Szj i between the ith and jth sites for the 2-site periodic case (a) in Fig. 1, where the site j is fixed as (d) j ¼ 35, (e) j ¼ 45 and (f) j ¼ 55.
ARTICLE IN PRESS 912
A. Yamamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 910–912
distinct Mott domains emerge, which are continuously connected via intermediate metallic domains. Note that we have calculated the variance of the local density [4] and also the compressibility for the reference system without confinement in order to distinguish the metallic and insulating domains. These quantities consistently specify the metallic and insulating domains in accordance with the result deduced from the local density. In Fig. 2(d)–(f), we show the spin correlation function hS zi S zj i for the 2-site periodic case. Before discussing the spin correlations, we briefly mention how the local spin moment is developed. In all the three superlattices discussed in Fig. 2, the local spin moment is well developed when the corresponding site has one localized atom approximately (ni 1). For example, in Fig. 2(a), the spin moment is well developed around j ¼ 35 (every two sites) and also around j ¼ 45, while it is smaller in the metallic domain around j ¼ 55. We can see that, in all the cases, the antiferromagnetic spin correlation is developed around the jth site, because localized spins are well developed there. An important point is that the spin correlation for j ¼ 35 in Fig. 2(d) is enhanced around i ¼ 85, although it is once suppressed in the domain of io85. This characteristic behavior is a new feature which has not been observed in homogeneous systems without confinement. This is indeed related to the formation of several distinct Mott insulating domains. In each Mott domain, a finite number of local spins are well developed, which can be quite sensitive to effective magnetic fields. Such spins belonging to the
different Mott domains can have enhanced correlations even if they are separated by the metallic domains. This is why we have observed the enhanced spin correlations. Note that this effect is significant if the number of spins in the Mott domain is odd, as is the case for Fig.2(d). We have studied the ground-state properties of fermions trapped in one-dimensional optical superlattices. It has been shown that the ground state is in the coexistence phase with the metallic and several distinct insulating domains. Also, it has been found that the enhanced spin correlation appears between the two Mott insulating domains which are spatially separated from each other. Systematic experimental studies on the one-dimensional optical superlattices may be within our reach soon. We expect that the characteristic properties discussed here will be observed in the near future. This work was partly supported by Grant-in-Aid for Scientific Research (No. 18043017 ) from MEXT.
References [1] [2] [3] [4] [5] [6]
G. Modugno, et al., Phys. Rev. A 68 (2003) 011601 (R). I. Bloch, Nature Phys. 1 (2005) 23. M. Ko¨hl, et al., Phys. Rev. Lett. 94 (2005) 080403. M. Rigol, et al., Phys. Rev. Lett. 91 (2003) 130403. S. Peil, et al., Phys. Rev. A 67 (2003) 051603 (R). S.R. White, Phys. Rev. Lett. 69 (1992) 2863.
Journal of Magnetism and Magnetic Materials 310 (2007) 910–912 www.elsevier.com/locate/jmmm
Ground-state properties of 1D Hubbard model in optical superlattice A. Yamamotoa,, T. Yamashitaa, M. Yamashitab, N. Kawakamia a
Department of Applied Physics, Osaka University, 2-1 Yamada, Suita, Osaka, 565-0871, Japan NTT Basic Research Laboratories, NTT Corporation, 3-1, Morinosato-Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan
b
Available online 30 October 2006
Abstract We study the ground-state properties of a cold Fermi gas trapped in a one-dimensional optical superlattice with periodically modulated lattice potentials. By applying the density matrix renormalization group method to the Hubbard model with harmonic confinement, we compute the density profile and the spin correlation function for several different choices of superlattice potential. The ground state is characterized by the coexistence phase with the metallic and several types of Mott-insulating domains. It is shown that the spin correlation can be enhanced between the two spatially separated Mott-insulating domains. r 2006 Elsevier B.V. All rights reserved. PACS: 32.80.P; 05.30.Fk; 71.30.+h Keywords: Optical lattice; DMRG; Spin correlation
Recent rapid advances in the laser cooling technology have made it possible to realize optical lattice systems. In particular, physics of cold atoms confined in an optical lattice has opened a new research area closely related to condensed matter physics. This includes, for example, a current hot topic of correlated fermions trapped in an optical lattice [1–3], which covers a wide variety of phenomena such as the BCS-BEC crossover, the Mott transition, etc. Since we focus on the repulsive Hubbard model in this paper, we do not go into the detail of superconducting correlations. We will address the problems related to superconducting correlations such as BCSBEC crossover by properly generalizing the model to the attractive cases in future studies. The behavior of a Fermi gas trapped in an optical lattice can be described by the Hubbard model with the confining potential. It is known that the metallic and insulating domains coexist in such confined systems [4]. Furthermore, a superlattice extension of the optical lattice has been demonstrated in the recent report [5]. It is expected that systematic studies on fermions in an optical superlattice will be done experimentally in the near future. Corresponding author. Tel.: +81 6 6879 7873.
E-mail address: [email protected] (A. Yamamoto). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.137
Motivated by these hot topics, we study the zerotemperature properties of the Hubbard model [5] in a onedimensional optical superlattice confined by harmonic potential, which may be a simplified model for multispecies cold fermions. By computing the density profile and the spin correlation function by means of the density matrix renormalization group (DMRG) [6], we study how the metallic state and the insulating states coexist and exhibit characteristic density/spin correlations. We consider spin-12 fermionic atoms confined in a onedimensional optical superlattice, which is modeled by the Hubbard Hamiltonian with harmonic confinement, X y X X H¼ t ðci;s ciþ1;s þ h:c:Þ þ U ni# ni" þ V i ni;s i;s
þ Vc
8 L=2
i L X
i;s
9 =
ði L=2Þ2 ni;s þ ði L=2 1Þ2 ni;s , :i¼1;s ; i¼L=2þ1;s
where cyi;s ; ci;s and ni;s ¼ cyi;s ci;s are, respectively, the fermion creation, annihilation and number operators relevant to the site labeled i with spin sð¼"; #Þ. Here t is the transfer integral, U the on-site repulsive interaction, V i the periodic superlattice potential, V c the curvature of confinement potential and L the number of sites. The potential V i is shown schematically in Fig. 1, where we
ARTICLE IN PRESS A. Yamamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 910–912
introduce the difference DV to specify the superlattice structure. We make use of DMRG [6], which allows us to deal with a large system with high numerical accuracy as required for the analysis of optical lattices. The essence of the idea is that we first diagonalize a given small system, and then enlarge the system size step by step via a renormaliza-
V
V unit cell
(a)
(c)
unit cell
V (b)
unit cell
Fig. 1. Schematic description of the optical superlattice: (a) 2-site periodic case; (b),(c) 3-site periodic case. These figures do not include a confining potential for simplicity.
911
tion procedure by retaining relevant low-energy states which have large eigenvalues of the density matrix. This procedure gives precise results in one-dimensional quantum systems. In Fig. 2(a)–(c), we show the results for the density profile hni i, which, respectively, correspond to the superlattices in Fig. 1(a)–(c). We can see characteristic sawtoothed behavior reflecting the multiple periodicities. Let us observe the results of the 2-site periodic superlattice in Fig. 2(a). It is seen that the local density alternates its amplitude between even and odd sites reflecting the 2-site periodic potential. Looking at the profile from left to right, we first encounter a small metallic domain around i ¼ 30, which is immediately followed by a kind of Mott insulating domain around i ¼ 35 (ni 0; 1 at every two sites). Then we come to another Mott insulating domain around i ¼ 45 (ni 1 at every site). In the center of the system, we have another metallic-like domain with localized atom at every two sites (i.e. ni 1 or ni 41 alternately), which is closely related to a heavy-fermion model for rare-earth compounds in condensed matter physics. Similar domain structures appear for the 3-site periodic superlattice cases shown in Figs. 2(b) and (c). In these cases, the local density oscillates periodically like 1,0,0 or 1,1,0. In this way, several
2 (d)
0.2 〈siz sjz〉
ni
(a)
1
j=35
0.1 0 -0.1
0 0
50
100
0
50
100
2 (e)
0.2 〈siz sjz〉
ni
(b)
1
j=45
0.1 0 -0.1
0 0
50
100
0
2
50
100
0.1 (c)
(f)
j=55
〈siz sjz〉
ni
0.05 1
0 -0.05
0 0
50
100 i
0
50
100 i
Fig. 2. The density profile hni i and the spin correlation function hS zi Szj i: (a), (b) and (c) show hni i, which respectively correspond to (a), (b) and (c) in Fig.1. We choose U=t ¼ 7:0, DV =t ¼ 3:0, L ¼ 120, and V c =t ¼ 0:01. The number of fermions is N f ¼ 60. In (d), (e) and (f), we represent the spin correlation function hSzi Szj i between the ith and jth sites for the 2-site periodic case (a) in Fig. 1, where the site j is fixed as (d) j ¼ 35, (e) j ¼ 45 and (f) j ¼ 55.
ARTICLE IN PRESS 912
A. Yamamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 910–912
distinct Mott domains emerge, which are continuously connected via intermediate metallic domains. Note that we have calculated the variance of the local density [4] and also the compressibility for the reference system without confinement in order to distinguish the metallic and insulating domains. These quantities consistently specify the metallic and insulating domains in accordance with the result deduced from the local density. In Fig. 2(d)–(f), we show the spin correlation function hS zi S zj i for the 2-site periodic case. Before discussing the spin correlations, we briefly mention how the local spin moment is developed. In all the three superlattices discussed in Fig. 2, the local spin moment is well developed when the corresponding site has one localized atom approximately (ni 1). For example, in Fig. 2(a), the spin moment is well developed around j ¼ 35 (every two sites) and also around j ¼ 45, while it is smaller in the metallic domain around j ¼ 55. We can see that, in all the cases, the antiferromagnetic spin correlation is developed around the jth site, because localized spins are well developed there. An important point is that the spin correlation for j ¼ 35 in Fig. 2(d) is enhanced around i ¼ 85, although it is once suppressed in the domain of io85. This characteristic behavior is a new feature which has not been observed in homogeneous systems without confinement. This is indeed related to the formation of several distinct Mott insulating domains. In each Mott domain, a finite number of local spins are well developed, which can be quite sensitive to effective magnetic fields. Such spins belonging to the
different Mott domains can have enhanced correlations even if they are separated by the metallic domains. This is why we have observed the enhanced spin correlations. Note that this effect is significant if the number of spins in the Mott domain is odd, as is the case for Fig.2(d). We have studied the ground-state properties of fermions trapped in one-dimensional optical superlattices. It has been shown that the ground state is in the coexistence phase with the metallic and several distinct insulating domains. Also, it has been found that the enhanced spin correlation appears between the two Mott insulating domains which are spatially separated from each other. Systematic experimental studies on the one-dimensional optical superlattices may be within our reach soon. We expect that the characteristic properties discussed here will be observed in the near future. This work was partly supported by Grant-in-Aid for Scientific Research (No. 18043017 ) from MEXT.
References [1] [2] [3] [4] [5] [6]
G. Modugno, et al., Phys. Rev. A 68 (2003) 011601 (R). I. Bloch, Nature Phys. 1 (2005) 23. M. Ko¨hl, et al., Phys. Rev. Lett. 94 (2005) 080403. M. Rigol, et al., Phys. Rev. Lett. 91 (2003) 130403. S. Peil, et al., Phys. Rev. A 67 (2003) 051603 (R). S.R. White, Phys. Rev. Lett. 69 (1992) 2863.