Magnetically Ordered State of Cold Fermions on a Decorated Square Lattice
ABSTRACT We study two-component ultracold fermions with repulsive interactions, which are loaded into a decorated square lattice. By
combining the real-space dynamical mean-field theory with the numerical renormalization group method, we discuss magnetic
properties in the system. It is clarified how the ferromagnetically ordered ground state, which is stabilized by a flat band
mechanism, is adiabatically connected to the ferrimagnetically ordered state expected in the strong coupling limit.
Journal of Low Temperature Physics - QFS2009 manuscript No.
(will be inserted by the editor)
Magnetically ordered state of cold fermions on a decorated
Kazuto Noda · Akihisa Koga, Norio
Kawakami, and Thomas Pruschke
Received: date / Accepted: date
Abstract We study two-component ultracold fermions with repulsive interactions,
which are loaded into a decorated square lattice. By combining the real-space dynam-
ical mean-field theory with the numerical renormalization group method, we discuss
magnetic properties in the system. It is clarified how the ferromagnetically ordered
ground state, which is stabilized by a flat band mechanism, is adiabatically connected
to the ferrimagnetically ordered state expected in the strong coupling limit.
Keywords optical lattice · fermion · ferromagnetism
PACS PACS code1 · PACS code2 · more
Recently, optical lattice systems have attracted much interest. One of the remarkable
examples is a fermionic optical lattice system, which is formed by loading fermionic
atoms in a periodic potential. Owing to its high controllability in the interaction
strength, the number of particles and other parameters, many remarkable phenomena
have been observed such as superfluidity , Mott transition [4,5], etc. Although a
magnetically ordered state has not been observed so far, it is theoretically predicted
that magnetic states should be realized in some fermionic optical lattice systems such
Department of Physics, Graduate School of Science, Kyoto University, Kyoto, Japan
Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
Department of Physics, Graduate School of Science, Kyoto University, Kyoto, Japan
Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen, G¨ ottingen, Germany
as a two-dimensional (2D) square lattice with confining potential and a honeycomb
lattice with p-orbitals. We here investigate how the magnetic state emerges for an op-
tical lattice with specific geometry that has a tendency to stabilize the ferromagnetism.
The model system we consider for this purpose is a decorated square lattice shown in
Fig. 1, which may be expected to be realized experimentally in the future. The lattice
is composed of three kinds of sites, A, B and C, whose characteristic structure gives
rise to a flat-band structure in the local density of states (LDOS): a delta-function-like
peak (flat band) appears in the LDOS for B and C sites (see Fig. 2). In the half-filled
case, an infinitesimal interaction lifts the degenerate electron states at the Fermi level,
which drives the system to the ferromagnetically ordered state, as guaranteed by Lieb’s
theorem. However, it is still nontrivial and interesting to elucidate how the magnet-
ically ordered state changes its character when the interaction strength is varied. To
address this problem, we study magnetic properties in the fermionic optical lattice
system on the decorated square lattice.
square lattice. Dashed line shows the
unit cell. Sketch of the RDMFT pro-
cedure, in detail see text.
1 (Color online).Decorated
B, C DOS
(LDOS) for noninteracting particles.
Up panel is LDOS of A site, and
down panel is that of B, C site.
2 Local densityof states
2 Model and Hamiltonian
We consider two-component ultracold fermions on the decorated square lattice, which
is described by the following Hubbard Hamiltonian,
iσ(cjσ) creates (annihilates) a fermion at the ith site with spin σ, and niσ=
iσciσ. tijis the nearest-neighbor hopping and U is a repulsive interaction.
To discuss ground-state properties of the system, we use the real-space dynamical
mean-field theory (R-DMFT), which has successfully been applied to some correlated
H = −
systems such as an interface between the band insulator and the Mott insulator,
repulsive [7,12] or attractive fermionic atoms with confining potential. In the R-
DMFT, the lattice model is mapped onto an effective impurity model, where local
correlations are taken into account precisely. In our decorated square lattice model, we
need to introduce three kinds of effective impurity models. The local Green function,
ˆGασ, for the effective bath is determined by the Dyson equation,
where µ is the chemical potential, k is the wave vector in the reciprocal lattice space,
andˆ Σα is the self-energy for the αth sublattices. Here,ˆG,ˆ Σ,ˆt are represented by 3×3
To solve the effective impurity models, we employ the numerical renormalization
group (NRG), where the effective bath is discretized on a logarithmic mesh, enabling
us to treat low-energy properties precisely. NRG is especially powerful to study our
system, since we should treat correlation effects including a singular behavior due to a
flat-band in DOS near the Fermi level. We can therefore access the small energy scales
in the system quantitatively. To ensure that the sum rules for dynamical quantities are
fulfilled, we use the complete basis set algorithm proposed recently. In this study,
we calculate the particle density nσ and the magnetization mα =
discuss the effects of the interaction on the magnetically ordered ground state.
ω + µ −ˆt(k) −ˆ Σασ(ω)
2(nα↑− nα↓) to
To discuss how the ferromagnetically ordered state is affected by repulsive interac-
tions, we show the magnetization for each sublattice in Fig.2. Upon introducing the
interaction U, the magnetization for the A sublattice increases gradually from zero,
while the magnetizations for the B and C sublattices appear discontinuously from zero
to m = −0.25. This singular behavior is caused by the flat band structure at the
0 5 10
Fig. 3 (Color online). Squares (triangles) represent spontaneous magnetizations for A (B and
C) sublattices. Crosses represent the total magnetizations per unit cell.
-10 -5 0 5 10
B, C DOS
U/t = 1.4
-10-5 0 5 10
U/t = 11.2
Fig. 4 (Color online). Solid (dashed) lines represent the LDOS for up (down) spin in the
system with U/t = 1.4(left) and 11.2(right).
Fermi level in the LDOS (see Fig. 2). The resulting ferromagnetism is often called
the flat-band ferromagnetism . As the interaction further increases, the magnetiza-
tion for each sublattice increases gradually and approaches the saturated values in the
strong coupling regime, where the ferrimagnetically ordered ground state is realized
with staggered magnetizations (mA∼ 0.5 and mB,C∼ −0.5). Therefore, there is a
smooth crossover from the flat-band ferromagnetism to the Heisenberg-type ferrimag-
netism when the strength of U is varied. We note that the total magnetization of the
unit cell is always preserved during the change of the interaction strength, which is
consistent with Lieb’s theorem.
What happens in the weak- and strong coupling regime can be more clearly seen in
the LDOS for each sublattice. We first recall that at U = 0 (Fig. 2), the LDOS has the
flat band in the B, C sublattices at the Fermi level, but not in the A sublattice. When
an infinitesimal interaction is introduced, the splitting of the flat band at the Fermi
level (Fig.3 (a) and (b)) leads to the sudden increase of magnetizations for the B and
C sublattices. The weight of the flat band is half of the total weight in DOS, resulting
in the total magnetization mtot= −0.5 per unit cell in accordance with the computed
results in Fig. 2. Therefore, we can see that the ferromagnetism in the weak coupling
regime is dominated by the electrons around the Fermi level in the B and C sublattices.
On the other hand, in the strong coupling regime, where the magnetization for each
sublattice is almost saturated, all the electrons in the entire energy region contribute
to the formation of the Heisenberg-type ferrimagnetic order (Fig. 3). Note that the
crossover occurs around U/W ∼ 1.
In summary, we have investigated the two-component fermionic repulsive Hubbard
model in a decorated square lattice. Using R-DMFT combined with NRG, we have
revealed how the system exhibits a crossover from the flat-band ferromagnetism in the
weak coupling regime to the Heisenberg ferrimagnetism in the strong coupling regime.
In order to observe the above magnetic state experimentally in an optical lattice, it
is crucial to examine how stable it is at finite temperatures. It is known that in the
pure 2D case addressed here, a phase transition should not occur at finite tempera-
tures. However, if we consider quasi-2D systems, we can expect a finite-temperature
phase transition. It can be shown that the corresponding transition temperature should
become highest in the crossover regime. This encourages experimental investigations
on a quasi-2D optical lattice system in the crossover regime. The detailed analysis of
quasi-2D cases is now under investigation.
In this paper, we have neglected, for simplicity, the effects of the energy-level split-
ting, the hopping imbalance, etc, which should be also important to discuss the stability
of the magnetically ordered state in the optical lattice. These effects will be addressed
Acknowledgements This work was partly supported by the Grant-in-Aid for Scientific Re-
search [20029013, 21540359 (N.K.) and 20740194 (A.K.)] and the Grant-in-Aid for the Global
COE Programs ”Next Generation of Physics, Spun from Universality and Emergence” and
”Nanoscience and Quantum Physics” from the Ministry of Education, Culture, Sports, Sci-
ence and Technology (MEXT) of Japan.
1. I. Bloch, J. Dalibard, and W. Zwerger: Rev. Mod. Phys. 80 885 (2008).
2. I. Bloch: Nature Physics 1 23 (2005).
3. J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu, and W. Ketterle:
Nature 443, 961 (2006).
4. R. J¨ ordens, N. Strohmaier, K. G¨ unter, H. Moritz, and T. Esslinger: Nature 455, 204 (2008).
5. U. Schneider, L. Hackerm¨ uller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes,
D. Rasch, and A. Rosch: Science 322 1520 (2008).
6. S. Trotzky, P. Cheinet, S. F¨ olling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov,
E. A. Demler, M. D. Lukin, and I. Bloch: Science 319 295 (2008).
7. M.Snoek, I. Titvinidze, C.Toke, K.Byczuk, and W.Hofstetter: New J. Phys. 10 093008
8. S. Zhang, H. Hung, and C. Wu: arXiv:0805.3031.
9. E. H. Lieb: Phys. Rev. Lett. 62 1201 (1989).
10. W. Metzner and D. Vollhardt: Phys. Rev. Lett. 62 324 (1989); E. M¨ uller-Hartmann: Z.
Phys. B 74 507 (1989); Th. Pruschke, M. Jarrell, and J. K. Freericks: Adv. Phys. 44 187
(1995) ; A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg: Rev. Mod. Phys. 68 13
(1996); G. Kotliar and D. Vollhardt: Physics Today 57 53 (2004).
11. S. Okamoto and A. J. Millis: Phys. Rev. B 70 241104 (2004); S. Okamoto and A. J. Millis:
Nature (London) 428 630 (2004).
12. R. W. Helmes, T. A. Costi, and A. Rosch: Phys. Rev. Lett. 100 056403 (2008)
13. A. Koga, T. Higashiyama, K. Inaba, S. Suga, and N. Kawakami: J. Phys. Soc. Jpn. 77
073602 (2008); A. Koga, T. Higashiyama, K. Inaba, S. Suga, and N. Kawakami: Phys. Rev.
A 79 013607 (2009).
14. K. G. Wilson: Rev. Mod. Phys. 47 (1975) 773; H. R. Krishna-murthy, J. W. Wilkins, and
K. G. Wilson: Phys. Rev. B 21 (1980) 1003; ibid 21 (1980) 1044; R. Bulla, T. A. Costi, and
Th. Pruschke: Rev. Mod. Phys. 80 (2008) 395
15. F. B. Anders, and A. Schiller: Phys. Rev. Lett. 95 196801 (2005); R. Peters, Th. Pruschke,
and F. B. Anders: Phys. Rev. B 74 245114 (2006)