Contents - Max-Planck-Institut für Physik komplexer Systeme

2.13 Quantum Spin Liquids in the Vicinity of Metal-Insulator Transitions
The Hubbard model is one of the central paradigms in
the field of strongly correlated systems for about five
decades, and new aspects of its extremely rich phase diagram
are still regularly unveiled. Even at half-filling,
the popular wisdom according to which the model has
only two phases, a metallic one at weak coupling and
an insulating one at strong coupling separated by a first
order transition [1], has been recently challenged. This
goes back to the numerical work of Morita et al. [2]
on the triangular lattice which revealed the presence
of a non-magnetic insulating phase close to the metalinsulator
transition. It is believed that this quantum
spin liquid (QSL) has no spin gap and algebraically decaying
correlations [3]. More recently, a different example
of a quantum spin liquid phase on the insulating
side of the metal-insulator transition has been identified
on the honeycomb lattice using Quantum Monte
Carlo simulations [4]. This spin liquid phase is reported
to have a small spin gap and no appreciable correlations
of any kind.
Given that there are now two different spin liquids on
the insulating side of the Mott metal-insulator transition,
we address here the conceptionally important
question whether a spin-only description of the spin
liquid is valid and if so, whether an accurate description
of the physics is possible, despite the intermediate
value of U/t and the vicinity of the Mott transition.
Triangular lattice The precise nature of the SL phase
of the Hubbard model on the triangular lattice is also
of direct experimental relevance for the 2D organic salt
κ-(BEDT-TTF)2Cu2(CN)3 [5]. As such, it has already
attracted a lot of attention, but fundamental questions
such as the appropriate low-energy effective theory remain
unanswered. Since the phase is insulating, an
effective model where charge fluctuations are treated
as virtual excitations should be possible. However,
whether a description in terms of a pure spin model
is possible is far from obvious, in particular since there
seems to be a jump in the double occupancy at the transition
from the three-sublattice Néel phase to the QSL.
In Ref. [6] we have now showd that the correct lowenergy
theory of both insulating phases, and in particular
of the QSL phase, is indeed a pure spin model. This
has been achieved by deriving an effective spin model
to high order (order 12) about the strong coupling limit
using perturbative continuous unitary transformations
(PCUTs) [7], and by showing that it gives a qualitative
and quantitative account of the transition from the
three-sublattice magnetic order to the SL state.
B. HETÉNYI, A. M. LÄUCHLI
Energy per site [t]
U 2 n d
-0.3
-0.4
-0.5
-0.6
5
4.8
4.6
4.4
4.2
4
(a) (b)
N=12, Spin model
N=21, Spin model
N=27, Spin model
N=36, Spin model
N=21, Hubbard
N=36, PIRG Ref. [6]
8 10 12 14 16
U/t
(c)
8 10 12 14 16
U/t
0.002
0.001
-0.001
-0.002
9.5 9.75 10
U/t
10.25 10.5
(d)
8 10 12
U/t
14
0
16
Figure 1: (a) Energy per site as a function of U/t for the Hubbard
model and the effective spin model. The agreement between the two
approaches is very good, despite the intermediate value of U/t. (b)
Level crossings in the effective spin model signalled by the energy
difference of the two lowest energies. This is an indicator for the
first order phase transition from the three sublattice Néel state to the
quantum spin liquid. (c) Double occupancy (times U 2 ) as a function
of U/t. Note the small jump at the transition, which is faithfully reproduced
in the spin model. (d) Magnetic structure factor at the 120 ◦
AFM ordering wave vector. In the Néel ordered phase the signal increases
with system size, as expected. In the quantum spin liquid
phase the structure factor saturates, indicating a magnetically disordered
phase.
In Fig. 1 we display selected observables obtained by
simulating the effective spin model including two-,
four- and six-spin interactions using large scale exact
diagonalizations (see caption of Fig. 1). We have also
obtained detailed low energy spectra which exhibit a
remarkable correspondence between the low-energy
sector of the Hubbard model and the spin-model and
which underline the first order nature of the transition
from the magnetically ordered phase at large U/t and
the quantum spin liquid at intermediate coupling. The
quantum numbers of the low-energy spectrum also
support the spin-Bose-metal description of the spin liquid
phase put forward in Ref. [3].
This spin-only description thus gives deep insights into
the nature of the QSL phase and clearly provides the
appropriate framework for further studies, for example
including the effect of hopping integral anisotropies
and the long range nature of the Coulomb interaction
in the spin liquid material κ-(BEDT-TTF)2Cu2(CN)3 [8].
66 Selection of Research Results
0
3
2
1
S[Q=(4π/3,0)]
GS Energy Crossing