Contents - Max-Planck-Institut für Physik komplexer Systeme

1 st excitation
S = 1 Γ−B2
S = 0 K −A1
S = 0
I
Fidelity minimum
V
Γ−E2
S = 1 M− σx= −1
Lowest triplet
K −A2
M− 1
σx= +1
Γ−E1
M−σx= −1
Γ−B1
0.8
J3
N = 24
N = 32
0.6
0.4
0.2
I
0
0 0.2 0.4
J2
0.6 0.8 1
Figure 2: Phase diagram of the frustrated S = 1/2 Heisenberg model
honeycomb lattice in the region J2, J3 ∈ [0, 1], based on a combination
of exact diagonalization results. The 5 regions identified here
correspond to: (I) a Néel ordered phase with staggered magnetization,
(II) a collinear magnetically ordered phase, (III) One or several
phases corresponding to short or long range ordered non-collinear
magnetic order, (IV) A different collinear magnetically ordered (or
disordered) phase and (V) a magnetically disordered phase forming
a plaquette valence bond crystal. The five phases are sketched in the
panels around the phase diagram.
Honeycomb lattice The exciting finding of a novel
quantum spin liquid in the half-filled Hubbard model
on the honeycomb lattice [4] leads us again to the question
of whether this spin liquid phase on the honeycomb
lattice can be described within a pure S =
1/2 spin model, despite the vicinity of the insulator
to semimetal transition. A high order derivation of
the corresponding spin model is currently in progress,
however the typical value of the expansion parameter
t/U ∼ 0.25 (U/t ∼ 4) relevant for the spin liquid phase
renders this task much more challenging in comparison
to the triangular lattice, where a typical value for
the spin liquid regime is about t/U ∼ 0.11 (U/t ∼ 9).
In the absence of an accurate prediction for a relevant
spin model, we start by exploring the effect of the
next-to-leading order correction to the nearest neighbor
Heisenberg model, which is a second neighbor Heisenberg
coupling J2 arising at fourth order in t/U. We thus
consider in the following a frustrated S = 1/2 Heisenberg
Hamiltonian on the honeycomb lattice, where we
also include a third neighbor coupling J3 for completeness.
The key finding of our numerical work [9] is the presence
of a sizable magnetically disordered phase adjacent
to the well studied Néel phase of the unfrustrated
honeycomb Heisenberg model (c.f. Fig. 2). We identify
this phase as a plaquette valence bond crystal (VBC).
Interestingly we find evidence for a possibly continuous
phase transition between the Néel phase and a
V
IV
II
III
II
III
IV
plaquette valence bond crystal. In addition the energy
and some of the key correlations of the frustrated spin
model in the transition region are well captured by
a simple Gutzwiller projected (GP) ”Dirac sea” wave
function (c.f. Fig. 3). These findings raise the possibility
of a continuous quantum phase transition beyond
the Ginzburg Landau paradigm in this honeycomb lattice
spin model.
P E
E P P E P P E
P P E P P E P P
E P
P P E
E
P E
P P
E P
P E
P P E P P E
E P P E
P P
P P E
Figure 3: Structure of the dimer dimer correlations in the Gutzwiller
projected Dirac sea wave function. This wave function has algebraic
Néel and dimer-dimer correlations and can possibly describe a spin
system at the Néel to spin liquid (here a valence bond crystal) transition.
Surprisingly, these correlations are in qualitative agreement
with the ones found in the spin liquid region of the Hubbard model
in Ref. [4].
In subsequent work we will address the role of the sixspin
terms which will enter the spin model at order six,
and investigate whether these additional terms indeed
reveal a gapped, featureless spin liquid as seen in the
Hubbard model simulations [4], or whether even more
extended spin interactions are required.
[1] A. Georges et al., Rev. Mod. Phys. 68, 041101 (1996).
[2] H. Morita, S. Watanabe, and M. Imada, J. Phys. Soc. Jpn. 71, 2109
(2002).
[3] O. L. Motrunich, Phys. Rev. B 72, 045105 (2005).
[4] Z. Y. Meng et al., Nature 464, 847 (2010).
[5] Y. S. Shimizu et al., Phys. Rev. Lett. 91, 107001 (2003).
[6] H.-Y. Yang, A. M. Läuchli, F. Mila, and K. P. Schmidt, Phys. Rev.
Lett. 105, 267204 (2010).
[7] C. Knetter, K. P. Schmidt, and G. S. Uhrig, J. Phys. A 36, 7889
(2003).
[8] H.C. Kandal et al., Phys. Rev. Lett. 103, 067004 (2009).
[9] A. F. Albuquerque, D. Schwandt, B. Hetenyi, S. Capponi, M.
Mambrini, and A. M. Läuchli, preprint arxiv:1102.5325.
2.13. Quantum Spin Liquids in the Vicinity of Metal-Insulator Transitions 67