Macroscopic degeneracy and emergent frustration in a honeycomb lattice magnet.
ABSTRACT Using a hybrid method based on fermionic diagonalization and classical Monte Carlo techniques, we investigate the interplay between itinerant and localized spins, with competing double- and superexchange interactions, on a honeycomb lattice. For moderate superexchange, a geometrically frustrated triangular lattice of hexagons forms spontaneously. For slightly larger superexchange a dimerized ground state is stable that has macroscopic degeneracy. The presence of these states on a nonfrustrated honeycomb lattice highlights novel phenomena in this itinerant electron system: emergent geometrical frustration and degeneracy related to a symmetry intermediate between local and global.
arXiv:1105.2469v1 [cond-mat.str-el] 12 May 2011
Macroscopic Degeneracy and Emergent Frustration in a Honeycomb Lattice Magnet
J¨ orn W.F. Venderbos,1, ∗Maria Daghofer,1Jeroen van den Brink,1and Sanjeev Kumar2
1IFW Dresden, P.O. Box 27 01 16, D-01171 Dresden, Germany
2Indian Institute of Science Education and Research (IISER) Mohali,
MGSIPAP Complex, Sector 26, Chandigarh 160019, India
(Dated: May 13, 2011)
Using a hybrid method based on fermionic diagonalization and classical Monte Carlo, we investi-
gate the interplay between itinerant and localized spins, with competing double- and super-exchange
interactions, on a honeycomb lattice. For moderate superexchange, a geometrically frustrated trian-
gular lattice of hexagons forms spontaneously. For slightly larger superexchange a dimerized ground-
state is stable that has macroscopic degeneracy. The presence of these states on a non-frustrated
honeycomb lattice highlights a novel phenomenon in this itinerant electron system: emergent geo-
metrical frustration and degeneracy.
PACS numbers: 71.10.-w , 75.10.-b , 71.27.+a , 71.30.+h
The Kondo lattice model (KLM) is probably the most
celebrated starting point for the investigation of the in-
terplay between localized spins and itinerant electrons
. It provides the canonical explanation for the Kondo
effect and for the heavy-fermion behaviour observed in
many materials . Motivated by the search for topolog-
ically non-trivial states of matter, several groups have
recently studied the itinerant KLM on frustrated lat-
tices, such as the triangular or the pyrochlore one, and
have shown that due to the strong geometrical frustration
scalar-chiral types of magnetic ordering emerge [3–7].
The physics of the KLM on non-frustrated lattices,
such as the square and cubic one, has been studied exten-
sively. In particular the limit of strong coupling and large
localized moments, where the KLM goes over into the
double-exchange (DE) model, is directly relevant to the
colossal magnetoresistance effect in perovskite mangan-
ites [8–10]. In such cases, the competition between DE
and antiferromagnetic (AFM) superexchange can lead to
canted spin states or phase separation . Although the
honeycomb lattice is also bi-partite, it has the small-
est possible coordination number for proper 2D lattices.
Moreover, its density of states vanishes at the Fermi level
at half filling, which allow for a fundamentally different
physical phenomena to emerge on a honeycomb lattice.
This is illustrated by recent Quantum Monte Carlo cal-
culations  that identify a novel spin-liquid phase for
the Hubbard model on the honeycomb lattice, a finding
supported by analytical studies [12–14] and very different
from the situation on the square lattice.
In this Letter, we investigate the consequences of the
competition between AFM superexchange and ferromag-
netic (FM) DE on the honeycomb lattice. We find that
two exotic ground states exist between the trivial, fully
FM and AFM phases. In the first, nearer to the FM
state, the spins self-organize into FM hexagons that are
coupled antiferromagnetically. Since the hexagonal rings
form a frustrated triangular lattice, their order is remi-
niscent of the Yafet-Kittel state . The competition
between isotropic magnetic interactions thus causes geo-
metric frustration to emerge in a non-frustrated lattice.
For slightly stronger AFM interactions, we find the
exact groundstate to consist of independent FM dimers
containing one electron each. Apart from the require-
ment that the alignment of adjacent dimers be AFM,
they are independent. The groundstate of this N-spin
system therefore has a high degeneracy ∝ 2
the macroscopic degeneracy ∝ eαNin (spin) ice is caused
by the local symmetry of the frustrated tetrahdra [16, 17],
mediate’ symmetry – a symmetry between local and
global . It is remarkable that this highly degener-
ate groundstate manifold arises as an emergent effect in
a Hamiltonian that itself does not have such a symmetry.
In a large number of material systems the essence of the
electronic structure is captured by interacting spins and
electrons on a honeycomb lattice. There has been a lot
of interest in the interactions between impurity magnetic
moments residing on the honeycomb lattice of graphene,
which have been studied in a Ruderman-Kittel-Kasuya-
Yosida (RKKY) framework  and using the KLM .
Going beyond RKKY is even more important in transi-
tion metal oxides, for example Bi3Mn4O12(NO3) [21–24]
or Li2MnO3, with Mn ions residing on a honeycomb
The Hamiltonian corresponding to the one-band DE
model in the presence of competing AFM superexchange
interactions on a honeycomb lattice is
√N exponent indicates the presence of an ‘inter-
H = −
iψj+ H.c.) + JAF
Si· Sj, (1)
nihilation operators, respectively.
the DE scheme these fermions have their spin aligned
with the on-site spins Si.
treated as classical spins with |Si| = 1, and thus can
be specified by their polar and azimuthal angles (θi,φi).
Both sums are over nearest neighbors, indicated by ?ij?.
iand ψiare the fermionic creation and an-
In accordance with
The on-site core spins are
FIG. 1. Schematic view of (a) the honeycomb lattice and
(b) the brick-wall lattice having the same topology as the
Due to the alignment of electron spin to the core spins,
the hopping amplitude depends on the direction of the
core spins, specifically tij = t0[cos(θi/2)cos(θj/2) +
AFM super-exchange is given by JAFand all energies are
in units of the hopping amplitude t0. To guarantee an
unbiased search for groundstate candidates, we employ a
well-established hybrid method of exact diagonalization
(ED) for the bilinear fermionic part of the Hamiltonian
and Monte Carlo (MC) for the classical spins . In this
scheme, a MC configuration is defined by a given core
spin texture and Markov chains are generated by diag-
onalizing the fermionic problem after each configuration
update. In addition, we make use of the travelling cluster
approximation (TCA) to be able to handle larger lattice
sizes . The TCA has proven its validity and success
in earlier studies investigating a similar class of models
[6, 10, 27]. The results in this work are based on calcu-
lations on a N = 122honeycomb lattice, using a cluster
of size Nc= 62. In the MC routine we use ∼ 104steps
for equilibration and the same number of steps for ther-
mal averaging. In this work we focus on the case of a
half-filled band. Note that half-filling refers to 1/2 an
electron per site, which is equivalent to quarter-filling in
the spinful problem. For selected parameter values, the
MC procedure was further refined by an optimization
routine that diminishes thermal fluctuations .
To identify the magnetically ordered states, we cal-
culate the spin structure factor S(q) =
Sj?eiq·(ri−rj), where ?...? is a thermal average and ri
is the position space vector of site i. For a clear un-
derstanding of the real-space structure of the magnetic
states it is helpful to look at S(q) on a square geom-
etry [see Fig. 1(b)].A specific long-range ordering is
expressed as the point in the Brillouin zone where the
structure factor shows a peak.
tronic properties we compute the density of states (DOS)
as D(ω) = ??
function by a Lorentzian with broadening γ.
In the absence of super-exchange interaction (JAF =
0), the spins order ferromagnetically, as expected from
the DE mechanism. The fermionic problem is then equiv-
alent to non-interacting spinless electrons on a honey-
The strength of the
To analyze the elec-
kδ(ω − ǫk)? and approximate the delta-
comb lattice, giving rise to a dispersion and DOS that
is well-known from graphene [see Fig. 2(a)]. Introduc-
ing a small JAF still leads to a FM ground state. At
JAF ≈ 0.14, the FM state becomes unstable and gives
way to a state with S(q) peaked at2
related to it by symmetry). and with the peculiar four-
peak DOS shown in Fig. 2(b). Real-space snapshots show
that a superlattice formed of hexagons emerges at low
temperatures T, as depicted in Fig. 3(a). This result was
corroborated by zero-temperature optimization of the
spin pattern. Spins within one hexagon are almost FM,
the allowed energies for electrons moving on a six-site
ring are −2t0cos0 = −2t0and −2t0cosπ/3 = −t0, with
twice as many states at −t0, which gives precisely the
DOS seen in Fig. 2(b). Coupling between the hexagons
is AF, but since they occupy a frustrated triangular lat-
tice, see Fig. 3(a), perfect AFM order is not possible.
The hexagons instead are at an angle of ≈ 2π/3, cor-
responding to the Yafet-Kittel state  well known for
the triangular lattice, leading to the signals at2
S(q). Thus a geometrically frustrated triangular lattice
emerges spontaneously from isotropic, competing inter-
actions on the non-frustrated honeycomb lattice.
For 0.18 ≤ JAF ? 0.25, a state consisting of classi-
cal dimers is stabilized. An individual dimer consists of
two spins aligned in parallel and the lattice is covered by
these dimers in such a way that the neighboring dimers
are anti-parallel with respect to each other. In Fig. 3(b)
and 3(c) we show two possible dimer configurations. In
this specific background spin texture, the electron kinetic
energy reduces to that of uncoupled two-level problems,
having only two eigenenergies ±t0. The DOS is there-
fore given by D(ω) = δ(ω − t0) + δ(ω + t0), in excellent
agreement with MC calculations [see Fig. 2(c)]. The
dimer state can be understood as a trade-off between the
FM ordering and the AFM ordering: the electrons are
allowed to populate all the −t0levels (which is more fa-
vorable compared to AFM) and the spins are anti-parallel
with respect to two of their nearest neighbors (which is
more favorable compared to FM).
Interestingly, the dimer ground state of this quantum
system has a macroscopic degeneracy, i.e., there is a
macroscopically large number of ways to cover the lat-
tice by dimers such that the neighboring dimers are anti-
parallel. One simple way to see the degeneracy is to start
covering each row of the lattice in Fig. 1(b) by dimers. It
is easy to see that having fixed the dimer pattern in the
1st row, there are two independent ways of covering each
subsequent row, hence leading to 2
being the number of lattice sites. The fact that there
is thus no long-range order along the y direction of the
brick-wall is also reflected in S(q), which becomes finite
along lines in momentum space, as also seen in compass
models [30–33]. In the 2D compass model, different de-
generate configurations can be reached by flipping a row
of spins. The corresponding√N operators commute with
3(π,0) (and the points
√N−1dimer states, N
-3-2-1 0 1 2 3
ω - EF
-3 -2-1 0 1 2 3
JAF = 0.02
JAF = 0.20
JAF = 0.16
JAF = 0.50
T = 0.001
T = 0.025
T = 0.05
ω - EF
FIG. 2. (a)-(d) Density of states (DOS) at low, intermediate
and high temperatures for different values of JAF.
the T = 0.001 curve shows the DOS of free fermions on a
honeycomb lattice in the thermodynamic limit.
the compass Hamiltonian and thus define an intermedi-
ate symmetry, i.e., between a local, gauge-like (∝ eN)
symmetry and global one (independent of N) . The
magnetic order parameter that obeys the intermediate
symmetry is consequently of nematic type. In the dimer
state, the minimal symmetry operations instead involves
translation of all spins in two adjacent zig-zag rows by
one lattice spacing, σij ?→ σij+1 [σij being the spin at
site (i,j)]. An example for two dimer configurations con-
nected by such an operation is given in Figs. 3(b) and
3(c), where the second and third rows were shifted. How-
ever, this operator does not commute with the Hamil-
tonian Eq. (1), and the intermediate symmetry is thus
rather a property that emerges in the system’s ground
state, similar to the case of striped phases at fractional
filling in the regime of narrow bandwidth and small Jahn-
Teller coupling in a model used for manganites .
For strong super-exchange coupling, there is a continu-
ous way in which the dimer state can approach the AFM
ordered state, captured by a canting angle θ [see Fig.
3(c)], which is the angle between the two spins form-
ing a dimer in the pure dimer phase. The spins remain
antiparallel to those of the neighboring canted dimers.
In this way, the two-level dimer systems remain uncou-
pled. The hopping amplitude between the two spins in
the dimer is renormalized by the double-exchange mech-
anism to t0cos(θ/2). The DOS for a canted dimer state
is consequently given by D(ω) = δ(ω − t0cos(θ/2)) +
δ(ω + t0cos(θ/2)), which tends towards a single peak at
ω = 0 as θ approaches π, as expected for AFM ordering.
The calculated DOS for JAF= 0.50, shown in Fig. 2(d),
FIG. 3. The real-space spin patterns at low temperatures, as
obtained in our Monte Carlo simulations for different values of
JAF. (a) Emergent triangular lattice formed by FM hexagons
at JAF = 0.14. The shown spin configuration is the result
of unbiased MC calculations supplemented with optimization
routines. Spins within each hexagon are almost FM, but there
is a small canting angle between groups of three, illustrated
by shading. Large black dots and yellow lines illustrate the
emergent triangular lattice, the colored spins illustrate the
2π/3-angle order of the Yafet-Kittel state. (b,c) Symmetry
related, degenerate spin states consisting of FM dimers. (d)
Canted dimer configuration.
shows the anticipated effect of canting on the electronic
properties. It is remarkable that this canted state retains
the macroscopic degeneracy inherent to the AFM dimer
state discussed in the preceding paragraph – also this
ordering is therefore of nematic type.
Our results are in good agreement with elementary
energy considerations.The energy per site varies as
3JAF/2 and −JAF/2 for the FM and the dimer states,
respectively.This would imply a phase transition at
JAF≈ 0.15, the FM state is indeed stable for JAF? 0.14
and the dimers for JAF ? 0.18. In between, the emer-
gent Yafet-Kittel state, with a more complex energy de-
pendence, is favorable, see Fig. 4(b). The energy per
site for the canted dimer state is −(2JAF− cos(θ)JAF+
t0cos(θ/2))/2.By differentiating with respect to the
canting angle θ, one easily obtains that canting becomes
favorable for JAF ≥ 0.25 and that the optimal energy is
then given by −3JAF/2 − t2
in behaviour of the ordering temperature for the dimer
state, which starts decreasing at JAF = 0.25 [see Fig.
The results are summarized in Fig. 4. In the finite-T
phase diagram Fig. 4(a), phase boundaries for the FM
and quasi-AFM regions are obtained by determining the
inflection point in the ?M?(T) and ?M?(T) (M denotes
staggered magnetization) curves. The onset of dimer and
other phases is determined by tracking the temperature
dependence of the spin structure factor and the charac-
0/(16JAF). This is reflected
0 0.1 0.2 0.3 0.4 0.5
Temperature vs. JAF
0 0.1 0.2 0.3 0.4 0.5 0.6
Energy / Site
FIG. 4. (a) T-JAF phase diagram at half-filling obtained by
MC for a 12 × 12 lattice. Non-trivial magnetic phases, with
insulating character and macroscopic degeneracy, appear be-
tween the FM (small JAF) and the AFM (large JAF) states.
(b) The energy of the various states: “alm. FM” refers to
a spiral with the longest wavelength supported by the lat-
tice, converging to FM in the thermodynamic limit. Simi-
lar finite-size effects are reported in doped 1D and 2D lat-
tices .“Hex” denotes the emergent Yafet-Kittel order
between hexagons depicted in Fig. 3(a), the energy was opti-
mized with respect to the canting angle within the hexagons.
“Dimers” and “C. Dim” are the highly degenerate FM and
canted dimer states, and “AF” denotes perfectly AF order.
The black crosses are energies obtained by unbiased MC and
a subsequent energy optimization.
teristic features in the DOS. The dimer state smoothly
crosses over to the AFM state via the canted-dimer state.
Figure 4(b) compares the ground-state energies of the
various phases and perfectly agrees with the unbiased
numerical data, indicating that we have identified the
ground states correctly.
We conclude that the isotropic double-exchange model
with competing super-exchange interactions on the non-
frustrated honeycomb lattice has a unexpectedly rich
phase diagram with exotic magnetic phases. In one of
these, FM rings become the essential building blocks,
which form a frustrated triangular lattice and are an-
frustrated spin states on a bipartite honeycomb lattice,
without explicit frustration, is so far unique and an exam-
ple of geometrical frustration emerging from competing
interactions. Another novel phase consist of FM dimers
ordered antiferromagnetically and has a 2
This is reminiscent of compass models, but in the present
case the corresponding symmetry is not a property of the
Hamiltonian given a priori, but rather a property that
emerges in the systems ground state [30–34]. These phe-
nomena are not only relevant in a theoretical context, im-
mediately raising the question which other models share
such features and how further residual interactions might
affect the degeneracy, but pertains in particular to honey-
comb manganese oxides, which form a promising class of
The stabilization of such
materials to realize these novel types of highly frustrated
states harboring macroscopic degeneracies.
This research was supported by the Interphase Pro-
gram of the Dutch Science Foundation NWO/FOM and
by the Emmy-Noether program of the DFG.
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