Entanglement renormalization techniques are applied to numerically
investigate the ground state of the spin-1/2 Heisenberg model on a kagome
lattice. Lattices of N={36,144,inf} sites with periodic boundary conditions are
considered. For the infinite lattice, the best approximation to the ground
state is found to be a valence bond crystal (VBC) with a 36-site unit cell,
compatible with a previous proposal. Its energy per site, E=-0.43221, is an
exact upper bound and is lower than the energy of any previous (gapped or
algebraic) spin liquid candidate for the ground state.Comment: 6 pages, 7 figures, RevTeX 4. Revised version with improved numerical
  results

Evenbly, G.

Vidal, G.

English

arXiv

G. Evenbly

G. Vidal

Crossref

Frustrated Antiferromagnets with Entanglement Renormalization: Ground State of the Spin-
                    
                      
                        1

Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-1/2 Heisenberg model on a kagome lattice. Lattices of N={36,144, infinity} sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal with a 36-site unit cell, compatible with a previous proposal. Its energy per site, E=-0.432 21, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state

University of Queensland eSpace

Frustrated Antiferromagnets with Entanglement Renormalization:
Ground State of the Spin- 12 Heisenberg Model on a Kagome Lattice
G. Evenbly and G. Vidal
School of Mathematics and Physics, the University of Queensland, Brisbane 4072, Australia
(Received 23 April 2009; revised manuscript received 10 December 2009; published 7 May 2010)
Entanglement renormalization techniques are applied to numerically investigate the ground state of the
spin- 12 Heisenberg model on a kagome lattice. Lattices of N ¼ f36; 144;1g sites with periodic boundary
conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be
a valence bond crystal with a 36-site unit cell, compatible with a previous proposal. Its energy per site,
E ¼ 0:432 21, is an exact upper bound and is lower than the energy of any previous (gapped or
algebraic) spin liquid candidate for the ground state.
DOI: 10.1103/PhysRevLett.104.187203 PACS numbers: 75.10.Jm, 02.70.c, 03.67.Mn, 05.30.d
Low dimensional spin- 12 quantum systems have long
been the focus of intense research efforts, largely fueled
by the search for exotic states of matter. An important
example of a geometrically frustrated quantum antiferro-
magnet [1] is the spin- 12 kagome-lattice Heisenberg model
(KLHM). Despite a long history of study, the nature of its
ground state remains an open question. Leading proposals
include valence bond crystal (VBC) [2–7] and spin liquid
(SL) [8–16] ground states. Interest has been further stimu-
lated by recent experimental work on herbertsmithite
ZnCu3ðOHÞ6Cl2, a possible physical realization of the
model [17].
Progress in our understanding of the KLHM has been
hindered, as with many other models of frustrated antifer-
romagnets, by the inapplicability of quantum Monte Carlo
methods due to the negative sign problem. Nevertheless,
systems with up to 36 sites have been addressed with exact
diagonalization [9,18], whereas the density matrix renor-
malization group (DMRG) has been used to explore latti-
ces of order N  100 sites [16]. Unfortunately, it is very
difficult to infer the nature of the ground state of an infinite
system from these results. The reason is that these lattices
are still relatively small given the 36-site unit cell of the
leading VBC proposal, or the algebraic decay of correla-
tions in some SL proposals. In larger systems, support for a
SL ground state has also been obtained with a SL ansatz
[13–15], whereas evidence for a VBC has been obtained
for an infinite lattice with a series expansion around an
arbitrary dimer covering [6,7], but both approaches are
clearly biased.
In this Letter we report new, independent numerical
evidence in favor of a VBC ground state for the KLHM
model. This is done with entanglement renormalization
[19–25], a real space RG approach that, through the proper
removal of short-range entanglement, is capable of provid-
ing an approximation to ground states of large 2D lattices
[24] by means of the multiscale entanglement renormal-
ization ansatz (MERA) [20]. After describing a scheme for
the kagome lattice with periodic boundary conditions, we
address lattices of N ¼ f36; 144;1g sites. Our simulations
converge to a VBC state compatible with that first pro-
posed by Marston and Zeng [2] and revisited by Nikolic
and Senthil [4], and by Singh and Huse [6,7]. For an
infinite lattice we obtain an energy per site E ¼
0:432 21. This energy corresponds to an explicit
(MERA) wave function and therefore provides us with a
strict upper bound for the true ground state energy.
Importantly, its value is lower than the energy of any
existing SL ansatz on a sufficiently large lattice, which
we interpret as strong evidence for a VBC ground state in
the thermodynamic limit. Our results are also the first
demonstration of the utility of entanglement renormaliza-
tion to study 2D lattice models that are beyond the reach of
quantum Monte Carlo techniques.
The present approach is based on the coarse-graining
transformation of Fig. 1, which is applied to a kagome
latticeL0 made of N sites. It maps blocks of 36 sites ofL0
onto single sites of a coarser latticeL1 made ofN=36 sites.
A Hamiltonian H0 defined on lattice L0 becomes an ef-
fective Hamiltonian H1 on lattice L1. Analogously, the
ground state j0i of H0 is transformed into the ground
state j1i of H1. The transformation decomposes into
three steps. First disentanglers u, unitary tensors that act
on 9 sites, are applied across the corners of three neighbor-
ing blocks. Then disentanglers v are applied across the
sides of two neighboring blocks; these tensors reduce ten
sites (each described by a vector space C2 of dimension 2)
into two effective sites (each described by a vector space
C~ of dimension ~). Finally isometries w map the remain-
ing sites of each block into a single effective site of L1.
Thus the tensors u, v, and w,
uy : C2
9 ! C29; uyu ¼ I29 ;
vy : C2
10 ! C~2; vyv ¼ I~2 ;
wy : C2
6  C~6 ! C; wyw ¼ I;
(1)
transform the ground state j0i of lattice L0 into the
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ground state j1i of lattice L1 through the sequence
j0i !u j00i !
v j000 i !
w j1i: (2)
The disentanglers u and v aim at removing short-range
entanglement across the boundaries of the blocks; there-
fore states j00i and j000 i possess decreasing amounts of
short-range entanglement. If state j0i only has short-
range entanglement to begin with, then it is conceivable
that the state j1i has no entanglement left at all. For a
finite lattice (N ¼ 144) we consider a state j0i that, after
the coarse-graining transformation, gives rise to an en-
tangled state j1i onN=36 ¼ 4 sites. For an infinite lattice
we make instead an important assumption, namely, that
j1i is a product (nonentangled) state [this is equivalent to
setting  ¼ 1 in Eq. (1)]. How short ranged must the
entanglement in j0i be for this assumption to be valid?
By reversing the transformation on a product state j1i, it
can be seen that each site in j0i is still entangled with at
least 84 neighboring sites.
The disentanglers and isometries (u, v, w) were initial-
ized randomly and then optimized so as to minimize the
expectation value of the KLHM Hamiltonian
H0 ¼ J
X
hi;ji
Si  Sj (3)
by following the algorithms of Ref. [25], with cost
Oð212 ~62Þ [26]. Specifically, for lattices with N ¼ 36
and 144 sites, the resulting (one-site and four-site)
Hamiltonian H1 is diagonalized exactly. Instead, for N ¼
1, we use the finite correlation range algorithm (Sec. V.D
of Ref. [25]). All computations led to highly dimerized
wave functions of the VBC type. In order to explain the
results, consider the exact ‘‘honeycomb’’ VBC state, de-
noted jh-VBCi, whose 36-site unit cell is shown in Fig. 2.
Each unit cell contains two ‘‘perfect hexagons’’ (resonat-
ing bonds around a hexagon) and a ‘‘pinwheel.’’ Three
different types of strong bonds can be identified: those of
the pinwheels (red), parallel bonds (green), and perfect
hexagons (blue). The pinwheel and parallel bonds are
singlets (energy per bond ¼ 0:75) while the perfect
hexagons are in the ground state of a periodic Heisenberg
chain of 6 sites (energy per bond ¼ 0:4671). The rest of
the links have zero energy. We call a ‘‘honeycomb’’ VBC a
state that has strong bonds according to the above pattern,
even though the rest of bonds (weak bonds) need not have
zero energy. The ‘‘honeycomb’’ VBC was originally pro-
posed by Marston and Zeng [2] (see also [4,6,7]). Our
simulations with N ¼ 144 and N ¼ 1 produce a VBC of
this type as the best MERA approximation to the ground
state.
The energies obtained for an infinite lattice are shown in
Table I. For each value of ~ [see Eq. (1)], the MERA is an
explicit wave function and therefore provides an upper
bound to the exact ground state energy. Energies computed
for the N ¼ 144 lattice matched those of the infinite lattice
to within 0.02% and have been omitted. OurN ¼ 1 energy
FIG. 2 (color online). The 36-site unit cell for the honeycomb
VBC, strong bonds are drawn with thick lines. Three different
types of strong bonds can be identified; the six bonds belonging
to the pinwheels (red), six bonds belonging to each ‘‘perfect
hexagon’’ (blue), and the parallel bonds between perfect hex-
agons (green). Dotted arrows indicate the axis where spin-spin
correlators have been computed. Bond-bond correlators have
been computed between the reference bond (1) and the other
numbered bonds.
FIG. 1 (color online). Coarse-graining transformation that
maps (i) a kagome lattice L0 of N sites into (vi) a coarser lattice
L1 of N=36 sites. (ii) The lattice is first partitioned into blocks of
36 sites. (iii) Disentanglers u are applied across the corners of
three blocks, followed by (iv) disentanglers v applied across the
sides of two neighboring blocks. (v) Isometries w map blocks to
an effective site of the coarse-grained lattice. (vii) Tensors u, v
and w, have a varying number of incoming and outgoing indices
according to Eq. (1).
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is also compatible with the energy E ¼ 0:433 0:001
for a VBC obtained with series expansion in Ref. [6], and is
lower than that obtained in Ref. [16] with DMRG (E ¼
0:431 60 for N ¼ 108) and in Ref. [14] with fermionic
mean-field theory and Gutzwiller projection (E ¼
0:428 63 for N ¼ 432). We further notice that, where
finite size effects are still relevant, such as in the N ¼ 108
case, they tend to decrease the ground state energy.
Figure 3 shows the distribution of bond energies ob-
tained for the N ¼ 1 lattice. With ~ ¼ 4, one observes an
energy increase per site over jh-VBCi of  0:08 in the
parallel (green) bonds and also in some of the hexagon
(blue) bonds, with the weak bonds having lower energy in
return. As ~ is increased, the energy of the ‘‘strong’’ bonds
becomes slightly larger and that of the ‘‘weak’’ bonds
continues to decrease. However, the dimerization clearly
survives: the bond energies are not seen to converge to a
uniform distribution as required for a SL.
Figure 4 shows spin-spin correlators evaluated along two
different lattice axis A and B (cf. Fig. 2) for N ¼ 1. These
correlators decay exponentially with well-defined ‘‘pla-
teaus,’’ where the correlation is the same with both spins
of a strong bond. Correlations along the line joining a
perfect hexagon and a pinwheel are seen to decay faster
than along the line joining two perfect hexagons, consistent
with the observation from Fig. 3 that the pinwheel bonds
remain almost exact singlets even for high values of ~.
Table II shows bond-bond connected and disconnected
correlators, C1; and D1;,
C1;  hð ~S  ~SÞ1ð ~S  ~SÞi; (4)
D1;  C1;  hð ~S  ~SÞ1ihð ~S  ~SÞi; (5)
between a reference bond ‘‘1’’ and a surrounding bond
 ¼ 1; . . . ; 14 (cf. Fig. 2). While disconnected correlators
decay exponentially with distance, some connected corre-
lators remain significant at arbitrary distances, demonstrat-
ing the long-range order of the VBC state.
Let us discuss the results for a lattice with N ¼ 36 sites.
When initialized with random tensors, the MERA pro-
TABLE I. Ground state energies as a function of ~.
~ N ¼ 1 N ¼ 36 N ¼ 36
(Randomly initialized) (jh VBCi initialized)
2 0:421 45 0:421 64 0:421 43
4 0:429 52 0:428 16 0:427 15
8 0:430 81 0:431 99 0:431 48
12 0:431 14 0:433 71 0:432 98
16 0:431 35 0:434 90 0:434 20
20 0:431 62 0:436 11 0:435 41
26 0:431 93
32 0:432 21
FIG. 3 (color online). (top) Bond energies for the 36-site unit
cell of infinite MERAwave functions, for two different values of
~, as compared to those of an exact honeycomb VBC, jh-VBCi,
and those of a spin liquid, which by definition has all equal
strength bonds. The MERA wave functions clearly match the
proposed honeycomb VBC; we identify (i) the six strong ‘‘pin-
wheel’’ bonds (red bonds), the six ‘‘parallel’’ bonds (green
bonds), and (iii) the 12 ‘‘perfect hexagon’’ bonds (blue bonds).
The (iv) remaining 48 bonds are the weak bonds of the unit cell.
(bottom) Bond energies for the 36-site lattice. Here a randomly
initialized MERA converges to a dimerized state that does not
match the honeycomb VBC pattern, but gives lower overall
energy than a honeycomb VBC initialized MERA of the same ~.
FIG. 4 (color online). Spin-spin correlators along arrows ‘‘A’’
and ‘‘B’’ of Fig. 2 for infinite lattice MERA of ~ ¼ 4 and ~ ¼
16. Although along both lattice directions considered the corre-
lators decay exponentially, the decay along arrow ‘‘A’’ (a line
joining two perfect hexagons) is seen to be slower than along
arrow ‘‘B’’ (a line joining perfect hexagon to pinwheel). The
plateaus marked (i), (ii), and (iii) show the correlation is the
same with both spins of a strong bond.
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duced VBC type configurations which typically did not
match the honeycomb VBC, although simulations initial-
ized in the state jh-VBCi retained the honeycomb VBC
pattern (see Fig. 3). Here the randomly initialized VBC
produced a lower energy (0.5% above the exact diagonal-
ization result E ¼ 0:438 377 of [9]) than the honeycomb
VBC type solution for an equivalent value of ~
(cf. Table I). These results strongly suggest that finite
size effects in the N ¼ 36 site lattice lead to a significant
departure from the physics of the infinite system.
To summarize, we have used entanglement renormaliza-
tion techniques to obtain new numerical evidence indicat-
ing that the ground state of the KLHM is of the honeycomb
VBC type. In order to assess the robustness of this result,
we briefly discuss some of the limitations of the present
approach.
First, the coarse-graining transformation of Fig. 1, which
maps 36 sites into one site, was designed to ensure com-
patibility with the 36-site unit cell of honeycomb VBC type
solutions. While our approach did not preclude solutions
with a smaller, compatible unit cell (such as a 12-site unit
cell or a fully translation invariant solution), we cannot rule
out the possibility that a state with an incompatible unit cell
might have a lower energy.
Second, the infinite lattice was investigated by restrict-
ing the range of entanglement in the ansatz to blocks of 84
spins, imposed through an unentangled state j1i in
Eq. (2). This restriction was only implemented after pre-
liminary simulations with ~ ¼ 12 had produced identical
energies irrespectively of whether j1i was allowed to be
entangled. However, it could still be that entanglement in
j1i would make a big difference for larger values of ~.
We find this scenario quite unlikely, but could not test it
due to computational limitations.
Finally, the MERA is an essentially unbiased method
provided that the candidates to be the ground state of the
system have all a relatively small amount of entanglement.
But when deciding between a VBC (which mostly has
short-range entanglement) and, e.g., the algebraic SL of
Refs. [14,15] (significantly more entangled at all length
scales), it might well be that the MERA is biased toward
the low entanglement solution. Therefore our results do not
conclusively exclude a SL ground state. We emphasize,
however, that the ground state energies obtained with the
MERA are lower than the SL energies of Refs. [14–16].
We thank P. Lee, R. Singh, X.-G. Wen, and S. White for
valuable conversations. Support from the Australian
Research Council (APA, FF0668731, DP0878830) is
acknowledged.
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[26] The computational cost of simulations can be significantly
reduced by incorporating symmetries of the system into
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symmetry is incorporated into the MERA to allow large
~ simulations that would otherwise be unaffordable.
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0907.2994v1.
TABLE II. Bond-bond correlators for ~ ¼ 16 (N ¼ 1).
Bond C1; D1; hð ~S  ~SÞi
1 0:386 39 0:228 15 0:397 80
2 0:294 90 0:034 96 0:653 42
3 0:004 22 0:001 15 0:013 49
4 0:095 03 0:046 44 0:122 11
5 0:224 88 0:066 09 0:399 18
6 0:006 09 0:014 26 0:020 54
7 0:103 72 0:033 76 0:345 61
8 0:260 83 0:012 52 0:624 21
9 0:003 36 0:003 68 0:000 82
10 0:000 01 0:000 17 0:000 42
11 0:291 13 0:001 96 0:726 93
12 0:000 90 0:000 36 0:003 21
13 0:166 32 0:000 26 0:417 44
14 0:156 74 0:000 37 0:394 96
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Frustrated antiferromagnets with entanglement renormalization: Ground state of the spin-1/2 heisenberg model on a kagome lattice

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Frustrated antiferromagnets with entanglement renormalization: ground state of the spin-1/2 Heisenberg model on a kagome lattice

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