# Decoupled phase of frustrated spin-1/2 antiferromagnetic chains with
and without long range order in the ground state

###### Abstract

The quantum phases of one-dimensional spin chains are discussed for models with
two parameters, frustrating exchange between second neighbors and
normalized nonfrustrating power-law exchange with exponent and distance dependence . The ground
state (GS) at has long-range order (LRO) for , long-range spin fluctuations
for . The models conserve total spin , have singlet
GS for any , and decouple at to linear Heisenberg
antiferromagnets on sublattices and of odd and even-numbered sites.
Exact diagonalization of finite chains gives the sublattice spin ,
the magnetic gap to the lowest triplet state and the excitation to the
lowest singlet with opposite inversion symmetry to the GS. An analytical model that
conserves sublattice spin has a first order quantum transition at from
a GS with perfect LRO to a decoupled phase with for
and no correlation between spins in different sublattices. The model with has
a first order transition to a decoupled phase that closely resembles the analytical model.
The bond order wave (BOW) phase and continuous quantum phase transitions
of finite models with are discussed in terms of GS degeneracy
where , excited state degeneracy where ,
and . The decoupled phase at large frustration
has nondegenerate GS for any exponent and excited states related
to sublattice excitations.

PACS numbers: 75.10.Pq, 75.10.Jm, 64.70.Tg, 73.22.Gk

Email:soos@princeton.edu,manoranjan.kumar@bose.res.in

## I Introduction

One-dimensional (1D) spin chains have provided a wealth of quantum many-body
problems over the years, starting with Bethe’s treatment r1 () of the linear Heisenberg
antiferromagnet (HAF) with exchange between adjacent sites. The linear
HAF has been extensively studied and generalized r2 (); r3 (); r4 (). Second-neighbor exchange
is frustrating in chains with either sign of . Magnetic frustration has recently been reported in
copper oxides that contain chains of Cu(II) ions with r4p () or r4pp ().
Increasing in the model generates a continuous quantum
transition from the HAF ground state to a bond order wave (BOW) phase that has been
established by multiple theoretical methods r5 (); r6 (); r7 () but to the best of our
knowledge, the BOW phase of spin chains has not been realized experimentally.
The characterization of the BOW phase is limited: The ground state is doubly
degenerate, inversion symmetry is broken and there is a finite energy gap
between the singlet ground state and lowest triplet state. We discuss in this
paper spin chains with frustrating and variable-range nonfrustrating
exchange that narrow and eventually suppress the BOW phase.

Laflorencie, Affleck and Berciu r8 () studied chains with nonfrustrating exchange between spins and . Power laws leads to 1D models whose ground state has long-range order (LRO) while leads to a spin liquid with long-range spin correlations. There are exact field theory resultsr9 () at . Sandvik r10 () combined frustrating second-neighbor exchange with variable-range in spin chains that are characterized by two parameters, frustration and exponent

(1) |

is always a linear HAF with unit exchange between neighbors in sublattices A and B of even and odd numbered sites,

(2) |

has nonfrustrating exchanges with normalization ,

(3) | |||||

in a chain of spins with periodic boundary conditions (PBC).

The model sketched in Fig. 1 is the short-range limit of
with . Using exact
diagonalization (ED) of finite systems, Sandvik r10 () constructed an
approximate quantum phase diagram of in the plane;
increasing leads to a first order transition for ,
a continuous transition for . We also use ED and focus on systems
with large frustration where we identify and characterize a decoupled phase for any
. We compare our phase
diagram in the plane with Sandvik’s
in Section III.

All models conserve total spin and have
inversion symmetry at sites. The ground state is always a singlet, .
Two excitations have special roles in the following: the magnetic gap
to the lowest triplet state, , and the gap
to the lowest singlet with opposite inversion symmetry to the GS. In finite
systems with fixed , the relation defines points
at which the GS is doubly degenerate and states with broken inversion
symmetry can readily be constructed. The BOW phase of extended systems has
over some interval [*,**]. Since
is known to open very slowly on entering a BOW phase r11 (); r6 (), phase
boundaries * in finite systems have been inferred from level
crossing, the excited-state degeneracy (*,) = *,.
For the model, Okamoto and Nomura r5 () obtained *=0.2411
for the BOW phase, also called the dimer phase r7 () or a
valence bond solid (VBS)r10 (). The excited-state degeneracy r12 ()
leads in the model to **= 2.02(3) for the boundary of the decoupled
phase that is the focus of this paper. We find narrower BOW phases when
has exchange beyond but no LRO. In Section II we solve exactly a model with
uniform exchange that undergoes a first order quantum transition to a decoupled
phase at . Models with variable allow a
more complete characterization of systems with strong frustration.

We note that
finite ** for the model disagrees with the field theory of White
and Affleckr7 () or of Itoi and Qin r13 (). Both start with small
and infer finite for arbitrarily large , albeit is exponentially
small and has different dependence in the two treatments. At large frustration,
it is natural to consider in Eq. 1 as an HAF on
each sublattice. The model in Fig. 1 then has
nearest-neighbor . Each spin
in sublattice A is coupled to two spins, and ,
in sublattice B, and vice-versa. In the two-leg spin ladder, each spin in one
leg (sublattice) is coupled to one spin in the other leg. Barnes et al. r14 ()
conclude that arbitrarily small at rungs opens a
finite gap in two-leg ladders, just as does any dimerization
along the chain r4 (); r15 (). Either or leads to two spins
per unit cell and breaks inversion symmetry at sites. We contrast in Section III
the magnetic gaps of the model and two-leg ladders.

In addition to the excitations and , we will focus on the GS expectation value of sublattice spin,

(4) |

where we have used and PBC. Correlation functions between spins in different
sublattices are governed by at and become vanishingly small at large .
Sublattice spin is an approximate or hidden symmetry since states with different ,
are orthogonal. The GS is a linear combination of states centered on
that shift to smaller with increasing frustration.

The paper is organized as follows. The model solved in Section II has uniform exchanges in and conserved and . The GS has perfect LRO for that depends weakly on size and goes to in the infinite chain. Increasing induces a first order transition to the decoupled phase that corresponds to noninteracting HAFs on sublattices. Accordingly, and are directly related to HAF excitations. In Section III we present ED results for models that do not conserve or . A first order transition to the decoupled phase is inferred for models with LRO at small frustration based on almost identical , and as in the uniform-exchange model. The BOW and decoupled phases of the model are related to degeneracies of ground states at and of excited states at . The BOW/decoupled boundary of is estimated in the plane. The magnetic gap in the decoupled phase is contrasted to of the two-leg ladder using the density matrix renormalization group (DMRG). In Section IV we briefly discuss the decoupled phase.

## Ii Frustrated chain with uniform exchange

In this Section, we solve a 1D model with spins that conserves sublattice spins and . Extensive results for finite and infinite linear HAFs are directly applicable in this case. The part, Eq. 3, is taken to have uniform AF exchange between spins in opposite sublattices, as in the Lieb-Mattis model r16p (), and also uniform exchange between all spins on the same sublattice. The exchanges per site are normalized to unity. Uniform exchange makes it straightforward to express for spins as

(5) |

The integer ranges are , , , and the
index has been omitted. The GS is evidently always in the
sector. In the absence of frustration, the GS is a linear combination of
sublattice states with and components
. The degeneracy of states with fixed , and
is lifted by with frustrating . There are two
contributions in Eq. 1, from and from .
All eigenstates of the model are products of HAF eigenstates on sublattices.
Increasing frustration generates energy shifts and numerous level crossings
that can be followed explicitly.

We classify the HAF eigenvalues in Eq. 2 as where , = 0,1, 2, .. are eigenvalues with sublattice spins and . The lowest energy in each sector is sufficient. The singlet GS for spins in the singlet sector has energy

(6) |

There is perfect ferromagnetic order with at small , where
, and the GS transforms as . The GS is doubly degenerate at ;
the GS in the sector with is odd under inversion. Repeating the argument
shows that the GS for is even, odd under inversion for even, odd . Since inversion symmetry
changes times between and , the GS for transforms as
and corresponds to the product , the singlet GS of each sublattice.

We find at which the ordered state is degenerate with the GS in the sector . The points are defined by

(7) |

where is an energy per site and . ED for -spin HAFs yields exact in Table 1 up to . Increasing leads directly from to at and then to at . Exact results in Eq. 7 for the infinite chain place the first-order transition at

(8) |

The Bethe ansatz was used by Hulthen r1 () to find the GS energy per site of the extended HAF; it has more recently been applied to finite systems of spins. Woynarovich and Eckler16 () found logarithmic corrections to the lowest energy per site in sector . To leading order in ,

(9) |

The entries in Table 1 illustrate the convergence of
and . It follows from substituting Eq. 9 into Eq. 7 that
the infinite chain also has , .

The degeneracy between the GS in the singlet sector with and 0 occurs at

(10) |

The singlet-triplet gap, , appears frequently in the following. ED returns the entries in Table 1. To leading order for large systems, is given by setting and multiplying Eq. 9 by , and then substituting in Eq. 10

(11) |

The absolute GS for is for chains of any length.
The eigenstate has , vanishing spin correlations
between sublattices in Eq. 4 and hence no possibility of a BOW phase.
The system of 400 spins in Table 1 shows slow convergence to .

16 | 0.47189 | 0.46789 | 0.51020 |
---|---|---|---|

20 | 0.45013 | 0.44710 | 0.49735 |

24 | 0.43544 | 0.43308 | 0.48873 |

28 | 0.42486 | 0.42299 | 0.48341 |

32 | 0.41689 | 0.41539 | 0.47756 |

36 | 0.41063 | 0.40944 | 0.47382 |

40 | 0.40570 | 0.40467 | 0.47058 |

400 | 0.36518 | 0.36510 | 0.44864 |

0.36067 | 0.36067 | 0.40528 |

The interval between and is not relevant in the
context of the infinite chain, in which
for arbitrarily large drops to at
and vanishes for . The discontinuity at marks a first
order transition to the decoupled phase. Spin correlations between the sublattices
vanish rigorously for . The degeneracy at involves states
of opposite inversion symmetry, as does degeneracy at for even . For odd ,
however, the GS in the sectors and 1 are both even under inversion. They are
mixed and lead to an avoided crossing in finite models that do not conserve .

Next we find and , the excitation energy to the lowest triplet and the lowest singlet with reversed inversion symmetry. It follows from Eq. 5 that the lowest triplet for is obtained by changing from 0 to 1 without changing or ; hence is constant, independent of up to . When the GS energy is , the lowest triplet has , . It is doubly degenerate, or in obvious notation, with a triplet on either sublattice. The magnetic gap is

(12) |

The second term is the contribution from Eq. 5.

The lowest singlet excitation for is

(13) | |||||

The excitations in Eq. 12 and in Eq. 13 are equal at . It is instructive to rewrite the excitations using Eq. 10

(14) |

The V-shaped dependence of on either side of
is evident, as are the related slopes with increasing .
These features are used in Section III to interpret and
in models that do not conserve . It is still convenient
to refer to products of or . Although no
longer exact eigenstates, the actual eigenstates can be
expanded in terms of sublattice eigenstates.

We conclude the discussion of the uniform exchange model by noting that the infinite chain has a first order quantum transition at modest frustration . The GS has perfect LRO up to . The decoupled phase has in the interval and for , when all spin correlation functions in Eq. 4 are zero. The infinite chain has for all .

## Iii Frustrated chains with variable range exchange

We present ED results for models that do not conserve sublattice spin.
The GS is degenerate under inversion at frustration where .
It is convenient to retain the labeling and in Eqs. 7 and 10
used for the model with uniform exchange. We start with a model with LRO at
and a first order transition to the decoupled phase that closely resembles the
uniform model. Next we consider the model without LRO at and
continuous transitions with increasing from spin liquid to BOW to decoupled phase.
We then consider intermediate to construct an approximate the GS phase
diagram in the (g, ) plane.

### iii.1 Model with LRO

The Hamiltonian in Eq. 1 has nonfrustrating exchanges in Eq. 3. It differs from the model studied by Sandvik only in the terms, which are double counted in Eq. 1 of ref r10 (). Since contributions decrease with size or increasing exponent , numerical difference due to double counting are limited to small and . The normalization condition leads to

(15) | |||||

Since frustrating is entirely in , the model with
and finite in Eq. 15 is slightly different from the
model with uniform exchange.

The following results are for ,
a model with r8 () LRO at .
Excitation energies and are shown in Fig. 2 as
a function of for sites (top panel) and 20 sites (bottom panel). As
anticipated for LRO systems with a first order transition, we have
at two points and when is even and one point when
is odd. Inversion symmetry at sites reverses twice for 24 sites, once for 20 sites.
The first order transition is at for 24 spins and the avoided crossing is
at for 20 spins. The degeneracy at corresponds to changing from
to 0 in the model with uniform exchange. The GS in the and -1 sectors cross at
for , in contrast to the avoided crossing in Fig. 2 of ref. r10 () for the
model with r17 () .

The separation for the
model is comparable to in Table 1 for uniform exchange.
In accord with Eq. 12 for uniform exchange, is almost constant
up to ; it is doubly degenerate for and linear with increasing .
The slopes are within of for 24 sites and
for 20 sites. Likewise for the lowest singlet excitation,
, the slope of between and
is slightly larger than in either case,
for 24 sites and 0.884 for 20 sites.

We define to quantify sublattice spin as the GS expectation value

(16) |

decreases with frustration and is double-valued at . Figure 3 shows with increasing for 20 and 24 sites. Dotted lines for refer to the excited state . The values in Table 2 at modest frustration are already close to , the exact result at . The eigenstates near the transition contain small admixtures of sublattice spin. The model requires matrix elements for all in Eq. 3, that make it tedious to evaluate . The results in Table 2 go to 24 spins for the

Model | |||||
---|---|---|---|---|---|

4n/state | |||||

16 | 0.092 | 0.106 | 0.161 | 0.150 | 0.158 |

20 | 0.081 | 0.088 | 0.132 | 0.123 | 0.130 |

24 | 0.062 | 0.075 | 0.110 | 0.104 | 0.109 |

28 | â | 0.067 | â | 0.093 | 0.094 |

model and to 28 spins for the model. Somewhat larger
systems are accessible in principle with current computational resources.

The model has almost perfect LRO at , ,
in agreement with larger systems studied in ref. r8 () using multiple methods.
As anticipated in Section II, is discontinuous when
and changes rapidly but continuously for 20 spins at the avoided crossing.
The size dependence of shown in Fig. 3 is consistent
with a discontinuity at in the infinite system and for .
Both results can be understood in terms of a GS with LRO at small
and a first order transition to the decoupled phase.

All properties of frustrated spin chains of spins are given at by exact HAF eigenstates for spins. The triplet with and energy has equal spin density at all sites. The spin density at site of the degenerate triplets and at is

(17) |

The plus sign corresponds to a triplet on the sublattice of even-numbered sites and for odd ; the minus sign has the triplet on the odd-numbered sublattice. The spin densities at are shown in the top panel of Fig. 4 for 24 spins. As expected, is constant on each sublattice, slightly less than on one and slightly positive on the other. The deviations from Eq. 17 are less at , less than at .

The decoupled phase found rigorously for uniform exchange readily
accounts for the model.

### iii.2 model

The linear HAF with between nearest neighbors and frustration is a prototypical model with a BOW phase r5 (); r6 (); r7 (). The model evolves from an HAF at to two HAFs at . Okamoto and Nomura r5 () placed the continuous transition to the BOW phase at *=0.2411 by extrapolation of the excited state degeneracy up to spins. The GS is nondegenerate for *. There is no LRO at but there are long-range spin correlation functions that go as r8 ()

(18) |

The exact values r18 () of and
are and ,
respectively, and all correlations between spins in opposite sublattices
are negative. It follows that for spins in Eq. 4 is of
order at .

The model has multiple GS degeneracies r12 ()
with increasing . The first one at is the Majumdar-Ghosh
point r19 () where the exact GS is known for any even number of spins.
Spin correlation functions are now limited to nearest neighbors, and the
infinite chain has , or 3/8 per site. In contrast to the transition
from to in models with LRO at , the symmetry
changes at , occur
sequentially with increasing up to 28 spins, the largest system we
solved. The upper panel of Fig. 5 shows and as a function
of frustration, with six arrows at . The magnitude of is
remarkably small between and . A finer energy scale is
needed to see in the BOW phase. The lower panel shows
and for spins, with five arrows at . The behavior at large frustration
is qualitatively similar to the model in Fig. 2. The triplet is
doubly degenerate for . The slopes and are
within and of and ,
respectively, between and .

We find that the relation is not limited to
finite models. On the contrary, Sandvik r10 () states that the
degeneracy is not exact except in the model at the special point
. The discrepancy is not due to ED but to motivation. ED
is necessarily performed at fixed values of that are typically on a grid.
The GS symmetry changes in calculations that keep track of inversion at sites.
It is then natural to search for the exact at which the GS is
degenerate by choosing more precisely. The lowest two
singlets are both even under inversion at the center of bonds, which suggests
an avoided crossing and less reason for refining in search of exact
degeneracy. The symmetry operators of course commute with
but not with each other. Hence inversion symmetry at sites or at bonds
leads to different linear combinations of degenerate eigenstates.

Degenerate GS at are products of singlet-paired spins on successive sites,
either or .
Such states are the familiar Kekulé diagrams of organic chemistry
or the degenerate GS of polyacetylene in the Su-Schrieffer-Heeger (SSH)
model r20 (). The gap that opens at *
is rigorously known r20p () to be finite in the infinite chain at .
It is already larger12 () at and remains large up
to , clearly exceeding finite-size effects in Fig. 5. The elementary excitations of
the BOW phase are topological spin solitons r20p () or domain walls
generated by spin correlations that closely resemble r12 () SSH
solitons generated by electron-phonon coupling. The BOW phase extends
beyond in Fig. 5, the GS degeneracy at the largest frustration.
We suppose that the BOW phase terminates at ** at the excited-state degeneracy
r12 () . As discussed below, finite-size effects are larger at ** than at *.

The evolution of with increasing frustration is shown in Fig. 6 for 20 and 24 spins; for the excited state is the dotted line for . As expected, decreases with increasing and is discontinuous when the GS is degenerate. Table 2 lists both values of . The excited state at is already within of the limit. The model has stronger but still modest mixing of states at than the model. The spin densities of the degenerate triplets at in Fig. 4 again follow Eq. 17 with slightly less than on one sublattice and slightly positive on the other. The spin densities at are and . At , the overlap of the VB diagrams for 4n spins is . Aside from finite-size effects that are already less than at , we obtain

(19) |

The infinite chain has continuous over the entire range .

### iii.3 Quantum phase diagram

Models with 2, 3 and 4 have nonfrustrating exchange intermediate
between and . We find the GS degeneracies
where and the excited state degeneracies * and ** where
that delimit the BOW phase.These points are used to construct
an approximate quantum phase diagram in the plane.

Fig. 7 shows the phase diagram of for to
4 and the model over the entire range
of frustration . The diagrams for 24 and 20 spins illustrate
the modest size dependence and differences between with even and odd .
Open points indicate GS degeneracy ; closed points are * and **;
solid lines mark first order transitions at for 24 spins and the
avoided crossing for 20 spins; dashed lines are the boundary between the BOW and decoupled
phases at large and between the spin liquid and BOW phase at small .
The BOW phase terminates at a multicritical point, and the dotted line separates
spin liquids from models with LRO. The BOW phase for closely resembles
the model, as might be expected since largest change is small, .
The width of the BOW phase, **-*, narrows for or 2 and
is satisfied at fewer than points. The model has a first order transition and no BOW phase.

Sandvik r10 () used additional values of to estimate the multicritical point
as , . We have not varied and took
in Fig. 7. It will be challenging to be more accurate as
long as ED is limited to about 30 spins. Our phase diagram in Fig. 7 is quite
similar at small frustration to Fig. 1 of ref. r10 ()
with VBS instead of BOW, AFM instead of LRO and QLRD() instead of spin liquid.
So far there is no consensus for naming phases. There are clear differences at large ,
however. The VBS or BOW phase in Fig. 1 of ref. r10 () does
not terminate at either large or at small and the line at large
between VBS and VBS+QLRD() separates phases with different .
A BOW phase is rigorously excluded at , the analytical model with
a first order transition to the decoupled phase.

We have given reasons for extending the decoupled phase
in Fig. 7 to first order LRO/decoupled transitions and
to continuous BOW/decoupled transitions without any
distinction for different . The sublattices of
have weak interactions at large
. The coupling is weak even in the
model and becomes arbitrarily small at large .
By continuity, we expect the same phase to be reached at
for any choice of . The decoupled
phase starts at the first order transition in systems with LRO
at and at ** in systems with a BOW phase
and continuous transitions.

### iii.4 Magnetic gap

Density matrix renormalization group (DMRG) has been extensively applied to 1D spin systems
r7 (); r21 (); r22 (); r23 (); r24 (). DMRG with open boundary conditions (OBC) breaks inversion symmetry
at sites at the outset: an even number of spins is required for a singlet GS and
even chains have inversion symmetry at the center of the central bond. As shown
explicitly for a half-filled band of free electrons r23 (), OBC generates corrections
to the bond order of the central bond.
OBC strongly breaks the GS degeneracies of the model at . At these points,
the lowest singlet excitation for 16, 20 or 24 spins is slightly higher than , thus reversing
the order of excitations in addition to lifting the degeneracy.

It is convenient to study in Eq. 1 for
. Then has decoupled HAFs on the sublattices for models with any exponent
. In particular, the frustrated model has exchanges between spins at adjacent
sites , in Fig. 1 while the non-frustrated two-leg ladder has
exchanges between sites , . The model has one spin
per unit cell and inversion symmetry at both sites and centers of bonds. The two-leg ladder
with has two spins per unit cell and inversion only at bond centers.

DMRG results for are compared in Fig. 8 for models and two-leg ladders, and are seen to be qualitatively different. The ladder has large at that decreases to at . Finite is expected r14 () for finite , nearly linear in small , just as for finite dimerization r4 (); r15 () in chains with alternating and . The model returns at , close to the BOW/decoupled boundary **, and at or 4, the limit r23 () of accuracy for DMRG with four spins added per step. Large in Fig. 8 brings out the contrasting behavior of . While an exponentially small cannot be ruled out in the model, DMRG is consistent with in the decoupled phase and suggests that ** is slightly larger than 2.0,

the value estimated from extrapolation of .

The relation for the BOW/decoupled boundary ** is less accurate due to finite size effects. Models with spins are used to obtain , which is then extrapolated; models of spins are needed for ** in order to decouple to HAFs with an even number of spins rather than two radicals with ground states. Even for large , weak coupling between open shell radicals with degenerate GS is quite different from weak coupling of closed shell systems with nondegenerate GS. Indeed, Fadeev and Takhtajan r26 () have pointed out that the HAF with odd and states with half-integer has unexpected and unexplored features. Different approaches are required for , including field theories designed for weakly coupled HAFs.

## Iv Discussion

The defining features of BOW phases are a doubly degenerate GS,
broken inversion symmetry at sites and finite magnetic gap
that opens slowly at a Kosterlitz-Thouless transition. The elementary excitations in
spin chains are solitons centered on sites in opposite sublattices.
The frustrated spin chains in Eq. 1 do not meet these
signatures at large . To be sure, we cannot discriminate between exponentially
small and zero . But the energy spectrum at is just the HAF spectrum.
The GS is not degenerate at and it is unlikely that arbitrarily small
will place below . Yet that is minimally required
for GS degeneracy at any . Conversely, the lowest triplet state at
small is not degenerate while the lowest triplet at large is doubly
degenerate, or with both
unpaired spins largely confined to one sublattice.

The principal goal of the present study is the identification
of a decoupled phase at large frustration g that is distinct from the
BOW phase of the model at intermediate . GS is not
degenerate in the decoupled phase; inversion symmetry is not broken;
the lowest triplet is doubly degenerate. The magnetic gap is zero within
numerical accuracy for ** and strictly so in the model with uniform
exchange. The properties of the decoupled phase at in
Table 2 or in Figs. [2-6] are already
close to the limit of decoupled HAFs on sublattices.

BOW phases occur in spin chains r5 (); r6 (); r7 (); r12 ()
with frustrated exchange or in half-filled 1D Hubbard models nearest-neighbor r11 ()
or long-range r27 () Coulomb interactions. Moderate interactions are required
to avoid a first order transition that for is related to
LRO at . Models without LRO have continuous transitions from
spin liquid to BOW to decoupled at * and **,
respectively. As shown in Fig. 7, all spin chains have a
decoupled phase at large . By contrast, Sandvikâs phase diagram r10 ()
at large distinguishes between models with and
without LRO and does not terminate the BOW phase.

Frustration ensures the existence of the correlated phases with variable sublattice spin . The model with uniform exchange has a first order transition at from to . The divergence of is linear or logarithmic, respectively, at in spin chains with and without LRO. The model has a first order transition from a GS with LRO to the decoupled phase. The model has at in the BOW phase and continuous transitions to a spin liquid phase * and the decoupled phase at **. The BOW phase narrows in models with intermediate and terminates for . Accurate phase boundaries pose major challenges for all models and especially for systems with continuous transitions and a BOW phase.

Acknowledgments: ZGS thanks A.W. Sandvik for a stimulating discussion of spin chains. We thank the National Science Foundation for partial support of this work through the Princeton MRSEC (DMR-0819860). MK thanks DST India for partial financial support of this work.

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