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Frustration-driven multi magnon condensates and their excitations Current trends in frustrated magnetism

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Frustration-driven multi magnon condensates and their excitations Current trends in frustrated magnetism
Frustration-driven multi magnon
condensates and their excitations
Oleg Starykh, University of Utah, USA
Current trends in frustrated magnetism,
ICTP and Jawaharlal Nehru University, New Delhi, India, Feb 9-13, 2015
Collaborators
Leon Balents,
KITP, UCSB
Andrey Chubukov,
Univ of Minnesota
not today but!
closely related findings:
spin-current state at the
tip of 1/3 magnetization plateau,
spontaneous generation of
orbiting spin currents
(ask me for details after the talk :))
Outline
• Frustrated magnetism (brief intro)
- emergence of composite orders from competing
interactions
• Nematic
vs
SDW in LiCuVO4
✓ spin nematic: “magnon superconductor”
✓ collinear SDW: “magnon charge density wave”
• Volborthite kagome antiferromagnet
- experimental status - magnetization plateau
- Nematic, SDW and more
- Field theory of the Lifshitz point
• Conclusions
in Sec.
III. Inout,
particular,
the
IsingIsing
the
BKT
consistent
with
universality.
Wetransitions
also discuss the quantum
phase diagram
finiteof
S.
clearly
indicated
by lattice.
aitsharp
peak
ofofthe
specific
heat
pointed
however,
in Ref.
25and
that
these
two
continuous
is divisible
by theforsize
the helix
pitchsquare
and
liesIninthe
the
limit
spinhave
S !repeated
1, there
is which
a zero
temperature
Lifshit
We
a similar
analysis
for several(T)
values
of
also
1
are considered
in merge
Secs.
III
A
and
III
B,
respectively.
The
transitions
can
into
a
single
first-order
one.
These
conJ
=J
and
the
complete
phase
diagram
is
shown
in
Fig.
5,0
is illustrated
inlong-range
Fig.
3 spiral
[10].spinThis
feature
to be
ordersharp
for J3 > is
range from 20
to numbers:
120. We
apply
the periodic
(toric)
boundary
3 at 1T ! 0
DOI: 10.1103/PhysRevLett.93.257206
PACS
75.10.Hk,
75.10.Jm
4 J1 . We present classical Monte
h Z2
where
we
have
plotted
T
versus
J
=J
.
We
find
that
T
neighborhood
ofnot
thedepend
Lifshitz
and theform
phase
diagram are
c spin rotation
3 1
clusions do
on point
the particular
of interactions
in
theory
forthe
T>
0 crossoversdisplayed
near the Lifshitz
point:
symmetry isc r
contrasted
the
broad
maximum
by
the
same
conditions as well as the cylindrical
onesto(i.e.,
with
periodic
f the
vanishes
linearly
for
J
=J
!
1=4;
a
theory
1 behav2
3
1
there is a1 broken lattice reflection symmetry for 0 " T < Tc # $J3for%this
the system
soon
the ground-state
degeneracy
remains the
discussed
in Sec.asIII
C. as
Section
IV contains
our conclusions.
0
4 J1 &S . Th
condition along the b axis
and the
one
along
the
a
ior
will
now
be
presented.
nded
quantity
for free
J3 =J
<
,
i.e.,
when
the
classical
ground
state
1
4
consistent
with
Ising
universality.
We
also
discuss
the
quantum
phase
diagram
same. They
are confirmed
by numerical
of the
models also suggest that discrete lattice symmetries may play a
Near the classical Lifshitz point, we can model quantum
Frustrated
antiferromagnets
havestudies
recently
attracted
axis). We have found thatdisplays
both conditions
to order.
the same
11,13–17,23,24
ordinarylead
Néel
In particular,
the maximum
and thermal fluctuations
by a continuum unit vector field0
mentioned
above.
role
near
other
quantum
critical
points
with
spiral
order
[6].
much interest in connection with the possibility of stabilizDOI:
10.1103/PhysRevLett.93.257206
PACS n
ansivalues
of
transition
temperatures
and
indexes.
In
contrast,
n"r;
%#,
where
r
!
"x;
y#
is spatial coordinate, % is imagiof the
specific
heat is consistent with a logarithmic
depenNevertheless,
the
situation
remains
contradictory
in
2D
II.
MODEL
AND
METHODS
PHYSICAL
REVIEW
B
85,
174404
(2012)
Broken
discrete
symmetries
have
also
been
discussed
ing
unconventional
low-temperature
(T)
phases,
with
novel
2
occur
nary time, and n ! 1 at all r, %. This field is proportional
values of Binder’s cumulants
andonthe
chiral-order-parameter
helimagnets
belonging
to
the
same
Z
⊗
SO(2)
class
as
dence
system
size
(see
the
inset
of
Fig.
corresponding
week
2
x &y
[7,8]
in
the
context
of
the
J
types
of
‘‘quantum
order’’
[1].
A
very
promising
candidate
%
J
model,
with
firstand
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T ENéel
R 3)
S order
1 (2004)
2
"%1#2004
n"rj ; %#.
toTthe
parameter with
S^ j /ending
sition We consider the model (1) of the classical XY magnet on a
17 DECEMBER
PRL
93,
257206
distribution
at
J
≈
0.309
depend
on
boundary
conditions
as
2
the
FFXY
model
and
the
antiferromagnet
on
the
triangular
4,5
for
a
spin-liquid
phase
is
the
J
second-neighbor
couplings
on
the
square
lattice.
However,
%
J
model
Chiral
spin
liquid
in
two-dimensional
XY
helimagnets
to
a
critical
exponent
"
!
0,
in
agreement
with
Ising
Spiral
order
will
therefore
appear
as
sinusoidal
1
3
er” square lattice. We set J = J 28
also suggest thatdependiscr
Frustrated
antiferromagnets
recently
attracted
= (see
1 foralso
simplicity,
the value
wemodel
discuss
below
in detail.
Standard
Metropolis
algorithm33havedence
X 29) and
1
bX
lattice.
Garel
and
Doniach
Ref.
considered
the
of n on r. The action for n is the conventional
this
has
only
collinear,
commensurate
spin
correlauniversality.
gonal
^ ' S^ j ( J3
^1,2,†
role near
much interest
in connectionwithin
with
the possibility
of sigma
stabilizS^ The
(1)
H^ 1,*
!
Ja1square
O(3)
nonlinear
model, expanded
to other
includequantum
quartic
i 'extra
j ; competing
Low-Temperature
Broken-Symmetry
Phases
of
Spiral
Antiferromagnets
A.
O. Sorokin
A.SV.i J
Syromyatnikov
of the simplest
extra exchange
interaction
is a variable.
Lifshitz
has and
been
The the
thermalization
was
maintained
helimagnet
onand
withSan
tions,
this used.
makes both
classical and quantum
theory
2lattice
eting
Broken
discrete
ing
unconventional
low-temperature
(T)
phases,
with
novel
This
critical
behavior
can
be
directly
related
to
the
hi;ji Institute, Gatchina,
hhi;jii St. Petersburg 188300, Russia
gradient
terms
(
h
!
!
k
!
lattice
spacing
!
1):
S n sym
!
tersburg
Nuclear
Physics
Institute,
NRC
Kurchatov
5
B
different
fromCarlo
that considered
here. As
will become
exchange
coupling
is described
by the quite
point
corresponds
to J2along
= 1/4one
in axis
this that
notation.
The system
4
×
10
Monte
steps
in
each
simulation.
Averages
have
1,2
R1=T 2,3 R 2
FIG.
4.
Bot
2
elong
Luca
Capriotti
and
Subir
Sachdev
[7,8]
in
the
context
of
types
of
‘‘quantum
order’’
[1].
A
very
promising
candidate
Department of Physics,^St. Petersburg State University, 198504 St. Petersburg, Russia
d rLn with
reflection
symmetry
by
studying
an appro0 d%
6 lattice
clear
below,
the spiral
order3.6
andbroken
associated
Lifshitz
point
1within
Hamiltonian
where
S
are
spin-S
operators
on
a
square
lattice
and
has
a
collinear
antiferromagnetic
ground
state
at
J
<
1/4.
To
been
calculated
×
10
steps
for
ordinary
points
and
i
2
Valuation
Risk
Group,
Credit
Suisse
First
Boston
(Europe)
Ltd.,
One
Cabot
Square,
London
E14
4QJ,
United
Kingdom
Eq.
(2)]
for
Received 7 November 2011; revised manuscript received 27 January 2012; published 3 May 2012)
model
fortheory
a spin-liquid
phase
is the
J1 % J3 model
second-neighbor coupld
2structurepriate
play
a
central
role
in
the
of
our
and
in
the
T
Ising
nematic
order
parameter.
From
the
symmetries
6
!
Kavli
Institute
for
Theoretical
Physics,
University
of
California,
Santa
Barbara,
California
93106-4030,
USA
J
;
J
)
0
are
the
nearestand
third-neighbor
antiferro1 3 transition from the (quasi-)antiferromagnetic
discuss
the phase
6 × 10 for
points close
to the critical
ones.
We have
used
alsoNewXHaven, Connecticut 06520-8120, this
X Box
shows
the dat
The
3
model
has only
co
carry
out Monte Carlomagnetic
simulations
to
discuss
critical
properties
of
a
classical
two-dimensional
XY
frustrated
Department
of
Physics,
Yale
University,
P.O.
208120,
USA
dependence
of
observables.
H
=
(J
cos(ϕ
−
ϕ
)
+
J
cos(ϕ
−
ϕ
)
^
^
^
^
^
1
x along
x+a
2 coordinate
x
x+2a
couplings
the
two
axes.
For
S
S
'
S
(
J
'
S
;
(1)
H
!
J
of
Fig.
2,
we
deduce
that
the
order
parameter
is
#
!
1
i
j
3
i
j
=
0
and
0.1
(see
phase
to
the
paramagnetic
one,
we
consider
J
(Received
23
September
2004;
published
14
December
2004)
tions,
and
this
makes
bo
Ising expone
spingnet on a square lattice. We find
two successive phase transitions upon2the temperature decreasing: the first
P
x
this model,
early large N computations, [2] and recent
hi;ji
hhi;jii
1=M"
# with
and the second
one is of
of a discrete
Z2 symmetry
from thd
We study
Heisenberg
with nearest- (J1 ) and third- (J3 ) neighbor exchange onquite
the different
Fig. with
1). breaking
The
ground
has ϕa helical
ordering
attheJ2Berezinskii-Kosterlitz> 1/4.
nassociated
3D
a #aantiferromagnets
temperature
large
scale
density
matrix
renormalization
Jb state
cos(ϕ
)), a narrow
(1) diagram
x−
x+b
with below, the spiral
square lattice. In the limit^ of spin S ! 1, there is a zero temperature (T) Lifshitz point at J3 ! 14 J1 , clear
ss (BKT) type at which
the −
SO(2)
symmetry
breaks.
Thus,
region exists on group
the phase
where
Si atare
operators
on classical
a square
0:5. The inse
1.2lattice and
1
(DMRG)
for S ! 1=2
have
the
present
Monte
Carlo simulations and a
long-range spiral
spin order
T !spin-S
0 for J3 >
n lines of the Ising and
the BKTcalculations
transitions that corresponds
to a [3]
chiral
spin suggested
liquid.
4 J1 . We
^
^
^
^
play
a
central
role intotht
#
!
"
S
(
S
&
S
(
S
#
;
(2)
ent.11
J
;
J
)
0
are
the
nearestand
third-neighbor
antiferrotheory
for
T
>
0
crossovers
near
the
Lifshitz
point:
spin
rotation
symmetry
is
restored
at
any
T
>
0,
but
a
1
3
2
4 a
whereexistence
the sum of
runs
over sites
x = (xastate
,xb ) with
of the
lattice, a =
sponding
a gapped
spin-liquid
exponentially
1 3
1
2
T
#
$J
%
J
&S
.
The
transition
at
T
!
T
is
there
is
a
broken
lattice
reflection
symmetry
for
0
"
T
<
T
dependence
of
observa
1 axes. For
c
3
c
0.1103/PhysRevB.85.174404
PACS
number(s):
64.60.De,
ature
4 1
magnetic couplings along the two
coordinate
spin correlations
and no
translation
sym- 75.30.Kz
(1,0) decaying
and b = (0,1)
are unit vectors
of broken
the
lattice,
the coupling
consistent with Ising universality. We also discuss the quantum phase diagram for finite S.
first
large N computations, [2] and recent
metryJ1,2
in the
of strong
frustration
0:5).they
are regime
positive.
Usingising
arguments(Jof3 =J
Ref.
constants
1 ’ 30,
ξspin~ S / T 1/2 this model, early
T
DOI:
10.1103/PhysRevLett.93.257206
PACS0.8numbers:group
75.10.Hk, 75.10.Jm
28
c
large
scale
density
matrix renormalization
s I.the
This Letter
properties
of
the above
model
257206-2
16
concluded
that atwill
lowdescribe
temperatures
vertices
bound
INTRODUCTION
model,
thethe
Coulomb
gasare
system
of half-integer charges,17
0.3
18 S. Our
13,19,20
(DMRG)
calculations
for
S
!
1=2
[3]
have
suggested
the
for
large
S
and
discuss
consequences
for
general
n by
two the
coupled
models, and
Ising-XY
and the
by strings,
which would
BKTXY
transition
make model,
0.25
0.6
ets have attracted
much attention
in recentinhibit
cS2 /of
T a gapped spin-liquid state with exponentially
21
c'S2 / T
results,
obtained
by
classical
Monte
Carlo
simulations
and
existence
chiral
generalized
fully
frustrated
XY
model.
And
surely,
the
most
0.2
ξ
~
e
ξspin~ e Frustrated antiferromagnets
thewhich
Ising have
transition
occurinfirst with the temperature increasing.
spin
Ising
nematic
order
liquid phases,
been found
also suggest
that discrete lattice symmetriesTmay play a
have recently
attracted
famous
of
them
is
the
fully
frustrated
XY
model
(FFXY
)
a
theory
described
below,
are
summarized
in
Fig.
1
for
the
decaying
spin correlations
nonear
broken
translation
sym- 0.15
31
hofthe
special Kolezhuk
interest.1 A noticed
chiral spin-liquid
role
other quantum
much interest in connection
with the possibility
of stabiliz- and
that those arguments are not
valid for a
0.4 critical points with spiral order [6].
22
introduced
by
Villain.
This
model
is
of
great
interest
because
0.1
limit
S
!
1.
There
is
a
T
!
0
state
with
long-range
spiral
metry
in
the
regime
of
strong
frustration
(J
=J
’
0:5).
Broken
discussed
ing unconventional
(T)Jphases, with novel
etems
of such an helimagnet,
exotic state of matter
in whichthat the KT
P H Ydiscrete
S I C3Asymmetries
L1 R E V I Ehave
W also
L E Tbeen
TER
S
and showed
Ising transition
temperature
J1 /low-temperature
4
1
it
describes
a
superconducting
array
of
Josephson
junctions
3
PRL
93,
257206
(2004)
0.05
spin
order
for
J
>
J
.
We
establish
that
at
0
<
T
<
T
#
[7,8]
in
the
context
of
the
J
types
of
‘‘quantum
order’’
[1].
A
very
promising
candidate
%
J
model,
with
first- and
This
Letter
will
describe
properties
of
the
above
model
uasi-long-range
nor
long-range
magnetic
3
1
c
1
2
Neel LRO
4 one at least near the Lifshitz point
y (see
0.2
23
Spiral
LRO
is larger than
the
BKT
an external transverse magnetic field. It was foundfor a Lifshitz
point
0
spin-liquid
phase is the
second-neighbor
couplings S.
on the
However,
% J3 model
12
×14SJj1⟩&Sis2 ,nonzero.
above this under
state there
is a phase with broken
$J⟨S
0.25 0.3
0.35 0.4
forJ1large
S and
discuss3 consequences
for3 general
Oursquare lattice.
3 i%
Jl 1order
-J2 parameter
that
the
temperature
of
the
Ising
transition
T
is
1%–3%
larger
4
4
J
=
J
/4.
It
was
found
by
Monte
Carlo
simulations
in
the
X
X
I
this
model
has
only
collinear,
commensurate
spin
correlaat
T=0
only
2
1 in context of onea phase is discussed
c'S2 / T
discrete symmetry
of lattice
reflections
about
the x andfory most of above-named
Monte
Carlo
simulations
and1 and1.510quantum
S^ i ' S^ j ; by classical
' S^ j ( J3 obtained
(1)
H^ ! J1 S^ i results,
0
ξ
~
e
than
that
of
the
BKT
transition
2
tions,
and
this
makes
both
the
classical
theory
^
0
0.5
2
2.5
3
spin
FIG.
1.
Phase
diagram
of
H
in
the
limit
S
!
1.
The
shaded
ted quantum magnetic
and itrotation
is
11,13,23,24is preserved. This
hi;ji
axes,systems,
while spin
invariance
areflections
theoryhhi;jii
described
below, are quite
summarized
in Fig.
1 for the here.J3As
systems.
/J 1 will become
different from
that considered
region
has
a
broken
symmetry
of
lattice
about the x
ly in174404-1
Ref. 3.
8
©2012
American
Physical
Society
25
phase
has
‘‘Ising
nematic’’
order.
We
present
strong
nuKorshunov argued that a phase transition,
un-where
below,
spiral order spiral
and associated Lifshitz point
limit
Son!atransition
1.
There
T ! 0clear
state
withthelong-range
S^ i are
spin-Sorder.
operators
square
lattice
and ydriven
axes, by
leading
to Ising
nematic
The Ising
is1is aand
sions, one of the systems in which the
FIG.
5. in
Critical
temperature
a 6function
theT frustration
play a central
the structure
of ourastheory
and inofthe
of kink-antikink
pairs on
associatedJ1 ; J3 ) 0 are the1 nearestmerical
evidence that
transition
at Tc is indeed
inthe
thedomain
andorder
third-neighbor
for J31>antiferrothatrole
hase can be found at
finite temperature
is a thebinding
The
spin
correlation
at walls
the temperature
Tc # $J3 % 4 J1 &S2 . spin
Neel LRO
1 ratio
2Jat=J0. < T < Tc #
4 J12. We establish
dependence
of
observables.
3
1
with
the
Z
symmetry,
can
take
place
in
models
similar
to
2D
Lif
1
magnetic
couplings
along
the
two
coordinate
axes.
For
2
2
class. Such Ising nematic order[4] was
Y ) helimagnet withIsing
Z2 ⊗ universality
SO(2) symmetry
%dependencies
above
is a phase with broken
4
length,
!Tspin , is(see
finite for all T > 0, with $J
the3 T
as this state there
4 J1 &S , [2]
(Q,Q)
(Q,−Q)
FFXY
one
at
temperatures
appreciably
smaller
than
this
model,
early
large
N
computations,
and
recent
BKT
originally
proposed in Ref. [2] for S ! 1=2 in a T ! 0
cal structure results
from a competition
shown,
with c=2oflarge
! c0 !scale
8"jJ3density
% 14 J1 j; matrix
the
crossovers
betweenofgroup
discrete
symmetry
lattice reflections about the x and y
also
Ref.
26).
Such
a
transition
could
lead
to
a
decoupling
renormalization
2
257
FIG.
1. Phase diagram
ctions between localized
spins.
Critical
spin-liquid
phase
described by a Z2 gauge theory [5]. Thus
the
differentinbehaviors
of calculations
!spin are atfor
theaxes,
dashed
attwo
Tdifferent
#
FIG.
2.lines
Thespin
minimum
energy configurations
with This
while
rotation
invariance
is
preserved.
(DMRG)
S!
1=2
[3]
have
suggested
the
phase
coherence
across
domain
boundaries,
producing
this
′
stems from
described
two diagram
region
has a broken sym
0
1 27 23. (Color online) Distribution
the issame
Ising by
nematic
order
canof
appear
whentransitions
spiral
spin
FIG.this
1.class
(Color
online)
Phase
the model
(1) that
is TBKT
of
the
in We
"Q; &Q# strong
with
magnetic
wave
vectors
! "Q;order.
Q# and
Q~ ? !present
existence
a gapped spin-liquid
state
with
jJ3 %
Spin
rotationofsymmetry
is phase
broken
only
atvalue
Texponentially
!
0EaQ~defined
0
0.1
0.2
0.3
0.4
0.5
way two
separate
bulk
with
< FIG.
has
‘‘Ising
nematic’’
nuI .1 jS It. was
4TJ
T
Aside from the conventional
magnetization
and
y
axes,
leadingT/Jto Isin
Q
!broken
2!=3,
corresponding
to J3 =J1 ! 0:5.
order
is destroyed either pointed
by thermal
fluctuations
(as 25
in the
decaying
spin
correlations
and
no
translation
sym=
0.5,
T
=
0.67
<
T
,
and
different
L.
found
in
this
paper.
Eq.
(10)
for
J
out,
however,
in
Ref.
that
these
two
continuous
where
!
!
1.
There
is
no
Lifshitz
point
at
finite
S
because
it
1
2
I
spin
merical evidence that the transition at Tc is indeed in the1/2 at the temperature T #
etry, one has to present
take intoLetter;
account
in the regime
of strong
frustration
(J3 =J
c
seealso
Fig. 1)transitions
or by quantum
fluctuations
(asfirst-order
in
1 ’ 0:5).
can merge
into a single
These conξspin~ Swas
/T
is one.
preempted
[13]metry
by quantum
effects
within
the
dotted
semiIsing
universality
class.
Such
Ising states
nematic
order[4]
rameter that is an Ising variable with Z2
Tc ,ofisthe
FIG.
3.
T
dependence
specific
This
Letter
will
describe
properties
of
the
above
model
length,
!
finite
for
We
begin
by
recalling
[9]
the
ground
of
H
at
S
!
spin
Ref. [2]). Our large S results
aredo
therefore
consonant
with form
clusions
not depend
on the particular
of interactions
circle:
here thereinis a T ! 0 spin gap ! #
S exp$%~
cS&proposed
and spin in Ref. [2] for S ! 1=2 in a T !
rameter characterizes the direction of the
Different
symbols refer to different0 clu
originally
0
for large S and discuss consequences
foris general
S. OurNéel order with magnetic wave
174404-2
1.
There
conventional
shown,
with
c=2 ! c !
the
system
as
soon
as
the
ground-state
degeneracy
remains
the
the possibility
of a spin-liquid phase at S ! 1=2 as derotation symmetryresults,
is preserved.
semicircular
region
extends
between L ! 24
stinguishes left-handed
and right-handed
cS and
/ T L ! 120. Data for J
c'S /1T
1Z gauge
obtainedThis
by classical
Monte
Carlo
simulations
and
~phase
spin-liquid
described
by
a
theory
[5].
Thus
X
ξ
~
e
ξ
~
e
vector
Q
!
"!;
!#
for
J
=J
$
.
For
J
=J
>
,
the
1
2
spin
same.
They
are
confirmed
by
numerical
studies
of
the
models
3
1
3
1
spin
the different
behaviors
o
4
4
scribed in Refs. [2,3]; we will discuss the11,13–17,23,24
quantum finite S
jS # !.
Furtherbelow,
detailsare
onsummarized
the physicsinwithin
over T # jJ3 % 4 Ja1theory
for comparison
(full dots and
dashed line)
described
Fig.
1 forincommensurate
the
~
~
groundIsing
state has
planar
antiferromagnetic
1 specific
the
same
nematic
order
can
appear
when
spiral
spin
2 . Spin
mentioned
above.
the
maximum
of
the
heat.
=
S
⇥
S
jJ
%
J
jS
rotati
phase the
diagram
further towards
this region appearlimit
at the
the isLetter.
i the end
j of the Letter. We
onal (3D) helimagnets,
phase transi3
S !end
1. of
There
a T ! 0 state with long-range spiral
4 1
J1 /the
4
order
at a wave vector
Q~ ! "Q; Q#, with Q
decreasing from
Nevertheless, the situation remains contradictory in 2D
order
is destroyed
fluctuations
(as in
3 ! 1. There i
where
!Jspin
tic and the chiral order parameters occur
spin order for J3 > 14 J1 . We
establish
that at 0 1< Teither
< Tc #by thermalNeel
LRO
!
as
J
=J
>
and
approaching
Q
!
!=2
monotonically
Spiral
LRO
triangle
helimagnets
belonging
to
the
same
Z
⊗
SO(2)
class
as
3
1
Lifshitz
point
2
1
2
4with
present
Fig.
1) or by quantum fluctuations
(as
in
was found numerically that the transition
there Letter;
is a phasesee
broken
$J3 % 4 J1 &S , above this state
where
a
labels
each
plaquette
of qu
th
is preempted
[13] by
1. The spiral order is incommensurate for 14 <
for J3 =J1 !
the FFXY model and the antiferromagnet
on the triangular © 2004 The American
4,5
0031-9007=04=93(25)=257206(4)$22.50
257206-1
Physical
Society
Ref.
[2]).
Our
large
S
results
are
therefore
consonant
with
"1;
2;
3;
4#
are
its
corners.
The
variabl
order or of the “almost-second-order”
discrete symmetry of lattice reflections about the x and y
circle: here there is a T !
where
Q !diagram
2!=3, corJ3 =J1 < 1, except at J3 =J1 ! 0:5
lattice. Garel and Doniach28 (see also Ref. 29) considered theaxes, while spin rotation invariance
1. Phase
of H^ in the
limit
S ! 1. The shaded
Néel
antiferromagnet,
while they
is preserved.
This % FIG.phase
erromagnets on a body-centered tetragonal
the
possibility
of
a
spin-liquid
at
S
!
1=2
as
derotation
symmetry
is assu
pres
hasspins
a broken
about
the x
responding to an angle of 120 region
between
(seesymmetry
Fig. 2). of lattice reflections
simplest
helimagnet
on
a
square
lattice
with
an
extra
competing
the
two
degenerate
ground
states
i
phase has ‘‘Ising nematic’’
order.
We
present
strong
numple cubic lattice with an extra competing
1
and
y axes,
leading
to there
Ising nematic
order.
IsingTtransition
is 4 J1 jS # !
scribed
in Refs.for
[2,3];
we will
discuss
the
quantum
finite
S Theover
# jJ3 %
~!
Interestingly,
each
spiral
state
with
Q
"Q;
Q#
is
exchange
coupling
along
one
axis
that
is
described
by
the
7
Consequently,
a
phase
with
Ising
nem
merical evidence that the transition at Tc is indeed in the
2
along one axis. These systems belong
$J % 1 JWe
spin correlation
at the
Tc #
1 &S . The
phase
diagram
further
towards
the temperature
end
Letter.
Hamiltonian
a distinct
but order[4]
equivalent
configuration
at Q~ ?of!the
"&Q;
Q#3 4 by
a h#a i this
! 0.region appear at the
Ising universality class. Such
Ising
nematic
was
do)universality class as, e.g., the model
1
3
σ
Emergent Ising order parameters
j
j
Tc
T
T
C
C max
15
10
5
1
Ising order: spin chirality
ular lattice8 and V2,2 Stiefel model.9 The
ence and stabilization of the chiral spin., Dzyaloshinsky-Moria interaction in 3D
ussed recently in Ref. 10.
ns (2D), the situation is rather different.11
2
!
H =
(J1 cos(ϕx − ϕx+a ) + J2 cos(ϕx − ϕx+2a )
x
− Jb cos(ϕx − ϕx+b )),
length, !
2
, is finite for all T > 0, with the T dependencies as
spin
be obtained
from the one
numerical results contain str
originally proposed in Ref. [2](for
forQS!!!).
1=2This
in astate
T !cannot
0
shown, with c=2 ! c0 ! 8"jJ3 % 14 JOur
1 j; the crossovers between
~
continuous
Isinglines
phase
betw
wave vector
a globalthe
spin
rotation.
Instead,ofthe
spin-liquid phase described by awith
Z2 gauge
theory Q
[5].byThus
different
behaviors
!spin are at the dashed
at Ttransition
#
0031-9007=04=93(25)=257206(4)$22.50
257206-1
©
20
with
h#i
!
0,
and
a
homogeneous
1
twoappear
configurations
arespin
connected
a 2 global rotation
the same Ising nematic order can
when spiral
jJ3 % by
4 J1 jS . Spin rotation symmetry is broken only at T ! 0
(1)order is destroyed either by thermal
h#i
!
0.
The
divergence
in
the
spec
combined
with a (as
reflection
or1.y There
axes. isThe
fluctuations
in the about
wherethe
!spinx !
no Lifshitz point at finite S because it
accompanied
a divergence
in the
symmetry
of the
classical
ground state
' effects
present Letter; see Fig. 1) or by global
quantum
fluctuations
(as in
is preempted
[13] isbyO"3#
quantum
within thebydotted
semi-
where the sum runs over sites x = (xa ,xb ) of the lattice, a =
Ising nematic in collinear spin system
~1 · N
~ 2 = ±1
=N
Outline
• Frustrated magnetism (brief intro)
- emergence of composite orders from competing
interactions
• Nematic
vs
SDW in LiCuVO4
✓ spin nematic: “magnon superconductor”
✓ collinear SDW: “magnon charge density wave”
• Volborthite kagome antiferromagnet
- experimental status - magnetization plateau
- Nematic, SDW and more
- Field theory of the Lifshitz point
• Conclusions
LiCuVO4 : magnon superconductor?
Letter
rystal structure of LiCuVO4 . Cu-O chains separated by VO4 tetrahedra and
estimates:
he b direction. ∠ Cu-O-Cu ∼ 90◦ indicates the ferromagnetic
interaction.
J1 = - 1.6 meV
J2 = 3.9 meV (subject of active debates)
J5 = -0.4 meV
tinger parameter.9) Recent numerical studies exhibit magnetization vs
nd the quadrupole phase in fact persists down to rather low magnetic
z z
High-field analysis: condensate of
bound magnon pairs
+
hS i = 0
+
+
hS S i =
6 0
Ferromagnetic J1 < 0 produces attraction in real space
Chubukov 1991
Kecke et al 2007
Kuzian and Drechsler 2007
Hikihara et al 2008
Sudan et al 2009
Zhitomirsky and Tsunetsugu 2010
Magnon binding
E-EFM = ε1 + h
1-magnon
2-magnon
bound state
E-EFM = ε2 + 2h
E
Sz=-2
Sz=-1
ε2 < 2 ε1 : “molecular”
bound state
h
Formation of molecular fluid
For d>1 at T=0 this is a molecular BEC
= true spin nematic
Hidden order
No dipolar order
+
hSi i
hSi+ Sj
Nematic order
nematic
director
=0
i⇠e
+ +
hSi Si+a i
|i j|/⇠
Sz=1 gap
=
6 0
Magnetic quadrupole moment
Symmetry breaking U(1) → Z2
can think of a fluctuating fan
state
LiCuVO4: NMR lineshape - collinear SDW along B
Hagiwara, Svistov et al, 2011
Buttgen et al 2012
LiCuVO4
No spin-flip
scattering above
~ 9 Tesla:
longitudinal
SDW state
SF = spin flip, ΔS = 1"
NSF = no spin flip, ΔS = 0
o
Geometry (motivated by LiCuVO4)
•
•
•
No true condensation [ U(1) breaking] in d=1.!
!
Inter-chain interaction is crucial for establishing!
symmetry breaking in d=2.!
!
Need to study weakly coupled “superconducting” chains
J1< 0 (ferro)
J2 >0, J’ > 0 (afm)
in magnetic field
Sato et al 2013
Starykh and Balents 2014
Inter-chain interaction
Hinter
chain
XZ
=
y
~y · S
~y+1 ⇠
dx S
XZ
z
dx Sy+ Sy+1 + Syz Sy+1
y
Superconducting analogy: single-particle (magnon) tunneling between magnon
superconductors is strongly suppressed at low energy (below the single-particle gap)
?
Hinter
=
Z
X
dx J
y
0
+
hSy (x)Sy+1 (x
+ 1)inematic
ground state
!0
Superconducting analogy: fluctuations generate two-magnon (Josephson coupling)
tunneling between chains. They are generically weak, ~ J1(J’/J1)2 << J’ , but responsible
for a true two-dimensional
nematic order
Z
02
Hnem ⇠ (J /J1 )
X
dx
+
[Ty (x)Ty+1 (x)
+ h.c.]
Ty+ (x) ⇠ Sy (x)Sy (x + 1)
y
At the same time, density-density inter-chain interaction does not experience any
suppression. It drives the system toward a two-dimensional
collinear SDW order.
p
Syz
z
Hinter
chain
=M
2npair = M
= Hsdw ⇠ J
0
X
y
z
Syz Sy+1
Ã1 e
⇠J
0
i
2⇡
XZ
'+
y (x) iksdw x
dx cos[
e
p
2⇡
('+
y
'+
y+1 )]
y
Away from the saturation, SDW is more relevant [and stronger, via J’ >> (J’)2/J1 ]
than the nematic interaction: coupled 1d nematic chains order in a 2d SDW state.
Simple scaling
Hnem ⇠ (J 02 /J1 )
Z
X
dx [Ty+ (x)Ty+1 (x) + h.c.]
y
• describes kinetic energy of magnon pairs, linear in magnon pair density npair
z
Hinter
chain
= Hsdw ⇠ J
0
X
z
Syz Sy+1
y
⇠J
0
XZ
dx cos[
p
2⇡
('+
y
'+
y+1 )]
y
• describes potential energy of interaction between magnon pairs on!
neighboring chains, quadratic in magnon pair density
npair
(J 0 )2
⇤
0 2
0 0
n
⇠
J
npair ⇠ J npair , hence npair,c
⇠ J /J/J
1 1
pair
J1
• Competition
• Hence:!
- Spin Nematic near saturation, for n
- SDW for n > n
pair
*pair
pair <
n*pair!
T=0 schematic phase diagram of weakly coupled
nematic spin chains
Spin Nematic
SDW
1/2 - O(J’/J)
cf: Sato, Hikihara, Momoi 2013
1/2
Fully Polarized
BEC physics
M
Cautionary remark: !
maybe impurity effect
Excitations (via spin-spin correlation functions)
• 2d SDW
z
hS (r)i = M + Re
⇣
e
iksdw ·r
⌘
• preserves U(1) [with respect to magnetic field] ->
hence NO transverse spin waves
• breaks translational symmetry -> longitudinal
phason mode at ksdw = π(1-2Μ) and k=0
(solitons (kinks) of massive sine-Gordon model!
which describes 2d ordered state)
phason
OS, Balents PRB 2014
Excitations (via spin-spin correlation functions)
!
•
+
+
0
2d Spin Nematic hS (r)S (r )i ⇠
6= 0
• breaks U(1) but ΔS=1 excitations are gapped
+
hS (r)i = 0
(magnon superconductor)
• gapless density fluctuations at k=0
-
( sector: solitons of !
massive sine-Gordon !
model describing!
1d zig-zag chain.)!
Energy scale J1
(+ sector: solitons of !
massive sine-Gordon !
model which describes !
2d ordered state.)!
Energy scale (J 0 )2 /J1
OS, Balents PRB 2014
Intermediate Summary
• Interesting magnetically ordered states: SDW and Spin
Nematic
-
Gapped ΔS=1 excitations (no usual spin waves!)
-
SDW naturally sensitive to structural disorder
-
analogy with superconductor/charge density wave
competition
Linearly-dispersing phason mode with ΔS=0 in 2d
SDW
Linearly-dispersing magnon density waves in 2d
Spin Nematic
Outline
• Frustrated magnetism (brief intro)
- emergence of composite orders from competing
interactions
• Nematic
vs
SDW in LiCuVO4
✓ spin nematic: “magnon superconductor”
✓ collinear SDW: “magnon charge density wave”
• Volborthite kagome antiferromagnet
- experimental status - magnetization plateau
- Nematic, SDW and more
- Field theory of the Lifshitz point
• Conclusions
n an
g the
f latconds to
nteration.
nical
spin
peritions
nical
model
constate
that
ually
y. As
cy of
Volborthite
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1875
a
H
Cu2
V
O5
c
b
a
b
c
z2 − r2
when
onds,
ds are
oxyween
s partions
the
Cu2 +
xis is
b
Cu2
2,
c
a
Cu1
Cu1
quantum spin liquid?!
J " Si • S j
impurity ordering at low T?
magnetization steps?
0.8
6
8
0.4
CD /T
● 0T
○ 1T
mol-Cu–1)
K– 2
150
60
40
1 4 3/2
1 6 9/2
150 K
30
magnetization
plateau
C/T (mJ
4
T (K)
40
1 4 7/2
magnetic order !
C/T (mJ K– 2 mol-Cu–1)
2
50
I 2/a (a × b
2014
T*
0.0
0
60
2012
0.1
0.6
70
Intensity (counts)
2009
M/H (10– 3 cm3
mol-Cu –1)
. Thus, Tp corresponds approximately to J / 4.24)
Another marked indication from the χ data is the absence of
any spin-gap behavior. Although χ would tend to zero as T
approaches zero, if a gap is opened, as illustrated in Fig. 1(a)
for the theoretical case of Δ = J / 4,8) the χ of volborthite can
remain large and finite at ~3 × 10-3 cm3 mol-Cu-1, implying
the absence of a gap or the presence of a very small gap.
Furthermore, we have extended our χ measurements down
to 60 mK, as shown in Fig. 1(b), and observed an almost
T-independent behavior with neither an anomaly nor any
indication of a downturn. Therefore, the spin gap can be
no more than J / 1500, which is muchNATURE
smallerCOMMUNICATIONS
than theoreti- | DOI: 10.1038/ncomms1875
cally predicted values.8, 9) This strongly suggests that the
ground state of volborthite is nearly gapless and probably a
spin liquid.
4
1.20
Spin glass transitions are observed even in our clean sample at Tg = 1.1 and 0.32 K at magnetic fields of 0.1 and 1 T,
1.10
respectively (Fig. 1(b)). It has been pointed out, however,
3
based on the previous NMR results, that this spin glass can
1.00
be associated with domains based around impurity spins
Ts
having local staggered moments and, therefore, is not intrinsic.19, 24) Fortunately, because the impurity-induced
spin
320
280
300
2
T (K)
glass disappears with increasing field, we can study the intrinsic properties of the kagome lattice at high magnetic
fields, above 2 T.
1
Hare
= 1 freT
Microscopic probes, such as µSR and NMR,
quently used to investigate the dynamics of spins. Polycrystal
The
Single
previous µSR study21, 22) revealed a significant increase
in crystal
relaxation rate λ at low temperatures below 3 0K, towards T ~
200
300
1 K (Fig. 2(a)), due to the slowing down of the0spin fluctua-100
T (K)
tions, which remain dynamic with a correlation time of 20
ns down to 50 mK.27) On the other hand, we observed a
sudden broadening of the 51V NMR line below 1 K, as
M/H (10– 3 cm3 mol-Cu–1)
2001
Cmag T -1 / mJ K-2 Cu-mol-1
of
es
its
re
as
on
or
his
ty”
m.
ds
ee
As
be
on
temperature was 320 K. Thus determined lattice
contribution CD/T is plotted with the broken line in
Fig. 3. A magnetic contribution is determined as Cm
= C -CD and is also plotted with solid circles in the
figure. Integrating Cm/T between 1.8 K and 60 K, we
find a value of 4.1 J/mol K which is about 30 % smaller
than the total magnetic entropy (Rln2) for S = 1/2.
The discrepancy is most likely due to crudeness of
the estimation of the lattice contribution particularly
at high temperature. To be noted here is that Cm/T
seems to show a broad maximum at 20-25 K and then
rather steep decrease below 3 K. Alternatively, one
can say that there is a second peak or shoulder around
3 K. The first maximum must be ascribed to shortrange AF ordering, because its temperature coincides
2
or
mé
is
2a
is
he
ly
he
ue
ce
he
es
ed
1).
re
all
in
J/
ro.
re
of
of
ly.
ite
S=
he
um
ns.
he
me
ns
improvements in crystallinity and particle size. Magnetization at moderately high magnetic fields was measured in a
Quantum Design MPMS equipment between 2 and 350 K
and in a Faraday-force capacitive magnetometer down to!60
sensitive to a phase transition than magnetic
25)
susceptibility especially in the case of quantummK.
AFMs.
High-field magnetization measurements were carThe specific heat of Volborthite exhibited
no using a pulsed magnet up to 55 T at T = 1.4 and 4.2
ried out
anomalies down to 1.8 K, which evidences absence
of
K. Specific
heat was measured in a Quantum Design
LRO above this temperature (Fig. 3). The anomaly
PPMS equipment down to 0.5 K.
seen at 9 K is an experimental artifact. It was
notmagnetic susceptibility χ shown in Fig. 1(a) in a wide
The
easy to extract a magnetic contribution Cm from
the
temperature
range exhibits a Curie-Weiss increase on coolmeasured data, because a nonmagnetic isomorph is
ing from high temperature, followed by a broad maximum at
not available at present from which the lattice
Tp ~ 22 K without any anomaly indicating LRO. From
contribution could be estimated. We fitted the
data
fitting to the theoretical model for the S-1/2 KAFM8) above
at 50 - 70 K to the simple Debye model assuming
150
K, the average antiferromagnetic interaction26) is deternegligible magnetic contributions at this
hight e m p e r a t u r e r a n g e . T h e e s t i m a t e d mined
D e b y eto be J = 86 K on the basis of the spin Hamiltonian
C/T (J/K mol Cu)
ue
des
he
by
nt
50
Volborthite’s timeline
100
120
80
40
0
0
0.5
1.0
1.5
2.0
T 2 (K2)
T*
△ 3T χ
Fig. 1. Temperature dependence of the magnetic susceptibility
20
▼ 5T
of
volborthite
measured
using
a
high-quality
polycrystalline
0.2
50
7T a
Cm /T
sample. (a) χ for the wide temperature range measured□with
Polycrystal
*
Quantum Design MPMS at µ0H = 0.1 T on heating,
after cooling at zero field. The solid curve above 150 K represents a fit
Single crystal
to the theoretical model for0the S-1/2 kagome antiferromag8)
0.0
0
0
10
20
30
40
50
60 net,70 which yields J = 86 K. 0The dotted
1 curve2is obtained
3
0
1
2
3
4
from
theoretical calculations on finite clusters for a spin gap of
T (K)
T/K
T (K)
Δ = J / 4 to open.8) The inset schematically shows a snapshot of
a long-range resonating-valence-bond state on a kagome lattice
Fig. 2. (a) Relaxation rates
λ (triangles)
previous
µSR and heat capacity of single crystals of
Fig.4
Figurefrom
2 | Magnetic
properties
51
made of Cu atoms shown by balls, which consists of various
measurements21) at µ0H = 0.01 T andvolborthite.
1/T1 from the
present
V dependence of magnetic susceptibilities
(a)
Temperature
Fig. 3. Specific heat of Volborthite. The open circlesranges
show of spin-singlet pairs, as indicated by broken ovals. (b) χ
NMR experiments at µ0H = 1 (circles)measured
and 4 (squares)
T.
The
using an assembly of randomly oriented single crystals in a
the raw data, while the magnetic contribution estimated
measured by the Faraday method with a dilution refrigerator on
*
-1 field
inset shows the temperature
evolution
of the
NMRof spectra
ta- 2 and 350 K upon cooling and heating. The
magnetic
is shown with closed circles. The estimated, Debye-type Fig. 4. Magnetic heat capacity of volborthite at low temperature
around
T
=
1
K
in
a
C
T
vs
T 1 T between
mag
heating from 60 mK and cooling to 100 mK at µ0H = 0.1 and 1
ken at µ0H = 1 T at frequencies between
8 and
14.5 MHz. sample
(b)
data for
a polycrystalline
measured under the same conditions are
lattice contribution is shown with the broken line. The
time = material quality
**
20
10
0
200
[101] *
[110]
250
T (K
Figure 3 | First-order structural phase tran
dependence of the intensity of superlattice
indices − 1 − 4 − 7/2, 1 − 4 − 3/2 and − 1 6
The inset shows a CCD image obtained at
marked by * and ** show major superlattice
1/2 and 5 10 1/2, respectively.
Further single-crystal structural a
150 and 323 K to investigate the struc
phase has a monoclinic structure, wit
tice parameters of a = 10.657(3) Å, b = 5
B = 95.035(8)°. Structural refinements
crystal structure as that reported by La
ity factors R [I > 2S(I)] = 4.13% and wR
of the LT phase was determined to b
parameters are a = 10.6418(1) Å, b = 5
and B = 95.443(1)° at 150 K, and the
[I > 2S(I)] = 2.93% and wR2 = 7.67%. Th
are provided in Supplementary Data 1
plementary Data 2 for the LT phase. Th
could not be determined in the presen
Of primary interest is to understa
transition modifies the Cu kagomé
around the Cu ions. The structures of
HT and LT phases are depicted in Fig
result of the structural transition, a mi
at the Cu1 site are lost with respect to t
causes a large change in the coordina
site, including the O3 site splitting int
2014: huge plateau!
H. Ishikawa…M.Takigawa…Z.Hiroi, unpublished, 2014
High-field magnetization
more different MH curves in a pile of 50
large “thick” arrowhead-shaped crystals
30 days growth
0.5
0.3
~2/5
Van Vleck
M(
a pile of thin crystals
B ab
0.2
0.1
T = 1.4 K
0.0
0
10
20
30
40
B (T)
50
Huge 1/3 plateau!
further optical meas.
@ Takeyama lab
It survives over 120 T!
B
/ Cu)
0.4
polycrystals
a pile of ~50 thick crystals
B ab
B // ab
60
70
Kagome plateau or
ferrimagnetic state?
coupled to lattice,
but already distorted
high-field mag. meas.
@ Tokunaga & Kindo labs
Phase diagram
T
1K
?
SDW
1T
1/3
plateau
?
26T
B
our interpretation
FIG. 2 (color H.
online).
(a) 51V NMR
Ishikawa
et al,spectra measured on a single-domain piece of a crystal in
magnetic fields between
15 and 30 T applied perpendicular to the ab plane at T = 0.4 K. (b)
unpublished
PHYSICAL
REVIEW B 82, 104434
I. INTRODUCTION
Cu-O !2010"
bonds; “2 + 4”"
while Cu!2" resides in a plaquette
formed by four short bonds !Fig. 1, top". Recently, densityfunctional theory !DFT" studies of CuSb2O6, implying the
2 + 4 local environment of Cu atoms, revealed that orbital
ordering !OO" drastically changes the nature of the magnetic
coupling from three-dimensional to one-dimensional !1D".18
The search for new magnetic ground states !GSs" is a
major subject in solid-state physics. Magnetic
monopoles in
experiment
the spin ice system Dy2Ti2O7 !Refs. 1–3", the metal-insulator
3.5
kagome
transition in the spin-Peierls compound TiOCl
!Ref. 4" and
!Refs.lattice
5 and 1
6"
the quantum critical behavior in Li2ZrCuON4=18
N=24
are among
thelattice
power 1
of
3 recent discoveries that demonstrate
=18
lattice
2
PHYSICAL REVIEWtechniques
B 82, 104434N
!2010"
combining precise experimental
with
modern
N=24
latticeex2
theory.
However, for a rather
large number
of problems
1
Coupled frustrated quantum spin- 2 chains with orbital order in volborthite Cu3V2O7(OH)2 · 2H2O
periment
2.5and theory do not keep abreast, since it is often
O. Janson, * J. Richter, P. Sindzingre, and H. Rosner
tricky to find Max-Planck-Institut
a real material
realization
well-studied
für Chemische
Physik fester Stoffe, for
D-01187aDresden,
Germany
Institut für Theoretische Physik, Universität Magdeburg, D-39016 Magdeburg, Germany
theoretical Laboratoire
model.de Physique
The most
example
is the
Théorique remarkable
de la Matière Condensée,
Univ. P. & M. Curie,
Paris, conFrance
!Received 9 August 2010; published 307September 2010"
2
J /|J1| = 1.1
Jic/|Jbond”
| = 2—a magnetic GS
cept of a “resonating
valence
1
We present 2
a microscopic magnetic model for
the spin-liquid
candidate volborthite Cu V O !OH" · 2H O.
formed The
byessentials
pairsof thisofdensity-functional-theory-based
coupled spin-singlets
longand
model are !i" thelacking
orbital orderingthe
of Cu!1"
3d
, !ii" three relevant couplings J , J , and J , !iii" the ferromagnetic nature of J , and !iv"
Cu!2" 3d
range magnetic
order
!LRO".
studies
revealed
a 150
0 governed
50 Subsequent
100
implies magnetism
of
frustration
by the next-nearest-neighbor
exchange
interaction
J . Our model
T
(K)
8,9
frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalfascinating
variety of disordered GS, commonly called
ization
1 studies reveal agreement with experiments.
“spin liquids”
in order
to emphasize
their dynamic nature
DOI: 10.1103/PhysRevB.82.104434
PACS number!s": 75.10.Jm, 75.25.Dk, 71.20.Ps, 91.60.Pn
kagome
c
and even
raised
the
discussion
of
their
possible
b
0.8 I.10INTRODUCTION
J
/|
J
|
=
1.1
a
Cu-O
bonds;
“2
+
4”"
while
Cu!2"
resides
in
a
plaquette
2
1
applications.
formed by four short bonds !Fig. 1, top". Recently, densityThe
search for new the
magnetic
is that
a
Jground
/|Jstates
| belief
=!GSs"
1.4
functional
theory !DFT" studies
CuSb O , implying the
Following
common
the spin-liquid
GSof may
1 monopoles
major subject
in solid-state physics. 2
Magnetic
in
0.6
2 + 4 local environment of Cu atoms, revealed that orbital
O !Refs.
the metal-insulator
the spin ice system
Dy Tithe
emerge
from
interplay
low dimensionality,
quantum
ordering !OO" drastically changes
the nature of the magnetic
J1–3",
/|
J
| =of1.6
transition in the spin-Peierls compound
TiOCl
2 1 !Ref. 4" and coupling from three-dimensional to one-dimensional !1D".
fluctuations,
frustration,
considerable effort has
5 and 6"
the quantum critical behavior
in Li ZrCuO !Refs.
0.4 and magnetic
are among recent discoveries that demonstrate the power of1
been
spent
the search
for
combining
precise on
experimental
techniques
withspinmodern2 Heisenberg magnets with
Jicof/|J1herbertsmithite
|=2
theory. However, for a rather large number of problems exkagome
geometry.
The
periment and
theory
do not keep abreast,
since it synthesis
is often
0.2
tricky to find a real material11realization for a well-studied
Cu
the example
first is inorganic
spin- 21 system with
3Zn!OH"
6Cl
2,remarkable
theoretical
model. The
most
the concept of a “resonating valence bond” —a magnetic GS
ideal
kagome
0 geometry and subsequent studies revealed beformed by pairs of coupled spin-singlets lacking the long0 !LRO". Subsequent
50 studies
150
200
range magnetic
order
revealed
a(T) LRO
sides
the
desired
absence
of
magnetic
!Ref.
12"
!i" inh
fascinating variety of disordered GS, commonly called
“spin liquids”
in order structural
to emphasize theirdisorder
dynamic nature
trinsic
Cu/Zn
and !ii" the
c presence of anand even raised the discussion of their possible
13 The
b
aphysics.
isotropic
interactions complicating the spin
applications.
Following
the common
belief that the spin-liquid
GS may
14 was
G. 4.
!Color
online"
Top:quantum
fits
to predicted
the experimental
!!T" !Ref.
recently
synthesized
kapellasite
to imply
emerge from the
interplay of low dimensionality,
b
fluctuations, and magnetic frustration, considerable effort has
kagome
physics
due
to
an
additional
relevant
Jic
he modified
solution
of
the
J
-J
-J
model
yields
an
improved
descripbeen spent on the search for spin- Heisenberg
1 2magnetsicwith
15
a
kagome geometry. The synthesis of herbertsmithite
coupling.
0
J1
Zn!OH"
Cl , K
the compared
first inorganic spin- system
with
ownCuidealto
50
to
the
kagome
model
!bold
gray
line".
Since
the and
search
system
kagome geometry
subsequent
studies a
revealed
be1 for
2 representing the pure
J2
sides the desired absence of magnetic LRO !Ref. 12" !i" inkagome
model
is
far
from
being
completed,
it
is
natural
to
m: magnetization
trinsic Cu/Zn structural disorder andcurves
!ii" the presence !N
of an- = 36 sites" for different solutions
isotropic interactions complicating the spin physics. The
consider
systems
with
where the distortion
recently synthesized
kapellasite
was lower
predicted tosymmetry
imply
b
J1-J
-J
model
in
comparison
to
the
kagome
model.FIG. 1. !Color online" Top: Cu!1"O2 dumbbells !yellow/gray",
16
modified
J
2 kagome
ic physics due to an additional relevant
-3
χ (10 emu / mol)
Frustrated ferromagnetism
1,
2
3
1,†
1
2
3
3
2
7
2
2
3z2−r2
x2−y 2
ic
1
2
1
2
m/ms
2
2
2
6
7
18
2
4
7
8,9
DFT gets it right!
10
1
2
3
6
2
11
J < 0, J > 0, J > 0
1
2
13
14
FM
Ferrimagnetic state
PHYSICAL REVIEW B 82, 104434 !2010"
1
Coupled frustrated quantum spin- 2 chains with orbital order in volborthite Cu3V2O7(OH)2 · 2H2O
O. Janson,1,* J. Richter,2 P. Sindzingre,3 and H. Rosner1,†
1Max-Planck-Institut
für Chemische Physik fester Stoffe, D-01187 Dresden, Germany
für Theoretische Physik, Universität Magdeburg, D-39016 Magdeburg, Germany
3Laboratoire de Physique Théorique de la Matière Condensée, Univ. P. & M. Curie, Paris, France
!Received 9 August 2010; published 30 September 2010"
2Institut
0
J1 < 0, J2 > 0, J > 0
We present a microscopic magnetic model for the spin-liquid candidate volborthite Cu3V2O7!OH"2 · 2H2O.
The essentials of this density-functional-theory-based model are !i" the orbital ordering of Cu!1" 3d3z2−r2 and
Cu!2" 3dx2−y2, !ii" three relevant couplings Jic, J1, and J2, !iii" the ferromagnetic nature of J1, and !iv"
frustration governed by the next-nearest-neighbor exchange interaction J2. Our model implies magnetism of
frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalization studies reveal agreement with experiments.
DOI: 10.1103/PhysRevB.82.104434
PACS number!s": 75.10.Jm, 75.25.Dk, 71.20.Ps, 91.60.Pn
I. INTRODUCTION
The search for new magnetic ground states !GSs" is a
major subject in solid-state physics. Magnetic monopoles in
the spin ice system Dy2Ti2O7 !Refs. 1–3", the metal-insulator
transition in the spin-Peierls compound TiOCl !Ref. 4" and
the quantum critical behavior in Li2ZrCuO4 !Refs. 5 and 6"
are among recent discoveries that demonstrate the power of
combining precise experimental techniques with modern
theory. However, for a rather large number of problems experiment and theory do not keep abreast, since it is often
tricky to find a real material realization for a well-studied
theoretical model. The most remarkable example is the concept of a “resonating valence bond”7—a magnetic GS
formed by pairs of coupled spin-singlets lacking the longrange magnetic order !LRO". Subsequent studies revealed a
fascinating variety of disordered GS,8,9 commonly called
“spin liquids” in order to emphasize their dynamic nature
and even raised the discussion of their possible
applications.10
Following the common belief that the spin-liquid GS may
emerge from the interplay of low dimensionality, quantum
fluctuations, and magnetic frustration, considerable effort has
been spent on the search for spin- 21 Heisenberg magnets with
kagome geometry. The synthesis of herbertsmithite
Cu3Zn!OH"6Cl2,11 the first inorganic spin- 21 system with
ideal kagome geometry and subsequent studies revealed besides the desired absence of magnetic LRO !Ref. 12" !i" intrinsic Cu/Zn structural disorder and !ii" the presence of anisotropic interactions complicating the spin physics.13 The
recently synthesized kapellasite14 was predicted to imply
modified kagome physics due to an additional relevant
J1 FM, J2 AF
Cu-O bonds; “2 + 4”" while Cu!2" resides in a plaquette
formed by four short bonds !Fig. 1, top". Recently, densityfunctional theory !DFT" studies of CuSb2O6, implying the
2 + 4 local environment of Cu atoms, revealed that orbital
ordering !OO" drastically changes the nature of the magnetic
coupling from three-dimensional to one-dimensional !1D".18
J’ AF
c
b
a
b
J
polarized chains?!
Phase diagram
1K
?
SDW
1T
spin nematic?
T
1/3
plateau
26T
B
may be a spin nematic??
FIG. 2 (color H.
online).
(a) 51V NMR
Ishikawa
et al,spectra measured on a single-domain piece of a crystal in
magnetic fields between
15 and 30 T applied perpendicular to the ab plane at T = 0.4 K. (b)
unpublished
tuations the primary spin order is lost, while a spinmultipolar order parameter survives. We demonstrate this
mechanism based on the magnetic field phase diagram of a
1
prototypical model, the frustrated S = 2 Heisenberg chain
with ferromagnetic nearest-neighbor and antiferromagnetic
next nearest-neighbor interactions. Furthermore we show
thatferromagnetic
this instability provides
a natural and unified understandFrustrated
chain
ing of previously discovered two-dimensional spinmultipolar phases.9,10
J1 FM
To be specific, we determine numerically the phase diaof the following Hamiltonian:
Jgram
2 AF
Spin chain redux
H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi ,
i
i
!1"
i
H/(|J1|+J2and
) we set J1 = −1, J2 " 0 in the following. Si are spin-1/2
operators at site i, while h denotes the uniform magnetic
z
S
field. The magnetization
is
defined
as
m
ª
1
/
L%
i i . We emFM
1098-0121/2009/80!14"/140402!4"
quasi-spin-nematic
0 1/5
'Sz = 2
uted to
domin
detaile
We
m / msa
presen
vector
tion w
Luttin
extend
m = 0+
crosso
relatio
spin-d
One a
the p =
140402-1
1 J2/(|J1|+J2)
m/msat
0.8
0.6
0.4
0.2
0
-4
SDW
-3.5
p=4
p=3
octupolar
144404-4
-3
-0.4
?
SDW
(p=3)
J1/J2
-2.5
-0.5
(p=4)
-2
-0.75
-1.5
-1
p=2
quadrupolar
0.025
Ψ “dominant”
0.02
SDW (p=2)
1
0.8
0.6
0.4
0.2
0
0
0
Vector Chiral Order
-1
FIG. 2. !Color online" Squared vector chirality order parameter
" $Eq. !2"% in the low magnetization phase for different values of
2
VC
0.015
0.01
0.2
m/msat
0.4
VC
ɸ “dominant”
(a)
(b)
0.6 0
0.1
m/msat
0.2
0.3 0
VC
0.2
metamagnetic
+
2
(S )
m/msat
p=2
-0.35
p=3
-0.3
p=4
SUDAN, LÜSCHER, AND LÄUCHLI
Ψ~
: spin-nematic
ɸ ~ Sz ei q x : SDW
J2/J1
metamagnetic
%p,k;&r1, . . . ,r p−1'# =
"
p
n
n
eikl /psl− %FM#,
)
*
(" l=1 n=1
1
1d J1-J2 chain is only quasi-spin-nematic
power-law correlations
$9!
FIG. 4. Schematic picture of antiferronematic quasi-long-range
order in the nematic phase. Ellipses represent directors of the nematic order on each bond.
-0.275
2
1
h/hsat
where
-0.25
κ
ure of the vector chiral order. The arrows
g of the parity symmetry by the vector
z
$1!
n! , which obeys the relation J1"# #
hexadecupolar
tion of the sz spin current, shown by the
ng, and there is no net spin current flow.
)
κ(2)
of the phase diagram by examining
polarized state. To that end, we nurgy dispersion of low-energy excitaber of magnons $down spins!. The
Hikihara et al, 2008
Sudan et al, 2009
AGNON INSTABILITY
bound states is the soft mode.
We calculate energy of p-magnon excitations using the
method we introduced in Ref. 6. The number of magnons p
and the total momentum k are good quantum numbers of
Hamiltonian $1!. We thus expand eigenstates in the sector of
p magnons with the basis
Quasi-1d nematic
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 80, 140402!R" !2009"
0.03
0.03
nnn Bond
0.025
0.02
nn Bond
0.005
0.4
0.015
0.01
0.005
(c)
0
0.6
tuations the primary spin order is lost, while a spinmultipolar order parameter survives. We demonstrate this
mechanism based on the magnetic field phase diagram of a
1
prototypical model, the frustrated S = 2 Heisenberg chain
with ferromagnetic nearest-neighbor and antiferromagnetic
next nearest-neighbor interactions. Furthermore we show
thatferromagnetic
this instability provides
a natural and unified understandFrustrated
chain
ing of previously discovered two-dimensional spinmultipolar phases.9,10
J1 FM
To be specific, we determine numerically the phase diaof the following Hamiltonian:
Jgram
2 AF
'Sz = 2
uted to
domin
detaile
We
m / msa
presen
vector
tion w
Luttin
extend
+
m
=
0
H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi ,
!1"
crosso
VECTOR CHIRAL AND MULTIPOLAR
PHYSICAL REVIEW B 78, 144
i
i ORDERS IN THE… i
relatio
0.8 and we set J = −1, J " 0 in the following.
spin-1/2
bosons whichSiareare
actually
two-magnon spin-d
bound s
1
2
total momentum k = ". The boson creationOne
operato
h/J
N site i, while h denotes the uniform
F
a1
operators
at
magnetic
− −
†
sponds
to
s
s
and
the
boson
density
n
=
b
b
#
l
0.6
l l+1
l l 2
z
thea fini
p=
em- costs
field. The magnetization is defined as
m ª 1a /two-magnon
L%iSi . We
breaking
bound-state
Hikihara et al, 2008
Multipolar phases
2
IN
0.4
SDW2
1098-0121/2009/80!14"/140402!4"
T
Q
(a)
0
−4
Is it an infinite progression?
SDW3
0.2
VC
−3
energy, the transverse-spin correlation #s+0 s−l $ is sh
where s+0 = sx0 + is0y . Being a TL liquid, the ground s
its power-law decaying correlations
of the si
140402-1
−
$, and the density fl
propagator, #b0b†l $ # #s+0 s+1 s−l sl+1
z
z z
z
#n0nl$ − #n0$#nl$ # #s0sl $ − #s0$#sl $. When the boson
decays slower than the density-density correlatio
propriate to call this phase the !spin" nematic ph
opposite case when the latter incommensurate den
lation is dominant, we call this phase the spin-de
−2
J1 / J2 −1
tuations the primary spin order is lost, while a spinmultipolar order parameter survives. We demonstrate this
mechanism based on the magnetic field phase diagram of a
1
prototypical model, the frustrated S = 2 Heisenberg chain
with ferromagnetic nearest-neighbor and antiferromagnetic
next nearest-neighbor interactions. Furthermore we show
thatferromagnetic
this instability provides
a natural and unified understandFrustrated
chain
ing of previously discovered two-dimensional spinmultipolar phases.9,10
J1 FM
To be specific, we determine numerically the phase diaof the following Hamiltonian:
Jgram
2 AF
A QCP parent?
H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi ,
i
i
!1"
i
H/(|J1|+J2and
) we set J1 = −1, J2 " 0 in the following. Si are spin-1/2
operators at site i, while h denotes the uniform magnetic
z
S
field. The magnetization
is
defined
as
m
ª
1
/
L%
i i . We emFM
“Lifshitz”
1098-0121/2009/80!14"/140402!4"
quasi-spin-nematic
QCP
0 1/5
'Sz = 2
uted to
domin
detaile
We
m / msa
presen
vector
tion w
Luttin
extend
m = 0+
crosso
relatio
spin-d
One a
the p =
140402-1
1 J2/(|J1|+J2)
I.
A.
NLSM
Classical limit
Lifshitz Point
onsider the Non-Linear sigma Model (NLsM)
uld describe the behavior near the Lifshitz
he J1 J2 chain. The action in 1+1 dimenZ
2
|@x m̂| +
dxd⌧ isAB [m̂]
•
•
+u|@x m̂|4
hm̂z .
v|@x m̂|4
K|@x2 m̂|2
hm̂z ,
(5)
Unusual QCP: order-to-order transition
(1)
the spin and AB is the Berry phase term dehose spins. It can be written in various ways,
e
Z 1
AB =
du m̂ · @⌧ m̂ ⇥ @u m̂,
(2)
where we defined v = u/K and h = hK/ 2 . We see
that when /K ⌧ 1, the action is large in dimensionless
terms, and we expect a saddle point approximation to
apply. This is precisely the classical limit! Note that this
is valid when u/K is fixed, and also h ⇠ 2 /K, which
fixed the overall field scale of the problem.
Effective action - NLσM
Z
0
Can we see this formally somehow? Let us try rescaling
p
to bring out the behavior for small . We let x ! K/ x
and ⌧ ! K2 ⌧ , where the second rescaling follows from
the linear derivative nature of the Berry phase term. The
magnetization itself does not rescale as m̂ is a unit vector.
Carrying out this rescaling, we find
r Z
K
S =
dxd⌧ isAB [m̂] |@x m̂|2 + |@x2 m̂|2
2
2
2
4
S
=
dxd⌧
isA
[
m̂]
+
|@
m̂|
+
K|@
m̂|
+
u|@
m̂|
B. Saddle
introduce a fictitious auxiliary coordinate
u
B
x
x
x point
m̂(u = 0) = ẑ and m̂(u = 1) = m̂ is the
alue, or equivalently,
Berry
m̂ @ m̂
m̂ @ m̂
A =
. phase(3)
1 + m̂
important point for us is that Aterm
contains a
vative of imaginary time ⌧ .
B
1 ⌧
2
2 ⌧
1
3
B
ion in Eq. (1) needs a condition for stability
ge gradients of m̂. To get it, we note that by
tion twice of m̂ · m̂ = 1 we obtain
|@x m̂|2
hm̂z
To find the actual saddle
point,symmetry
we make an assumptwo
tunes
tion that it is of the form of an umbrella state (I tried
also to look for a planar
state, but
it seemed to be
allowed
interactions
QCP
energetically unfavorable). To avoid having to rescale,
4 Let m̂ =
we work in the original
variables
of
Eq.
(1).
p
(' cos qx, ' sin qx, 1 '2 ). Then the action is just the
integral of the energy density
p
2 2
4 2
4 4
"=
q ' + Kq ' + uq '
h( 1 '2 1), (6)
at O(q )
All properties near Lifshitz point obey “one parameter
where we chose to add a constant h factor so that " = 0
= m̂ · @ m̂  |@ m̂|,
(4)
universality” dependent
upon
u/Kover
ratio
when ' = 0. This is easily
minimized
wavevector
2
x
2
x
S=
Z
•
Lifshitz Point
2
dxd⌧ isAB [m̂] + |@x m̂| +
S=
+ u|@x m̂|
4
hm̂z
Intuition: behavior near the Lifshitz
point should be semi-classical, since
“close” to FM state which is classical
x!
r
2
2
K|@x m̂|
K
Z
s
K
x
| |
⌧!
K
⌧
2
dxd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + v|@x m̂|4
Large parameter:
saddle point!
u
v=
K
hm̂z
h=
hK
2
S=
r
K
Z
Saddle point
dxd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + v|@x m̂|4
hm̂z
v derives from quantum fluctuations
By a spin wave analysis, one
finds v ~ -3/(2S) < 0
h
first order
hc =
8K
p
|v|(1
1<v<
local instability of FM state
(1-magnon condensation)
FM
IC cone
0
2
spiral
1
4
p
|v|)
Phase diagram
VECTORchain
CHIRAL AND MULTIPOLAR ORDERS
Frustrated ferromagnetic
First order
metamagnetic
transition near
Lifshitz point
0.8
h / J2
N
F
0.6
IN
0.4
Higher
dimensions?
SDW2
T
SDW3
0.2
Q
Hikihara et al, 2008
(a)
0
−4
VC
−3
−2
J1 / J2 −1
S=
Z
d>1
dxdd
•
1
yd⌧ isAB [m̂] + |@x m̂|2 + c|@y m̂|2 + K|@x2 m̂|2 + u|@x m̂|4
Rescaling:
x!
S=
p
K d cd 1
d 1/2
Z
dxdd
s
1
K
x
| |
⌧!
K
⌧
2
y!
p
cK
hm̂z
y
yd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + |@y m̂|2 + v|@x m̂|4
∴ Similar theory applies in d>1, and very
similar conclusions apply
hm̂z
Phase diagram
ECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE…
0.8
h / J2
N
F
0.6
IN
0.4
SDW2
T
SDW3
0.2
Q
(a)
0
−4
VC
−3
−2
J1 / J2 −1
bosons w
multipolar phases
total mom
from QCP?
sponds to
breaking
energy, th
+
where s0
its powe
propagato
#n0nl$ − #n
decays sl
propriate
opposite
lation is d
Origin of multipolar
phases
VECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE…
0.8
h / J2
N
F
0.6
IN
0.4
SDW2
T
SDW3
0.2
Q
(a)
0
−4
VC
−3
−2
J1 / J2 −1
M
0.4
0.2
(b)
PHYSICAL REVIEW B 78, 144404 !2008"
bosons which are actually two-magnon bound states with
total momentum k = ". The boson creation operator b†l corre−
sponds to s−l sl+1
and the boson density nl = b†l bl # 21 − szl . Since
breaking a two-magnon bound-state costs a finite binding
energy, the transverse-spin correlation #s+0 s−l $ is short ranged,
where s+0 = sx0 + is0y . Being a TL liquid, the ground state exhibits power-law decaying correlations of the single-boson
−
$, and the density fluctuations,
propagator, #b0b†l $ # #s+0 s+1 s−l sl+1
#n0nl$ − #n0$#nl$ # #sz0szl $ − #sz0$#szl $. When the boson propagator
decays slower than the density-density correlation, it is appropriate to call this phase the !spin" nematic phase. In the
opposite case when the latter incommensurate density correlation is dominant, we call this phase the spin-density-wave
!SDW2" phase. The SDW2 phase is extended to the antiferromagnetic side J1 $ 0 across the decoupled-chain limit J1
= 0; it is called even-odd phase in Ref. 25. The boundary
between the SDW2 phase and the nematic phase is shown by
a dotted line in Fig. 1.
In the semiclassical picture we can write s−l = e−i%l, where
%l is the angle of the two-dimensional vector !sxl , sly" measured from the positive x direction, 0 & %l ' 2". The product
−
= e−i!%l+%l+1" can be represented by the vector Nl+1/2
s−l sl+1
= !cos (l,2 , sin (l,2" with (l,2 = −!%l + %l+1" / 2. We now realize that we need to identify Nl+1/2 with −Nl+1/2 because of the
physical identification !%l , %l+1" % !%l + 2" , %l+1" % !%l , %l+1
+ 2"". We can thus consider Nl+1/2 as a director representing
the nematic order. We will show in Sec. VI that the nematic
phase has antiferronematic quasi-long-range order of the director, as shown schematically in Fig. 4. The ground state is
not dimerized in this phase as opposed to the initial proposal
of Chubukov.1
Incommensurate nematic phase. The incommensurate
First order transition:
partially polarized state
coexists with plateau one
With enough quantum fluctuations,
“bubbles” of partially polarized phase may
become many-magnon bound states and
form multipolar phases
0
−4
:N
: SDW2
: IN
:T
: SDW3
:Q
: VC
−3
−2
J1 / J2 −1
FIG. 1. !Color online" Magnetic phase diagram of the spin-1/2
zigzag chain with ferromagnetic J1 and antiferromagnetic J2 !a" in
the J1 / J2 versus h / J2 plane and !b" in the J1 / J2 versus M plane.
Crosses show the transition and crossover points obtained from the
magnetization curves and correlation functions. In !a", symbols VC,
N, IN, T, Q, and F indicate the vector chiral !*Sztot = 1", nematic
!*Sztot = 2", incommensurate nematic !*Sztot = 2", triatic !*Sztot = 3",
Origin of multipolar
phases
VECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE…
0.8
h / J2
N
F
0.6
IN
0.4
SDW2
T
SDW3
0.2
Q
(a)
0
−4
VC
−3
−2
J1 / J2 −1
M
0.4
0.2
(b)
PHYSICAL REVIEW B 78, 144404 !2008"
bosons which are actually two-magnon bound states with
total momentum k = ". The boson creation operator b†l corre−
sponds to s−l sl+1
and the boson density nl = b†l bl # 21 − szl . Since
breaking a two-magnon bound-state costs a finite binding
energy, the transverse-spin correlation #s+0 s−l $ is short ranged,
where s+0 = sx0 + is0y . Being a TL liquid, the ground state exhibits power-law decaying correlations of the single-boson
−
$, and the density fluctuations,
propagator, #b0b†l $ # #s+0 s+1 s−l sl+1
#n0nl$ − #n0$#nl$ # #sz0szl $ − #sz0$#szl $. When the boson propagator
decays slower than the density-density correlation, it is appropriate to call this phase the !spin" nematic phase. In the
opposite case when the latter incommensurate density correlation is dominant, we call this phase the spin-density-wave
!SDW2" phase. The SDW2 phase is extended to the antiferromagnetic side J1 $ 0 across the decoupled-chain limit J1
= 0; it is called even-odd phase in Ref. 25. The boundary
between the SDW2 phase and the nematic phase is shown by
a dotted line in Fig. 1.
In the semiclassical picture we can write s−l = e−i%l, where
%l is the angle of the two-dimensional vector !sxl , sly" measured from the positive x direction, 0 & %l ' 2". The product
−
= e−i!%l+%l+1" can be represented by the vector Nl+1/2
s−l sl+1
= !cos (l,2 , sin (l,2" with (l,2 = −!%l + %l+1" / 2. We now realize that we need to identify Nl+1/2 with −Nl+1/2 because of the
physical identification !%l , %l+1" % !%l + 2" , %l+1" % !%l , %l+1
+ 2"". We can thus consider Nl+1/2 as a director representing
the nematic order. We will show in Sec. VI that the nematic
phase has antiferronematic quasi-long-range order of the director, as shown schematically in Fig. 4. The ground state is
not dimerized in this phase as opposed to the initial proposal
of Chubukov.1
Incommensurate nematic phase. The incommensurate
First order transition:
partially polarized state
coexists with plateau one
With enough quantum fluctuations,
“bubbles” of partially polarized phase may
become many-magnon bound states and
form multipolar phases
0
−4
:N
: SDW2
: IN
:T
: SDW3
:Q
: VC
−3
−2
J1 / J2 −1
FIG. 1. !Color online" Magnetic phase diagram of the spin-1/2
zigzag chain with ferromagnetic J1 and antiferromagnetic J2 !a" in
the J1 / J2 versus h / J2 plane and !b" in the J1 / J2 versus M plane.
Crosses show the transition and crossover points obtained from the
magnetization curves and correlation functions. In !a", symbols VC,
N, IN, T, Q, and F indicate the vector chiral !*Sztot = 1", nematic
!*Sztot = 2", incommensurate nematic !*Sztot = 2", triatic !*Sztot = 3",
Summary
•
Spin chains keep showing up in
unexpected places
✓ Nematic physics of frustrated
ferromagnets
✓ Explored Lifshitz point as a “parent”
for multipolar states and
metamagnetism
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