Download Two-magnon instabilities and other surprises in magnetized quantum antiferromagnets Oleg Starykh

Two-magnon instabilities and other
surprises in magnetized quantum
antiferromagnets
Credit: Francis Pratt / ISIS / STFC
Oleg Starykh
University of Utah
!
Andrey Chubukov, U Wisconsin
Conference on Field Theory Methods in Low-Dimensional Strongly Correlated Quantum Systems,
August 25-29, 2014, ICTP, Trieste, Italy
Outline
•
Emergent Ising orders - a very brief history
!
•
UUD magnetization plateau and its instabilities
!
•
High-field phase diagram of a triangular antiferromagnet
system supports solitons (domain
quence, the
symplanar' Q model on a triangular lattice reveal a wealth of interal antiferromagnetic
walls) as additional elementary excitations. An
orderto a
transition
LETTERS
model
REVIEW
From
this
simple
PHYSICAL
critical
phenomena.
NUMBER
6 1.arise a zero-field
ing
VOLUME
52,
in Fig.
is shown
example of a soliton
ed, Kosand a critical point
for with
disordering,
new mechanism
spin the
te of
long-range Theorder,
associated
&3
order aparameter
classithe
class.
new universality
sociated
possible
where
is given by pl,
the comperiodicity
ar (XY') with a W3
Breaking and Novel Critical Phenomena
Discrete-Symmetry
us
&&
6
FEBRUWRV
1984
in
I
as 70.Jk, 75.
is 70.
defined
vector
from
an
plex
Ak
Planar (XY) Model in Two Dimensions
an 10.
64. 60.
Antiferromagnetic
numbers:
Fh, 05.
CS
Cn, g05.
ered phase
-=— exp(-iq B,.)s,
D, H. Iee, J. D. Joannopoul. os, and J. W. Negel. e
eas the
03139
Institute
Cambridge, Massachusetts
of Technology,
of Physics,of Massachusetts
As a consestates.
the tmo ground
nal. spin systems cannot exhibitDepartmentordering
partite
solitons (domain
the system supports and
quence, comcontinuous
der
symHere a
e ferro0) and (~~,
q=(+~m,
by breaking
(~ are orthogonal
' Although
parallel and perpendicular
of ( chosen
case
for
ponents
walls) as additional elementary excitations. An
ordertemperature.
D. P. Landau
H o 0. In terms of g,a
fundamenfield
when
to
the
magnetic
in Fig. 1.
shown
soliton
is
6 FEBRUARY 1984
Kos- PHYSICAL
of
example
LETTERS
REVIEW
itions are' VOLUME
excluded,
thereby
6
52, NUMBER
Department
x 1m(()
of Physics, University of Georgia, Athens, Georgia 30602
the staggered helicity' is given by rt =Be(g)
rustratiori.
with1983)
the &3
associated
order parameter
classi(KT)'= showed
that thelattice
ouless
14 November
(Received
ref l. ection The
is a discrete
pl' sing. Since
ilson
spin-disincreases,
As
the temperature
(XY')
is given analyses
W3 periodicity
onal.
ferromagnetic
planar
by pl, andwhere
calculations for the clasCarlocomMonte the
heliandau-Ginzburg-Wilson
can haveItwo
symmetry
the
long-range
arlo (MC)
system
symmetry,
i
of
the
8
unbinding
mechanisms
compete:
ordering
antiferromagnetic
as on a triangular lattice reveal a wealth of intersical
defined
vector g isplanar'
transition
Q model
a unique phase
plex
without from
Mermin-Wagner
antiferroviolatingan the
city order
and
and the
both Re(ICI)
vortex pairs
the theorem.
"
' Theinzero-field
Im(()
transition to a
From
this simple model arise a zero-field
critical
phenomena.
esting
low-temperature
phase
X
Hy
FTls& = 0
to a disordered
ordered
&,
phase
, &(s,. ,"s phase
critical
and
between
point
correlation
a
for
the
Re(ICI)
disordering,
of
mechanism
phase
new
unlocking
a
spin
of
state
long-range
order,
=
~i ⇥ S
~j
is pairs.
characterized
thereforeWhereas
atticeof and
( g) 0 and (rl) =+1
=
S
. )s, and discrete spin chirality
ng
vortex
the bytakes
Order
parameters:
continuous
B,
class.
new exp(-iq
universality
with
associated
a
possible
through
latter
place
The
process
and
Im(ICI).
so that although both Be(() and Im(p) have no longal behavonofasol.
ic planarthe model.
bipartite
triangle
transition
occurs
the PACS
itons
creation
areand
locked
in phase
64. 60.Cn, 05.70.Fh, 05.70.Jk, 75.10.Ak
range
order,
numbers:
they
(y).
the
1.2
freeferrothe surface
0.4 orthogonal comto the
Herewith
energy associated
equivalent
0) and (~~, (~ are
q=(+~m,
e lattice
triangu- iswhen
/J
argu-of
simple
soliton vanishes. Using
and ordering
chosen
perpendicular
case
stence
l. , theofantiferromagnetic
for stability
ponents
of the tmo ground states. As a conseTmo-dimensional.
cannot parallel
exhibit
spin systems
thatfundamenthe pair unbinding temperawe find is
the ments" l. attics
Two possibilities:
cy
the system
In terms
of g, supports solitons (domain
fieldH~&when
ordertobythe
a continuous
sym- H o 0.quence,
magnetic
l.ong-range
breaking
.g. ,in triangular)
assois
twice
the
temperature
ture
roughly
T„
T,
'
partite
as additional
walls)
orderx 1m(()elementary excitations. An
metry at finite temperature.
is given by
helicity'
the with
because of
inherent
frustratiori.
rt =Be(g)
staggeredAlthough
8
transition. Thisthesuggests
ciated
a soliton
on two
a soliton is shown in Fig. 1.
example a of
disorder transitions pl'aresing.
thereby excluded, Kosdiscrete
me
•
that helicity order will be lost first, = with
a spon- Since Hylattice ref l. ection is order
s. use
For Landau-Ginzburg-Wilson
the &3 transition: both
The
parameter associated with single
terlitz and Thouless (KT)' showed that the classiAs
verified
of
helisolitons.
can have long-range
and generation
the system (XY')
Monte Carlo
by
ry analyses
symmetry,
(MC)
onsists
of taneous
spinstheand
chiralities order,
comW3 periodicity is given by pl, where
cal iwo-dimensional.
ferromagnetic
planar
solitons
the
shown
the
screen
MC
results
below,
0'
angles
without
the
Mermin-Wagner
antiferroorder
violating
the
the
o study
phases of
model exhibits acity
unique phase transition from an
plex vector g is defined as
iminteraction
and induce
unbinding
vortex
pair
"
'
ipartite
=
"s
zero-fieldphase
The
low-temperature phase
ar (AFP) model, X
theorem.
a disordered
ordered
phase to
&,algebraical.
, &(s,. , forly spin
disordering
•
l possibl. e mediately. This mechanism
two separate transitions:
=—
. )s,
=+1
via the from
characterized
unbinding isoftherefore
vortex
Whereas
the by ( g) = 0 and (rl)
exp(-iq
and
a
and
ar
lattice
pairs.
triangul.
square
B,
H, '
unvortex
different
is fundamentally
pair
flection
Ising (chirality) transition
a bipartite
model. on both
antiferromagnetic
planar
so
behavin
their
differences
critical
Be(() and Im(p) have no longuninew although
binding and leads to what seems to be athat
Fig. 1,
com-by the BKT (spins)
lattice is equivalent to the ferroHere
0) and (~~, (~ are orthogonal
(e.g. , square)
is followed
ar latphysics.
l.ying
versality class of critical
phenomena.range order, they are locked in phaseq=(+~m,
(y).
case for
model.
antiferromagnetic
ponents of chosen parallel and perpendicular
, thestaggered
other
In the
an
in-plane
field
ofmagnetic
magnetic
by distinguishing
the
triangufeature
states
of opposite
I. Two ground
FIG.presence
fundamen- 0. 2 to the0.magnetic
when H o 0. In terms of g,
4 T
a tripartite
0.field
6
(e.
g.The
,&3triangular)
&v3
perithe
still
mmetry
statesexistence
of l. attics is
the helicity
domain
wall.
helicity
ground
by a preserve
is separated
the
of
e AFP models
'
the staggered helicity' /J is given by rt =Be(g) x 1m(()
tal. l.y ground-state
different because
of the inherent frustratiori.
is indicated.
each triangle
hei. icity
magnetizaThe sublattice
odicity.
= pl'for
ell as continuous
in the
degeneracy
Landau-Ginzburg-Wilson
In this
sing.
me all
usei and
FIG. 2. Phase diagrams
AFP Since
models lattice
on the ref l. ection is a discrete
. l = 1 for
tions m„m» m, satisfy
m,work,
and triangular
square
(bottom) the
lattices.
(top)(MC)
can have long-range helia bipartite
state.
ForSociety
P The
Carlo
systemComground
symmetry,
(LGW) symmetry analyses and
433Monte
84
American
Physical
of the order parameters
ponents
are violating
zero are the Mermin-Wagner
without
the antiferroto study the phases
city order which
of spins calculations
on two
nd state consists
(2) of indicated
m,. =H/3Z.
The
circle
illustrate
explicitly.
diagrams
the
"
'
)=1
. ,"s
theorem. The zero-field low-temperature
phase
&, , &(s,
directions.magnetic
For pl. anar (AFP) model, X = spin configurations
gned in opposite
in the ordered phases. The dots
+though
is therefore
characterized
a square
and
arhasandnoleads
a triangul.model
by ( g) = 0 and (rl) =+1
s,. 's,. '),the
around
circles
the number
represent
of additionaland
exceedingly
simple
tolattice
athesurprising
richness
of
phases
critical
behavior.
!
stateeven
l. attice, aSurprisingly,
consists
ofonHamiltonian
ground “An
and
so
that
both
behavhave
longno
in their critical
although
dramatic
differences
find there
Im(p)
Be(()
exist
continuous symmetry,
continuously
distinct and degenerate spin configurations obtained
sublattiees forming + 120'iorangles
arewhich
locked
range order,
and
underl.
they in
physics.
ying
The manner
the in phase (y).
permuting the sublattices.
can
solutions
to
it
be
Moreover,
by
degenerate The
(2). triangular lattice and the
underlying
associated
degeneracy
a
crucial
role in this physics.”!
play
3&& v 3 periodicity.
a
In
bipartite
boundaries merge at & is not determined
thephase
trianguAn essential.as feature
distinguishingthree
P is increased
shown that at zero temperature
in this
e (2) change
spin rotation
AFP models
is theprecisely
existence
of work.
larpossibl.
and bipartite
abovegenerates
from
to
H, =3J, theallsolutions
discrete (corresponding
as mell as continuous
while an two
extra
l. attice reflection
manifolds
to two degeneracy in the
disconnected
[unfortunately,
incorrect identification of spin configurations]
triangular
ground state.to For a bipartite
hei. icity
states) to
(corresponding
lattice.
In one
the triangular
Fig.manifold
1, AFP
a ground
of =[0.
on two is associated with spontaneous genlattice,
spins495(5)]Z
Above
9J all consists
a state
the
helicity).
II, = state
ar latof of
thezero
triangul.
round states
in opposite directions.
paramagnetic
eading to thealigned
sol. itons as described earlier.
spins are aligned, l.sublattices
eration ofFor
g
I
&&
I
-=—g
I
I
(
'
!
I
&&
I
!
Jg
- g
(
I
l
g
Jg
I
Classical isotropic triangular AFM in magnetic field
•
Zero field: co-planar spiral (120 degree) state!
•
Magnetic field: accidental degeneracy!
! =J
H
!
!
⃗i · S
⃗j −
S
i,j
!
⃗h · S
⃗i
i
⃗h #2
1 !"! ⃗
Si −
H! = J
2
3J
!
△
i∈△
•
⃗i1 + S
⃗i2 + S
⃗i3
all states with S
•
Accidental degeneracy!
120o state
co-planar supersolid non-coplanar
superfluid
⃗h
form the lowest-energy manifold!
=
3J
– O(2) spins: 3 angles, 2 equations => 1 continuous angle undetermined!
– O(3) spins: 6 angles, 3 equations => 2 continuous angles (upto global U(1) rotation
about h)
that in two-dimensional systems with a continuous Abelian symmetry an arbitrary small
anisotropy proves to be relevant if the temperature is small in comparison with the
constant in the gradient energy term (Pokrovsky and Uimin 1973a, b, JosC et a1 1977).
Let us now consider what kind of states have the minimal free energy of the spin
waves. The configurations of spins in different sublattices maximising S(
q52,r#J3) with
constraints (5) being taken into account are shown in figure 3. For h < hcl the three
J. Phys. C: Solid State Phys. 19 (1986) 5927-5935. Printed in Great Britain
Phase diagram of the antiferromagneticXY model with a
triangular lattice in an external magnetic field
S E Korshunov
L D Landau Institute for Theoretical Physics, Academy of Sciences of the USSR,
Kosygina 2,117940 Moscow, USSR
Antiferromagnetic XY model with triangular lattice
5933
Received 16 December 1985
Abstract. The ordered states of a planar antiferromagnet with a triangular lattice are
investigated in the presence of a magnetic field. The spin wave free energy is taken into
account and proves to be important for determining the properties of the system. The phase
diagram is constructed. It contains four different phases with rigorous long-range order.
Three of them are characterised by different configurations of mean magnetic moments for
the three sublattices. The existence of one more non-trivial phase with an algebraic decay
of the correlation functions is very probable.
51
t
Figure 3. Configuration of spins of the three sublattices with the minimal spin wave free
energy: ( a ) , 0 < h < hcl;( b ) ,h = hCl;( c ) ,h,, < h < hc2;( d ) , in the case of the opposite sign
of the anisotropic part of the free energy.
sublattices are non-equivalent. On one of them the spins are antiparallel to the field,
and on the two others they have the perpendicular-to-field components of opposite signs
P
(figure 3(a)). This state is six-fold degenerate in accordance with a number of possible
In the exchange approximation a planar antiferromagnet can be described with the
permutations of non-equivalent sublattices.
Hamiltonian:
With increasing h the angle between the spins, which are not antiparallel to the
field,
diminishes,
and at h = hcl vanishes (figure 3(b)). Two of the sublattices become
H =J
m i mi!= J
cos(qj - qj,)
(1)
(11')
(ii')
equivalent, the spins on them being parallel to the field. The degeneracy multiplicity of
state
4 qj, sin qj)are unit planar vectors defined on lattice sitesthis
where mi= (cos
and
theis equal to three.
For
hcl < h < hc2the equivalence of two of the sublattices is retained, but the symsummation is performed over pairs of nearest neighbours. In the case of a flat triangular
in the direction perpendicular to the field becomes broken (figure 3(c)). That
lattice (and this is the case we are interested in) the ground state consistsmetry
of three
leads
to
the degeneracy multiplicity being increased up to six, as for 0 < h < hcl.The
sublattices. The magnetic moments
(spins) mibelonging to different sublattices form the
,I
anharmonicities being taken into account, the magnetic moment in this state is not
angles 120" with respect to each other:
parallel to the field (at finite temperatures).
Thus(2)
we have considered which states are preferred by the spin wave free energy at
@* = @ I ? 120"
= @ I 7 120".
the lowest temperatures.
Here Figure
q$ ( I = 4.
1 ,Phase
2 , 3 ) denote
theofcommon
values
of q iininan
each
of themagnetic
three sublattices.
In
diagram
the AF XY(t)
model
external
field. In phases
a, b
addition
continuous
degeneracy
caused
the invariance
of form
the Hamiltonian
with
andtoc the
mean magnetic
moments
of by
different
sublattices
configurations
similar to
respect
to homogeneous
rotation
of all
the ground state
also dpossesses
two-fold order
those
shown in figures
3(a),3(b)
andspins
3(c) respectively.
In phase
only the long-range
The order parameter degeneracy
discrete
degeneracy
(upper and
lower signs
in p(2)).
with
respect to helicities
is retained.
Phase
is paramagnetic.
space R is, accordingly, a pair of circumferences:
1. Introduction
hlJ
Order of helicities (chiralities) only
-
/id\
R = Z 2 x SI.
1977). So the
curve convenient
BA (or attoleast
partthisofadditional
it) should
split into
two curves
joining in the
It proves
describe
(discrete)
degeneracy
by introducing
Phase diagram of the Heisenberg (XXX) model in the field
Seabra, Momoi, Sindzingre, Shannon 2011
Z2 vortex (chirality ordering) transition
Gvozdikova, Melchy, Zhitomirsky 2010
can merge
into with
a single
first-order
one.
These
con-the quantum
in2 Sec.transitions
III. In particular,
the
Ising
and
the BKT
consistent
Ising
universality.
Wetransitions
also
discuss
phase diagram
finiteof
S.
clearly
indicated
by lattice.
aitsharp
peak
ofofthe
specific
heat
is divisible
by theforsize
the helix
pitchsquare
and
liesIninthe
the
th Z
limit
spinhave
S !repeated
1, there
is which
a zero
temperature
Lifshit
We
a similar
analysis
for several(T)
values
of
clusions
do
not
depend
on
the
particular
form
of
interactions
in
1
are
considered
in
Secs.
III
A
and
III
B,
respectively.
The
J
=J
and
the
complete
phase
diagram
is
shown
in
Fig.
5,0
is illustrated
inlong-range
Fig.
3 spiral
[10].spinThis
feature
to be
of the
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
the
system
as
soon
as
the
ground-state
degeneracy
remains
the
where
we
have
plotted
T
versus
J
=J
.
We
find
that
T
neighborhood of the Lifshitz point and the phase diagram are
c spin rotation
3 1
theory
forthe
T>
0 crossoversdisplayed
near the Lifshitz
point:
symmetry isc r
anded
contrasted
the
broad
maximum
by
the
same
conditions as well as the cylindrical
onesto(i.e.,
with
periodic
vanishes
linearly
for
J
=J
!
1=4;
a
theory
1 behav2
same.
They
are
confirmed
by
numerical
studies
of
the
models
3
1
there is a1 broken lattice reflection symmetry for 0 " T < Tc # $J3for%this
discussed in Sec. III C.11,13–17,23,24
Section IV contains our conclusions.
0
4 J1 &S . Th
condition
along
the
b
axis
and
the
free
one
along
the
a
ior
will
now
be
presented.
quantity
for
J
=J
<
,
i.e.,
when
the
classical
ground
state
3
1
mentioned
above.
PHYSICAL REVIEW B 85, 174404 (2012)
4
consistent
with
Ising
universality.
We
also
discuss
the
quantum
phase
diagram
ransiNear the classical Lifshitz point, we can model quantum
also
suggest
symmetries
may play
a to order.
Frustrated the
antiferromagnets
have contradictory
recently attracted
axis).
We that
havediscrete
foundlattice
thatdisplays
both
conditions
lead
the same
Nevertheless,
situation
remains
in
2D
ordinary
Néel
In
particular,
the maximum
and
thermal fluctuations
by a continuum unit vector field0
occur
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
values
of
transition
temperatures
and
indexes.
In
contrast,
helimagnets
belonging
to
the
same
Z
⊗
SO(2)
class
as
Chiral
spin
liquid
in
two-dimensional
XY
helimagnets
n"r;
%#,
where
r
!
"x;
y#
is spatial coordinate, % is imagi2
of the specific heat is consistent with a logarithmic
depenII. MODELlow-temperature
AND METHODS
sition
Broken discrete symmetries have also been discussed
ing unconventional
(T) phases, with novel
2
nary time, and n ! 1 at all r, %. This field is proportional
the FFXY model and the antiferromagnet on the triangular
values of Binder’s cumulants
andonthe
chiral-order-parameter
dence
system
size
(see
the
inset
of
Fig.
corresponding
week
der”4,5
x &y
1,* order’’
1,2,†
[7,8]
in
the
context
of
the
J
types
of
‘‘quantum
[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
28V. Syromyatnikov
1 (2004)
2
"%1#2004
n"rj ; %#.
toTthe
parameter with
S^ j /ending
A. O.
Sorokin
and
A.
17 DECEMBER
PRL
93,
257206
lattice.
Garel
and
Doniach
(see
also
Ref.
29)
considered
the
We
consider
the
model
(1)
of
the
classical
XY
magnet
on
a
distribution
at
J
≈
0.309
depend
on
boundary
conditions
as
2
gonal
for
a
spin-liquid
phase
is
the
J
second-neighbor
couplings
on
the
square
lattice.
However,
%
J
model
to
a
critical
exponent
"
!
0,
in
agreement
with
Ising
Spiral
order
will
therefore
appear
as
sinusoidal
tersburg Nuclear Physics Institute, NRC Kurchatov Institute,
St. Petersburg 188300, Russia
1 Gatchina,
3
also suggest thatdependiscr
Frustrated
antiferromagnets
recently
attracted
simplest
helimagnet
square
lattice
with
anPetersburg,
extra
competing
square
lattice.
We
set
J1 =onJaState
=University,
1 for
simplicity,
and the
value this
2
wemodel
discuss
below
in detail.
Standard
Metropolis
algorithm33havedence
X St.
bX
of n on r. The action for n is the conventional
has
only
collinear,
commensurate
spin
correlaDepartment
of Physics,
St. Petersburg
198504
Russia
peting
universality.
^ i ' S^axis
^ !along
role near
much interest
in connectionwithin
with
the possibility
of sigma
stabilizS
S^ i described
(aJanuary
Jthat
' S^published
(1)
H
J1 received
exchange
one
is
the
O(3)
nonlinear
model, expanded
to other
includequantum
quartic
3
j;
Low-Temperature
Broken-Symmetry
Phases
of
Spiral
Antiferromagnets
Received
7 November
2011;coupling
revisedinteraction
manuscript
27
2012;
3 by
May
2012) tions,
of the
extra
exchange
J2 jis
variable.
The
Lifshitz
has and
been
The the
thermalization
was
maintained
this used.
makes both
classical and quantum
theory
elong
Broken
discrete
ing
unconventional
low-temperature
(T)
phases,
with
novel
This
critical
behavior
can
be
directly
related
to
the
hi;ji
hhi;jii
Hamiltonian
gradient
terms
(
h
!
!
k
!
lattice
spacing
!
1):
S n sym
!
5
B
quite
fromCarlo
that considered
here. As
will become
corresponds
J2 =critical
1/4 properties
in thisofnotation.
The system
4 ×different
10 Monte
steps in each
simulation.
Averages
have
model
1,2
R1=T 2,3 R 2
carrypoint
out Monte
Carlo simulationsto
to discuss
a classical two-dimensional
XY frustrated
FIG.
4.
Bot
Luca
Capriotti
and
Subir
Sachdev
[7,8]
in
the
context
of
types
of
‘‘quantum
order’’
[1].
A
very
promising
candidate
!
d rLn with
reflection
symmetry
by
studying
an appro9gnet on a square lattice.
0 d%
6 lattice
^ itwo
clear
below,
the spiral
order3.6
andbroken
associated
Lifshitz
point
1within
Weantiferromagnetic
find
successive
phase
transitions
upon
the
temperature
decreasing:
the
first
where
S
are
spin-S
operators
on
a
square
lattice
and
The
has a collinear
ground
state
at
J
<
1/4.
To
been
calculated
×
10
steps
for
ordinary
points
and
2
Valuation
Risk
Group,
Credit
Suisse
First
Boston
(Europe)
Ltd.,
One
Cabot
Square,
London
E14
4QJ,
United
Kingdom
Eq.
(2)]
for
H of
= a discrete
(J1 Z
cos(ϕ
) +second
J2 cos(ϕ
− ϕ Berezinskii-Kosterlitz)
fortheory
a spin-liquid
phase
is the
J1 % J3 model
second-neighbor coupld
x − ϕand
x+athe
2structurepriate
play
a
central
role
in
the
of
our
and
in
the
T
one isxof the x+2a
associated
with breaking
Ising
nematic
order
parameter.
From
the
symmetries
6
2 symmetry
Kavli
Institute
for
Theoretical
Physics,
University
of
California,
Santa
Barbara,
California
93106-4030,
USA
J
;
J
)
0
are
the
nearestand
third-neighbor
antiferrospin1 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
3
model
has only
co
x symmetry breaks. Thus, a narrow region exists on the phase diagram
ss (BKT) type at which
the
SO(2)
Department
of
Physics,
Yale
University,
P.O.
208120,
USA
dependence
of
observables.
^
^
^
^
^
magnetic
couplings
along
the
two
coordinate
axes.
For
S
S
'
S
(
J
'
S
;
(1)
H
!
J
of
Fig.
2,
we
deduce
that
the
order
parameter
is
#
!
in
3D
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
2 spin liquid.
n lines of the Ising and the BKT
that
to a chiral
P
− Jtransitions
− corresponds
ϕx+b
(1)
b cos(ϕ
this model,
early xlarge
N )),
computations,
[2] and recent
hi;ji
hhi;jii
1=M"
# with
from thd
We study
Heisenberg
with nearest- (J1 ) and third- (J3 ) neighbor exchange onquite
the different
Fig.
1). The
ground
has a helical
at J2 >group
1/4.
a #aantiferromagnets
temperature
11
large
scalestate
density
matrix ordering
renormalization
1
0.1103/PhysRevB.85.174404
PACS number(s): 64.60.De,
75.30.Kz
ent.
!
J
,
with
square
lattice.
In
the
limit
of
spin
S
!
1,
there
is
a
zero
temperature
(T)
Lifshitz
point
at
J
3
1
where
the sum runs over sites x = (x ,x ) of the lattice, a =
4
clear below,
where
S^ i atare
operators
on classical
a square
0:5. the
Thespiral
inse
1.2lattice and
1
(DMRG) calculations for S ! 1=2 a[3]bhave suggested the
present
Monte
Carlo simulations and a
long-range spiral
spin order
T !spin-S
0 for J3 >
4 J1 . We
rature
^
^
^
^
play
a
central
role
in
(1,0)existence
and b = of
(0,1)
are
unit
vectors
of
the
lattice,
the
coupling
#
!
"
S
(
S
&
S
(
S
#
;
(2)
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
sponding totht
a gapped spin-liquid
state with exponentially
1 3
1
2
T
16
17
eI.first
INTRODUCTION
#
$J
%
J
&S
.
The
transition
at
T
!
T
is
there
is
a
broken
lattice
reflection
symmetry
for
0
"
T
<
T
dependence
of observa
are
positive.
Using
arguments
of
Ref.
30,
they
constants
J
model,
Coulomb
gas system
of half-integer charges,
1 axes. For
c
3
c
1,2 spin correlations
4 1
magnetic couplings along the two
coordinate
decaying
and nothe
broken
translation
sym18
13,19,20
consistent
with
Ising
universality.
We
also
discuss
the
quantum
phase
diagram
for
finite
S.
28
s
the
two coupledthe
XY vertices
models, are
and the
concluded
atrecent
low oftemperatures
bound21model,
large N computations, [2] and recent
ets have attracted
much
in
metryattention
in that
the regime
strong
frustration
(J3 =J1 ’ Ising-XY
0:5).
ising
ξspin~ S / T 1/2 this model, early
generalized
fully
frustrated
XY
model.
And
surely,
the
most
en
by
T
DOI:
10.1103/PhysRevLett.93.257206
PACS0.8numbers:group
75.10.Hk, 75.10.Jm
bywhich
strings,
which
would
inhibit
the
BKT
transition
and
make
liquid phases,
have
been
found
in
c
large
scale
density
matrix renormalization
This Letter will describe
properties
ofisthe
above
model XY model (FFXY )
257206-2
famous
of
them
the
fully
frustrated
0.3
1
chiral
of special interest.
A
chiral
spin-liquid
the Ising
transition
firstconsequences
with
the
temperature
increasing.
(DMRG)
calculations
for
S
!
1=2
[3]
have
suggested
the
22
for large
S andoccur
discuss
for
general
S.
Our
introduced by Villain. This model is of great interest because
0.25
0.6
ehof the
such an exotic
state ofnoticed
matter
in
31which
cS2 /of
T a gapped spin-liquid state with exponentially
c'S2 / T
Kolezhuk
that
those
arguments
aresimulations
not validarray
for ofa Josephson junctions
results,
obtained
by classical
Monte aCarlo
and
existence
it
describes
superconducting
0.2
ξ
~
e
ξspin~ e Frustrated antiferromagnets
spin
Ising
nematic
order
uasi-long-range
nor long-range magnetic
also suggest
that discrete lattice symmetriesTmay play a
have recently
attracted
23
stems
a
theory
described
below,
are
summarized
in
Fig.
1
for
the
an external
transverse
magnetic field. It was foundmuch interest in connection
decaying
spin correlations
nonear
broken
translation
sym- 0.15
helimagnet,
theKT
Ising
transition
temperature
role
other quantum
with the possibility
of stabiliz- and
⟩ is showed
nonzero. thatunder
l order parameter
⟨Si × Sjand
0.4 critical points with spiral order [6].
ya (see
that
the
temperature
of
the
Ising
transition
T
is
1%–3%
larger
0.1
limit
S
!
1.
There
is
a
T
!
0
state
with
long-range
spiral
I
metry
in
the
regime
of
strong
frustration
(J
=J
’
0:5).
is
larger
than
the
BKT
one
at
least
near
the
Lifshitz
point
Broken
discussed
ing unconventional
(T)Jphases, with novel
phase is discussed in context of oneP H Ydiscrete
S I C3Asymmetries
L1 R E V I Ehave
W also
L E Tbeen
TER
S
J1 /low-temperature
4
1
3
PRL
93,
257206
(2004)
than
that
of
the
BKT
transition
for
most
of
above-named
0.05
2
J
-J
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
2
This
Letter
will
describe
properties
of
the
above
model
3
1
c
ted1 quantum
systems,
and
it
is
Jmagnetic
=
J
/4.
It
was
found
by
Monte
Carlo
simulations
in
the
1
2
Neel LRO
4
2
1
0.2
11,13,23,24
Spiral
LRO
point
0
for a Lifshitz
spin-liquid
phase is the
second-neighbor
couplings S.
on the
However,
% J3 model
12
ly in Ref. 3.
state there is a phase
$J3 % 14 J1 &S2 , above this systems.
0.25 0.3
0.35 0.4
forJ1large
S and
discuss3 consequences
for3 general
Oursquare lattice.
25 with broken
Korshunov
argued
that
a
phase
transition,
driven
by
un4
4
X
X
this
model
has
only
collinear,
commensurate
spin
correlaat
T=0
only
sions, one of thediscrete
systems symmetry
in which the
c'S2 / T
of lattice
reflections
about
the
xonand
ydomain walls associated
Monte
Carlo
simulations
and1 and1.510quantum
S^ i ' S^ j ; by classical
' S^ j ( J3 obtained
(1)
H^ ! J1 S^ i results,
0
ξspin
174404-1
©2012
American
Physical
Society
binding
of
kink-antikink
pairs
the
tions,
and
this
makes
both
the
theory
^
0
0.5classical
2
2.5 ~ e3
FIG.
1.
Phase
diagram
of
H
in
the
limit
S
!
1.
The
shaded
hase can be found at
finitewhile
temperature
a
hi;ji
axes,
spin isrotation
invariance
is preserved.
Thisin models similar to 2D
a theoryhhi;jii
described below, are quite
summarized
in Fig.
1 for the here.J3As
/J 1 will become
with
the Z2 symmetry,
can take place
different from
that considered
region has a broken symmetry of lattice reflections about the x
8
Y ) helimagnet with Z2 ⊗ SO(2) symmetry
phase has ‘‘Ising nematic’’
order.
We
present
strong
nubelow,
spiral order spiral
and associated Lifshitz point
FFXY one at temperatures appreciably smaller
TBKT
(seewhere
limit
Son!atransition
1.
There
T ! 0clear
state
withthelong-range
S^ i are
spin-Sorder.
operators
square
lattice
and than
y axes,
leading
to Ising
nematic
The Ising
is1is aand
cal structure results from a competition
FIG.
5. in
Critical
temperature
a 6function
theT frustration
play a central
the structure
of ourastheory
and inofthe
Ref. 26).atSuch
could
to athe
decoupling
ofJ1 ; JT
merical spins.
evidence
that thealso
transition
Tc aistransition
indeed in
the lead at
2 spin
0 $J
are %the1 Jnearestandorder
third-neighbor
for J31>antiferrothatrole
3 c)#
temperature
ctions between localized
Critical
Neel LRO
1 ratio
2Jat=J0. < T < Tc #
4 J12. We establish
3
4 1 &S . The spin correlation
dependence
of
observables.
3
1
phase
coherence
across
domain
boundaries,
producing
in
this
Lif
1
magnetic
couplings
along
the
two
coordinate
axes.
For
2
Ising
universality
class. Such Ising nematic order[4] was
stems from this class
is described
by two
%dependencies
J1 &S , above
is a phase with broken
4
length, !spin
the3 T
as this state there
27 , is finite for all T > 0, with $J
4
(Q,Q)
(Q,−Q)
way
two
separate
bulk
transitions
with
T
<
T
.
It
was
this
model,
early
large
N
computations,
[2]
and
recent
BKT
I
originallymagnetization
proposed in Ref. [2] for S ! 1=2 in a T ! 0
Aside from the conventional
shown,
with c=2 large
! c0 !scale
8"jJ3density
% 14 J1 j; matrix
the
crossovers
betweenofgroup
discrete
symmetry
lattice reflections about the x and y
renormalization
pointed
out,
however,
in
Ref.
25
that
these
two
continuous
2
257
FIG.
1. Phase diagram
phase also
described by a Z2 gauge theory [5]. Thus
etry, one has to spin-liquid
take into account
the
different
behaviors
of
!
are
at
the
dashed
lines
at
T
#
FIG.
2.
The
two
different
minimum
energy
configurations
with
axes,
while
spin
rotation
invariance
is
preserved.
This
(DMRG)
calculations
for
S
!
1=2
[3]
have
suggested
the
spin
transitions
can
merge
into
a
single
first-order
one.
These
con′
rameter FIG.
that is1.an(Color
Ising
variable
Z2 diagram
region
has a broken sym
0
1
the
same
Isingwith
nematic
order
canofappear
when
spiral
spinis form
23. (Color
online)
Phase
thedepend
model
(1)particular
that
FIG.
online)
Distribution
of
the
in We
"Q; &Q# strong
with
magnetic
wave
vectors
! "Q;order.
Q# and
Q~ ? !present
ofsymmetry
a gapped
spin-liquid
state
with
jJof
rotation
is phase
broken
only
atvalue
Texponentially
!
0EaQ~defined
0
0.1
0.2
0.3
0.4
0.5
clusions do
not
on the
interactions
inexistence
has
‘‘Ising
nematic’’
nu3%
4 J1 jS . Spin
T
rameter characterizes
the
direction
of
the
and
y
axes,
leadingT/Jto Isin
Q
!broken
2!=3,
corresponding
to J3 =J1 ! 0:5.
order
is destroyed either the
by system
thermal
fluctuations
(as in the degeneracy
decaying
spin
correlations
and
no
translation
sym=
0.5,
T
=
0.67
<
T
,
and
different
L.
found
in
this
paper.
Eq.
(10)
for
J
where
!
!
1.
There
is
no
Lifshitz
point
at
finite
S
because
it
as
soon
as
the
ground-state
remains
the
1
2
I
spin
merical evidence that the transition at Tc is indeed in the1/2 at the temperature T #
stinguishes left-handed
andLetter;
right-handed
in the regime
of strong
frustration
(J3 =J
c
present
see Fig. 1)same.
or byThey
quantum
fluctuations
(as in studies
1 ’ 0:5).
are confirmed
by numerical
of the models
ξspin~ Swas
/T
is preempted
[13]metry
by quantum
effects
within
the
dotted
semiIsing
universality
class.
Such
Ising states
nematic
order[4]
Tc ,ofisthe
FIG.
3.
T
dependence
specific
This
Letter
will
describe
properties
of
the
above
model
11,13–17,23,24
length,
!
finite
for
We
begin
by
recalling
[9]
the
ground
of
H
at
S
!
spin
Ref. [2]). Our large S results
are therefore
mentioned
above. consonant with
circle: here there is a T ! 0 spin gap ! #
S exp$%~
cS&proposed
and spin in Ref. [2] for S ! 1=2 in a T !
onal (3D) helimagnets, the phase transiDifferent
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 !
Nevertheless,
the
situation
remains
contradictory
in
2D
a spin-liquid phase at S ! 1=2 as derotation symmetryresults,
is preserved.
semicircular
region
extends
between L ! 24
cS and
/ T L ! 120. Data for J
tic and the chiral the
orderpossibility
parameters of
occur
c'S /1T
1Z gauge
obtainedThis
by classical
Monte
Carlo
simulations
and
~
spin-liquid
phase
described
by
a
theory
[5].
Thus
ξ
~
e
ξ
~
e
vector
Q
!
"!;
!#
for
J
=J
$
.
For
J
=J
>
,
the
1
2
spin
3
1
3
1
spin
to the
same
⊗ SO(2)
class
as 1 jS # !. Further details on the physics within
the different
behaviors
o
4
4
scribed
Refs.
[2,3]; wehelimagnets
will discussbelonging
the quantum
finite
S Z2 over
T # jJ
for comparison
(full dots and
dashed line)
3 % 4J
was found numerically
thatinthe
transition
a theory described below, the
are summarized
Fig.
1 forincommensurate
the
groundIsing
stateinhas
planar
antiferromagnetic
1 specific
same
nematic
order
can
appear
when
spiral
spin
2 . Spin
the
FFXY
model
and
the
antiferromagnet
on
the
triangular
the
maximum
of
the
heat.
4,5
jJ
%
J
jS
rotati
phase diagram further towards the end of the Letter.
We
this region appearlimit
at the
the isLetter.
3
order or of the “almost-second-order”
S !end
1. of
There
a T ! 0 state with long-range spiral
4 1
J1 /the
4
order
at a wave vector
Q~ !by
"Q;thermal
Q#, with Q
decreasing from
lattice. Garel and Doniach28 (see also Ref. 29) considered thespin order for J > 1 J . We
order
is destroyed
either
fluctuations
(as in
3 ! 1. There i
where
!Jspin
establish
that
at
0
<
T
<
T
#
erromagnets on a body-centered tetragonal
3
c
1
Neel LRO
4 1
!
as
J
=J
>
and
approaching
Q
!
!=2
monotonically
Spiral
LRO
simplest
helimagnet
on
a
square
lattice
with
an
extra
competing
3
1
Lifshitz
point
1
2
4with
present
Fig.
1) or by quantum fluctuations
(as
in
there Letter;
is a phasesee
broken
$J3 % 4 J1 &S , above this state
mple cubic lattice with an extra competing
where
a
labels
each
plaquette
of qu
th
is preempted
[13] by
1. The spiral order is incommensurate for 14 <
for J3 =J1 !
exchange
coupling
along
one
axis
that
is
described
by
the
257206-1
©
2004
The
American
Physical
Society
7 0031-9007=04=93(25)=257206(4)$22.50
Ref.
[2]).
Our
large
S
results
are
therefore
consonant
with
"1;
2;
3;
4#
are
its
corners.
The
variabl
along one axis. These systems belong
discrete symmetry of lattice reflections about the x and y
circle: here there is a T !
where
Q !diagram
2!=3, corJ3 =J1 < 1, except at J3 =J1 ! 0:5
1. Phase
of H^ in the
limit
S ! 1. The shaded
Hamiltonian
Néel
antiferromagnet,
while they
axes, while spin rotationthe
invariance
is preserved.
This % FIG.phase
do)universality class as, e.g., the model
possibility
of
a
spin-liquid
at
S
!
1=2
as
derotation
symmetry
is assu
pres
hasspins
a broken
about
the x
responding to an angle of 120 region
between
(seesymmetry
Fig. 2). of lattice reflections
!
8
9
the
two
degenerate
ground
states
i
phase has ‘‘Ising nematic’’
order.
We
present
strong
nu1
ular lattice and V2,2 Stiefel model. The
and
y axes,
leading
to there
Ising nematic
order.
IsingTtransition
is 4 J1 jS # !
scribed
in Refs.for
[2,3];
we will
discuss
the
quantum
finite
S Theover
# jJ3 %
~!
H =
(J1 cos(ϕx − ϕx+a ) + J2 cos(ϕx − ϕx+2a )
Interestingly,
each
spiral
state
with
Q
"Q;
Q#
is
Consequently,
a
phase
with
Ising
nem
merical evidence that the transition at Tc is indeed in the
2
ence and stabilization of the chiral spin$J % 1 JWe
spin correlation
at the
Tc #
1 &S . The
x
phase
diagram
further
towards
the temperature
end
Letter.
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
., Dzyaloshinsky-Moria interaction in 3D
ussed recently in Ref. 10.
ns (2D), the situation is rather different.11
sitions were observed with the temperature
iral order appears as a result of the first
of the Ising type. Another one is the
1
3
σ
Emergent Ising order parameters
j
j
Tc
T
T
C
C max
15
10
5
1
2
− Jb cos(ϕx − ϕx+b )),
2
length, !spin , is finite for all T > 0, with the T dependencies as
(1)originally proposed in Ref. [2](for
be obtained from the one
numerical results contain str
forQS!!!).
1=2This
in astate
T !cannot
0
shown, with c=2 ! c0 ! 8"jJ3 % 1 JOur
1 j; the crossovers between
~ byThus
wave vector
a global spin rotation. Instead, the
spin-liquid phase described by awith
Z gauge
theory Q
[5].
4
continuous
Ising phase transition betw
2
the different behaviors of !spin are at the dashed lines at T #
where the sum runs over sites x = (xa ,xb ) of the lattice, a =the same Ising nematic order
0031-9007=04=93(25)=257206(4)$22.50
257206-1
© 20
with
h#i ! 0, and a homogeneous
1
twoappear
configurations
arespin
connected
by
a 2 global rotation
can
when spiral
jJ
3 % 4 J1 jS . Spin rotation symmetry is broken only at T ! 0
(1,0) and b = (0,1) are unit vectors of the lattice, the couplingorder 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
constants J1,2 are positive. Using arguments of Ref. 30, theypresent Letter; see Fig. 1) or by global
accompanied
by
a
divergence
in
the
symmetry
of the
classical
ground state
' effects within the dotted semiquantum
fluctuations
(as in
is preempted
[13] isbyO"3#
quantum
concluded28 that at low temperatures the vertices are bound
Ising nematic in collinear spin system
~1 · N
~ 2 = ±1
=N
this talk:
Emergent Ising orders in quantum two-dimensional
triangular antiferromagnet at T=0
hsat H
H=
X
hi,ji
D
~i · S
~j
Jij S
U(1)*U(1)
U(1)*U(1)*Z2
U(1)*Z2
U(1)*Z3
b
hc2 G
Z3
hc1 F
40 ⇣ J J 0 ⌘2
=
S
3
J
Z3*Z2
B
U(1)*Z3*Z2
C2
C
C1
A
a
U(1)*Z3
U(1)*U(1)
O
0
U(1)*Z2
U(1)*U(1)*Z2
1
E
3
δcr
4
δ
Spatially anisotropic model: classical vs quantum
J
H = ∑ J ijSi ⋅ S j − h ∑ S
〈 ij 〉
S=1
′
J ̸= J
1
S=
2
0
1/3-plateau
J’
i
hsat
0
z
i
hsat
h
h
Umbrella state: !
favored classically;!
energy gain (J-J’)2/J
Planar states: favored by !
quantum fluctuations;!
energy gain J/S
The competition is controlled by
′ 2
2
δ
=
S(J
−
J
)
/J
dimensionless parameter
h
X
hi,ji
hsat
hsat H
~i · S
~j
Jij S
h
H=
D
hsat
U(1)*U(1)
planar
hc1 F
?
b
B
Z3
A
U(1)*U(1)
0
C
C1
U(1)*Z2
a
U(1)*Z3
O
C2
1
E
3
δcr
4
40 ⇣ J J 0 ⌘2
δ= S
3
J
0
1/3-plateau
hc2 G
cone
U(1)*Z3
0
h
hsat
0
Emergent Ising orders in quantum two-dimensional
triangular antiferromagnet at T=0
UUD-to-cone phase transition
Z3 ! U (1) ⇥ Z2 or Z3 ! smth else ! U (1) ⇥ Z2 ?
hsat H
H=
X
hi,ji
D
~i · S
~j
Jij S
U(1)*U(1)
U(1)*U(1)*Z2
U(1)*Z3
b
hc2 G
C2
Z3 (UUD)
hc1 F
40 ⇣ J J 0 ⌘2
=
S
3
J
Z3*Z2
B
C
C1
U(1)*Z2 (cone)
A
a
U(1)*Z3
U(1)*U(1)
O
0
U(1)*Z2
U(1)*U(1)*Z2
1
E
3
δcr
4
δ
Low-energy excitation spectra
✏d2
9Jk 2
= hc2 h +
4
for δ < 3
-k2
d2
Magnetization plateau is
collinear phase: preserves
O(2) rotations about magnetic field -no gapless spin waves.
Breaks only discrete Z3.
Hence, very stable.
vacuum of d1,2
hc2
40 S
=
(1
3
d1
✏d1 = h
3Jk 2
hc1 +
4
hc1
0.6
0.6
=
hsat =
(9JS)
2S
2S
J 0 /J)2
Bose-Einstein condensation
of d1 (d2) mode at k =0 leads to
lower (upper) co-planar phase
for δ < 1
Alicea, Chubukov, OS PRL 2009
Low-energy excitation spectra
near the plateau’s end-point
40 S
=
(1
3
J 0 /J)2 parameterizes anisotropy J’/J
extended symmetry:
4 gapless modes at the
plateau’s end-point
Out[24]=
d2
-k
-k22
+k2
Out[25]=
δ=4
k1 = k2 = k0
vacuum of d1,2
d1
-k0
+k0
k0 =
r
3
10S
S>>1
Out[19]=
=
40 S
(1
3
J 0 /J)2
Magnetization plateau is
collinear phase: preserves
O(2) rotations about magnetic field -no gapless spin waves.
Breaks only discrete Z3.
-k1
+k1
Alicea, Chubukov, OS PRL 2009
Bosonization of 2d interacting magnons
d†1,k0 +p d†2, k0 p d†1, k0 +q d†2,k0 q
k0 p d1, k0 +q d2,k0 q
}
⇣
3 X
=
(p, q) d†1,k0 +p d†2,
N p,q
}
(4)
Hd1 d2
( 3J)k02
(p, q) ⇠
|p||q|
}
†
1,p
†
1,p
2,q
⌘
+ h.c.
†
2,q
singular magnon interaction
magnon pair
operators
1,p = d1,k0 +p d2, k0
2,p = d1, k0 +p d2,k0
Out[25]=
p
p
1
2
2
1
Obey canonical Bose commutation relations in the UUD ground state
[
1,p ,
2,q ] =
1,2 p,q
⇣
⌘
1 + d†1,k0 +p d1,k0 +p + d†2,k0 +p d2,k0 +p !
†
†
In the UUD ground state hd1 d1 iuud = hd2 d2 iuud = 0
1,2 p,q
★ Interacting magnon Hamiltonian in terms of d1,2 bosons =
non-interacting Hamiltonian in terms of Ψ1,2 magnon pairs
Chubukov, OS PRL 2013
Two-magnon instability
Magnon pairs Ψ1,2 condense before single magnons d1,2
Equations of motion for Ψ - Hamiltonian h
†
1,p
h
†
2,p
`Superconducting’ solution with
imaginary order parameter
Instability = softening of twomagnon mode @ δcr = 4 - O(1/S2)
no single particle condensate
6Jfp2 3 X 2
f h
1,p i =
⌦p N q q
6Jfp2 3 X 2
f h
2,p i =
⌦p N q q
†
2,q
2,q i
†
1,q
1,q i
⌥
h 1,p i = h 2,p i ⇠ i 2
p
1 1 X
k0
p
1=
SN p
|p|2 + (1
/4)k02
hd1 i = hd2 i = 0
Chubukov, OS PRL 2013
hc2
Two-magnon condensate = Spin-current nematic state
distorted !
Υ<0
Υ>0
umbrella
uud
hc1
spin-!
current
J
J’
distorted !
umbrella
δcr 4
J’
domain wall
δ
no transverse magnetic order
0i
hS
·
S
hSx,y
i
=
0
r
r
r
is not affected
Finite scalar (and vector) chiralities. Sign of ⌥determines sense of spin-current circulation
hẑ · SA ⇥ SC i = hẑ · SC ⇥ SB i = hẑ · SB ⇥ SA i / ⌥
Spontaneously broken Z2 -- spatial inversion [in addition to broken Z3
inherited from the UUD state]
!
Leads to spontaneous generation of Dzyaloshisnkii-Moriya interaction
Chubukov, OS PRL 2013
Spontaneous generation of Dzyaloshinskii-Moriya interaction
(4)
Hd1 d2
1 X
p
/
N
(k
k2+k0
ik0
k0 )2 + (1
†
†
(d
d
1,k
2, k
/4)k02
d1,k d2,
k)
X
p2 k0
p
ik0
†
†
(d
d
1,p
2,
2
(p + k0 )2 + (1
/4)k0
continuum limit of DM in triangular lattice
X
ẑ · Sr ⇥ (Sr+a1 + Sr+a2 )
a1
p
d1,p d2,
B || z
r
a2
Mean-field approximation:
(4)
Hd1 d2
!D
X
k
k0
k0
†
†
(
+
)(d1,k d2,
|k k0 | |k + k0 |
k
d1,k d2,
D⇠⌥
spin currents appear due to spontaneously generated DM
(similar to Lauchli et al (PRL 2005) for Heis.+ring exchange model;
also ‘chiral Mott insulator’, Dhar et al, PRB 2013; Zaletel et al, 2013 )
k)
p)
PHYSICAL REVIEW B 87, 174501 (2013)
Chiral Mott insulator with staggered loop currents in the fully frustrated Bose-Hubbard model
Arya Dhar,1 Tapan Mishra,2 Maheswar Maji,3 R. V. Pai,4 Subroto Mukerjee,3,5 and Arun Paramekanti2,3,6,7
1
Indian Institute of Astrophysics, Bangalore 560 034, India
International Center for Theoretical Sciences (ICTS), Bangalore 560 012, India
3
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
4
Department of Physics, Goa University, Taleigao Plateau, Goa 403 206, India
5
Centre for Quantum Information and Quantum Computing, Indian Institute of Science, Bangalore 560 012, India
6
Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
7
Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8
(Received 9 July 2012; revised manuscript received 15 March 2013; published 3 May 2013)
2
Motivated by experiments on Josephson junction arrays in a magnetic field and ultracold interacting atoms in an
optical lattice in the presence of a “synthetic” orbital magnetic field, we study the “fully frustrated” Bose-Hubbard
model and quantum XY model with half a flux quantum per lattice plaquette. Using Monte Carlo simulations and
the density matrix renormalization group method, we show that these kinetically frustrated boson models admit
three phases at integer filling: a weakly interacting chiral superfluid phase with staggered loop currents which
spontaneously break time-reversal symmetry, a conventional Mott insulator at strong coupling, and a remarkable
“chiral Mott insulator” (CMI) with staggered loop currents sandwiched between them at intermediate correlation.
We discuss how the CMI state may be viewed as an exciton condensate or a vortex supersolid, study a Jastrow
variational wave function which captures its correlations, present results for the boson momentum distribution
across the phase diagram, and consider various experimental implications of our phase diagram. Finally, we
consider generalizations to a staggered flux Bose-Hubbard model and a two-dimensional (2D) version of the
CMI in weakly coupled ladders.
DOI: 10.1103/PhysRevB.87.174501
I. INTRODUCTION
The effect of frustration in generating unusual states of
matter such as fractional quantum Hall fluids or quantum spin
liquids is an important and recurring theme in the physics of
condensed matter systems.1,2 Recently, research in the field
of ultracold atomic gases has begun to explore this area,
gapped single particles;
but
spontaneously broken time-reversal
PACS number(s): 67.85.Hj, 03.75.Lm, 75.10.Jm
= spontaneous circulating
fluctuations which can “sense” the local flux on a plaquette.
In a recent paper, we have found numerical evidence for the
currents
existence of such a remarkable intermediate state in frustrated
two-leg ladders of bosons for the so-called fully frustrated
Bose-Hubbard (FFBH) model which has half a flux quantum
per plaquette. We call this state a “chiral Mott insulator” (CMI)
since it is fully gapped due to boson-boson interactions, exactly
Phases of a triangular-lattice antiferromagnet near saturation
hsat H
H=
X
hi,ji
~i · S
~j
Jij S
D U(1)*U(1)*Z2
U(1)*U(1)
U(1)*Z2
U(1)*Z3
b
hc2 G
C2
Z3
hc1 F
40 ⇣ J J 0 ⌘2
=
S
3
J
Z3*Z2
B
C
C1
A
a
U(1)*Z3
U(1)*U(1)
O
0
U(1)*Z2
U(1)*U(1)*Z2
1
E
3
δcr
4
δ
OS, Jin Wen, Andrey Chubukov, PRL 2014
High-field phases: from cone to incommensurate planar at h = hsat
fully polarized state
h
A
1
>
2
1
B
<
2
U(1)*U(1)
U(1)*Z3
~
~
hsat
incommensurate
planar
U(1)*Z2
cone
D
commensurate
planar (V)
U(1)*U(1)*Z2
double cone
C
plateau, Z3
0
J 0 /J
1
Low-density expansion
E0 /N =
=
(1)
1|
µ(|
2
1
2
+|
=
2|
2
)+
(0)
1
2
+
1 X ⇣ (J0 + 5Jk )2
=
16S
J0 Jk
k2BZ
1 (| 1 |
4
(1)
+|
2|
4
)+
2| 1|
9( J)2
=
J
2
|
2|
2
1.6J
S
(J0 4JQ+k )2 ⌘ 3J
+
⇡
JQ+k JQ
8S
1.6J
S
Phases of a triangular-lattice antiferromagnet near saturation
U (1) ⇥ Z3 ! U (1) ⇥ U (1)
Q = (4⇡/3, 0) ! Qi = incommensurate
hsat H
H=
X
hi,ji
~i · S
~j
Jij S
D U(1)*U(1)*Z2
U(1)*U(1)
U(1)*Z2
U(1)*Z3
b
hc2 G
C2
Z3
hc1 F
40 ⇣ J J 0 ⌘2
=
S
3
J
Z3*Z2
B
C
C1
A
a
U(1)*Z3
U(1)*U(1)
O
0
U(1)*Z2
U(1)*U(1)*Z2
1
E
3
δcr
4
δ
OS, Jin Wen, Andrey Chubukov, PRL 2014
High-field phases: commensurate-incommensurate transition
h
A
Q = (4⇡/3, 0) ! Qi = incommensurate
fully polarized state
B
hsat
U(1)*U(1)
U(1)*Z3
~
~
incommensurate
planar
U(1)*Z2
cone
D
commensurate
planar (V)
plateau, Z3
B
U(1)*U(1)*Z2
δ1
double cone
C
1
δ3
A
J0
δ2
A
0
J
C
J 0 /J
Low-density expansion -> classical sine-Gordon model of the (relative) phase fluctuations
e
i3Q·r
=1
first calculation of
3
3 X ⇣ (5Jk + J0 )(5JQ+k + J0 )JQ
=
32S 2
(J0 Jk )(J0 JQ+k )
k2BZ
k
(5Jk + J0 )(Jk + J0 ) ⌘
3J0
+
⇡
2(J0 Jk )
64S 2
0.69J
S2
Phases of a triangular-lattice antiferromagnet near saturation
U (1) ⇥ U (1) ! U (1) ⇥ Z2 or U (1) ⇥ U (1) ! smth else ! U (1) ⇥ Z2 ?
hsat H
H=
X
hi,ji
~i · S
~j
Jij S
D U(1)*U(1)*Z2
U(1)*U(1)
U(1)*Z2
U(1)*Z3
b
hc2 G
C2
Z3
hc1 F
40 ⇣ J J 0 ⌘2
=
S
3
J
Z3*Z2
B
C
C1
A
a
U(1)*Z3
U(1)*U(1)
O
0
U(1)*Z2
U(1)*U(1)*Z2
1
E
3
δcr
4
δ
OS, Jin Wen, Andrey Chubukov, PRL 2014
High-field phases: from cone to incommensurate planar at finite density
fully polarized state
h
A
>
1
2
1
B
<
2
U(1)*U(1)
U(1)*Z3
~
~
hsat
incommensurate
planar
U(1)*Z2
cone
D
commensurate
planar (V)
U(1)*U(1)*Z2
double cone
plateau, Z3
C
0
1
J 0 /J
Elementary excitation spectrum of the cone phase: Goldstone mode at k = +Q, gapped at
Transition at BD: softening of the mode at -Q’
-Q’ -Q
+Q
1.45(hsat h)
p
Q =Q+
hsat S
0
Q0 6=
Q
High-field phases, XXZ model: commensuratecommensurate transitions
HXXZ =
X
x x
J(Si Sj
+
y y
Si Sj )
+
z z
J z Si Sj
h
fully polarized state
B
C
A
~
~
U(1)*Z3
V state
U(1)*Z3
state
co-planar
c2
= 0.45/S
= 0.53/S
3
3J
=
2S
hsat
U(1)*Z2
cone
=1
c2
c1
c1
z
Sj
j
hi,ji
h
X
0.69J
S2
commensurate !
non-coplanar
3JS 2
E✓ =
(@x ✓)2 + S
4hsat
Jz /J?
µ3
3 3 cos[6✓]
hsat
Solid-Solid transition due to the sign change of Γ
OS, Jin Wen, Andrey Chubukov, PRL 2014
Conclusions!
!
Emergent Ising orders
!
Two-dimensional chiral spin-current phase Z3*Z2
!
High-field phases of triangular antiferromagnet U(1)*U(1)*Z2
“An exceedingly simple model leads to a surprising richness of phases and critical behavior. !
The underlying triangular lattice and the associated degeneracy play a crucial role in this physics.”!
HB2U, Alyosha!
two-dimensional
Schematic phase diagram for spin-1/2 triangular lattice AFM
h/J
0.8
fully polarized
4
plateau
width
0.6
0.4
0.2
3
0.0
0.0
IC planar
C planar
2
0.4
C planar
R 0.6
0.8
1.0
cone
SDW
1/3
1
0.2
plateau for
all J’/J
(crystal of spindowns; end-point
at J’=0)
SDW
IC planar
0
0.0
quasi-collinear
0.2
0.4
0.6
R=1-J’/J
0.8
1.0
spin-current state does not
apply directly to s=1/2 model
Compare with:
Cluster mean-field theory:!
direct 1st order transition!
between the UUD and the cone phases.!
!
Can there be an intermediate!
spin-current (chiral Mott) phase!
with broken Z3*Z2 ?
High-field phases, J-J’ model
Out[24]=
h
A
fully polarized state
B
2
1
-Q
Q
hsat
U(1)*U(1)
U(1)*Z3
~
~
incommensurate
planar
cone
D
commensurate
planar (V)
plateau, Z3
0
U(1)*Z2
B
U(1)*U(1)*Z2
δ1
double cone
C
A
1
J
δ3
A
J0
δ2
C
J 0 /J
OS, Jin Wen, Andrey Chubukov, PRL 2014