Download Magnon collapse near the Lifshitz point and Leon Balents, KITP, UCSB

Magnon collapse near the Lifshitz point and
multipolar phases of frustrated magnets
Oleg Starykh, University of Utah
Leon Balents, KITP, UCSB
Edward Parker, UCSB
New Phases and Emergent Phenomena in Correlated Materials
with Strong Spin-Orbit Coupling, KITP, July 24, 2015
Exotic ordered phases
spin
nematic
composite order parameter
ordered
states
quantum
spin
liquids
Usual
Exotic
Outline
• Motivation: usual unusual ordered states
- emergence of composite orders from competing
vs
interactions
• Nematic/SDW states in LiCuVO
4
✓ spin nematic: “magnon superconductor”
✓ collinear SDW: “magnon charge density wave”
• Volborthite kagome antiferromagnet
- experimental status - magnetization plateau
- UUD, SDW and more
✴ Field theory of the Lifshitz point
Magnon binding and pair
condensation route to nematic phases
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
High-field analysis: condensate of
bound magnon pairs
+
hS i = 0
+
+
hS S i =
6 0
Ferromagnetic J1 < 0 produces attraction in real space
quasi-1d
Chubukov 1991
Kecke et al 2007
Kuzian and Drechsler 2007
Hikihara et al 2008
Sudan et al 2009
Zhitomirsky and Tsunetsugu 2010
2d
Shannon, Momoi, Sindzingre PRL 2006
uples directly to the ferroelectricity.8
In this Rapid Communication we provide evidence for the
stence of yet a different instability of4 spiral states toward
n-multipolar phases. The basic idea is that many spinltipolar order parameters are finite in the magnetically
Letter
ered spiral state, but that under a suitable amount of fluctions the primary spin order is lost, while a spinltipolar order parameter survives. We demonstrate this
chanism based on the magnetic field phase diagram of a
1
totypical model, the frustrated S = 2 Heisenberg chain
h ferromagnetic nearest-neighbor and antiferromagnetic
xt nearest-neighbor interactions. Furthermore we show
t this instability provides a natural and unified understandof previously discovered two-dimensional spinltipolar phases.9,10
To be specific, we determine numerically the phase diam of the following Hamiltonian:
and an exactly known sin
diagram at finite field h
attention,15–17 triggered
dimensional cuprate helim
liar features of the finite
appearance of elemen
'Sz = 2 , 3 , 4 in certain J2 a
uted to the formation of bo
dominant spin-multipolar
detailed phase diagram is
We present our numeric
m / msat plane in Fig. 1. A
present. The low magneti
vector chiral phase !gray"
tion we confirm the pres
Luttinger liquid phases !re
extends up to J2 → (,16
m = 0+ in that limit. All t
estimates:as a function of
crossover
J1 = - 1.6 meV
relations
change from spi
J2 = 3.9 meV
spin-density
J5 = -0.4 meVwave !SDW"
One also expects a tiny in
the p = 3 phase,17 which w
LiCuVO : spin nematic?
rystal structure of LiCuVO4 . Cu-O chains separated by VO4 tetrahedra and
H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Si ,
z
he b direction. ∠ Cu-O-Cu ∼ 90◦ indicates the ferromagnetic interaction.
i
i
!1"
i
+ weak
interchain
coupling
J
…
5are
dtinger
we parameter.
set
J1 =9)−1,
J
"
0
in
the
following.
S
spin-1/2
2
i
Recent numerical studies exhibit magnetization
vs
erators
at site phase
i, while
denotes
uniform
magnetic
nd the quadrupole
in facthpersists
down the
to rather
low magnetic
emd. The magnetization is defined as m ª 1 / L% Sz. zWe
z
LiCuVO4 experiment: collinear SDW along B
Hagiwara, Svistov et al, 2011
Buttgen et al 2012, 2014
LiCuVO
longitudinal
SDW4
No spin-flip
scattering above
~ 9 Tesla:
longitudinal
SDW state
SF = spin flip, ΔS = 1"
NSF = no spin flip, ΔS = 0
T=0 schematic phase diagram of weakly coupled
nematic spin chains
Spin Nematic
SDW
1/2 - O(J’/J)
Sato, Hikihara, Momoi 2013; Starykh, Balents 2014
1/2
Fully Polarized
BEC physics
M
Cautionary remark: !
an impurity effect !
Cold reality
“Our results suggest that the theoretically predicted spin-nematic phase, if it exists in
LiCuVO4, can be established only within the narrow field range 40.5<H<41.4 T.”
• so far, extensive experimental evidence for
longitudinal SDW
• Spin Nematic phase is constrained to field
interval < 1 T right below the saturation field
(of the order 40 T)
Have to look elsewhere
Outline
• Motivation: usual unusual ordered states
- emergence of composite orders from competing
vs
interactions
• Nematic/SDW states in LiCuVO
4
✓ spin nematic: “magnon superconductor”
✓ collinear SDW: “magnon charge density wave”
• Volborthite kagome antiferromagnet
- experimental status - magnetization plateau
- UUD, SDW and more
✴ Field theory of the Lifshitz point
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
Huge 1/3 magnetization plateau !
How far in field does the plateau extend?
Meeting abstracts of the Physical Society of Japan 2014
Translation:!
== Intro ==!
!
- S=1/2 kagome antiferromagnet is a known candidate spin liquid. !
Under external field, five magnetic plateaus have been theoretically !
found in S=1/2 antiferromagnet: m=0, 1/9, 1/3, 5/9, and 7/9. !
Among them, first two are spin liquids, and the later three are long-range orders (Ref. 1).!
- Experimentally, a large 1/3 magnetization plateau was recently observed in Volborthite, !
which extends above 75T.!
!
== Method & Results ==!
!
- We study the magnetic phases of Volborthite in high-field, above 100T, !
using Faraday rotation.!
- The figure shows rotation angle observed at 11K using a laser of wave length 532nm; !
it is compared to magnetization curve at 1.3K. We used single crystal samples and !
the field is applied perpendicular to the ab plane.!
- The rotation angle show increase around 25T, where rapid increase in !
magnetization curve is observed. The rotation angle become nearly !
constant corresponding to the 1/3 magnetic plateau, which extends up to 120T.
this is the best plateau ever:
28 T < plateau < 120 T
Phase diagram
small plateau’s onset field of 27 Tesla,
relative to J ~ 100 K,
suggest the presence of ferromagnetic
exchange interactions
T
1/3
plateau
N?
1K
?
SDW
1T
FIG. 2 (color online). (a)
51
1/3
plateau
26T
B
may be a spin nematic??
V NMR spectra measured on a single-domain piece of a crystal in
H. Ishikawa et al, PRL 2015
spin nematic?
SDW
magnetic fields between 15 and 30 T applied perpendicular to the ab plane at T = 0.4 K. (b)
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?!
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)
Numerics: nematicity and
st
1 order seem connected?
PHYSICAL REVIEW B 8
SUDAN, LÜSCHER, AND LÄUCHLI
SDW
0.4
?
-0.5
-1
0.025
p=2
0.02
quadrupolar
octupolar
SDW (p=2)
SDW
(p=3)
(p=4)
(a)
0
0
-3
-2.5
J1/J2
-2
-1.5
-1
1st
2nd
order
FIG.order
1. !Color online" Phase
diagram
of the frustrated ferromag-
2d square lattice
0.015
VC
0.005
Vector Chiral Order
-3.5
VC
0.01
Sudan
et al, 2009
1
0.8
0.6
0.4
0.2
0
0.2
0
-4
-0.75
p=3
p=3
-0.4
p=4
0.6
m/msat
1d chain
-0.35
metamagnetic
0.8
hexadecupolar
p=4
-0.3
2
-0.275
κ
1
0.03
h/hsat
-0.25
J2/J1
R
netic chain !1" in the J1 / J2 vs m / msat plane. The gray low-m region
exhibits vector chiral long range order. The colored regions denote
spin-multipolar Luttinger liquids of bound states of p = 2 , 3 , 4 spin
flips. Close to saturation the dominant correlations are multipolar,
while below the dashed crossover lines, the dominant correlations
are of SDW!p" type. The tiny cyan colored region corresponds to an
incommensurate p = 2 phase. The white region denotes a metamagnetic jump. Finally the scribbled region close to the transition
J1 / J2 → −4 has not been studied here, but consists most likely of a
low field vector chiral phase, followed by a metamagnetic region
extending up to saturation magnetization. The inset shows the same
diagram in the J1 / J2 vs h / hsat plane.
study. Finally
the multipolar
Luttinger
liquids
are separated
Shannon,
Momoi,
Sindzingre
PRL
2006
from the vector chiral phase by a metamagnetic transition,
0.2
(b)
0.4
m/msat
0.6 0
0.1
0.2
m/msat
0.3 0
FIG. 2. !Color online" Squared vector chir
" $Eq. !2"% in the low magnetization phase f
J2 / 'J1' as a function of m / msat: !a" J2 / 'J1' = 0
and !c" J2 / 'J1' = 0.38. The order parameter van
lar Luttinger liquids.
2
multipolar phases at larger m seems to oc
metamagnetic behavior !cf. left and righ
For the parameter set in the middle panel
behavior, but it can’t be resolved based o
used.
Hamiltonian !1" presents unusual ele
$Sz ! 1 in some extended J2 / 'J1' and m d
is independent of the system size.15 Thi
been explained based on the formation
p = $Sz magnons in the completely satu
finite m / msat a description in terms of a
Luttinger liquid of bound states has bee
We have determined the extension of the
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
Quantum Lifshitz point
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)
It is time for a blackboard discussion
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 .
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
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 for unit vector m
Z
0
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,
hm̂z
To find the actual saddle
point,symmetry
we make an assumpBerry
two
tunes
tion that it is of the form of an umbrella state (I tried
m̂ @ m̂
m̂ @ m̂
A =
. phase(3)
also to look for a planar
state, but
it seemed to be
allowed
interactions
1 + m̂
QCP
energetically unfavorable). To avoid having to rescale,
4 Let m̂ =
we work in the original
variables
of
Eq.
(1).
important point for us is that Aterm
contains a
at
O(q
)
p
/
|Jqx,
4J21 ' ). Then the action is just the
(' cos
vative of imaginary time ⌧ .
1 |' sin qx,
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
2
integral of the energy density
p
1), (6)
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
"=
2
2
4
2
4
q ' + Kq ' + uq '
4
h(
1
'2
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
-1 < v < -1/4 derives from quantum fluctuations
Large S >>1: v ~ -3/(2S) < 0
S=1/2: v = -5/8
h
order
parameter discontinuity
p
(
4|v|
1)/|v| ⌧ 1 for v ⇡
1/4
FM
first order
spiral
hc =
8K
p
|v|(1
1<v<
local instability of FM state
(1-magnon condensation)
IC cone
0
2
1
4
p
|v|)
A natural speculation
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
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