Download Spin-current and other unusual phases in magnetized triangular lattice antiferromagnets

Spin-current and other unusual
phases in magnetized triangular
lattice antiferromagnets
Oleg Starykh, University of Utah
collaborators:
Jason Alicea, Leon Balents, Andrey Chubukov, Zhihao Hao,
Ru Chen, Hyejin Ju, Hong-Chen Jiang
Instabilities of 1/3 magnetization plateau
Oleg Starykh, University of Utah
collaborators:
Jason Alicea, Leon Balents, Andrey Chubukov, Zhihao Hao,
Ru Chen, Hyejin Ju, Hong-Chen Jiang
Outline
• Context: interesting experiments and theories
• Magnetization curve
• classical antiferromagnet in a field: entropic
selection
• Quantum spins - plateau due to quantum
fluctuations
• Instability: two-particle condensation at large S
• Instability: transition to half-metal on lowering U
★ Conclusions
Experiments S=1/2
Ono et al 2005
Tsuji 2007
Susuki 2013
s=1/2, Ba3CoNb2O9
s=1, Ba3NiNb2O9
Shirata et al, PRL 2012
Hwang et al, PRL 2012
Experiments S >> 1
plateau
RbFe(MoO4)2:
S=5/2 Fe3+
Svistov et al PRB (2003)
Smirnov et al PRB (2007)
Phase diagram contains
only co-planar states
(while classically - only
non-planar cone state!)
Two antiferromagnetically coupled layers
Plateau width increases with T
plateau is signaled
by depression in
dM/dH
Theory: Magnetization plateau in one dimensional J1-J2 chain (zig-zag ladder)
Okunishi, Tonegawa JPSJ (2003)
Hikihara et al PRB (2010)
S=1/2
M=1/3 plateau
agrees with Oshikawa, Yamanaka, Affleck
argument (PRL 2007):
p S (1 - M) = integer
S=1,3/2
Heirich-Meisner et al PRB (2007)
p = period, S = spin,
M = magnetization:
M=1/3, p=3
possible for all S
Theory: Connections
with interacting bosons
• surprisingly complex phase diagram of spatially anisotropic
triangular lattice antiferromagnet
• connections with interacting boson system
– Superfluids (XY order)
– Mott insulators
– Supersolids
Andreev, Lifshitz 1969
Nikuni, Shiba 1995
Heidarian, Damle 2005
Wang et al 2009
Jiang et al 2009
Tay, Motrunich 2010
VOLUME 87, NUMBER 9
PHYSICAL REVIEW LETTERS
Magnetization Plateaus of SrCu2 !BO3 " 2 from a Chern-Simons Theory
theory
G. Misguich,1 Th. Jolicoeur,2 and S. M. Girvin3,4
1
Service de Physique Théorique, CE Saclay, 91191 Gif-sur-Yvette, France
Département de Physique, LPMC-ENS, 24 rue Lhomond, 75005 Paris, France
3
Department of Physics, Indiana University, Bloomington, Indiana 47405-7105
4
Institute for Theoretical Physics, UCSB, Santa Barbara, California 93106-4030
(Received 21 February 2001; published 13 August 2001)
proposals of
liquid-like
plateau states
2
The antiferromagnetic Heisenberg model on the frustrated Shastry-Sutherland lattice is studied by a
mapping onto spinless fermions carrying one quantum of statistical flux. Using a mean-field approximation these fermions populate the bands of a generalized Hofstadter problem. Their filling leads to the
magnetization curve. For SrCu2 !BO3 "2 we reproduce plateaus at 1#3 and 1#4 of the saturation moment
and predict a new one at 1#2. Gaussian fluctuations of the gauge field are shown to be massive at these
plateau values.
PHYSICAL REVIEW B 75, 144411 !2007"
DOI: 10.1103/PhysRevLett.87.097203
Two-dimensional (2D) quantum spin systems that do
not
magnetically at zero temperature are currently
Critical spin liquid at 3 magnetization in a spin- 2 triangular order
antiferromagnet
a subject of great theoretical and experimental interest.
The recently discovered [1] compound SrCu2 !BO3 "2 is an
1
2
Jason Alicea and Matthew P. A. Fisher
1
1Physics Department, University of California, Santa Barbara, California
antiferromagnet
93106, USA (AF) with localized spins S ! 2 , a gap
the ground
state
[2], and the unique property that its
2
Kavli Institute for Theoretical Physics, University of California, Santa Barbara,above
California
93106,
USA
magnetization curve has plateaus at 1#3, 1#4, and 1#8 of
!Received 10 February 2007; revised manuscript received 23 February 2007; published 10 April 2007"
the full saturation moment [3]. The spin system may be
1
describedCsby
a 2D Heisenberg model on a square lattice
Although magnetically ordered at low temperatures, the spin- 2 triangular antiferromagnet
2CuCl4 exhibits
0
with
exchange
remarkable spin dynamics that strongly suggest proximity to a spin-liquid phase. Here we askconstant
whetherJ a and additional diagonal bonds
J imprint
on half on
of the
plaquettes (see inset of Fig. 1).
proximate spin liquid may also occur in an applied magnetic field, leaving a similar
the square
dynamical
This lattice was
1 studied many years ago by Shastry and
spin correlations of this material. Specifically, we explore a spatially anisotropic Heisenberg spin- 2 triangular
Sutherland [4] who noted that there is an exact eigenstate
1
a “critical”
spin-liquid
antiferromagnet at 3 magnetization from a dual vortex perspective, and indeed find
which
is obtained
by putting singlets on all diagonal J
phase described by quantum electrodynamics in !2 + 1"-dimensions with an emergent
SU!6"
symmetry.
bonds. This eigenstate isAthe ground state for a wide in0
number of nontrivial predictions follow for the dynamical spin structure factor in this
“algebraic
For Jliquid”
#J smaller than $0.7 [5], the system
terval
of J 0 #J.vortex
phase, which can be tested via inelastic neutron scattering. We also discuss how well-studied
“up-up-down”
has dimer long-range
order, and for larger J 0 it has convenmagnetization plateaus can be captured within our approach, and further predicttional
the existence
of a stable
Néel long-range
AF order. There may be additional
between such
as a plaquette singlet phase [6–8]
gapless solid phase in a weakly ordered up-up-down state. Finally, we predict phases
several in
anomalous
“roton”
but
they
are
apparently
not
minima in the excitation spectrum in the regime of lattice anisotropy where the canted Néel state appears. realized in SrCu2 !BO3 "2 where
J 0 #J is estimated to be smaller than 0.65 [9,10]. This dimer
ground 75.10.Jm,
state explains
the spin gap as seen in experiments.
DOI: 10.1103/PhysRevB.75.144411
PACS number!s":
75.40.Gb
However, the existence of plateaus has no immediate explanation since the simplicity of the ground state does not
extend toorder
the excited
states
peculiar lattice. In this
The quest for an unambiguous experimental realization of
Low-temperature magnetic
develops
in of
thethis
presence
Letterleading
we usetoa amean-field
quantum spin liquid remains a central pursuit in condensed
of a magnetic field as well,
rich phaseapproximation
diagram.15 with a ChernSimons
(CS)
field-theoretic
approach
to quantum magnets
Motivated in part by the observed zero-field phenomenology,
matter physics, despite a long history dating back to Andersuggested
some
time
ago
[11,12],
and
we obtain an excelwe will explore the following question here. Can spin-liquid
on’s suggestion that the nearest-neighbor Heisenberg trianlent quantitative fit of the magnetization curve [see Fig. 1]
phases appear at nonzero
magnetic field, which influence the
ular antiferromagnet may realize a “resonating valence
for realistic values of the exchange constants. As we dis1 Quite generally, the theoretical search
at intermediate
energies, just as it
dynamics of Cs2CuCl4cuss,
ond” ground state.
this may be evidence for unconventional character of
appears
to
be
the
case
in
the
absence
of
a field? Several
or models realizing such exotic quantum ground states has
the plateau ground states.
15
makefrom
thisthe
scenario
plausible
arefirst excited state
experimental features that
ocused primarily on frustrated magnets in zero magnetic
Starting
pure dimer
state, the
worth
noting.
First,
a
broad
temperature
range
characterized
eld. The central goal of this paper is to take a first step at
can be constructed by first breaking a singlet bond into
by
short-range
order
persists
to sizable
around J 0 , such a state
a triplet up
state.
Becausefields
of theofexchange
nalyzing the situation when a finite magnetic field is present
1
27 AUGUST 2001
1
PACS numbers: 75.10.Jm, 73.21. –b, 75.40.Cx, 75.50.Ee
forbidden at low orders in perturbation theory [2]. As a
consequence, the triplet band is very flat, a striking fact
observed by neutron scattering experiments [13]. Since
the triplets are very massive particles, it is natural to expect that they can crystallize at finite density, and it has
been proposed that the plateaus are Wigner crystals of
triplets [14–16]. There exists a spin model which is derived from the Shastry-Sutherland Hamiltonian [17] for
which the plateaus are demonstrated to originate from such
ordered states. However, in this model there are plateaus at
1#4, 1#2, and 3#4, and the overall shape of the magnetization curve is not in agreement with experiments. A closely
related physical picture is obtained by describing the magnetized triplets by hard-core bosons [15]. Then the repulsion may favor charge-density wave states that are among
the known insulating phases of the lattice Bose gas.
FIG. 1. Comparison between the magnetization curve of
Outline
• Context: interesting experiments and theories
• Magnetization curve
• classical antiferromagnet in a field: entropic
selection
• Quantum spins - plateau due to quantum
fluctuations
• Instability: two-particle condensation at large S
• Instability: transition to half-metal on lowering U
★ Conclusions
Magnetization curve and plateau
Kawamura, Miyashita: classical Heisenberg model+Monte Carlo, 1985
Chubukov, Golosov: quantum large-S, 1991
square
triangular
?
A. Honecker , J. Schulenburg and J. Richter (2004)
Classical isotropic triangular AFM in magnetic field
•
No field: spiral (120 degree) state
•
Magnetic field: accidental degeneracy
H=J
!
!i · S
!j −
S
!
i
i,j
!h · S
!i
!h #2
1 !"! !
Si −
H= J
2
3J
!
•
i∈!
exists at single h
value at T=0
!h
form the lowest-energy manifold
=
3J
!i1 + S
!i2 + S
!i3
all states with S
– 6 angles, 3 equations => 2 continuous angles (upto global U(1) rotation about h)
Planar
Umbrella (cone)
No plateau possible
Phase diagram
Finite T: minimize F = E - T S
Planar states have higher entropy!
Head, Griset, Alicea, OS 2010
Phase diagram of the classical model
why so stable?
Collinear UUD state does not
break U(1) symmetry
(rotations about field axis).
Z3 U(1)
para
Hence, no gapless Goldstone modes.
All spin excitations are gapped.
Z3
Z3 U(1)
Seabra, Momoi, Sindzingre, Shannon 2011
Gvozdikova, Melchy, Zhitomirsky 2010
14
Quantum fluctuations, S >> 1, T=0.
J’ = J: Quantum fluctuations select co-planar and collinear phases
UUD plateau is due to interactions between spin waves
hc2 - hc1 = (0.6/2S) hsat
Exp: M=1/3 magnetization plateau in Cs2CuBr4
★ Observed in Cs2CuBr4 (Ono 2004, Tsuji 2007)
J’/J = 0.75
S=1/2
J’
J
★ first observation of “up-up-down” state
in spin-1/2 triangular lattice antiferromagnet
★ total of 9 phases -- instead of 3 expected!
Important: the lattice is strongly anisotropic, J’/J = 0.75
Spatially anisotropic model: classical prediction fails
T=0
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
Our semiclassical approach: treat spatial anisotropy (J-J’)
as a perturbation to interacting spin waves
fully polarized state
V incommensurate: fan
h
V
commensurate
distorted
umbrella (2)
planar
hc2
UUD plateau
BEC k = 0
BEC k != 0
2 low-energy gapped modes
hc1
commensurate
planar
distorted
umbrella (1)
incommensurate
zero-field spiral
1
Cs2CuBr4
2
Alicea, Chubukov, Starykh PRL 102, 137201 (2009)
3
Cs2CuCl4
subject to significant DM
4
!
δ ~ S(1 - J’/J)2
“Exact” dilute boson calculation
incomm. planar
h
commensurate
planar
distorted
umbrella (2)
hc2
UUD plateau
hc1
BEC k != 0
2 low-energy gapped modes
planar
distorted
umbrella (1)
incommensurate
1
2
3
Variational wave function calculation
Tay, Motrunich PRB 2010
4
!
Outline
• Context: interesting experiments and theories
• Magnetization curve
• classical antiferromagnet in a field: entropic
selection
• Quantum spins - plateau due to quantum
fluctuations
• Instability: two-particle condensation at large S
• Instability: transition to half-metal on lowering U
★ Conclusions
S=1/2
Model
A
B
C
y=1
J’
y=3
Winding
vector
J
A
✤
y=2
y=1
B
Periodic boundary conditions
along y
in numerical studies
C
Hamiltonian
H=
uesday, June 5, 2012
X
x,y
h
0
JSx,y · Sx+1,y + J (Sx,y · Sx,y+1 + Sx,y · Sx
X
x,y
z
Sx,y
1,y+1 )
Cs2CuCl4: J’/J = 0.34
Cs2CuBr4: J’/J = 0.5-0.7
ey V. Chubukov, and Oleg A. Starykh
ornia Institute of Technology, Pasadena, CA 91125
University of Wisconsin, Madison, WI 53706
University of Utah, Salt Lake City, UT 84112
ted: September 22, 2008)
Isotropic case: quantum fluctuations select coplanar states (agrees with semi-classical S>> 1
ally anisotropic 2D triangular antiferromagnet in a magnetic field.
for calculations)
all fields, but we show that the quantum phase diagram is much
eau, two commensurate planar states, two incommensurate chiral
ate separating the two chiral phases. Our analysis sheds light on
0.6
spin-1/2 system Cs2 CuBr4 .
Fully polarized
Width of
4
Ms/3 plateau
0.4
0.2
3
IC planar
Ising Q≠(4π/3,0)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
isotropic case:
Chubukov+Golosov,
1991;
Alicea, Chubukov,
Starykh, 2008
Magnetic field h
d quanh
distorted
planar
Ising
cone umbrella (2)
ting or- Q=(4
π/3,0)
C planar
hc2
UUD preserves
um fluccone
2
UUD
plateau
U(1) symmetry.
s exemSDW
materi-Ms/3 plateau hc1
Gapped spin waves
(4π/3,0)
magnetsQ=
C planar
distorted
1 Ising
planar
umbrella (1)
nd weak
SDW
e latter, Ising QIC
≠(4π/3,0)
0
2
! ⇠ S(1 J /J)
oplanar
1
2
3
4
Interacting magnons perturbed by spatial anisotropy
0 a
ates in
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
FIG.
2:
Proposed
phase
diagram
for
the
anisotropic
nearestetization
R=1-J’/J
neighbor Heisenberg model near 1/3 magnetization (full field range
Plateau phase
Fully polarized
0.6
Width of
Ms/3 plateau
M/Ms=1/3 plateau:0.4
0.5
uud state
0.2
4
IC planar
Ising Q≠(4
0π/3,0)
3
Ms/3 plateau
0.8
R = 0.0
1.0
cone
SDW
R = 0.2
0
0.5
Ising Q≠(4π/3,0)
0.1
0.6
-0.25
C planar
Ising
0
0.0
0.4
cone
0.25
Q=(4π/3,0)
0.2
0.5
2
1
0.0
0.0
-0.25
Ising
Q=(4
π/3,0)
C planar
z
Sy
Magnetic field h
0.25
0.2
0.25
0
0.3
-0.25
SDWx
0
0.4
0.5
0.6
R=1-J’/J
0.7
120
0.8
0.9
1.0
R = 0.4
R = 1- J’/J
Weakly coupled chains
Fully polarized
4
0.6
Width of
Ms/3 plateau
0.4
Magnetic field h
0.2
IC planar
Ising Q≠(4π/3,0)
3
0.0
0.0
Ising
Q=(4
π/3,0)
C planar
0.2
0.4
0.6
0.8
1.0
cone
2
cone
SDW
Ms/3 plateau
weakly
coupled
chains
Q=(4π/3,0)
1
C planar
Ising
SDW
Ising QIC
≠(4π/3,0)
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
R=1-J’/J
0.7
0.8
0.9
J’ << J
1.0
R = 1- J’/J
Ideal J-J’ model in magnetic field
OS, Balents 2007
• Two important couplings for h>0
• Quantum phase transition between SDW and Cone states
Magnetic field relieves frustration!
dim 1/2πR2: 1 -> 2
dim 1+2πR2: 2 -> 3/2
spiral “cone” state
“collinear” SDW
kF ↓ − kF ↑ = 2δ = 2πM
• “Critical point”: 1+2πR2 = 1/2πR2
gives
at M = 0.3
0.4
Tc
cone
0.3
0.2
1
sdw
0.1
0.0
0.1
0.2
0.3
0.4
M
also: Kolezhuk, Vekua 2005
1/2
h/hsat
3-leg ladder: dominant SDW phase
Ru Chen, Hyejin Ju, Hongchen Jiang, OS, Leon Balents (2012)
Fully polarized
4
0.6
Width of
Ms/3 plateau
0.4
Magnetic field h
0.2
IC planar
Ising Q≠(4π/3,0)
3
0.0
0.0
Ising
Q=(4
π/3,0)
C planar
0.2
0.4
0.6
0.8
1.0
cone
2
cone
SDW
Ms/3 plateau
KT
Q=(4π/3,0)
1
C planar
Ising
SDW
Ising QIC
≠(4π/3,0)
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
R=1-J’/J
0.7
0.8
0.9
1.0
SDW phase
agrees with
analytical
predictions
for J’ << J
J-J’ model: magnetization plateaux via
commensurate locking of SDW
• “Collinear” SDW state locks to the lattice at low-T
-“irrelevant” (1d) umklapp terms become relevant once SDW order is present (when
commensurate): multiparticle umklapp scattering
-strongest locking is at M=1/3 Msat
Observed in Cs2CuBr4 (Ono 2004, Tsuji 07, Fortune 09)
• down-spins at the centers of hexagons
T
“collinear” SDW
polarized
momentum
locking
!
Ψ†R ΨL
"n
Cs2CuBr4 Fortune et al 2009
“cone”
0.9
uud
1
h/hsat
→ (π − 2δ)n = 2πm → 2M = 1 − 2m/n
n
3
4
5
5
6
m
1
1
1
2
1
2M
1/3
1/2 3/5 1/5
2/3
naively
thinking
1/3
2/3
Plateau more carefully
OS, Katsura, Balents PRB 2010
• Umklapp must respect triangular lattice symmetries
fy(x) ! fy(x + 1)
– translation along chain direction
– spatial inversion
Z
(n)
Humk
n
dx tn cos[ fy]
R
=Â
y
0
width ⇠ J /J
n
3
8
5
10
12
m
1
2
1
4
2
2M
1/3
1/2
3/5
1/5
2/3
2d)
R(p
⌘n2/(4(4pR2
same parity
condition
1))
large n leads to exponential
suppression
• Ladder: Kosterlitz-Thouless transition to the plateau state @ R=0.7±0.1 (J’/J = 0.3)
because umklapp operator becomes irrelevant for J’/J -> 0
2m ⌘
n
2d)/2
n = m (mod 2)
and
⇣
• n-th plateau width (in field)
R(p
fy(x) ! fy+1(x + 1/2)
fy(x) ! pR fy( x)
– translation along diagonal
⇣
1
M (n,m) = 1
2
Plateau endpoint (3-leg ladder)
• Ladder: Kosterlitz-Thouless transition to the plateau state @ R=0.7±0.1 (J’/J = 0.3)
because umklapp operator becomes irrelevant for J’/J -> 0
Fully polarized
4
0.6
Width of
Ms/3 plateau
0.4
Magnetic field h
0.2
IC planar
Ising Q≠(4π/3,0)
3
0.0
0.0
Ising
Q=(4
π/3,0)
C planar
0.2
0.4
0.6
0.8
1.0
cone
2
cone
SDW
Ms/3 plateau
KT
Q=(4π/3,0)
1
C planar
Ising
SDW
Ising QIC
≠(4π/3,0)
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
R=1-J’/J
0.7
0.8
0.9
1.0
Width of 1/3 Plateau
Boundaries of 1/3 Plateau
DMRG: From 3- to 6-leg ladder
2.1
R=0.0
R=0.2
R=0.4
R=0.6
1.8
1.5
1.2
(a) Ny=6
0.9
0.00
0.8
0.02
0.04 1/Nx 0.06
0.08
Ny=6
Ny=3
0.6
plateau expands
in a wider ladder:
stable phase in two
dimensions
0.4
0.2
0.0
0.0
(b)
0.2
0.4
R 0.6
0.8
1.0
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
R 0.6
C planar
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
special feature of s=1/2 model
S>>1
?
S=1/2
0.8
4
0.6
0.4
0.2
3
0.0
0.0
2
out of plateau transitions non-coplanar states
0.4
0.6
0.8
1.0
SDW
1
0
0.0
0.2
SDW
0.2
0.4
0.6
out of plateau transitions collinear SDW state
0.8
1-J’/J
1.0
Low-energy excitation spectra
near the plateau’s end-point
S>>1
Out[24]=
d2
-k
-k22
+k2
Out[25]=
δ=4
k1 = k2 = k0
d1
-k0
+k0
k0 =
r
3
10S
Out[19]=
-k1
+k1
A. Chubukov, OS, PRL 2013
Interaction between plateau’s eigen-modes
(4)
Hd1 d2
⇣
3 X
(p, q) d†1,k0 +p d†2,
=
N p,q
d†1,k0 +p d†2, k0 p d†1, k0 +q d†2,k0 q
k0 p d1, k0 +q d2,k0 q
†
1,p
2,q
†
1,p
†
2,q
!
k02
3J
|p||q|
⌘
+ h.c.
Out[25]=
magnon pairs
interact strongly
near the plateau’s
end-point δ=4
1/2,p
= d1,±k0 +p d2,⌥k0
(p, q) =
k0
3J p
p2 + (1
`Superconducting’ solution with
imaginary order parameter
no single particle condensate!
Instability = softening of twomagnon mode @ δcr = 4 - O(1/S2)
p
/4)k02
p
k0
q2
+ (1
/4)k02
⌥
h 1,p i = h 2,q i ⇠ i
|p|
hd1/2,k i = 0
k0
1 1 X
p
1=
SN p
|p|2 + (1
=
+
=
+
/4)k02
A. Chubukov, OS, PRL 2013
Two-magnon condensate = Spin-current state
hc2
uud
hc1
Υ<0
Υ>0
distorted
umbrella
spincurrent
J
J’
J’
distorted
umbrella
δcr 4
domain wall
δ
no transverse magnetic order
hSx,y
r i=0
Finite scalar (and vector) chiralities. Sign Y determines sense of spin-current circulation
hẑ · SA ⇥ SC i = hẑ · SC ⇥ SB i = hẑ · SB ⇥ SA i / ⌥
but
hSr · Sr0 i
is not affected
Spontaneously broken Z2 , in addition to broken Z3 inherited from the UUD state
A. 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 |
D⇠
2
⌥k0
ln(
4
4
k
d1,k d2,
)
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 )
k)
p)
“This could be the discovery of the century. Depending,
of course, on how far down it goes”
Magnetization plateau in itinerant electron systems
• Heisenberg = large-U Hubbard model
• What is the fate of M =1/3 plateau when U/t
is reduced?
simple mean-field analysis
of the UUD state
Zhihao Hao and OS, PRB 2013
Band structure and geometry, U=0
M=1/3
M=1/2
n" = 2/3
n" = 3/4
n# = 1/3
n# = 1/4
Brillouin zone of triangular lattice
van Hove points
n" + n# = 1
M/Msat = (n"
n# )
Z. Hao, OS, PRB 2013
Band structure and geometry, U=0
M=1/3
M=1/2
n" = 2/3
n" = 3/4
n# = 1/3
n# = 1/4
Strong-coupling instability.
Result = half-metal.
Magnetization plateau state.
Weak-coupling instability:
arbitrary weak e-e interaction gaps out
up-spins Fermi surface completely;
down-spins FS reconstructs but remains
gapless.
Result = half-metal, electric current is
carried by minority (down) spin electrons.
Magnetization plateau state.
Extensions
reconstructed Fermi surface
Other lattices: almost 1/2-filled square lattice
Conclusions
• Magnetization plateau are interesting states
– collinear
– commensurate
– highly stable
– allow access to novel magnetic phases
• collinear SDW (s=1/2)
• spin-current nematic (s > 1/2)
• interaction-induced half-metals (works for many
lattice geometries)
Thank you!