The spin-1 kagome antiferromagnet

Transcription

The spin-1 kagome antiferromagnet
Wei Li
Arnold Sommerfeld Center For Theoretical Physics,
Physics Department, LMU Munich, Germany.
04. 09. 2014
Benasque
Samstag, 6. September 14
why spin-1 kagome antiferromagnet (KAF)?
✎ Spin-1/2 kagome antiferromagnets have been intensively studied.
5
S. Yan, D.A. Huse, & S.R. White, Spin
liquid ground state of the S=1/2 kagome
Heisenberg antiferromagnet. Science
332, 1173–1176 (2011)
4
Entropy S(Ly)
3
2
1
J2=0.10
J2=0.15
0
H.-C Jiang, Z.H. Wang & L. Balents, Nature
Physics 8, 902-905 (2012)
S. Depenbrock, et. al, PRL, 2012
-1
-ln2
0
2
4
6
Ly
8
10
12
Figure 3: The entanglement entropy S(Ly ) of the kagomé J1 -J2 model in Eq.(2
RESEARCH
4 ⇠ 12 at Lx = 1. By fitting S(LLETTER
, we get = 0.698(8) at J2
y ) = aLy
= 0.694(6) at J2 = 0.15. Inset: kagomé lattice with Lx = 12 and Ly = 8.
0
a
5
10
15
20
3
0
0.15
0.3
0.45
3
d
phase. Fortunately,
for a given D, there are only finitely many distinct topologica
2
2
T.-H. Han,
et.topological
al, Fractionalized
for small values of D, a complete classification
of all
phases is know
1
2
excitations in the spin-liquid state of
kagome-lattice antiferromagnet,
constraints0such as time-reversal symmetrya (if
present) further constrain the possib
Nature 492, 406–410 (2012)
1
order. Fore 3example, there are only two time-reversal invariant phases consistent
1
(–K, K, 0)
b
0
0
found20here2 for the kagomé Heisenberg model: the Z2 phase, and a doubled s
c 2
1
Herbertsmithite
0
–2
1
10
It will be interesting to develop methods to distinguish these in the future, and
0
–1
0
(H, H, 0)
1
2
0
–2
–1
0
1
2
H, 0)
the topological ground (H,state
splitting. Identifying topological order by combini
Figure 1 | Inelastic neutron scattering from the spin excitations, plotted in
reciprocal space. a–c, Measurements were made at T 5 1.6 K on a singlecrystal sample of ZnCu3(OD)6Cl2. The dynamic structure factor, Stot(Q, v), is
plotted for Bv 5 6 meV (a) and Bv 5 2 meV (b) with Ef 5 5.1 meV and
over 1 # Bv # 9 meV. e, Calculation of the equal-time structure factor,
Smag(Q), for a model of uncorrelated nearest-neighbour dimers. The intensity
corresponds to 1/8 of the total moment sum rule S(S 1 1) for the spins on the
kagome lattice. The data presented in a–c are expressed in barn sr21 eV21 per
classification results with numerical simulation is a major step in the developm
✎ Spin-1 KAF are less well studied....
3× 3
or
hexagon valence bond solid state?
[K. Hida, 2000]
Magnetically ordered?
True for large S limit!
[D.A. Huse et. al, PRB (1992)]
[C.L. Henley et. al, JMMM. (1995)]
Stable in spin-1 KAF [C. Xu et. al,
PRB (2007)]
or
fully trimerized crystal?
[Z. Cai, 2009]
Experimental studies of the spin-1 KAF
material
comments
(a) gapped antiferromagnet
(a) N. Wada, et. al, JPSJ (1997).
I
m-MPYNN·BF4
II
III
IV
reference
(b) T. Matsushita, et. al, JPSJ (2010).
(b) magnetization curve (0, 1/2 & 3/4
plateaus)
Ni3V2O8
G. Lawes, et. al, PRL (2004).
kgaome staircase
KV3Ge2O9
S. Hara, et. al, JPSJ (2012).
BaNi3(OH)2(VO4)2
D.E. Freedman, et. al, Chem.
Commun. (2012).
AF & F couplings
Experiments show:
(a) No magnetic susceptibility peak [I & III] ➙ no magnetic transition, no SO(3) symmetry
breaking
(b) Low-T susceptibility collapse to zero [I] ➙ gapped states
(c) Specific heat has a maximum around J/2 (J the AF coupling) [I] ➙ phase transition?
Numerical simulations of the spin-1 KAF was ... few
✎ Coupled cluster calculation [Götze et. al., PRB (2012)]
Long-range magnetic order appears only for S>1.
➜ No 3 × 3 magnetic order in spin-1 KAF!
Spin-1 KAF GS energy estimated as E0 = −1.4031
✎ Until this year:
H.J. Changlani and A.M. Läuchli, Ground state of the spin-1 antiferromagnet on the kagome
lattice, arXiv:1406.4767v1 (2014).
T. Liu, WL, A. Weichselbaum, J. von Delft, and G. Gu, Simplex valence-bond crystal in the spin-1
kagome Heisenberg antiferromagnet, arXiv:1406.5905 (2014).
T. Picot, D. Poilblanc, Nematic and supernematic phases in Kagome quantum antiferromagnets
under a magnetic field, arXiv:1406.7205 (2014).
The Resonating AKLT-loop State
✎ Topological resonating AKLT-loop states and its PEPS representation
✎ The RAL family and its variational energies
related paper:
WL, S. Yang, M. Cheng, Z.-X. Liu, and H.-H. Tu
Topology and criticality in the resonating Affleck-Kennedy-LiebTasaki loop spin liquid states
Phys. Rev. B 89, 174411 (2014).
Spin-1 system: Affleck-Kennedy-Lieb-Tasaki (AKLT) string state
S=1
S=1
S=1
virtual particle s=1/2
s=1/2
S=1
singlet pair
Bm,n
S=1
S=1
On-site tensor A
Ai
✎ AKLT-string state has a natural Matrix-Product State representation.
⊗N
s=1
AKLT = P
∏↑
2i−1
Matrix-Product State, D=2
↓2i − ↓2i−1 ↑2i
m=1
i
A
i=1
i labels site where physical spin-1 locates
⎡ 1 0 ⎤ m=−1 ⎡ 0 0 ⎤ m=0
2⎡ 0 1 ⎤
=⎢
=⎢
⎥ , Ai
⎥ , Ai =
⎢
⎥
0
0
0
1
2
1
0
⎣
⎦
⎣
⎦
⎣
⎦
B2i−1,2i
2⎡ 0 1 ⎤
=
⎢
⎥
2 ⎣ −1 0 ⎦
✎ Symmetry-Protected Topological Order: spinon carries spin-1/2, symmetry fractionalization
I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799
(1987); Commun. Math. Phys. 115, 477, (1988).
uids of spin-1/2 systems protected by space group symmetry, based on the projective symmetry group (PSG)
approach to study the variational ground state wave function obtained from slave-particle construction. The connection between the PSG property of ground states and
the physical quantum number of quasiparticles is not perfectly understood. Moreover, the distinction between
spin liquids protected by spatial symmetry is not robust
against disorder, whereas our study dealing with timereversal symmetry is.
Resonating valence loops (RVL): For simplicity,
we shall consider a spin-1 system on the honeycomb latα
biν , where σ α are
tice first. On each site i, Siα = 12 b†iµ σµν
Pauli matrices and biµ are Schwinger bosons with the
constraint b†i↑ bi↑ + b†i↓ bi↓ = 2. In the spirit of AKLT,
each of two bosons on each site can form a singlet bond
with another boson on its nearest neighbor site - each
spin-1 can participate two singlet bonds. A spin singlet
†
bond on the link !ij" is created by Bij
= "µν b†iµ b†jν ("µν is
the Levi-Civita symbol). For reasons which will be clear
later, we are interested in the following states:
" †
!
|c" ; |c" = (−1)nc
Bij |0" ,
|Ψ"RVL =
(1)
of
of
co
th
sin
qu
im
of non-intercepting loops and which touches every site
by one and only one loop and nc counts the number of
singlet bonds of the loop configuration c on those vertical links with upper site in sublattice A. |Ψ"RVL is
an equal weight superposition of all loop-covering configurations, which we propose to call “resonating valence
loop” (RVL) state. A typical loop-covering configuration is shown in Fig. 1. all
Forloop
the configs
honeycomb (or any
other trivalent) lattice, there is a one-to-one correspondence between loop-covering and dimer-covering config-
wh
tio
lar
po
is
th
th
Al
th
of
ze
spin-1 systems: Resonating AKLT-loop (RAL) State
project 1/2⊗1/2 => spin-1
singlet
1
( ↑,↓ −c ↓,↑ )
#ij$∈c
2 c is a loop-covering configuration
where
which consists
∑
RAL =
i
L
i
=N(i)
=N2(i)
5
6
1
6
3
1 2
a
i
2
=N1(i)
The singlets form a
closed loop called
AKLT loop.
5
4
a
(a)
i
=N3(i)
4
3
(b)
Hong
Yao,
and Xiao-Liang
Qi
FIG. 1: (a)
The Liang
schematicFu,
representation
of the honeycomb
lattice and a typical loop-covering configuration. The black
arXiv:1012.4470
(2010).
(white) sites are in A (B) sublattices.
Thick bond represent a
†
spin singlet created by Bij
(see text) and they forms a loopcovering on the honeycomb lattice. A hexagon plaquette is
flippable if it has three thick and three thin bonds. For in-
fir
sta
co
an
co
H
wh
of
sit
Fi
th
loo
gle
pa
cle
#a
S
all
po
co
low
is
y-direction
The spin-1 RAL on the kagome lattice
𝛽
α
P
0 ⊕ 1/2
T
G
𝜇
Loop config. on kagome lattice
x-direction
T
𝜇
T
T
0 ⊕ 1/2 ⊕ 1
The RAL states on the kagome lattices:
gapped spin liquid
0
correlation functions
10
10
10
10
iPEPS
Dc=54, spin spin
dimer dimer
quad quad
5
10
Z2 spin
liquid?
15
0
5
10
|i j|
15
20
✎ No magnetic, dimer, or quadrupolar order.
The kagome RAL states: the topological sectors
spinon line
“electric” Wilson loop
flux (vison) line
“magnetic” Wilson loop
0
0
spinon on the boundary
0
V
0
0
V=diag(1,-1,-1,1,1,1)
One can constructed 4 topological sectors on infinite cylinder, they are
labeled by {Pv=even/odd, Gv=±1}.
Variational energies of four topological sectors
0.95
NN correlator
1
Even, Gv=+1
Odd, Gv=+1
1.05
Even, Gv= 1
1.1
Odd, Gv= 1
1.15
iPEPS,
1.2
=−1.2158
0
1.25
1.3
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1/L
y
✎ Four constructed states are degenerate in the thermodynamic limit.
D. Poilblanc, et al, PRB 86, 014404 (2012)
D. Poilblanc, et al, PRB 87, 140407 (2013)
Topological Entanglement Entropy
SE = α L − γ + ...
von Nuemann entropy
6
5
4
Leading term, proportional
to entanglement surface.
3
2
1
Even sector, Gv=+1
0
Odd sector, Gv=+1
−0.83
1
0
1
2
3
L
4
5
6
sub-leading term, owing to
long range entanglement.
y
✎ 𝛾≈ln(2) verifies the existence of
ℤ2 topological order.
Alexei Kitaev and John Preskill, Phys. Rev. Lett. 96, 11404 (2006).
Michael Levin and Xiao-Gang Wen, Phys. Rev. Lett. 96, 110405 (2006).
Hong-Chen Jiang, ZhenghanWang and Leon Balents, Nat. Phys. 8, 902 (2012).
representation has a redundancy
which
be revealed
After removing
thiscan
redundancy,
we find that
virtual particle
⌫ is in
thetwo
SU(2)
representation
by the following procedure:
we group
the
virtual
and
has triangle
dimension(see
D =
particles belonging to 1the
same
e.g.6 (see
, ⌧ Fig.
in 3(b)),
D = into
9 from
a naivevirtual
counting
of the dimensi
Fig. 3(a)) and block them
a single
particle
product
(0 1/2)
1/2). Let us
(see ⌫ in Fig. 3(b)). tensor
It turns
out that
not ⌦
all(0virtual
this procedure
exact and
the removal FIG.
of re
degrees of freedom survive
after this is
blocking
procedure.
allows us to reduce computational costs inkagom
our
After removing this redundancy,
we
find
that
the
new
spin-1/2 dimer
based
calculations.
virtual particle ⌫ is in the SU(2) representation 0 1/2
Furthermore, in our framework it is straig
1 and has dimension D
= 6 (see
3(b)), instead
of
↑↓aFig.
− ↓↑
to introduce
one-parameter
family of PEPS,
D = 9 from a naive counting
the dimension
of the
terpolates of
between
the RAL and
the spin-1 RV
tensor product (0 1/2)
Let us the
note
that represent
This⌦is(0
done1/2).
by extending
virtual
this procedure is exactin and
removal
redundancy
Fig. the
3 (a)]
from 0 of1/2
to 0 1/2 1. Thu
allows us to reduce computational
inmodified
our numerical
tual bonds spin-1
incosts
(1) are
as |✏mix i = |0, 0i
dimer
p
calculations.
|2, 1i + (|3, 5i |4, 4i + |5, 3i)/ 3, where |3, 4, 5
+1,−1
−straightforward
0,0
+ spin-1
−1,+1space. Ac
three states
virtual
Furthermore, in ourthe
framework
itinisthe
the projector
P of
in PEPS,
(2) has to
be modified
as
to introduce a one-parameter
family
which
interpolates between the RAL and the spin-1 RVB states.
0
P
= (1 ↵)P +
This is done by extending the virtual representation
[ ,↵W,
⌧
in Fig. 3 (a)] from 0 1/2 to 0 1/2 1. Thus, the virThemodified
mixed
as+awhich
PEPS:
where
W as
isRAL
a|✏localistate
projector
maps one vir
tual bonds in (1) are
=
|0,
0i
|1,
2i
mix
p of the four virtual spin-1 particles
1 state, out
|2, 1i + (|3, 5i |4,bond
4i +physical
|5, 3i)/
3, where
|3, 4, 5i denotes
state
forspin-1
mix
RAL
Hilbert space. Its explicit form
the three states in the by
virtual spin-1 space. Accordingly,
= 0,0
+ 1,2
− 2,1 as
+ 4,6 − 5,5 + 6, 43
mix has
the projector P inε(2)
to be
modified
The mixed RAL states:
allow spin-1 dimer (L=2 loop) to appear
mixed RAL configuration
wn in Fig. 1 also becomes transparuration contains di↵erent patterns of
✎ mRAL
state interpolate
AKLT
loops covering
the wholethe
latRAL
spin-1the
RVBsame
states
ojector
in and
(2) gives
weight
s of combining
spin-1/2’s 𝛼).
into a
(controlledtwo
by parameter
e wave function (1) can be viewed as
erposition
(up14to a sign depending on
Samstag, 6. September
4
X
FIG.
the iP
l=1
tract
where W X
is a local
Xprojector which maps one virtual spinMPS
m
1 state,
spin-1 particles,
the
W1 = out of the fourCµvirtual
|mihµonto
|.
1 , µ2 , µ3 , µ4lation
1 ,0 µ2 ,0 µ3 ,0 µ4 ,0
physical spin-1
space. Its explicit form is given
funct
m µ1 µHilbert
2 µ3 µ4
by
boun
(6)
m
1
0 cludi
Here Cµ,0
store the trivial
CG
coefficients
C
=
C
=
3,0
4,0 i(j).
4
X
0
P
= (1
+ ↵W,as W =
where, for instance,
W1 ↵)P
is defined
Wl (4)
,
spin-1 kagome Heisenberg
model. Periodic
boundary condistates is observed
at ✓c ⇡ 0.04⇡.
ical on
tions are assumed in both
directions,
and
the
cluster
has
2
⇥
3
Model and Method.— We consider the quantum sp
Variational
Energies
of
the
RAL
states
lattice
unit cells.
KAH model with only nearest-neighbor isotropic excha
P
variational energy (per site)
the rea
interactions, whose Hamiltonian is H = hi, ji Si · S j ,
clarified
spin
operator
S
on
sites
i,
while
hi,
ji
stands
for
the
nea
i
−1.03
the top
sign convention
1,
Dc=24
neighbor link on a lattice. We use the projected entang
sign convention 1, Dc=32
entangl
−1.07
pair state
sign convention
2, Dc=24 (PEPS) as a wavefunction ansatz [26], and inv
conside
sign convention 2, Dc=32
an imaginary-time evolution (through the Trotter-Suzuki
ing bet
−1.11
composition [27]) to seek the best variational ground s
symme
The initial tensor network [Fig. 1(a)] is set up as a honeyc
−1.15
fied alo
lattice, with two kinds of tensors, T A and T B , associated
the RA
−1.19
all A and all B triangles of the lattice, respectively. T A (
ergy fo
circle) has three physical and three geometric indices. T B
−1.23
εmingeometric
=−1.2696
circle) only has three
indices. Such a tensor
−1.27
work ansatz is also called projected entangled simplex
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
We t
[28].
α
You,
Zi
After each step of the imaginary time evolution, we
cussion
✎ By exploring to
𝛼 in
the family
of RAL
states,indices
we provide
reallocate
the three
physical
(from T A to
T B, o
Planck
an upper
bound
for spin-1
kagome
model.
other
way round)
and HAF
truncate
the
stateRAL
space dimensio
FIG.
14. (Color
online)
Variational
energy
of the
mixed
work
h
states for spin-1 kagome
Heisenberg
model,
computed
using
Here we use the single-triangle (ST) and double-triangle (
Tai-Ka
iPEPS. Convergence
has
been
checked
against
di↵erent
D
.
c
Bethe-lattice approximations for truncations [23, 24], fol
visit in
The RAL states: conclusions
✎ Resonating AKLT-loop state: a family of states with natural PEPS
representation.
✎ The kagome RAL states have no magnetic order, no dimer crystal
order, or any other symmetry breaking orders, but they are
topologically ordered.
✎ The RAL states serve as variational wavefunctions of the spin-1
KAF.
✎ However, the variational energy is still high, Eg = -1.2696.
✎ Tune more parameters in the RAL family ➜ lowest Eg ~-1.38
PEPS simulations of the spin-1 KAF
✎ Simple update & (double-triangle) cluster update
✎ Non-abelian symmetry in PEPS
✎ Ground state properties
Related paper:
T. Liu, WL, A. Weichselbaum, J. von Delft, and G. Su
Simplex valence-bond crystal in the spin-1 kagome Heisenberg antiferromagnet
arXiv:1406.5905 (2014).
am, find a quantum phase transition between the SVBC and a
Tensor network states
rly
(a)
(b)
has
assly
B
TB
D
roTA
A
tuinysuid
he
atFIG. 1. (Color online) (a) Kagome lattice (dotted lines) and the initial
setup of the tensor-network wavefunction (solid lines). T A and T B
[9]
✎ Take the RAL state as a starting point of further optimizations.
are triangle tensors and D is the bond dimension. (b) Illustration
agof the simplex valence-bond crystal. The two kinds of triangles or
ng
“simplexes” [of type A (blue) and B (pink)] have di↵erent energies,
ea
Update the tensors
✎ Adopt the same tensor-network structure as the RAL state.
✎ Take imaginary time evolution for updating the tensors.
Tb
B
X
B
Ta
Y
A
Tb
B
Z
Tb
H.-C. Jiang, Z.-Y. Weng, and T. Xiang, PRL, 2008.
WL, J. von Delft, and T. Xiang, PRB, 2012.
Tb
Z.-Y. Xie, J. Chen, J.-F. Yu, X. Kong, B. Normand, T. Xiang, PRX, 2014.
T. Liu, S.-J. Ran, WL, X. Yan, Y. Zhao and G. Su, PRB, 2014.
Variational energy (per site)
✎ eg ≃ -1.409 [with D=14, dc=40]
✽ Lower than the CC result [-1.4031, Götze et al. PRB (2012)]
Variational energy (per site)
✎ Roll the wavefunction on a cylinder and perform exact contractions.
❋ Variational energy on cylinder.
Trimerization Order
B
A
✎ The ground state breaks the lattice symmetry.
❋ A simplex (triangle) valence bond crystal (SVBC).
the bond dimension and get a variational energy close to the
“true” ground state value.
Bilinear-biquadratic Heisenberg model.— We also studied
the spin-1 BLBQ Heisenberg model with Hamiltonian
The bilinear biquadratic model:
X
H=
[cos ✓ (Si · S j ) + sin ✓ (Si · S j )2 ],
(1)
which im
in a tran
namely
tained m
The D⇤ =
geometr
fore), an
The SVBC phase
<i j>
0.5
0.37
AFQ
π
4
Trimerization
FM
-0 .0 4
-
7
0.
5
FQ
✎ The SVBC order is rather robust, and extends to a phase.
SU(3) point with trimerization order
[P. Corboz, et al, PRB 2012]
Qz Qz
i iz
charge
SU(2)spin ⌦ U(1)charge ⌦ SU(3)channel
SU(2)spin ⌦ Sp(6)
respective
Liecharge+channel
algebra
10 i=1•
4
detailed
information
als
260
> 30
61 bonds,
> 250
evaluated
based on the
respective
Lievar
a
• explicitly evaluated based on
the • explicitly
the
obtained
QSpace
tensor
library
i.e. the generators of the symmetry
(thusthe
generic).
) drastic
reduction
of e↵ective
dimensions
in multiplet
space!
i.e.
generators
of the
symmetry
(thus generic
&
the
SU(2)
PEPS
algorithm
• got decomposition of full matrix
elements
which and
allows
for
) •generic,
efficient,
transparent
of
got decomposition
of full implementation
matrix
elements
whic
5
Outlook
2
arbitrary tensor contractions, non-abelian
A. Weichselbaum,
of Physics
2972 library.
–3047 (2012).
symmetries Annals
through
QSpace327,
tensor
arbitrary
tensor contractions,
m
Improved CGC performan
O
m
A⌫ ⌦ {C}⌫ ⌘ A⌫ ⌦
Cs;⌫
O
Applications
to fermionic
i=1
A⌫ ⌦ {C}⌫ ⌘ A⌫ ⌦
s=1
QSpace
tensor library
Elementary ingredients: (i) state2space
decomposition:
|Q0n0; Q0z i =
X
X
Elementary
ingredients:
setz of m compact
AQQGiven:
0 nn0 · CQ Q0 |Qn; Qz i|Q̃ñ; Q̃z i
[Q̃]
[ñ]
[Q̃ ]
z
Cs;⌫
s=1
References
(i) state
space decompositi
particle number
spin SU(2)
CGC
z
CGC
[1] K. G. Wilson, Rev. Mod. Phys. 47, 773–840 (1975).
channel SU(3)
X
X
[2] [A.
Weichselbaum,
Annals[Q̃
of Physics
327, 2972 –3047 (2
simple symmetries
{S
},
Q̃]
[ñ]
z]
0
0
0
…
qn
qn
[3] F. Verstraete
and
J.C
I. Cirac, arXiv:cond-mat/0407066v1
|Q
n ; Qz i =
A
·
|Qn;
Qz i|Q
0
0
0
m
nn
QQ
Q
Q
z
j
z
O theorem): i
(ii) operator representation (cf. Wigner-Eckart
[4] N. Schuch, I. Cirac, and D. Pérez-Garcı́a,
Annals of Phys
Qn;Qz Q̃ñ;Q̃z [5] I. A✏eck, T. Kennedy, E. H. Lieb, and H. Tasaki, Comm
S
⌘
S
.
[Q̃] [1]
[Q̃z ]
[6] T. Liu, W. Li, A. Weichselbaum, J. von Delft, and G. Su,
0 0 0 Q̃
Qn;Qz Q̃ñ;Q̃z
hQ n ; Qz |F̂ |Qn; Qz i = FQQ0 nn0 · CQ Q0 ,
Q̃z
=1 z z representation
(ii) operator
(cf. Wigner-Eckart theo
[7] H. J. Changlani and A. M. Läuchli, arXiv:1406.4767 [con
[8] H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett.
[9] P. Corboz, R. Orús, B. Bauer, and G. Vidal, Phys. Rev. B
qn
State space decomposi[Q̃] [1]
[Q̃z ]
Q̃
0
0
0
n ; Qz |F̂ |Qn; Qz i =CGCFQQ0 nn0 · CQ Q0 ,
tion via composite hQ
index
z z
Q̃z
|ii ⌘ |Qn; Qz i, with

The benefits of using SU(2)
spin symmetric
multiplet
labels Q ⌘tensors:
[q1, . . . , qm] and z-labels (weights) Qz ⌘
[z1, . . . , zm] simulation
(implicitly present
only by the index
the gener✎ A clearly more efficient numerical
[the numerical
cost into
of PEPS
alized 10~12
Clebsch-Gordan coefficients (CGCs),
on a 2D lattice scales like O(D
)!]
[Q̃ ]
CQ zQ0 ⌘
z
z
n
Y
i=1
0 )
(qiqiz ; q̃iq̃iz |qi0qiz
• explicitly
evaluated
basedstructure
on the respective
Lie bonds,
algebra
✎ Detailed information also
about the
multiplet
along the
i.e. update
the generators
of the symmetry (thus generic).
obtained variationally through
methods.
• got decomposition of full matrix elements which allows for
(a) X-cylinder
Ly
P
TB
ay=1
TA
V
(b) D*=3
0⊕1⊕1
ax=1
(c) D*=4
1
P
0⊕1⊕2
0⊕0⊕1⊕2
Lx
1
P
0⊕1⊕1⊕2
Energy & Entanglement3 on cylinders
(a)
Energy per site
−1.4
−1.41
−1.43
−1.44
Entanglement Entropy
−1.45
0
6
4
eg=−1.4104
XC, D*=3
XC, D*=3, DT
XC, D*=4
iPEPS, D=10
iPEPS, D=12
−1.42
0.1
0.2
In agreement with previous PEPS
calculation without symmetry
implementation.
0.3
1/Ly
0.4
0.5
D*=3, even
D*=3, odd
D*=3, DT, uniform
D*=4, even
D*=4, odd
𝛾≈0 suggests a topologically trivial state.
2
0
0
(b)
1
2
3
Ly
4
5
6
Conclusions
• We employ the tensor network algorithm (with/without SU2
symmetry) to perform a variational study of the spin-1 KAF model.
☀1st (?) example that SU2 symmetry pays off in PEPS calculations.
Just in its beginning!
• The ground state has a simplex valence-bond crystal order, with GS
energy determined as Eg ≃ -1.409.
☀ SVBC picture is consistent with experimental
observations: gapped, non-magnetic, &
symmetry breaking low-T phase
Collaborators of the RAL project
Shuo Yang
MPQ, Germany
moving to PI, Canada
Zheng-Xin Liu
Tsinghua
Hong-Hao Tu
MPQ, Germany
Meng Cheng
Station-Q, US
spin-1 kagome numerical project
University of Chinese
Academy of Sciences,
Beijing, China
Tao Liu
Gang Su
LMU Munich,
Germany
Andreas Weichselbaum
Jan von Delft
!anks for y"r a#ention!

The spin-1 kagome antiferromagnet

Transcription

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