千葉大学理学部集中講義

Symmetry protectionsfortopological phases
Y. Hatsugai
Institute of Physics, Univ. Tsukuba

2015年7月9日-10日

Plan
Why topological ?
Novel phases without symmetry breaking
Why Symmetry ?
Symmetry protection of gap nodes
Dimension & co-dimension
Anisotropic superconductivity/fluidity & graphene
Topological order parameter by quantum interference
Berry connection: Z2 Berry phases & Chern number
Successful examples

Plan
Why topological ?
Novel phases without symmetry breaking
Why Symmetry ?
Symmetry protection of gap nodes
Dimension & co-dimension
anisotropic superfluidity & graphene
Topological order parameter by quantum interference
Berry connection: Z2 Berry phases & Chern number
Successful examples
Zoo of edge states as topological order parameters
Bulk-edge correspondence
Examples : Zoo of what we care.

Why do we care topological phases ?
Characterization of phases
too much success

Ginzburg-Landau theory

Local order parameter: hS(r)i
Symmetry breakinghS(r)i 6= 0

Magnetism, superconductivity, charge/orbital ordering ...

Is thisTosatisfactory
?
pol
ogi
calliquids
Quantum/Spin!

Absence of symmetry breaking
need something more:Quantum/Spin Liquids ?
Quantum Liquids in Low Dimensional Quantum Systems
Low Dimensionality, Quantum Fluctuations
No (fundamental) Symmetry Breaking
No Local Order Parameter

New Type of Order
Topological Order!

Quantum Liquids in Condensed Matter

X.-G.Wen ’89

Integer & Fractional Quantum Hall States
Dimer Models of Fermions and Spins
Half filled Kondo Lattice
Kitaev model & Levin-Wen model
Anisotropic superfluids/superconductors (ABM, BW, p-wave )Graphene, Weyl semi-metal
Topological insulators : quantum spin Hall states
Photonic crystals & Some of cold atoms ..

Quantum Liquids ?
Quantum Liquids in Low Dimensional Quantum Systems
Low Dimensionality, Quantum Fluctuations
No (fundamental) Symmetry Breaking

GappedGapless

New Type of Order
Topological Order!

No Local Order Parameter

X.-G.Wen ’89

Quantum Liquids in Condensed Matter

GappedInteger & Fractional Quantum Hall States
Dimer Models of Fermions and Spins Gapped
Half filled Kondo Lattice

GappedKitaev model & Levin-Wen model

Gapped/Gapless

Anisotropic superfluids/superconductors (ABM, BW, p-wave)
Graphene, Weyl semi-metal

Gapless

Topological insulators : quantum spin Hall states
Photonic crystals & Some of cold atoms ..

GappedQuantum Liquids are Featureless !!
A phase without symmetry breaking is interesting ?
Are there something to be learned ?

Too much general is boring !
Nothing to be characterized
in sufficiently high dimensions

SYMMETRY & DIMENSION constrains !
Symmetry protection ofTopological Phases
without symmetry breaking

“TRULY GENERIC” phase without any symmetry breaking
topologically single phase (too simple ?)With some symmetry A , B , C

YH, ’06
1.Discrete symmetry
Time reversal
Charge conjugation
Space inversion
Reflection
2.Gauge symmetry
U(1) : QHE (TR ×)
Sp(1) : QSHE (TR ○)Chen-Gu-Wen, ’10
Pollmann et al., ’10

How to characterize the phase
Without Symmetry Breaking ?
Try toshow overview
Topological!
Stability against for perturbation !
Gapless✓Nodes structurespoint nodes, line nodes,...

Gapped✓Adiabatic invariants ✓Chern numbers, ZQ Berry phases

Protected by symmetry✓Bulk-edge correspondencegeometrically induced gapless excitations in gapped phasePlan
Why topological ?
Novel phases without symmetry breaking
Why Symmetry ?
Symmetry protection of gap nodes
Dimension & co-dimension
Gapless
Anisotropic superconductivity/fluidity & graphene
Topological order parameter by quantum interference
Berry connection: Z2 Berry phases & Chern number
Successful examples
Zoo of edge states as topological order parameters
Bulk-edge correspondence
Examples : Zoo of what we care.

Topological !

Gapless

Single particle problem (mean field)

Nodes structurespoint nodes, line nodes,...
protected by symmetry
gapless : generic 2 levels near the gap von Neumann-Wigner ’29
H(k) = R(k) ·E = ±|R(k)|

=

✓

Rz
Ry + iRy

gapless point

Rx

iRy
Rz

R=0

◆

Berry ’84

expanded by Pauli matrices
3 parameters(Rx , Ry , Rz )To be gapless: 3 parameters to be tuned
co-dimension=3 (3 conditions)ex.

2D Brillouin zone
2

2D Torus

T2:periodic in kx & ky

1map

2-D closed surface in 3D

2

R(T )

2D examples
d-wave superconductivity

2D Brillouin zone2

1

p-wave superconductivity

H(k) = R(k) ·

=

✓

Rz
Ry + iRy

Rx

iRy
Rz

◆

YH-Ryu, ’02

ABM states & Dirac mono pole
2D Brillouin zone2

Anderson-Brinkman-Morel (ABM) phase of He

1

3rd momentum: time line

co-dimension 3
In 3D, 3--3=0 : point nodes
topological stability

H(k) = R(k) ·

=

✓

Rz
Ry + iRy

Rx

iRy
Rz

◆

YH-Ryu-Kohmoto, ’04

Geometrical meaning of Chiral symmetry
H(k) = R(k) ·=Chiral symmetry

=

⇢

z

{He↵ ,✓Rz
Ry + iRy

9

Rx

iRy
Rz

: bipartite lattice & hopping between them

Generically=n ·{He↵ , } = 0

Zero gap condition: Dirac dispersion

R(k)
Y

2

=1

Rz = 0

Ry = 0n ?RHe↵ ! 0, k ! k0

Chiral SymmetryX

E = ±|R(k)|3D (Rx , Ry , Rz )} = He↵ + He↵ = 0

y : real He↵ : Time reversal & Inversion

n

◆

{H,

} = 0,

2=1

co-dimension of Dirac cones=2graphene, d-wave superconductor in 2D

Topological stability of the Doubled Dirac cones

2D Nielsen-Ninomiya theorem

H(k) = R(k) ·

=

✓

Rz
Ry + iRyRx

2D Brillouin zone :periodic in kx & ky
2D Torus

T2

2

1

map

c.f. 4D graphene & chiral symmetry, M. Creutz ’08◆
also with TR inv. 5D YH, ’10

iRyRz

3D (Rx , Ry , Rz )Generically 2-D closed surface in 3D“balloon”

2

R(T )Chiral symmetry{H, } = 0=n ·R(T 2 ) is collapsed on the planen ?RR(k) is on a plane normal to n=(0,0,0)“collapsed balloon”

Topologically stable“collapsed balloon”
nγ

nγ

R

doubled Dirac cones

R

Gepped :perturbation is too large

YH-Fukui-Aoki, ’06

Graphene with deformation
2D Brillouin zone2

co-dimension 2

1

YH-Fukui-Aoki, ’06

deformation of the system: time line

topological stability in 2D
In 2D with chiral symmetry, 2--2=0 Dirac cones of graphened-wave superconductor

c.f. Blount’85

YH-Ryu-Kohmoto, ’04

Gapless

Topological !

Nodes characterize the phase topologically

Generic
co-dimension 3

Volovik ’97

In 3D, 3--3=0 : point nodes :ABM state of He YH-Ryu’02
Weyl semi-metal YH-Ryu-Kohmoto ’04
topological stabile Dirac point
Burkov-Balents ’11
with TR invariance/chiral symmetry
co-dimension 2
Blount’85

In 3D with TR invariance, 3--2=1 : line nodes super

Wallace’47

In 2D with chiral symmety 2--2=0 : Dirac cones of graphenewith TR invarianced-wave superconductor
YH-Ryu & Ryu-YH ’02

Plan
Why topological ?
Novel phases without symmetry breaking
Why Symmetry ?
Symmetry protection of gap nodes
Dimension & co-dimension
Gappedanisotropic superfluidity & graphene
Topological order parameter by quantum interference
Berry connection: Z2 Berry phases & Chern number
Successful examples
Zoo of edge states as topological order parameters
Bulk-edge correspondence
Examples : Zoo of what we care.

Gapped Are insulators boring ??
Insulators
Non: metal,
gappedGapped: Nothing
in the :gap
cf. Nambu-GoldstonebosonBand insulators
No low lying excitations
Superconductors
No Response
againstsmall
perturbation
Integer & Fractional Quantum Hall States
gapless modes:
Integer spin chains (Haldane)
acoustic phonons
zero sounds
？？？
Dimer
Models(Shastry-Sutherland)??
spin waves
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators

Absence of low energy excitations
Energy gap above the ground state

Lots of variety
Absence of fundamental symmetry breaking (mostly)
No responses against for small perturbation

Topological !

GappedAdiabatic invariants Chern numbers, ZQ Berry phases
protected by symmetry
Parameter dependent hamiltonian

Berry connection

Intrinsically quantized
Chern numbers: 1st, 2nd, 3rd,.... C1 =QHE ...

Z

Symmetry protected quantization
Berry phases & generalization:
Quantum spin chains,
Spin-QHE ...

Z2

1
2⇡i

Z

F
M2

...,-2,-1,0,1,2,...
Z
1
A
1 =
2⇡i M 10 or 1/2

GappedTopological !

Adiabatic invariants Chern numbers, ZQ Berry phases
protected by symmetry
Fractional QHE
Adiabatic heuristic (Wilczek)

Connect states byadiabatic process1✓d( + ) = 0⇡ ⌫Flux attachment

Classical vs Quantum“Classical” Observables in Quantum Physics

Ocl : H(energy), p(momentum), n(r)(charge density) · · ·quantization

Ocl ⇥ O : Hermite Operator⇥O⇤G = ⇥G|O|G⇤ = ⇥G |O|G ⇤ = ⇥O⇤G|G ⇤ = |G⇤ei:Independent of the phasewith N fold degeneracy= (|G1 ⇥, · · · , |GN ⇥)
1
1
†
⇥O⇤ =
⇥Gi |O|Gi ⇤ = Tr O
N i
NUnitary Invariant ?
1†
O = ⇥O⇤= Tr
YES!
N= U , U : UnitaryBerry connection as quantum interference“Quantum” Observables !
No classical Correspondences
Quantum Interferences between 2 different states
Aharonov-Bohm Effects
Geometrical Phases⇥G(tA )|G(tB )⇤ = ⇥G (tA )|G (tB )⇤Berry connection & Phases|G(t)⇤ = |G (t)⇤e

G|G + dG⇥ = 1 + G|dG⇥
A = G|dG⇥ :Berry Connectioni =A:Berry Phase

Unitary Invariant ? NO ! Phase dependenti (t)Berry Connection?
parameter space

Berry ’84

Eigenvectors ( space )with ParametersxyH(x) Y(x) =% (x) Y(x)H(x) and H(y) are independent

Information between nearby states
Berry connection : A
gauge potential(Abelian)

Fiber Bundle

=

|d

=

d
| dx

dx.

Gauge Transformation phase change=gauge transformations

| (x)
A

= | (x) ei (x)phase fix = gauge fixd= A + id = A + i dx
dxBerry phase and its gauge dependence
Parameter Dependent Hamiltonian
H(x)| (x) = E(x)| (x) ,H(x) Y(x) =% (x) Y(x)(x)| (x) = 1.

dBerry Connections A = |d = | dxdx.
i C (A ) =A
Berry Phases
C(Abelian)
Phase Ambiguity of the eigen state| (x)= | (x) ei (x)d= A + id = A + i dx
dx

AGauge Transformation

Berry phases are not well-defined without
C (A

)=

C (A

)+
C

Well Defined up to mod 2
C (A

)

C (A

specifying the gauged2) mod 2(integer) if ei is single valued

Anti-Unitary Operator and Berry Phases
Anti-Unitary Operator
K:= KU ,
U :

|

=

|

=

J

(Time Reversal, Particle-Hole)Complex conjugate
Unitary(parameter independent)
CJ CJ =

CJ |J

J

=

|

J

CJ |J

,

=

|J

|

=1

|J

Berry Phases and Anti-Unitary Operation
A =
A

=

|d
|d

C (A

=

CJ dCJJ

dCJ CJ +J=J

CJ dCJ =

A

J

)=

C (A

CJ dCJ =0)

Anti-Unitary Invariant State andQuantized Berry Phases
Anti-Unitary Symmetry
Invariant State,

ex. Unique Eigen State[H(x), ] = 0

=

|

|

|

To be compatible with the ambiguity,

=|

i

e

Gauge
Equivalent(Different
Gauge)

the Berry Phases have to be quantized as
C (A
C (A

)=
)=

0
C (A

mod 2
)

C (A

), mod2

Z2 Berry phase as a topological order parameter
Generic Heisenberg Models with possible frustration

H=

Jij S i · S jij

Time Reversal Invariantj, S j

1
T

Sj

T =

Sj

U(1) twist as a Local Probe to define Berry Phases
Si · Sj

1(e i Si+ Sj + e+i Si Sj+ ) + Siz Sjz
2

H(x = ei )Parameter dependent HamiltonianC = {x = ei | : 0

2 }Define Berry Phases by the Entire Many Spin Wavefunction

Quantization
C =⇥Excitation Gap!

A =

C

⇥

Time Reversal Invariance⇤|d⇤⇥ =C

0:⇥

Z2 Berry phasemod 2⇥

Topological order parameter at the link <ij>

YH, J. Phys. Soc. Jpn. 75, 123601, ’06

Quantum liquids Short range entangled states

Ex.1) AKLT state(1,1)

gapped integer spin chain

Ex.2) Collection of singlets
Something complicatedbut gapped

many-body gap
smallQuantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open
Something complicatedbut gapped

many-body gap

Quantum liquids Short range entangled states

Adiabatic deformation !
gap remains open

Decoupled !

Something very simple& gappedbig !
many-body gap

Quantum liquids Short range entangled states

Adiabatic process to be decoupled: gap remains open

Collectionoflocal quantum objects

Def. of short range entangled stateHow to characterize local object ?
Consider a gauge transform at some siteHow to characterize local object ?
If decoupled,
the twistby the transformation
is gauged
Considera gaugetransform at somesite away !

zyx
It characterizes locality of the quantum object !
Question
Answer !?
How
Calculateto see thisa topologicallocality by invariant
skipping the
as anadiabatic
adiabatic
deformation
invariant ?

Z2 Berry phase of Singlet 0Pair

=

A

QA

B

QB

cos ✓
y
x
HAB = (SA, SA
, SZz ) @ sin ✓

sin ✓
cos ✓
1

10
A@

1
+
+
z z
= (e i✓ SA
SB + ei✓ SA SB) + SA
SB
2

x
SB
y
SB
z
SB

1
A

Local Singlet Pair with the twist

1 i✓/2| i = p (e|"A #B i2e i✓/2 |#A "B i)

Berry phase of the twisted singlet pair
A=
=
=

i

⇥

†

d

⇤
i
⌃
1
iArg
(a
be
)
⌃
⇥
i
e
⇧
2

i
⌃
1
iArg
(b
ae
)
⌃
⌅ i ⇥2 e

A=⇥⇥|a| > |b||a| < |b|singlet pair = ⇥ mod 2

1
e i
ei
1

⇥

⇥

|a| > |b||a| < |b|
: a, bC(gauge parameters)

A singlet does not carry spin
but does the Berry phase ⇡Introduce interaction between singlets

Adiabatic Continuation & the Quantization

Quantization of the Berry phases protects fromcontinuous change
Adiabatic Continuation in a gapped system
Renormalization Group in a gapless system

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons

Sorry if I’m wrong

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons
hopping

make energy band
metal

Sorry if I’m wrong

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons
hopping

Peierls instability
Opening gap
stabilize

Sorry if I’m wrong

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons
hoppingchemists

form molecules first

Peierls instability
Opening gap
stabilize

Sorry if I’m wrong

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons
hoppingchemists

form molecules first

Peierls instability
Opening gap
stabilize

Sorry if I’m wrong

Metal/gapped : physicists & chemists ?
physicists

itinerant electrons

Peierls instability
stabilize

chemists

form molecules firstmake bands of
non orthogonality
short range entanglement

chemist way: BETTER !

molecules
Adiabatic process
EF
Insulator

Think locally when gapped!
molecules, singlet pairs, bonds

Sorry if I’m wrong

Validity of our general scheme
Examples in 1D, 2D, 3D and ...
Integer spin chains with dimerization
Random hopping models
Orthogonal dimers in 2D
BEC-BCS crossover at half filling
Dimerization transition on Kagome & Pyrochlore

1D S=1/2 chains with dimerization
H=

X
hii

Ji S i · S i+1

Y.H., J. Phys. Soc. Jpn. 75 123601 (2006)

AF-AF case
AF-AF

テキスト

Strong bonds
:

bonds

F-AF caseFerro-AFHida

AF bonds:

bonds

Heisenberg Spin Chains with integer S
S=1 (S i ) = S(S + 1), S = 1
2

H=J
ij⇥

Haldane phaseP

Si · Sj + D

z 2
(Si )
i
Y.H., J. Phys. Soc. Jpn. 75 123601 (2006)

P

P

P

P

P

P

P















D < DC

Large D phaseCharacterize the Quantum Phase Transition

D > DCS=1,2 dimerized Heisenberg model
T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08

2(a)ian Berry connection obtained by the single(2,0) , J2 = sin (1,1)(0,2)
ormalized ground state |GS(φ)⟩ of H(φ) as J1 = cos⟨GS(φ)|∂φ |GS(φ)⟩. This Berry phase is real
2Red line :Berry phasentized to20 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

⇡

P

P

P

P

P

P

P

P

















S=1 & 2

Sequential transitions among gapped phases

set of two numbers as (n, m). The phase boundaries in theS=1,2 dimerized Heisenberg model
T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08

2(a)ian Berry connection obtained by the single(2,0) , J2 = sin (1,1)(0,2)
ormalized ground state |GS(φ)⟩ of H(φ) as J1 = cos⟨GS(φ)|∂φ |GS(φ)⟩. This Berry phase is real
2Red line :Berry phasentized to20 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
(3,1)
local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this
(4,0)
+ −
− +
(c)
(b) the following(3,1)
n twist,
term
S
S
+
S
i j
i Sj in(1,1)(3,1)(2,2)+ − (1,3)+(4,0)
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
(2,2)
j ,
±
= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
(0,4) (3,1) (2,2)
the(2,0)local spin twists extracts a (0,2)
local structure of(1,3)
tum system. By this quantized Berry phase, one
(4,0)
(c)
(0,4)
e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases (1,3)
on the local link of (a) the S = 1: S=1/2
singlet: Symmetrization
0-bond”, “π-bond”, or “undefined”.
It hasa re- state
periodic N = 14and(2,2)(b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
ss against the small perturbations unless the en(0,4) Heisenberg chain with single-ion anisotropy. The Berry phaseis π on the bold line
while that is 0 on the other line. We la(1,3)
between the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase⇡

Reconstruction of valence bonds!

set of two numbers as (n, m). The phase boundaries in the

2

S=2
Heisenberg
model
with
D
term
(a)
btained by the single-|GS(φ)⟩ of H(φ) ass Berry phase is realπ) if the Hamiltonian-unitary operation Θ,at the Berry phase isground state and theying the parameter φ.
k as a generic paramphase [1]. Under thisrm Si+ Sj− + Si− Sj+ in
eiφ Si+ Sj− + e−iφ Si− Sj+ ,hase defined by the reacts a local structure of
tized Berry phase, one
ch link has one of threendefined”. It has a rephase has topologicalrbations unless the enand the excited states
rry phase numerically,
erry phase[1, 33]. It iseter space of φ into N

S=2

(2,0)

(1,1)

(0,2)

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08

(b)
(4,0)

(3,1)

(2,2)

(1,3)

(0,4)

Red line :Berry phase
(c)
0

⇡

S = 2 N = 10
0.5

1

1.5

2

FIG. 1: The Berry phases on the local link of (a) the S = 1
periodic N = 14 and (b) the S = 2 periodic N = 10 dimerized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phaseis π on the bold line while that is 0 on the other line. We la:0 magnetization
bel the region of the dimerized Heisenberg chains using theset of two numbers as (n, m). The phase boundaries in thefinite size system are θc1 = 0.531237, θc2 = 0.287453 andθc3 = 0.609305, respectively. The Berry phase in (a) and (b)has an inversion symmetry with respect to θ = π/4.

Reconstruction of valence bonds!

boson representation of the spin operaors as Si = ai bi ,&as † |φ⟩
†
†
†
†
†
−
z
t
(ai bi+1[36].
has
averageofvectorspinchiralSi This
= aGeneric
Si nonzero
=
(a
a
−
b
b
)/2.
a
and
b
satisfyi bstateiiiii , andii
AKLT (VBS)
modelswheretha|k††obviousitythe⟨Scommutation
the[atwist= 0 allor π.
= [bi , bj ] = δφijijwith
relationi × S j · ẑ⟩ unless
i , aj ] parameter
triplet
b
since
the
st
T.Hirano,
H.Katsura&YH,
Phys.Rev.B77
094431’08
†
This
state
is a zero-energy
groundstateof the following'
othercommutators
vanishing[37]. The
constraint
ai ai +
orthogonal
t
(1/2)
†
Hamiltonian:Twist themodel the dimension of thebi =of2Stheis generic
imposedAKLT
to reproduce
bilinkthe (B
Berry
pk
−
12
spin S Hilbert spaceN at eachsite. In general, the ground
2Bi,i+1
Next
we
sha
$
$
of
the
c
state of the chiral AKLT model having B
valence
bonds
J ij
B
H({φi,i+1 }) =AJ Pi,i+1[φi,i+1 ],(5) arbitraryIm⟨k|l⟩on the link (ij) is written as
with a loca
i=1 J=Bi,i+1 +1# VB† or odd)"#s and t† a!
Bij
iφij /2 † †−iφij /2 † †parities.
|{φ
}⟩
=
e
a
b
−
e
b
a
|vac⟩,
J(4)
i,j
where
(· · · )ijijwhere AJ is the arbitrary positive coeﬃcient. Pi,i+1 [0]
state
rep
⟨ij⟩
using
the
b
is the polynomial in S i · S i+1 and act as a projection
we
can
o
as
|φ⟩
=
Berry
phase
on
a
link
(ij)
operatorprojecting
the bond
spinofJvector
S ichiral+ S i+1[36]. This
state has nonzero
averagespin
i,i+1 =
using
th
where
|k⟩
=
onto
the
subspaceofspinmagnitudeJ.
The
replacement
ity ⟨S i × S j · ẑ⟩ unless the twist parameter φij = 0 or π.
S=1/2
ij
ij
phase
of
tripletbonds+This− state −
+
+
−
−
+
iφ
−iφ
i,i+1
i,i+1
zero-energy
groundSi Si+1 + SiisSai+1→e
Si Si+1state+ e of the following
Si Si+1 in'B12 (
the k=0edg(1/2)
JHamiltonian:
S i · S i+1 produces Pi,i+1 [φi,i+1 ] in Eq. (5).
(B12is−trunck) ta
2Bshow
i,i+1
N
Now we shall explicitlythat the Berry phase of of the conne$$
property
The BerryphasecountsthenumberJ of the valence bonds!
H({φ], valence(5)=
the VBS
state
extracts
the localAnumberof theJ Pi,i+1 [φi,i+1
i,i+1 })
Im⟨k|l⟩= 0
stressed
i=1 J=Bi,i+1 +1
bonds Bij as Bij π(mod2π).
Let us now consider the or odd) and
restricte
local twist of the parameters φij = φδij,12 andJ rewrite parities. Thi
S=1/2
objects
arearbitrary
fundamental
in integerPi,i+1
spin[0]chainsgenerali
where
AJ is thepositive coeﬃcient.
state represe
theis ground
state
|{φ
}⟩
as
|φ⟩.
To
calculate
the
Berry
i,j
the polynomial in S i · S i+1 and act as a projection
longobtai
as t
we
can
phase
of theVBS state,
relation
operatorprojecting
the thebondfollowing
spin J i,i+1= S i is+ useful:
S i+1
Now,
using therel

= B ⇥ mod 2⇥

Other systems applied
Spin ladders with ring exchange

I. Maruyama, T. Hirano, and Y. H.,Phys. Rev. B 79, 115107 (2009)
M. Arikawa, S. Tanaya, I. Maruyama, Y. H.,Phys. Rev. B 79, 205107 (2009)

BEC-BCS crossover at half filling
M. Arikawa, I. Maruyama, and Y. H., Phys. Rev. B 82, 073105 (2010)

Orthogonal dimers

Sr Cu2(BO3)2

discovery
H. Kageyama et al. , Phys. Rev. Lett. 82, 3168 (1999)

Theory: spin gap & magnetic plateaus
B. S. Shastry and B. Sutherland, Physica, 108B, 1069 (1981).

H=J

X
hiji

Si · Sj + J

0

X
hiji

Si · Sj

S. Miyahara & K. Ueda , Phys. Rev. Lett. 82, 3701 (1999)
T. Momoi and K. Totsuka, Phys. Rev. B 61, 3231 (2000)

Gapped to gapped transition
Plaquette singlet phase=

Dimer phaseJ >> J

0

J ⇡J

H=J

X

+

0

Si · Sj + J

0X

Si · Sjhiji
hiji
A. Koga & N. Kawakami, Phys. Rev. Lett. 84, 4461 (2000)

Z2 Berry phaseDgauge twist for singlet pair
0.661<J’/J<0.665

2π

6
5

D

:⇥

2

Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê

1

Ê

4

π 3

0 0

0.0

0.2

0.4

0.6

:0

D

Ê

0.8

1.0

1.2

1.4

J /J
I. Maruyama, S. Tanaya, M.Arikawa & YH. , arXiv:1103.1226

Z2 Berry phasePgauge twist for plaquette singlet
6
5

P: 0

⇥

P

⇥
⇥

:⇥

⇥

4
3

⇥

2

⇥
⇥

1

⇥

0
0.0

0.2

0.4

0.6

J /J 0.8

1.0

1.2

1.4

I. Maruyama, S. Tanaya, M.Arikawa & YH. , arXiv:1103.1226

Local Order Parameters of Bonds

2D Extended SSH ( Su-Schrieffer-Heeger) Model
Strong Coupling Limit has a gapped unique ground state.

Distribution of hoppingst’/t=0.0

Vij = 0
Non-Abelian Connection
for the Fermi-Sea
Large System isavailable
YH, JPSJ. 73, 2604 (2004),
74, 1374 (2005)

Local Order Parameters of Dimer Pairs
2D Extended SSH ( Su-Schrieffer-Heeger) Model
Strong Coupling Limit has a gapped unique ground state.

Distribution of the Quantized Berry Phasest’/t=0.6

C =Local Order Parameters of Dimer Pairs
2D Extended SSH ( Su-Schrieffer-Heeger) Model
Strong Coupling
R Limit has a gapped unique ground state.

econstruction of bonBerryDistribution of the Quantizedds Phasest’/t=0.7

Quantum Phase Transition
with (local) Gap Closing

C =BEC-BCS crossover
as a local quantum phase transition
Switching on attractive interaction among particles

spin up electrons
spin down electrons

BEC-BCS crossover
as a local quantum phase transition

BEC-BCS crossover
as a local quantum phase transition
BEC : strong coupling

BCS : weak coupling

Crossoveradiabatically connected
Making bosons in real spacethen condense

Cooper pairingin momentum space

BCS Model at half filling

H=

X †
t
ci cj
,i,j

|U |

X

ij ci" ci#Arikawa-Maruyama-YH, 2010ij

modify only at special (local) order parameter Z ij !
2⇡

to calculate the Berry phase

=

i

0

1D

s-wave Gapi✓ij e (✓ : 0 ! 2⇡)d✓ h |@✓ i| i manybody state

Crossover of the bulk
by Quantum Phase transition with local gauge twist
crossover : gapped always
weak coupling (BCS)=0strong coupling (BEC)= ⇡p|UC |/t = 2/ 3 = 1.15, 1.25, 1.6
1D

2D

3D