Corner states of Kagome lattice & related · 2018-10-19 · Plan TTCM2018, Geneva, Oct.8-10 (2018) Berry phase as a quantum order parameter Symmetry protection & quantization Bulk-edge

TTCM2018, Geneva, Oct.8-10 (2018)

Corner states of Kagome lattice & related

Yasuhiro Hatsugai

Univ. Tsukuba

Trends in Theory of Correlated Materials 2018

TTCM2018, Geneva, Oct.8-10 (2018)

Group members in TTCM2018

Hiromu Araki

Tsuneya Yoshida

Tomonari Mizoguchi

(PhD student)

c.f. Session 6, Norio Kawakami

arXiv:1809.09865, H. Araki, T.Mizoguchi & Y. Hatsugai

　

Phase Diagram of Disordered Higher Order Topological Insulator: a Machine Learning Study

　

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TTCM2018, Geneva, Oct.8-10 (2018)Plan

Berry phase as a quantum order parameter Symmetry protection & quantization Bulk-edge correspondence

Dimerization/Breathing Molecules or Peierls instability Series of materials Breathing Kagome

Corner states of Kagome Breathing Kagome : Machine learning

Bulk-edge correspondence of Ising models

2006 —

arXiv:1809.09865, H. Araki, T.Mizoguchi & YH

T.Mizoguchi & YH, in preparation

TTCM2018, Geneva, Oct.8-10 (2018)Adiabatic principle for gapped systems

Gapped quantum (spin) liquids No symmetry breaking No low energy excitations (Nambu-Goldstone)

Topological characterization for gapped system Example: Adiabatic principle: a lesson from the QHE

flux attachment (Jain)

Adiabatic heuristic argument (Wilczek)

Collect gapped phases and classify into several classes by adiabatic continuation

Label of the Class : Adiabatic invariant (topological number)

Adiabatic heuristic (Wilczek)

Flux attachment (Jain)

d(✓

⇡+

1

⌫) = 0

Connect states by a rule

TTCM2018, Geneva, Oct.8-10 (2018)Short range entangled states

Ex.1) AKLT state

Ex.2) Collection of singlets

(1,1)

Something complicated but gapped

many-body gapsmall

gapped integer spin chain



many-body gap

Adiabatic deformation ! gap remains open



many-body gap




many-body gap



Something very simple & gapped

many-body gap


Decoupled !

big !


Adiabatic process to be decoupled: gap remains open

Collection of

local quantum objects

“Short range entangled state”

TTCM2018, Geneva, Oct.8-10 (2018)How to characterize local object ?

| (✓)i = U(✓)| (0)iU(✓) = ei(S�Sz)✓

If decoupled, the twist by the transformation is gauged away !z

x

y

It characterizes locality of the quantum object !

Consider a gauge transform at some site

How to see this locality by skipping the adiabatic deformation ?

Question ?

Calculate a topological invariant as an adiabatic invariant

Answer !

TTCM2018, Geneva, Oct.8-10 (2018)Gauge transformation & Berry phase

If gauged away, the Berry phase is trivially obtained z

x

y

| (✓)i = U(✓)| (0)iU(✓) = ei(S�Sz)✓

A = h |d i = Sd✓

� = 2⇡S

S = 1/2� = 2⇡S = ⇡S = (odd integer)/2

SpinsZ2

Fermions with filling ⇢ = P/Q, (P,Q) = 1

� = 2⇡⇢ = 2⇡P

Q ZQ

Hirano-Katsura-YH, Phys. Rev. B 78, 054431 (2008)


Adiabatic process to be decoupled: gap remains open

TTCM2018, Geneva, Oct.8-10 (2018)Example: Heisenberg model with local twist

H(x = ei�)

C = {x = ei�|� : 0� 2�}

Si · Sj �12(e�i�Si+Sj� + e+i�Si�Sj+) + SizSjz

Calculate the Berry phases using the many spin wave function

Only link <ij>

Define a many body hamiltonian by local twist as a periodic parameter

i�C =

Z

CA =

Z 2⇡

0h |@

@✓i d✓

H(✓)| (✓)i = E(✓)| (✓)i Lanczos diagonalization

Topological order parameter YH, J. Phys. Soc. Jpn. 75, 123601, ’06

= ⇡, 0Time-reversal

Z2 Berry phase

H0 =X

hiji

JijSi · Sj

TTCM2018, Geneva, Oct.8-10 (2018)Bulk-Edge Correspondence (BEC)

of the short-range entangled states

TTCM2018, Geneva, Oct.8-10 (2018)Bulk-edge correspondence






3 localized states3 localized statesMystery ! I know !

Aha !Hidden


of the short-range entangled states3 localized states at the red sites and no-more

Why these 3 sites are special ?

It is determined by the bulk (before making the boundary)

Q:

A:

M. Arikawa, S. Tanaya, I.Maruyama, and Y.Hatsugai, Phys. Rev. B 79, 205107 (2009)T. Kariyado and Y.Hatsugai, Phys. Rev. B 90, 085132 (2014)

…S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002) SSH & ladders, …


of the short-range entangled statesHow the system is adiabatically decoupled ?Q:See edge states (local modes) by making various boundaries !

A:


of the short-range entangled statesHow the system is adiabatically decoupled ?Q:See edge states (local modes) by making various boundaries !

A:

edge states : “topological order parameter”

TTCM2018, Geneva, Oct.8-10 (2018)

Not that simple, of course

Need symmetry protection or something else for the Berry phase to be a topological invariant The symmetries can be approximate Residual interactions between the edge states

It’s a “dream”

TTCM2018, Geneva, Oct.8-10 (2018)

With some symmetry A ,B ,C

SPT and symmetry breakingGeneric

Symmetries are destined to be broken (Y. Nambu)

With symmetry breaking perturbation

“SPT” !

“No SPT” ? Boring ?

Boring !

(finite but small)

Edge states are still there if the gap is finite reflecting the SPTs’

at least as a one particle theory, fermion zero modes are special as a manybody theory

TTCM2018, Geneva, Oct.8-10 (2018)

Why (breathing) Kagome?

Z2 to ZQ

Y. Hatsugai & I. Maruyama, EPL 95, 20003 (2011), arXiv:1009.3792

(Q=d+1)

Series of lattice models in d-dimensions

...d=1

d=3d=2

d=4

ZQ Berry phases

pyrochlore

SSH

kagome

TTCM2018, Geneva, Oct.8-10 (2018)

Insulators : physicist’s view vs chemist’s ?

Covalent molecular orbital

physicist itinerant electrons

TTCM2018, Geneva, Oct.8-10 (2018)


make energy bandmetal


hopping


TTCM2018, Geneva, Oct.8-10 (2018)


Peierls instability


Opening gap stabilize

hopping


TTCM2018, Geneva, Oct.8-10 (2018)


Peierls instability



chemistform molecules first

hopping


TTCM2018, Geneva, Oct.8-10 (2018)


Peierls instability




hopping


TTCM2018, Geneva, Oct.8-10 (2018)

Peierls instability


make bands of

molecules

stabilize


Adiabatic processInsulator

Dimer & Moleculequantum objects to be respected

non orthogonalityshort range entanglement

EF


TTCM2018, Geneva, Oct.8-10 (2018)Hyper-Pyrochlore in D-dimensions

D=3PyrochloreD=2

Kagome

D=1Polyacetylene

Also in Any D-dim

TTCM2018, Geneva, Oct.8-10 (2018)

H =X

hiji

tijc†i cj + h.c.� µ

X

i

ni

+VX

ninj

One may include interaction if the energy gap remains open

tij =

⇢tR hiji 2tB hiji 2

Tetramerization

3D pyrochlore

“Breathing”

H =X

hiji

tijc†i cj + h.c.� µ

X

i

ni

Trimerization

2D kagome

tij =


“breathing kagome”

TTCM2018, Geneva, Oct.8-10 (2018)

Diagonalizable within

h = Q(tBpB + tRpR) = �D�† Sum of 2 projections

Hamiltonian in momentum space

LB = {c�B

��c � C} LR = {c�R

��c � C}LB + LR

pBLB = LB

pRLR = LR

is invariant for any linear operationLinear space:

LB

LRLB + LR dim (LB + LR) 2

Non zero energy bands are at most 2.Q� 2

L? : null

Non zero energies are eigen states of h� = O1/2hO1/2 2⇥ 2

deth� = Q2tBtR detO Trh� = Trh = QTr (tBpB + tRpR) = Q(tB + tR)

E(k) = (Q/2)�tB + tR ±

p(tB � tR)2 + tBtR|�(k)|2

�Energy bands :

At least zero energy flat bands

If , one of the 2 bands degenerate with the flat bandsk = 0 : touching momentum

detO = 0

tB 6= tR tB = tRE

Eg = |tB � tR|(Q/2)

E

Massless Dirac, CriticalQuantum Phase Transition

tB < tRtB > tR

d=2 Kagome tB = tR tB = tR

Dirac fermions + flat bands with d-1 fold degeneracy

Q=d+1=3

TTCM2018, Geneva, Oct.8-10 (2018)

TTCM2018, Geneva, Oct.8-10 (2018)Ex.) ZQ=3 quantized Berry phases for

fermions on Kagome

✓1✓2

✓3

✓3 = �✓1 � ✓2

d=2, Q=3

|⇥(�)�

Many body state

filling 1/Q

i� =

Z

LA

A = �⇥(�)|d⇥(�)⇥

� ⌘ 2⇥n

Q, mod2⇥, n 2 Z ZQ quantization

modify phases locally (in some way)

periodic boundary condition

Global ZQ symmetry with twists ⇥

⇥ = (✓1, ✓2, ✓3)

TTCM2018, Geneva, Oct.8-10 (2018)Topological order parameter for Breathing “Kagome/Pyrochlore”

tB = tRE

Massless Dirac, Critical

E|tB | > |tR||tB | < |tR|

E

EF EF1/Q filling

Quantum Phase Transition

tR =

Q-Multimerization

d=2, Q=3, Kagome

tij =


tB = �1,

� = 0� =2⇡

Q

Y. Hatsugai & I. Maruyama, EPL 95, 20003 (2011), arXiv:1009.3792

TTCM2018, Geneva, Oct.8-10 (2018)

Higher order topological insulators and related

Kagome higher order topological insulator model

Motohiko Ezawa, PRL (2018)

Frank Schindler, Zhijun Wang, Maia G. Vergniory, Ashley M. Cook, Anil Murani, Shamashis Sengupta, Alik Yu. Kasumov, Richard Deblock, Sangjun Jeon, Ilya Drozdov, Hélène Bouchiat, Sophie Guéron, Ali Yazdani, B. Andrei Bernevig, and Titus Neupert, Nat. Phys. (2018)

Wladimir A. Benalcazar, B. Andrei Bernevig, Taylor L. Hughes, Science (2017)

Frank Schindler, Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, B. Andrei Bernevig, and Titus Neupert, Science Advances (2018)

Haoran Xue, Yahui Yang, Fei Gao, Yidong Chong, Baile Zhang, arXiv:1806.09418

Marc Serra-Garcia, Valerio Peri, Roman Süsstrunk1, osama R. Bilal, tom Larsen, Luis Guillermo Villanueva, and Sebastian D. huber, Nature (2018)

Stefan Imhof, Christian Berger, Florian Bayer, Johannes Brehm, Laurens W. Molenkamp, Tobias Kiessling, Frank Schindler, Ching Hua Lee, Martin Greiter, Titus Neupert, and Ronny Thomale , Nat. Phys. (2018)

Max Geier, Luka Trifunovic, Max Hoskam, and Piet W. Brouwer, PRB (2018)

Shin Hayashi, Commun. Math. Phys. (2018)

Yichen Xu, Ruolan Xue, and Shaolong Wan, arXiv:1711.09202

TTCM2018, Geneva, Oct.8-10 (2018)

Quantized Berry phases for the breathing Kagome model

H = �P

hiji tpc†i cj

where the hopping strengths tp = t1(2)<latexit sha1_base64="10D3F9ZwbGpb6tcg0VcalO0BCI8=">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</latexit><latexit sha1_base64="10D3F9ZwbGpb6tcg0VcalO0BCI8=">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</latexit><latexit sha1_base64="10D3F9ZwbGpb6tcg0VcalO0BCI8=">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</latexit><latexit sha1_base64="10D3F9ZwbGpb6tcg0VcalO0BCI8=">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</latexit>

➤ Z3 Berry phase for k-space

➤ Kagome HOTI model

Z3 Berry phase for ▽ triangleZ3 Berry phase for △ triangle

Phase diagram. γ=(γ△, γ▽)

・1/3 filling for HOTI1 and HOTI2 phases

・2/3 filling for the trivial phase

-1.5 -1.0 -0.5 0.5 1.0 1.5

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0.7

HOTI 1 HOTI 2Trivial

t1/t2

2/3�/2⇡

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-1.5 -1.0 -0.5 0.5 1.0 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t1/t2

2/3

1/3

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・Unit cell is the ▽ triangle.

・Corresponding with the Γ2 corner states

・Unit cell is the △ triangle.


MetallicHOTI 1

HOTI 2Trivial

t2

t1

γ=(2π/3, 0)

γ=(0, 4π/3)

γ=(4π/3, 0)

Bulk dispersion

Bulk dispersion

edge state

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TTCM2018, Geneva, Oct.8-10 (2018)

Quantized Berry phases for the breathing Kagome model

H = �P

hiji tpc†i cj

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➤ Z3 Berry phase for k-space

➤ Kagome HOTI model

Z3 Berry phase for ▽ triangleZ3 Berry phase for △ triangle

Phase diagram. γ=(γ△, γ▽)

・1/3 filling for HOTI1 and HOTI2 phases

・2/3 filling for the trivial phase

-1.5 -1.0 -0.5 0.5 1.0 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7


t1/t2

2/3�/2⇡

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-1.5 -1.0 -0.5 0.5 1.0 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t1/t2

2/3

1/3

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・Unit cell is the ▽ triangle.


・Unit cell is the △ triangle.


MetallicHOTI 1

HOTI 2Trivial

t2

t1

γ=(2π/3, 0)

γ=(0, 4π/3)

γ=(4π/3, 0)

Bulk dispersion

Bulk dispersionedge state

� = 0<latexit sha1_base64="O0PEUluHPYIn12qpGFOQEVpg2I0=">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</latexit><latexit sha1_base64="O0PEUluHPYIn12qpGFOQEVpg2I0=">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</latexit><latexit sha1_base64="O0PEUluHPYIn12qpGFOQEVpg2I0=">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</latexit><latexit sha1_base64="O0PEUluHPYIn12qpGFOQEVpg2I0=">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</latexit>

TTCM2018, Geneva, Oct.8-10 (2018)

Machine Learning for Kagome HOTI ModelsAveraged wave functions of the occupied states with W=1.0

H. Araki, T. Mizoguchi, Y. Hatsugai:arXiv:1809.09865

with randomness

TTCM2018, Geneva, Oct.8-10 (2018)

kagome J1-J2-J3 Ising model

Conservation law (Gauss’ law)

“Bulk-boundary correspondence” in classical spin system!total charge flux

σ=+1

σ=-1

Q=+3

Q=+1

Q=-1

Q=-3

53

第9章カゴメ格子上のJ1-J2-J3イジング模型

×

(a) (b)

A

B

C

x

y

上向き下向き

図 9.1: (a) カゴメ格子および本模型の模式図。赤の点線はカゴメ格子の単位胞を表す。(b) 各三角形の取りうる状態のリスト。赤い点は σz = +1、青い点は σz = −1を表す。三角形の色は電荷を表し、オレンジは Q = +3、黄色は Q = +1、緑は Q = −1、青はQ = −3である。

図 9.2: 隣り合う上下の三角形の配置。

本章では本研究で取り扱うカゴメ格子上の J1-J2-J3イジング模型について説明する。なお、以後は kB = 1とし、長さの単位はカゴメ格子の単位胞の辺の長さを 1とするようにとる。[図 9.1(a)

参照。]

EðαÞc ¼

!1

2− J

"NðαÞ

△

− JnðαÞi þ J2nðαÞb ; ð5Þ

where nðαÞi and nðαÞb are, respectively, the number of internaland boundary sites of the cluster [Fig. 1(f)], which satisfy3NðαÞ

△

¼ 2nðαÞi þ nðαÞb . The term ðJ=2ÞnðαÞb expresses theinteraction between clusters, and the factor 1=2 accountsfor double counting. Any spin configuration can bedecomposed as a paving of the lattice by same-chargeclusters, where every cluster is surrounded by clusters ofopposite charges. The total energy is thus Etot ¼

PαE

ðαÞc .

The Gauss’ law (4) imposes the inequality, NðαÞ△

≤ nðαÞb ,which provides a lower bound to the total energy,Etot ≥ 1

2 ð1 − 3JÞN△

. N△

is the total number of triangles.This lower bound is reached when each and every clusterforms a closed ring, with potential sprouting branches[Fig. 1(f)]. For more details, see the Supplemental Material[39]. By construction, more than one closed ring in asame-charge cluster is impossible. This ground state ismacroscopically degenerate and on average, each chargeneighbors exactly two charges of the same sign.This argument gives rigorous proof that the ground state

is a degenerate set of spin configurations where the entirelattice is covered with same-charge clusters containing oneand only one ring. As long as this constraint is satisfied, eachconfiguration is realized with equal probability. In principle,rings have no upper-size limit. However, any large ringnecessarily encircles other cluster(s), thereby limiting thenumber of configurations hosting large rings. Low-temperature classical Monte Carlo simulations confirm thisscenario. We find that most of the clusters have hexamerrings, composed of six triangles. We call this novel degen-erate ground state the hexamer CSL phase.It is not possible to pave the entire system with hexamer

rings, because this paving resides on a non-bipartitetriangular lattice. It means that branches are necessary toaccommodate the clusters on the lattice. Then, what is thedensity ρhex of hexamer rings defined per number ofhexagons in the lattice, Nhex? Monte Carlo simulationsfind ρhex ∼ 0.08 [Fig. 2(f)], and thus an average number oftriangles per cluster, N

△

=ρhexNhex ∼ 25. On average, thereis an equal number of positive and negative hexamers, butsmall fluctuations exist.When increasing J, the hexamer CSL shows an insta-

bility to triple charge creation. By extending the discussionabove, one can rigorously show [39] that the phaseboundary is at J ¼ 1=3, with a 12-site order for J > 1=3[Fig. 1(e)].Thermodynamic quantities.—The hexamer CSL for 0 <

J < 1=3 gives characteristic features in thermodynamicquantities, especially when compared to the other disor-dered regime, kagome ice, for −1 < J < 0. To see this, wechoose J ¼ −0.1 and 0.1, and perform a Monte Carlosimulation and Husimi-cactus calculations [Fig. 1(g)] toobtain theT dependence of the specific heat (C), entropy (S),

and susceptibility (χ) [Figs. 2(b)–2(e)]. The two methodsmatch quantitatively well.C has two peaks both for J ¼ −0.1 and J ¼ 0.1. The high-

temperature broad peak, around T ∼ 1.5, corresponds to theentropy release due to the vanishing triple charges. Belowthis peak, S takes a value close to the residual entropy of theNN Ising model, S ∼ SNN ¼ 0.502.At low temperatures, however, thermodynamic behaviors

are quite different between these two regions. For J ¼ −0.1,C and χ diverge at Tc ≈ 0.163 due to the long-range chargeorder with ferrimagnetic spin ordering. While a previousstudy interpreted this transition to be of the three-state Pottsuniversality class [20], recent state-of-the-art analyses in

(b) (c)

0

0.2

0.4

0.6

0 1 2 3 4

(d)

(f)

00.5

11.5

22.5

33.5

44.5

-1 -0.5 0 0.5

Paramagnet

FMKagome Ice

12-site order

Hexamer CSL

(a)

1/3

0

0.02

0.04

0.06

0.08

0.1

0 0.2 1 0 2 4 6 8 10

(g)

0

0.2

0.4

0.6

0.8

1

00.20.40.60.81

0 10 20 30

0

0.4

0.8

1.2

0 1 2 3 4

0

200

400

600

0.14 0.16 0.18 0.2

0

0.1

0.2

0 1 2 0

0.04

0.08

0.12

0 1 2 3 4

0.4 0.6 0.8

0

0.2

0.4

0.6

0 1 2 3 4

(e)

FIG. 2. (a) Finite-temperature phase diagram. Circles andcrosses are, respectively, phase transitions and crossovers. Blackstars denote the exact phase boundaries at T ¼ 0. (b)–(e) Temper-ature dependence of (b) C and (c) S at J ¼ −0.1, and that of (d) Cand (e) S at J ¼ 0.1. Insets of (b) and (d) are the magneticsusceptibilities χ. Red lines are results of Husimi-cactus calcu-lations, and orange, green, and blue dots are, respectively, theresults of Monte Carlo simulations with L ¼ 32, 64, and 84. S iscomputed for L ¼ 64. (f) Temperature dependence of the densityof hexamers ρhex for J ¼ 0.1 for L ¼ 64. (g) Autocorrelation,RspinðtÞ≡

Piσ

zi ð0Þσzi ðtÞ=Nsite, for J ¼ 0.1, T ¼ 0.03. Red,

green, and blue dots are for the single spin flip, the loop-updatealgorithm, and the new worm algorithm, respectively.

PRL 119, 077207 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

18 AUGUST 2017

077207-3

2

general combinations of J2 and J3, various magnetically or-dered states have been discovered at T = 0 [12, 13]. In thepresent work, we focus on the case of J2 = J3(≡ J), wherea charge representation is available [19]. To introduce thisrepresentation, we classify the triangles on the kagome latticeinto “upward” (△) and “downward” (▽), depending on theirorientations. We, then, define a charge at each triangle asQp = ηp

!i∈p σ

zi with ηp = +1(−1) for p ∈ △(▽), which is

an analog of magnetic charge in spin ice. In fact, the chargethus defined has a fractional character; flipping a single spinchanges Qp by ±2 on a pair of triangles the spin belongs to,i.e., the change of spin quantum number is fractionalized intoa pair of positive and negative charges. We list possible spinconfigurations on upward and downward triangles and corre-sponding values of Qp in Fig. 1(b).

In terms of Qp, the Hamiltonian can be rewritten as follows,up to a constant term [19]

H ="

12− J

#$

p

Q2p − J

$

⟨p,q⟩QpQq. (2)

Here, the summation is take over the triangles, p, in the firstterm, and the neighboring pairs of upward and downward tri-

(a) (b)

(d)

(e)

(c)

(f)

FIG. 1. (Color online) Schematic picture of (a) the Hamiltonian, and(b) possible charge states on triangles. The red (blue) dots representσ =↑ (↓). The colors of triangles represent the charges; orange, yel-low, green and cyan represent Q = +3,+1,−1, and −3, respectively.The snapshot of a ground state for (c) J < −1, (d) −1 < J < 0, (e)0 < J < 1

3 , and (f) J > 13 .

angles, ⟨p, q⟩, in the second term. The first term representsthe self-energy of a charge. The second term represents theinteraction between charges, introduced by farther-neighborcouplings, J2 = J3 = J; the opposite charges attract for J < 0,while they repel for J > 0.

Since the triangles form a bipartite honeycomb lattice, theground state of (2) seems trivial at first sight. However, pos-sible configurations of Qp are constrained by the underlyingspin structure. The definition of Qp leads to an analogue ofGauss’ law;

$

p∈DQp =

$

i∈∂Dσz

iηi(D) (3)

for a set of triangles, D, and the sites on its boundary, ∂D.Here, i(D) is a triangle in D, which includes the site i.This constraint strictly limits possible charge configurations.Firstly, if we take D as all triangles on a lattice, it directlyleads to a global conservation law:

!p Qp = 0. Secondly, for

arbitrary region D, the total amount of contained charges isbounded by at most the number of boundary sites, implyingthat a long-wavelength charge fluctuation is suppressed.

Phase diagram.- First, we consider the phase diagram atT = 0. At J = 0, the model is reduced to the Ising model withAF nearest-neighbor coupling, for which an exact solution isavailable [10]. At this limit, the ground state is disorderedwith a macroscopic entropy, S = 0.502kB. In terms of thecharge representation, the ground state manifold consists ofthe charge configurations with Q = ±1 on all the triangles.

If finite J is introduced, the ground state degeneracy islifted, and various ordered and disordered phases show up.To see this, it is instructive to rewrite the Hamiltonian (2), as

H = 12

(1 + J)$

p

Q2p −

J2

$

⟨p,q⟩(Qp + Qq)2. (4)

For J < 0, the first and second terms can be easily mini-mized simultaneously, leading to the following rules to con-struct ground-state configurations: (i) Qp + Qq = 0 for all thenearest-neighbor triangle pairs, and (ii) |Qp| = 1 (= 3) on allthe triangles for −1 < J < 0 (J < −1).

The condition (i) shows there is a staggered charge order-ing, or equivalently, uniform spin ordering in the ground state.For J < −1, the ground state is a triple charge crystal, or afully polarized ferrromagnet [Fig. 1(c)].

Similarly, for −1 < J < 0, the system shows a single chargeordering. In this phase, the charge configuration is uniquelydetermined [Fig. 1(d)]. However, the underlying spin con-figurations are left disordered, even showing macroscopic de-generacy. In this regard, the system shows a CSL ground statein this region. This CSL state can be described by a simpleconstraint: all the triangles are occupied by, say, 1-up 2-downconfigurations, accompanying ferrimagnetic spin order. ThisCSL can be mapped to the kagome ice state observed in the[111] magnetization plateau for Dy2Ti2O7. Equivalent stateshave been discussed in artificial spin ice [16] and itinerant sys-tems [19].

Mizoguchi, Jaubert, and Udagawa, PRL 119, 077207 (2017).

Phase diagram

Classical spin liquid with&without charge orders

[学位論文]

以下のハミルトニアンを考える。

H = H1 +H2 +H3 = J1!

⟨i,j⟩n.n.

σzi σ

zj + J2

!

⟨i,j⟩2nd

σzi σ

zj + J3

!

⟨i,j⟩3rd

σzi σ

zj . (9.1)

J1, J2, J3はそれぞれ、最近接、次近接、次々近接のイジング相互作用の結合定数である。ただし、次々近接については、六角形を跨ぐような相互作用は入れないものとする。[図 9.1(a)参照。]

本模型を解析するメリットは、あるパラメーター領域で分数励起間の短距離相互作用という描像が見やすい形にハミルトニアンを書き換えることができる点にある。具体的には、パラメーターを

(J1, J2, J3) = (1, J, J), (9.2)

のようにとる。（すなわち、次近接と次々近接の相互作用が等しくなる場合に着目する。）また、カゴメ格子を構成する各三角形 p上に、以下のように「電荷」を定義する。

Qp =!

i,j,k∈pζp(σi + σj + σk), (9.3)

ここで ζp = ±1は三角形の「向き」に依存する符号因子である。カゴメ格子は上向きの三角形と下向きの三角形が頂点を共有した構造をとる。そこで上向きの三角形 (△と表記する) に対して

ζp = +1, p ∈ △, (9.4)

とし、上向きの三角形 (▽と表記する) に対して

ζp = −1, p ∈ ▽, (9.5)

とする。この電荷を用いてハミルトニアンを書き直す。まず最近接項H1は

H1 =1

2

!

p

(σp1 + σp2 + σp3)2 − (const.)

=1

2

!

p

Q2p − (const.) (9.6)

と書くことができる。ここで pは三角形のラベルである。次に、H2+H3を導出するために、図 9.2

のような、ある上向き三角形とそれに隣接する下向き三角形のペアを考えよう。σ0−-σ4とQ1, Q2

は図中にあるように定義する。このとき、隣り合う電荷の積に −J をかけたものは、

−JQ1Q2 = J(σ1 + σ2)(σ3 + σ4) + Jσ0(σ1 + σ2 + σ3 + σ4) + (const.), (9.7)

となる。この式を見ると、第 1項はちょうど次近接と次々近接の相互作用を足し合わせたものになることがわかる。また、第 2項は最近接相互作用に J をかけたものになっており、この項は先ほどと同様にQの 2乗で書くことができる。第 3項の定数は、2つの三角形が共有するスピン σ0の 2乗に由来する。以上から、すべての隣り合う三角形のペアについて電荷の積に −J をかけた

54

“magnetic charges” on triangles (Castelnovo, et al., 2008)

Tomonari Mizoguchi

TTCM2018, Geneva, Oct.8-10 (2018)

Thank you

Corner states of Kagome lattice as edge states

controlled by the bulk

Documents

Corner states of Kagome lattice & related · 2018-10-19 · Plan TTCM2018, Geneva, Oct.8-10 (2018) Berry phase as a quantum order parameter Symmetry protection & quantization Bulk-edge