Sine square deformation

Chisa Hotta

Komaba, Univ. of Tokyo

C. H, S. Nishimoto, N. Shibata, PRB 87 115128 (2013)C. H & N. Shibata, PRB 85, 041108R (2012)

C. H, Asano, arxiv /1809.05200/, PRB rapid comm in press.

S. Nishimoto, N. Shibata, C. H. Nature Comm. 4, 2287 (2013)

What is SSD ?

thermodynamic potential

" sine square deformation "

"sine-square-deformation"= SSDNishino, et al. proposed it as a smooth boundary condition.

...L

envelope function

L

Are we allowed to spatially modify the full Hamiltonian as we like ?

- It is not like putting O(1) potential at the edges.- What happens to the bulk O(N) properties ?

not that simple!

→　Yes!

Low energy structures of critical systems are properly renormalized.→　Ej (N) ∝ 1/N2

Quasi-particle trapped near the center.→　fine evaluation of the energy gap

To have a spatially uniform wave functionfor all the imaginary time τ,

H should be spatially nonuniform.

Poincare disc

real space

Finite size correction is 1/N 2Ground state energy of

2

Why we started SSD.

Jt t

1electron/2site

small J : Dimer long range order ? or Tomonaga Luttinger liquid ?

severe boundary effect

2003

Shibata, C. H, PRB 84, 115116 (2011).”Boundary effects in density matrix renormaliation group calculation”

0 5 10 15 20 25 30

L = 32sin2-deformed OBCOBC

i

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Sz iSz i+

1

L = 32sin2-deformed OBCOBC

-0.88

-0.86

-0.84

-0.82

-0.8

-0.78

-0.76

-0.74

-0.72

c ic i

+1+

h.c.

+

SSD can suppress the boundary effect, but else...

→ Grand canonical analysisQuasi-particles are localized at the edge.

Why we started SSD.

33

isite index

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Ng= 37363534

32313029

deformation

L=50 μ=1

2/3"bulk"

L=50 μ=1

Ng=

32313029

3736353433

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

periodic boundary

Ng/L

translational symmetry

deform

isite index

......L

Grand canonical analysis

C. H - Shibata, PRB 85 041108R (2012)

Ng :total particle number

=1

=2

exact solutionL=50

0 1 2 30

0.1

0.2

0.3

0.4

0.5

h/J

M/L 3250

100

L=

1.51.0

0.1

0.15

Heisenberg

XXZ

Grand canonical analysis

Heisenberg S= 1/2 spin chain

Regardless of whether there is an interaction or not, particles are fermions or bosons, electrons,L~O(10) small system reproduces properties (particle density, magnetization, energy... )

at L= within 10-4 accuracy in 1D.

S=1/2 zigzag chain

0 1 2 3

J2/J1=0.5

H/J1

J2/J1= 1

EO-Chiral

Chiral-EO

0.5

0.4

0.3

0.2

0.1

0

M/L

L=50,100

Majumdar-Gosch

J1J2

spin-1/2 Heisenberg zig-zag chain

PWFRG

Okunishi-Hieda-Akutsu(1999)

3250

100

L=

1.85 1.900.24

0.25

0.26

0.27

0.28

J2/J1=0.5

3250

100

L=

1.85 1.90

0.24

0.25

0.26

0.27

0.28

J2/J1= 0.5

TL1

TL2

EO

P

TL2

D

chiralchiral

00 0.5 1

h/hs

J2/J1

J1/J2

0

1 Okunishi (2008)DMRG N< 384

1

0

-1

0.8 0.9 1 1.1 1.2

1/2-filled

filling

L= 64

hubbard gap(exact)

ρ

-μ

1D Hubbard model

Kagome heisenberg antiferromagnet

Q=3

NNiisshhiimmoottoo--SShhiibbaattaa--CCHH NNaattuurree CCoommmm.. 44,, 22228877 ((22001133))

0 2 31

N=27364554

h/hsat

Honecker,et.al.(‘03)

M/Msat

0

0.5

1

0 1 2 3

1/9

1/3

5/9

7/9M

/Msa

t

H/J

0-plateauZ2 spin liquid

1/2

0

1/2

1

1/2

2

Long range ordered plateauswidth= spin gap Δ~0.05(5)

Z3 spin liquid

Applications :Checkerboard Heisenberg model

Morita-Shibata, PRB 94, 140404(R) (2016) SSD

PBC &OBC

How SSD works ~ Free fermion

C. H, Nishimoto, Shibata, PRB 87 115128 (2013)see alsoI. Maruyama, H. Katsura, T. Hikihara,PRB 84 65132(2011)Fourier

transform

0

ε(k)

μ

k

eigen levels

1D free fermion

εl

leigenvalue index

L= 50μ= 0.5

μmix

0

ε(k)

μ

k

eigen levels

1D free fermion

How SSD works

eigen states = plane waves

k :good quantum number

...

C. H, Nishimoto, Shibata, PRB 87 115128 (2013)

εl

leigenvalue index

L= 50μ= 0.5

μ

SSD

mix

mix

eigen states = wave packets

edge states are formednear μ

SSD energy scale

light mass heavy masszero-energy

1

0

real space

single impurity Kondo problem

logerithmic discretization

=

...0 1 2 3 N

impurity

scaling relation

-1 -2 -2 -1-1 1

HK= dk ak ak -J A σA τ

ｘΛ　

Real space renormalization

Okunishi-Nishino PRB 82,144409(2010)

application to real space:

cut

SSD

hopping/energy scale depends onthe location (in real space)

hopping/energy scale depends on"n" (location in k-space)

wave function is a wave packet = does not feel the system size

0.10.05 0.20.150

2050 12 10 8 6

-2

-1

0

1

2

1D Free fermion:

Finite size scaling behavior ~ PBC

uniform & periodic boundary (PBC)

~1/N1D quantum criticalexcitation energy

C. H, Asano, arxiv /1809.05200/

0.02

2050 12 10 8 6

0 0.030.01

-2

-1

0

1

2

~1/N 2excitation energy

SSD

-2 -1 0 1 2

60

0.5

0.15

0

0.2

0.1

0.05

8

10

12

20

50

Finite size scaling behavior ~ SSD

PBC(periodic uniform sys) SSD

0 1-1

particle distribution in space

usual discrete DOS ∝ 1/energy spacing (system-center particle #)/energy spacing

C. H, Asano, arxiv /1809.05200/

-2 -1 0 1 2

60

0.5

12

10

80.02

0

0.03

0.01

Thermodynamic properties

Could we determine this curve free of parameter-tuning/bias?and without size effect?

C~T aC~exp(-Δ/T )

when magnetically gapped

when gapless

~J~J High T expansion

Bernu, Misguich PRB 63 134409 (2001)

0

c TT dT ln 2S 1

0c T dT e T e T 0

Entropy method

interpolate

SSD will allow us to.

S= 1/2 1D XX chain ~free fermion

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4brokenline : exact solution

0 0.2 0.4 0.6 0.8 1

C. H, Asano, arxiv /1809.05200/

4

58

6 45

8

6

SSD PBC

0 1

20304050

100

4

5816

N= 456

10121620304050

100

8

10

SSD PBC

10-1

10-5

10-6

10-3

10-4

10-2

0 0.2 0.4 0.6 0.8 1

error

10-1

10-5

1

10-6

10-3

10-4

10-2

10

0 0.2 0.4 0.6 0.8 1

0

0.1

0.18

1D Heisenberg chain

0

0.1

0.18

0 1 2 3 0 1 2 3

exact solution

C. H, Asano, arxiv /1809.05200/

SSD PBCN= 4

6456

81012

7

N=

0 1 2 310-6

10-4

10-3

10-2

10-1

10-5

10-7

1

10-6

10-4

10-3

10-2

10-1

10-5

0 1 2 3

:broken line

error fromN= ∞

SSD PBC

N=16 SSD

0

0.1

0.18

0 1 2 3

1D Heisenberg chain

0 0.2 0.4 0.6 0.8 1

0

0.1

0.18

seems Gapped! = WRONG

10-6

10-4

10-3

10-2

10-1

10-5

10-7

1

10-6

10-4

10-3

10-2

10-1

10-5

0

0.1

0.18

0 0.2 0.4 0.6 0.8 1

SSD PBC

error fromN= ∞

456

81012

7

N=

C. H, Asano, arxiv /1809.05200/

Previous reports on 2D Heisenberg models

Quantum Monte Carlo (QMC)

Exact diagonalization ~ finite T lanczos, TPQ

High temperature expansion + Pade

Coupled cluster method (CCP)

size effect, not reliable at kBT < J

kBT < J not available at all.~

~

short range correlation only?

sign problem.

extrapolation (highT + ground state data) physical "intuition" required.Details are not precise.

What shall we beleive as a reference ?

0

0.1

0.2

0.3

0.4

0.5

0 1 20

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2

M. S. Makivic, H.-Q. Ding, PRB 43, 3562 (1992)Okabe, Kikuchi JPSJ 57, 4351 (1988) Bernu Misguish extrapolation

HighT-E

M. Takahashi, Modified SW

QMC

high-temperature expansion

QMC N=128x12812x12

0

0.1

0 1 2 3

2D S= 1/2 Heisenberg antiferromagnets

square lattice

Makivic, Ding, PRB 43, 3562 (1992)

Okabe, Kikuchi JPSJ 57, 4351 (1988)

SSD

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5

High-T expansion

entropy method(gapped g.s.)

Transfer matrix Monte Carlo

0 1 2 30

0.1

2D S= 1/2 kagome Heisenberg antiferromagnets

kagome lattice

entropy method

High-T expansion

Transfer matrix Monte Carlo

Lohmann, Schmidt, Richter, PRB 89, 014415 (2014)

Bernu-Lhuillier, PRL 114, 057201 (2015)

T. Nakamura, Miyashita, PRB52, 9174 (1992)

SSD

Summary

SSD ~sine square deformation.- real space renormalization effect

→ Low energy sectors of the thermodynamic limit are well reproduced..

Ground state properties, thermodynamic properties. ..almost free of size effect, boundary effect.

Collaborators:Naokazu Shibata (Tohoku Univ. Jpn)Satoshi Nishimoto (Dresden, Germany)Kenichi Asano (Osaka Univ. Jpn)