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December, 2018 @ RIKEN-CCS, Kobe Tensor network approaches to Disordered quantum systems Yu-Cheng Lin / National Chengchi University Disordered systems spin glass high-T superconductor Anderson localization many-body localization disorder-induced spin liquid t=10000 t=10000

Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

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Page 1: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

December, 2018 @ RIKEN-CCS, Kobe

Tensor network approaches to Disordered quantum systems

Yu-Cheng Lin / National Chengchi University

Disordered systems

✤ spin glass

✤ high-T superconductor

✤ Anderson localization

✤ many-body localization

✤ disorder-induced spin liquid

t=10000

t=10000

Page 2: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Numerical challenges

✤ long relaxation time

✤ tendency to get stuck in local minima

✤ disorder average

✤ broken translational invariance (iDMRG)

✤ difficult even in 1D

Heisenberg chain

spin-spin correlations at T=0

entanglement entropy

H = JX

i

~Si · ~Si+1 (clean chain)

C(r) = (�1)rh~Si · ~Si+ri ⇠ln1/2(r)

r

S1(`) =1

3log2 `+ c1

`

Page 3: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Disordered Heisenberg chain

✤ mean spin-spin correlation at T=0

✤ mean entanglement entropy

H =X

i

Ji~Si · ~Si+1

C(r) ⇠ 1

r2

`

S1(`) =ln 2

3log2 `+ c1

Refael & Moore (2004)

(random chain)

D. Fisher (1995)

✤ Project the strongest coupled pair of spins to singlet

Strong-disorder RG(SDRG)

strongest coupling

effective coupling

Random-singlet state: a non-crossing valence-bond state

J̃ =JJ 0

2⌦< ⌦

Ma, Dasgupta, Hu(1979) Dasgupta, Ma (1980)

D. Fisher, (1994)

1p2(|"#i � |#"i)

Page 4: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Random-singlet state (1D)

✤ fixed-point distribution of

✤ bond-length distribution

� = � ln(⌦), ⌦ = max{J̃}

� ! 1

P (J̃)infinite-randomness

P (J̃) =1

�J̃�1+1/�

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P (`) ⇠ `�2<latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">AAAFNHicfZTfb9MwEMe9EdgoP7aBeOLFIkLakJiSrNrG2wAh8TgQ+yE1ZXLca2rVdjLbYZQsfwyv8M7/gsQb4pW/ASfNurVLsdT0eve5712ds6OUM2087+fC4g3n5q2l5dutO3fv3V9ZXXtwqJNMUTigCU/UcUQ0cCbhwDDD4ThVQETE4Sgavi7jR59AaZbID2aUQleQWLI+o8RY18nqo/31EDjfCDUTuLQ+5s+D4mTV9TZf7G4H7W3sbXrejh/4pRHstLfa2LeecrmoXvsna85S2EtoJkAayonWHd9LTTcnyjDKoWiFmYaU0CGJoWNNSQTobl71X+Cn1tPD/UTZjzS48l7NyInQeiQiSwpiBno2VjqbYp3M9He7OZNpZkDScaF+xrFJcLkZuMcUUMNH1iBUMdsrpgOiCDV2y6aqRKJotUIJZzQRgsheHka0sA+Imcyp/degimIagBIAazWHIzbJZwYE+wLXBFgtMCcewUQBTrPqhT67pgG1xlwiglN5RUcSpcioQaeixkpzmCGYouN385BD35zj3PWLULF4YEJFZFxNwVRhRS7xkFdIlYTHWeczvB6mRSco+StoWSYomgu8aZAPB8SUif/tDOrEyzrNnFHMtrQ1jZ6XDdnH1pwkEula3p5L4/qhKr9noJcXUMRiPiHtD9WIv4d+zRv4bKrzkxsS25dU5Ot25+xsp0bbEcK2wY1rEyDYeGPHMyCYZKmd+aLjWkU3qDTPWM+esdl9EhcTOskpr46L+wHPNw6DTd/a79ru3qv6EllGj9ETtI58tIP20Fu0jw4QRTn6ir6h784P55fz2/kzRhcX6pyHaGo5f/8BmxrY3w==</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit>

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Random-singlet state (1D)

✤ Length-energy relation Unconventional dynamical scaling

✤ Mean spin-spin correlation function:

✤ Magnetic susceptibility:

⌧ ⇠ exp(p⇠)

z = 1dynamical exponent

C(r) ⇠1

r2

D. Fisher (1994)

conventional:

⌧ ⇠ ⇠z

� ⇠ T�1|lnT |�2

L ⇠ ln2 E

Page 5: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

More examples for 1D random singlet phase

✤ bond-disordered J1-J2 chain at the Majumdar-Ghosh point

✤ Heisenberg chain with additional multi-spin couplings

YR Shu, DX Yao, CW Ke, YCL, A Sandvik, PRB 94, 174442 (2016)spinon

A. Lavarelo & G. Roux, PRL 110, 087204 (2016)

amorphous valence-bond solid

SDRG for quantum Ising chain

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hxmax

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H = �X

i

Ji�zi �

zi+1 �

X

i

hxi �

xi

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GS

Page 6: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Fixed-point of random Ising chain

✤ fixed-point distributions of J and hx (infinite randomness)

✤ length-energy scaling

✤ entanglement entropy

P (J̃) =1

�J̃�1+1/�

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P (h̃) =1

�h̃�1+1/�

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� = � ln(⌦)<latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">AAAFMnicfZTfb9MwEMe9EdgoP7axR14iIqQOianpkOAFaYAQvDEQ3SbV1eS419Sq7WS2wyiZ/xZe4Z1/Bt4Qr/wROGnWrW2KpaZX3+e+d72cHaWcadNq/VxZveZdv7G2frNx6/aduxubW/cOdZIpCh2a8EQdR0QDZxI6hhkOx6kCIiIOR9HoVeE/+gRKs0R+NOMUeoLEkg0YJcZtnWxu4zdECPL8Meayid8JiMnOyWbQ2m2Vy180wsoIULUOTra8NdxPaCZAGsqJ1t2wlZpeTpRhlINt4ExDSuiIxNB1piQCdC8vq7f+Q7fT9weJch9p/HL3akROhNZjETlSEDPU875is87XzczgWS9nMs0MSDpJNMi4bxK/aIXfZwqo4WNnEKqYq9WnQ6IINa5hM1kiYRsNLOGMJq5Zsp/jiFr3gJjJnLp/DcraWQAKAJxV747YNJ4ZEOwLLAiwSmCJP4KpApxm5et8tKABlcZSIoJTeUVHEqXIuEanpCZKS5gRGNsNeznmMDDnfh6EFisWDw1WRMblFMwkVuQSx7xEyiB/EnU+x+tRarvtgr+CFmnatj7B6xp5PCSmCPxvZVAFXuap54xirqS9WfS8KMg99pYEkUhX8u5UmiDEqvieg15cQBGL+ZR0P1Qt/gEGFW/gsynPT25I7F6SzZuuc262U6PdCPmuwJ2FCRBs0tjJDAgmWepm3nYDpxi0S80z1ndnbL5P4mJCpzHWXR3h/EWxaBy2d0Nnv38S7L+sLpF1dB89QE0UoqdoH71FB6iDKBqjr+gb+u798H55v70/E3R1pYrZRjPL+/sPDMHXsQ==</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">AAAFMnicfZTfb9MwEMe9EdgoP7axR14iIqQOianpkOAFaYAQvDEQ3SbV1eS419Sq7WS2wyiZ/xZe4Z1/Bt4Qr/wROGnWrW2KpaZX3+e+d72cHaWcadNq/VxZveZdv7G2frNx6/aduxubW/cOdZIpCh2a8EQdR0QDZxI6hhkOx6kCIiIOR9HoVeE/+gRKs0R+NOMUeoLEkg0YJcZtnWxu4zdECPL8Meayid8JiMnOyWbQ2m2Vy180wsoIULUOTra8NdxPaCZAGsqJ1t2wlZpeTpRhlINt4ExDSuiIxNB1piQCdC8vq7f+Q7fT9weJch9p/HL3akROhNZjETlSEDPU875is87XzczgWS9nMs0MSDpJNMi4bxK/aIXfZwqo4WNnEKqYq9WnQ6IINa5hM1kiYRsNLOGMJq5Zsp/jiFr3gJjJnLp/DcraWQAKAJxV747YNJ4ZEOwLLAiwSmCJP4KpApxm5et8tKABlcZSIoJTeUVHEqXIuEanpCZKS5gRGNsNeznmMDDnfh6EFisWDw1WRMblFMwkVuQSx7xEyiB/EnU+x+tRarvtgr+CFmnatj7B6xp5PCSmCPxvZVAFXuap54xirqS9WfS8KMg99pYEkUhX8u5UmiDEqvieg15cQBGL+ZR0P1Qt/gEGFW/gsynPT25I7F6SzZuuc262U6PdCPmuwJ2FCRBs0tjJDAgmWepm3nYDpxi0S80z1ndnbL5P4mJCpzHWXR3h/EWxaBy2d0Nnv38S7L+sLpF1dB89QE0UoqdoH71FB6iDKBqjr+gb+u798H55v70/E3R1pYrZRjPL+/sPDMHXsQ==</latexit>

⌦ = J̃max or h̃max<latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">AAAFXXicfZTPb9MwFMe9sbJRxtjgwIGLRYQ0OExNhwQSAg0QEuLCQOyHVFeV476kVmMnsx22kvm/4a/hChdO/Cs4adb1R4oPybP9ed/38vLsII25Nq3Wn5XVG2uNm+sbt5q3N+9s3d3euXesk0wxOGJJnKjTgGqIuYQjw00Mp6kCKoIYToLhu2L/5BsozRP51YxS6AoaSR5yRo1b6m2/Jp8ERPQVMTzuQ/7R9oiBC5MLemHJS1xOcJ4obHGFDKaR3rbX2muVAy8afmV4qBqHvZ21ddJPWCZAGhZTrTt+KzXdnCrDWQy2STINKWVDGkHHmZIK0N28/FCLH7uVPg5dOmEiDS5Xpz1yKrQeicCRgpqBnt8rFuv2OpkJX3RzLtPMgGTjQGEWY5Pgomq4zxUwE4+cQZniLlfMBlRRZlxtZ6IEwjabRMI5S4Sgsp+TgFn3gIjLnLmvBmXtLAAFAM6q3w74xJ8bEPw7LAjwSmDJfgATBTjLyj//dEEDKo2lRABnckpHUqXoqEanpMZKS5ghGNvxuzmJITSXOPd8SxSPBoYoKqOyC2YCK3qNk7hESic89rqc4/UwtZ12wU+hRZi2rQ/wvkaeDKgpHP+bGVSO13HqOaO4S2l/Fr0sEnKP/SVONNCVvDvAxvOJKt5z0JsrKOBRPCHdRNXiXyCs+OIIl+cnNzRyP8nmu65yrrdTo10LYZfgk4UOEHxc2HEPCC556nredjyn6LVLzXPed2dsvk7iqkMnPsXV4c9fFIvGcXvPd/bnZ97B2+oS2UAP0SO0i3z0HB2gD+gQHSGGfqCf6Bf6vfa30WhsNrbG6OpK5XMfzYzGg3+gvud7</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit>

` ⇠ �2 ) z = 1<latexit sha1_base64="/mFl7xSLEE4bDxTs6/yJbF0ewXA=">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</latexit><latexit sha1_base64="/mFl7xSLEE4bDxTs6/yJbF0ewXA=">AAAFXHicfZTPb9MwFMe90bLRMehA4sIlIkIaHKamQ4LLpAFCcBwT+yHVZXKcl9Sq7WS2w+iy/DX8NVzhxIW/BefHurVNsdTkxe/zvs99frafcKZNr/dnZfVOq313bf1eZ+P+5oOH3a1HxzpOFYUjGvNYnfpEA2cSjgwzHE4TBUT4HE788fvCf/INlGax/GImCQwFiSQLGSXGTp119zBwjjUTDv5IhCBf+w4+T0mAD1k0MkSp+KL8dnCZK1MQ5NnlHmYyNJP8rOv2dnrlcBYNrzZcVI+Ds63WGg5imgqQhnKi9cDrJWaYEWUY5ZB3cKohIXRMIhhYUxIBepiVuXPnuZ0JnDBW9ieNU87ejsiI0HoifEsKYkZ63ldMNvkGqQnfDDMmk9SApFWiMOWOiZ2iaE7AFFDDJ9YgVDG7VoeOiCLU2NLOZPFF3ulgCRc0ttWUQYZ9mtsHRExm1P5rUHk+C0ABgLWa3T6bxjMDgl3CggCrBZb4fZgqgN3LYuNfLmhArbGU8OFc3tKRtjfIpEGnpCqlJcwYTD7whhnmEJorJ3O9HKui3bAiMiq7YCaxIjc45iVSBjlV1NUcr8dJPugX/C20SNPPmxN8aJDHI2KKwP+uDOrAmzzNnFHMLml3Fr0qFmQfu0uCiK9reXt+jethVbznoLfXkM8iPiXth2rEDyGseQPfTXWiDYnsJuXZtq2c7e3EaNtCjl3gi4UOEKwqbNUDgkmW2J7PB65VdPul5gUL7Bmbr5O47tBpTHF1ePMXxaJx3N/xrP35lbv/rr5E1tFT9AxtIw+9RvvoEzpAR4iiH+gn+oV+t/62W+2N9maFrq7UMY/RzGg/+QcprObH</latexit><latexit sha1_base64="/mFl7xSLEE4bDxTs6/yJbF0ewXA=">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</latexit><latexit sha1_base64="/mFl7xSLEE4bDxTs6/yJbF0ewXA=">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</latexit>

S1(`) =ln 2

6log2 `+ c1

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D. Fisher (1994)Refael & Moore (2004)

Entanglement entropy

100 101 102 103

0.6

0.8

1

1.2

1.4

1.6L = 32L = 64L = 128L = 256L = 512L = 1024

0 0.2 0.4 0.6 0.8 10.60.8

11.21.41.61.8

2

S(ℓ

)

ℓ/L

S+

ln2

6lo

g 2(5

12/ℓ

)

verified by numerics via free fermion representation

S` 'ln 2

6ln

L

⇡sin

✓`⇡

L

◆�+ const.

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Igloi & YCL, JSTAT (2008)

Page 7: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Griffiths singularities

✤ in the paramagnetic phase

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✓J

hx

◆1/z

= 1<latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit>

F. Igloi, Phys. Rev. B 65, 064416 (2002)

hx > J

Extension of SDRG

Page 8: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Extension of SDRG

✤ Basic idea: refine the perturbative approximation in SDRG by including higher energy states

HBB0

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

HBB0 = HB +HB0 +HCBB0

HCBB0 = JC ~SR

B · ~SLB0

Extension of SDRG

✤ Compute the energy spectrum of each pair of nearest-neighbor blocks

�� �� ��

HBB0

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

�� ��

Page 9: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Extension of SDRG

✤ Merge the pair of blocks with the largest energy gap into a block using the lowest eigenstates

fusion

�max �

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

�max�

✤ Formulate blocks in terms of Matrix Product Operators

Implementation using MPO

Goldsborough & Römer, PRB 89 (2014)

W W W W W W

work with Yu-Ping Lin, Zheng-Lin Tsai, Ying-Jer Kao, Pochung Chen

Page 10: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Tree tensor network SDRG (tSDRG)

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)Goldsborough & Römer, PRB 89 (2014)

�max� isometry

V †V = I

RG

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Tree tensor network SDRG (tSDRG)

Goldsborough & Römer, PRB 89 (2014)

disordered tree tensor network

Page 11: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Energy gaps (at the IRFP)

-25 -20 -15 -10 -5 0x = ln(ε1)

10-5

10-4

10-3

10-2

10-1

100

P(x)

L = 16L = 32L = 64L = 128

-5 -4 -3 -2 -1 0x = ln(ε1)L-1/2

10-3

10-2

10-1

100(a) (b)

quantum Ising chain

at criticality

z = 1

Ji 2 (0, 1], hxi 2 (0, h0]

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Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Energy gaps (Griffiths phase)

-5 0 5

x = ln(ε1Lz)

10-3

10-2

10-1

P(x

)

L = 16

L = 32

L = 64

L = 128

-2 0 2 4 6

x = ln(ε2Lz)

10-3

10-2

10-1

-10 -5 0

x = ln(ε1)

10-3

10-2

10-1

P(x

)

(a) (b)

Frechet distribution

z > 1

in the PM phase

Ji 2 (0, 1], hxi 2 (0, h0]

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Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Page 12: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

2-point correlations

(a) (b)

Bulk spin correlations

101 102

L10-4

10-3

10-2

10-1

C(L/2)

δ = 1 (tSDRG) δ = 2 (tSDRG)

~ 1/L2

Heisenberg AF S=1/2 chain

⇡(J) =1

�J1/��1

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0 < J 1<latexit sha1_base64="mr3PLjcuudTXVvcoVMoQhdUgnpw=">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</latexit><latexit sha1_base64="mr3PLjcuudTXVvcoVMoQhdUgnpw=">AAAFJXicfZTPb9MwFMe9EdgoP7bBkUtEhDQ4TE2HBAcOA4SEOA3EfkhNNTnua2rVdjL7hVEy/xNc4c5fww0hceJfwUmzbm1TLCV58fu873txnh1nghtst/+srF7zrt9YW7/ZunX7zt2Nza17hybNNYMDlopUH8fUgOAKDpCjgONMA5WxgKN49Lr0H30CbXiqPuI4g56kieIDzii6qeP2i3eRAD882QzaO+1q+ItGWBsBqcf+yZa3FvVTlktQyAQ1phu2M+wVVCNnAmwryg1klI1oAl1nKirB9IqqYOs/cjN9f5Bqdyn0q9mrEQWVxoxl7EhJcWjmfeVkk6+b4+B5r+AqyxEUmyQa5MLH1C+/3u9zDQzF2BmUae5q9dmQasrQrdFMlljaVitScMZSKanqF1HMrLtBwlXB3FeDtnYWgBIAZzW7Yz6N5wiSf4EFAV4LLPHHMFWA07z6g08WNKDWWErEcKqu6CiqNR036FTURGkJMwK03bBXuBYa4LlfBKGNNE+GGGmqkqoLZhJreolHokKqIH8SdT7Hm1Fmu52Sv4KWaTq2OcGbBvloSLEM/G9lUAde5mnmUHNX0u4sel4W5G67S4JobGp5txExCCNdPueglxdQzBMxJd2LbsQ/wKDmET5jtX8KpIn7SbbYdivnejtD41rIdwU+XugAyScLO+kByRXPXM/bbuAUg06lecb7bo/Nr5O86NBpjHVHRzh/UCwah52d0NnvnwZ7r+pDZJ08IA/JNgnJM7JH3pJ9ckAYEeQr+Ua+ez+8n94v7/cEXV2pY+6TmeH9/QcnFNLR</latexit><latexit sha1_base64="mr3PLjcuudTXVvcoVMoQhdUgnpw=">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</latexit><latexit sha1_base64="mr3PLjcuudTXVvcoVMoQhdUgnpw=">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</latexit>

Page 13: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

End-end spin correlations

101 102

L

10-2

10-1

C1,L

Heisenberg (binary J=2, J=1/2)Ising (critical point)

~ 1/L

End-end spin correlations

-30 -25 -20 -15 -10 -5 0x = ln(C1,L)

10-5

10-4

10-3

10-2

10-1

100

P(x)

L = 16L = 32L = 64L = 128

-5 -4 -3 -2 -1 0x = ln(C1,L)L-1/2

10-3

10-2

10-1

100(a) (b)

quantum Ising chain

at criticality

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Page 14: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

High-order moments

✤ n-th moment of an observable O of the form which requires a multinomial expansion

(LX

i=1

Oi)n =

X

k1+k2+···+kL=n

n!

k1! k2! · · · kL!

LY

j=1

Okj

j

On =

LX

i=1

Oi

!n

High-order moments

✤ MPO for the n-th moment

On =

LX

i=1

Oi

!n

= W [1]OnW

[2]On · · ·W [L]

On

W[i]On =

0

BBBBBBB@

1

Oi 1

O2i

�21

�Oi 1

.... . .

On

i

�n

1

�O

n�1i

�n

2

�O

n�2i

· · · 1

1

CCCCCCCA

Yu-Ping Lin, Y.-J. Kao, P. Chen & YCL, PRB 96, 064427 (2017)

Page 15: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Binder cumulant

0.8 0.85 0.9 0.95 1 1.05

Γ0

0.4

0.5

0.6

0.7

0.8

0.9

U

L = 8

L = 16

L = 32

L = 64

L = 128

0 0.05 0.11/L

0.85

0.9

0.95

1

Γ0

c

h0random quantum Ising chain

Ji 2 (0, 1], hxi 2 (0, h0]

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Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Binder cumulant

quantum Ising AF chain

add a weak longitudinal field h

0 0.2 0.4 0.6 0.8 1

Γ0

0.2

0.4

0.6

0.8

1

U

L = 16

L = 32

L = 64

L = 128

0 0.5 1Γ0

0

0.5

1

<ms2>

h0hx

h=0.02Ji 2 (0, 1], hx

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Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Page 16: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Phase diagrams

0 0.5 1 1.5 2

h

0

0.2

0.4

0.6

0.8

1

Γ

DMRG

QMC

1-0.19h2

1.4(2-h)CMP

AFM

PM

FM

hx

clean case

2

1 IRFP

AFM

PM

FM

PM

h

Γ0hx

random case

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96, 064427 (2017)

H =X

i

Ji�zi �

zi+1 �

X

i

hxi �

xi � h

X

i

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Heisenberg S=1 chain

gapless Haldane phase random singlet phase

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strength of bond randomness

Refael & Moore J. Phys. A (2009)P. Lajko, E. Carlon. H. Rieger, F. Igloi, PRB (2005)

Hyman & Yang, PRL (1997)Months, Golinelli & Jolicoeur, PRL (1997)

Page 17: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Heisenberg S=1 chaingapless Haldane phase random singlet phase

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101 102

L

10-2

10-1

100

C1,L

binary J=1.5, J=1/1.5binary J=2.2, J=1/2.2 ~ 1/L

Preliminary

end-end spin correlation

r=L/2

Heisenberg S=1 chaingapless Haldane phase random singlet phase

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string order parameter

Preliminary

Oz(r) = �

*Szj exp

2

4i⇡j+r�1X

k=j+1

3

5Szj+r

+

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Page 18: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

r=L/2

Heisenberg S=1 chaingapless Haldane phase random singlet phase

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string order parameter

Preliminary

Oz(r) = �

*Szj exp

2

4i⇡j+r�1X

k=j+1

3

5Szj+r

+

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Higher dimensions

SDRG in high dimensions

Bhatt & Lee, PRL (1982)Westernerg et al, PRB (1997)

Melin , EPJB (2000)YCL, et al., PRB (2003)

taken from Kimchi, Nahum & Senthil, PRX 8, 031028 (2018)

Page 19: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

SDRG in higher dimensions

RG

square lattice (no frustration)

F

F

AF

no frustration

SDRG in higher dimensions with geometrical frustration

RG

formation of large spins

Page 20: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

2D: AF order

0 0.01 0.02 0.03 0.04 0.05 0.06 0.071/L

0

0.002

0.004

0.006

0.008

m2

square latticetriangular lattice

2D SDRG

2D: AF order

0 1 2 3 4 5 6δ

0

0.1

0.2

0.3

0.4

mAF

QMC~ exp (-0.3 δ)spin-waves~ exp (-0.26 δ)

0 1 2 3 4 5 6δ

0.1mAF

strength of disorderN. Laflorencie, et al, PRB 73, 060403(2006)

square lattice

finite AF order !

AF order is robust against strong bond randomness.

⇡(J) =1

�J1/��1, for 0 < J 1

Page 21: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

2D random-singlet phase

JQ model with random J or Q

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

i j

Jxi

j

Jy

i j

k l

m n

Qx

i

j

k

l

m

n

Qy

2D random-singlet phase

8 12 16 24 32 48

10-4

10-3

10-2

Cs(L

/2,L

/2)

J/Q=0

Q/J=4

Q/J=2

128 16 24 32 48L

10-4

10-3

10-2

10-1

Cd(L

/2,L

/2)

J/Q=0

(a)

(b)

C(L) ⇠ 1/L2<latexit sha1_base64="RE33agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE33agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE33agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE33agLjNkhIN9M48fWdg/HPze4=">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</latexit>

D(L) ⇠ 1/L4<latexit sha1_base64="kllCNRJ/aOFmGeGyKxdS0xgfC88=">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</latexit><latexit sha1_base64="kllCNRJ/aOFmGeGyKxdS0xgfC88=">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</latexit><latexit sha1_base64="kllCNRJ/aOFmGeGyKxdS0xgfC88=">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</latexit><latexit sha1_base64="kllCNRJ/aOFmGeGyKxdS0xgfC88=">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</latexit>

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

Page 22: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

2D random-singlet phase

10-2

100

T/Q

10-2

10-1

χu ,

χlo

c

L=48, χloc

L=48, χu

10-2

10-1

100

T/J

10-2

10-1

χu ,

χlo

c

L=64, χloc

L=64, χu

Q/J=4

J/Q=0

� ⇠ T�1+2/z<latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">AAAFMnicfZTNbtNAEMe3xdASvlp65GJhIRUQJU6R4FhASBwLatpKcajWm4mzinft7o4pqbPPwhXuvAzcEFcegrXjpk3isFKcyc5v/jMZz26Yxlxjs/lzZfWac/3G2vrNxq3bd+7e29i8f6iTTDFosyRO1HFINcRcQhs5xnCcKqAijOEoHL4t/EefQWmeyAMcpdAVNJK8zxlFu3WysRWwAQ80F+7Bp/yZ/7T1/NycbHjNnWa53EXDrwyPVGv/ZNNZC3oJywRIZDHVuuM3U+zmVCFnMZhGkGlIKRvSCDrWlFSA7uZl9cZ9ZHd6bj9R9iPRLXevRuRUaD0SoSUFxYGe9xWbdb5Ohv1X3ZzLNEOQbJKon8UuJm7RCrfHFTCMR9agTHFbq8sGVFGGtmEzWUJhGo1AwhlLhKCylwchM/YBEZc5s/8alDGzABQAWKveHfJpPEcQ/BwWBHglsMQfwlQBTrPydT5Z0IBKYykRwqm8oiOpUnRUo1NSE6UlzBDQdPxuHsTQx7Gbe74JFI8GGCgqo3IKZhIreokHcYmUQe4kajzH62FqOq2Cv4IWaVqmPsG7GvlgQLEI/G9lUAVe5qnnUHFb0u4sOi4Kso/dJUE01JW8PZXo+YEqvueg1xdQyKN4Stofqhb/CP2KR/iC5fnJkUb2JZl823bOznaK2o6Qawt8vDABgk8aO5kBwSVP7cybjmcVvVapecZ79ozN90lcTOg0prg6/PmLYtE4bO341v7wwtt7U10i6+QBeUi2iU9ekj3ynuyTNmFkRL6Sb+S788P55fx2/kzQ1ZUqZovMLOfvP7ln158=</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit>

z>2

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

Experiment

cf. M. Watanabe, N. Kurita, et cl. PRB 98, 054422 (2018)

1 10 T [K]

2.5

3.0

3.5

χu [

10

-3 e

mu/m

ol]

1 10T [K]

2.5

3.0

3.5

4.02.0

2.2

2.4

2.6

2.5

3.0

3.5x = 0.2 x = 0.3

x = 0.4 x = 0.5

a1 = 0.82

a2 = 0.70

a1 = 0.81

a2 = 0.73

a1 = 0.78

a2 = 0.73

a1 = 0.73

a2 = 0.62

Sr2CuTe1-xWxO6

J1-J2 on the square lattice

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Page 23: Disordered systems - Kobe Universityquattro.phys.sci.kobe-u.ac.jp/kobe_2018/Slides/Lin.pdf · More examples for 1D random singlet phase bond-disordered J1-J2 chain at the Majumdar-Ghosh

Future work

✤ dMERA (Goldsborough & Evenbly PRB (2017))

✤ tSDRG in 2D

✤ tSDRG for dynamics

taken from PRB 95, 024205 (2017)✤ Vosk & Altman, PRL (2013)✤ Pekker et. al. PRX (2014)