Ribhu Kaul University of Kentucky Collaborators Nisheeta Desai (U. - - PowerPoint PPT Presentation

Designer Hamiltonians for quantum spin liquids & exotic valence bond ordered states

Ribhu Kaul

University of Kentucky

Collaborators

• Dr. Jon Demidio (U. Ky Lausanne)
• Prof. Matt block(U. Ky CSU-Sacramento)

Nisheeta Desai (U. Ky)

Designer Models Model RVB: Topological Order Haldane Nematic

Designer Models Model RVB: Topological Order Haldane Nematic

Monte Carlo - Field Theory

S = Z (rφ)2 + rφ2 + uφ4

H = −J X

hiji

σiσj

Quantum Spin Chains

HJ = J X

hiji

Si · Sj

d=1 DMRG, CFT, bosonization, Bethe ansatz

S = 1 g Z d2x(∂µˆ n)2 + i2SπQ

Designer Models

simple models of quantum many body physics sign problem free interesting physics study connections with QFT/topological terms in d>1 universality

R.K. Kaul, R.G. Melko A. W. Sandvik, Ann. Rev. Cond. Mat. (2013)

Designer Models Model RVB: Topological Order Haldane Nematic

Anderson RVB (1973)

RVB wavefunction

Savary, Balents (2016)

Assign one of N color to each site Many body Hilbert space N

site

N

The Model

The Model

|Si = 1 p N

N

X

α=1

|ααi

define two-site “bond” wave function e.g. For N=3

+ +

i i i

| | |

|Si

1 √ 3

=

( (

call this a “singlet” wave function

• R. K. Kaul PRL (2015)

The Model

HJ = J X

hiji

|SijihSij|

|Si = 1 p N

N

X

α=1

|ααi

Model is a sum of “singlet” projectors

• R. K. Kaul PRL (2015)
• cf. Affleck; Sachdev & Read; Harada, Kawashima & Troyer.

Notes on Model

• In spirit of Anderson’s original proposal
• symmetry: SO(N) singlets
• Apart from sign - Heisenberg model

HJ = J X

hiji

|SijihSij|

|Si = 1 p N

N

X

α=1

|ααi

Hilbert Space

|SihS| = 1 N X

α,β

|ααihββ|

Quantum Dynamics

|SihS| = 1 N X

α,β

|ααihββ|

Quantum Dynamics

|SihS| = 1 N X

α,β

|ααihββ|

Quantum Dynamics

|SihS| = 1 N X

α,β

|ααihββ|

Quantum Dynamics

|SihS| = 1 N X

α,β

|ααihββ|

Quantum Dynamics

|SihS| = 1 N X

α,β

|ααihββ|

Competition in model

HJ = J X

hiji

|SijihSij|

Competing tendency: Equal colors vs singlet formation

Quantum, Classical: Loop model

Tr(e−βH) =

• n

(−β)nTr(Hn) n!

typical term in an expansion for d=1 forms a tightly packed loop model!

• cf. Nahum, Chalker, Serna, Somoza, Ortuno

SSE review, Sandvik (2010)

Quantum Phase Diagram

loops v/s wavefunctios (small-N) : long loop phase — “quadruploar phase” (large-N) : short loop phase — “singlet phase” what is the nature of singlet phase?

Kagome

Block, D’Emidio, Kaul (arxiv:2019)

HJ = J X

hiji

|SijihSij| Hb = − J N X

hiji

⇣ b†

iαb† jα

⌘ ⇣ bjβbiβ ⌘

Arovas, Auerbach; Read, Sachdev; Affleck Block, D'Emidio, Kaul (arxiv:2019)

nb = 1

With forms new SO(N) spin “bosonic” representations

nb 6= 1

Large-N

Phase Diagram

Block, D’Emidio, Kaul (arxiv:2019)

Large-N

Block, D’Emidio, Kaul (arxiv:2019)

Lb = b∗

iα∂τbiα + N

J |Qij|2 + Q∗

ijbiαbjα + c.c. + λi(b† iαbiα − κbN)

Uniform saddle : only gauge fluctuations! Stable to confinement.

Z2

Access saddle with nb = κbN

κb

quadrupolar QSL

Z2

κc = 0.148 . . . hbαi 6= 0

Phase Diagram

Block, D’Emidio, Kaul (arxiv:2019)

Phase Diagram

Block, D’Emidio, Kaul (arxiv:2019)

Phase Diagram

Block, D’Emidio, Kaul (arxiv:2019)

Phase Diagram

Block, D’Emidio, Kaul (arxiv:2019)

• Topological Order

Z2

Spectral gap ; e and m excitations — anyons!

Topological Entanglement Entropy

Levin & Wen; Kitaev & Preskill (2006)

S = aL − γ γ = ln(2)

Z2

for

Topological Entanglement Entropy

D’Emidio (2019); Block, D’Emidio & Kaul (2019)

• cf. Melko, Hastings et al

Topological Entanglement Entropy

Block, D’Emidio, Kaul (arxiv:2019)

Models

Kitaev toric code Balents, Fisher Girvin Kagome (cf Isakov et al) Kitaev honeycomb model Quantum Dimer Model (Mossner & Sondhi; Misguich et al)

Z2

Designer Model for QSL

HJ = J X

hiji

|SijihSij|

|Si = 1 p N

N

X

α=1

|ααi

New simple model for topological order

Kaul (2015); Block, D’Emidio, Kaul (arxiv:2019)

Z2

Designer Models Model for Spin Liquid Haldane Nematic

2+1 d: S=1/2 Néel-VBS

Senthil & Fisher; Tanaka & Hu

~ = (nx, ny, nz, Vx, Vy)

S = Z d3x1 g (r~ )2 + i2⇡k vol(S4) Z d3xdu "abcdearubrxcrydrτe

Senthil Vishwanath, Balents, Sachdev, Fisher

S=1/2 Néel VBS

L = 1 g

• ⇣

~ r i~ a ⌘ zα

• 2

+ 1 e2 ⇣ ~ r ⇥ ~ a ⌘2

S=1 phase transition

Néel ?? S=1 ?? What are the natural disordered phases? What is the nature of the phase transition?

S=1 phase transition

Néel Haldane nematic ?? S=1 “natural transition” to 2-fold lattice nematic Read & Sachdev complex state with large entangled objects

S=1 phase transition

Néel Haldane nematic CP + doubled monopoles

O(4) model at θ = π O(3) × Z2

+

1

OR

?? S=1 Need a J-Q model to study this transition!

Wang, Kivelson, Lee

exotic phase transition?

Sign Problem with S>1/2

HJ = J X

ij

• Si · Sj − S2

Heisenberg model General Strategy? sign free rotationally symmetric spin-S models

Hbq = − X

hiji

(Si · Sj)2

A few other examples, for S=1: Harada & Kawashima

Split-Spin

Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004)

Si = X

a

sa

i

Z = TrS h e−βH(S)i = Trs h e−βH(s)P i i

a = 1 a = 2

Split-Spin idea

Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004)

Si = X

a

sa

i

Z = TrS h e−βH(S)i = Trs h e−βH(s)P i

Hij

2 = Si · Sj − 1 = −

X

a,b

(1 4 − sa

i · sb j)

e.g. S=1

(1 4 − s · s)

i

a = 1 a = 2

Split-Spin idea

Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004)

Si = X

a

sa

i

Z = TrS h e−βH(S)i = Trs h e−βH(s)P i

Hij

2 = Si · Sj − 1 = −

X

a,b

(1 4 − sa

i · sb j)

e.g. S=1

(1 4 − s · s)

Hij

bq = 1 − (Si · Sj)2

i

a = 1 a = 2

S=1 designer

Desai & Kaul, PRL (2019)

S=1 Phase Diagram

Desai & Kaul, PRL (2019)

H = X

hiji

Hij

2 + g

X

p

Hp

3⇥3

Order Parameters

ψ = X

r

(Bx(r) − By(r))/Ns φ = X

r

ei(π,0)·rBx(r)/Ns Bi(r) ≡ JSr · Sr+ei

Desai & Kaul, PRL (2019) Okubo, Harada, Lou, Kawashima (2015)

Finite-Size Scaling

Desai & Kaul, PRL (2019)

Phase Transitions

Desai & Kaul, PRL (2019)

Phase Transitions

Desai & Kaul, PRL (2019)

Designer Model

Desai & Kaul, PRL (2019)

Néel Haldane nematic ?? S=1 Sign free model to realize Haldane nematic Interesting VB phase with long range singlets First order transition New designer models for S>1/2

THE END