Designer Hamiltonians for quantum spin liquids & exotic valence bond ordered states Ribhu Kaul University of Kentucky
Collaborators Nisheeta Desai (U. Ky) Dr. Jon Demidio (U. Ky Lausanne) Prof. Matt block(U. Ky CSU-Sacramento)
Designer Models Model RVB: Topological Order Haldane Nematic
Designer Models Model RVB: Topological Order Haldane Nematic
Monte Carlo - Field Theory X H = − J σ i σ j h ij i Z ( r φ ) 2 + r φ 2 + u φ 4 S =
Quantum Spin Chains X S i · S j H J = J h ij i d=1 DMRG, CFT, bosonization, Bethe ansatz S = 1 Z n ) 2 + i 2 S π Q d 2 x ( ∂ µ ˆ g
Designer Models R.K. Kaul, R.G. Melko A. W. Sandvik, Ann. Rev. Cond. Mat. (2013) simple models of quantum many body physics sign problem free interesting physics study connections with QFT/topological terms in d>1 universality
Designer Models Model RVB: Topological Order Haldane Nematic
Anderson RVB (1973) RVB wavefunction Savary, Balents (2016)
The Model Assign one of N color to each site N Many body Hilbert space N site
The Model R. K. Kaul PRL (2015) define two-site “bond” wave function N 1 X p |S i = | αα i N α =1 e.g. For N=3 ( ( |S i 1 | | i i | i = + + √ 3 call this a “singlet” wave function
The Model R. K. Kaul PRL (2015) Model is a sum of “singlet” projectors X H J = � J |S ij ih S ij | h ij i N 1 X |S i = p | αα i N α =1 cf. Affleck; Sachdev & Read; Harada, Kawashima & Troyer.
Notes on Model X H J = � J |S ij ih S ij | h ij i • In spirit of Anderson’s original proposal • symmetry: SO( N ) singlets N 1 X |S i = p | αα i N • Apart from sign - Heisenberg model α =1
Hilbert Space |S ih S| = 1 X | αα ih ββ | N α , β
Quantum Dynamics |S ih S| = 1 X | αα ih ββ | N α , β
Quantum Dynamics |S ih S| = 1 X | αα ih ββ | N α , β
Quantum Dynamics |S ih S| = 1 X | αα ih ββ | N α , β
Quantum Dynamics |S ih S| = 1 X | αα ih ββ | N α , β
Quantum Dynamics |S ih S| = 1 X | αα ih ββ | N α , β
Competition in model X H J = � J |S ij ih S ij | h ij i Competing tendency: Equal colors vs singlet formation
Quantum, Classical: Loop model ( − β ) n Tr( H n ) � Tr( e − β H ) = n ! n SSE review, Sandvik (2010) typical term in an expansion for d=1 forms a tightly packed loop model! cf. Nahum, Chalker, Serna, Somoza, Ortuno
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)
Large- N Arovas, Auerbach; Read, Sachdev; Affleck Block, D'Emidio, Kaul (arxiv:2019) X H J = � J |S ij ih S ij | h ij i H b = − J ⇣ ⌘ ⇣ ⌘ X b † i α b † b j β b i β n b = 1 j α N h ij i With forms new SO( N ) n b 6 = 1 spin “bosonic” representations
Phase Diagram Block, D’Emidio, Kaul (arxiv:2019)
Large- N Block, D’Emidio, Kaul (arxiv:2019) Access saddle with n b = κ b N i α ∂ τ b i α + N J | Q ij | 2 + Q ∗ ij b i α b j α + c . c . + λ i ( b † L b = b ∗ i α b i α − κ b N ) κ c = 0 . 148 . . . Uniform saddle : only gauge fluctuations! Z 2 Stable to confinement. QSL quadrupolar Z 2 κ b h b α i 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 Z 2 Spectral gap ; e and m excitations — anyons!
Topological Entanglement Entropy Levin & Wen; Kitaev & Preskill (2006) S = aL − γ for Z 2 γ = ln(2)
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 Z 2 Kitaev toric code Balents, Fisher Girvin Kagome (cf Isakov et al) Kitaev honeycomb model Quantum Dimer Model ( Mossner & Sondhi; Misguich et al )
Designer Model for QSL Kaul (2015); Block, D’Emidio, Kaul (arxiv:2019) X H J = � J |S ij ih S ij | h ij i N 1 X |S i = p | αα i N α =1 New simple model for topological order Z 2
Designer Models Model for Spin Liquid Haldane Nematic
2+1 d: S=1/2 Néel-VBS Néel VBS S=1/2 Senthil Vishwanath, Balents, Sachdev, Fisher 2 ⌘ 2 L = 1 + 1 � � ⇣ ⌘ ⇣ ~ ~ r � i ~ r ⇥ ~ a z α a � � e 2 g � � Senthil & Fisher; Tanaka & Hu ~ � = ( n x , n y , n z , V x , V y ) d 3 x 1 i 2 ⇡ k Z Z � ) 2 + g ( r ~ d 3 xdu " abcde � a r u � b r x � c r y � d r τ � e S = vol( S 4 )
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 Read & Sachdev Néel Haldane nematic S=1 ?? “natural transition” to 2-fold lattice nematic complex state with large entangled objects
S=1 phase transition Néel Haldane nematic S=1 ?? exotic phase transition? 1 CP + doubled monopoles OR + O (3) × Z 2 O (4) model at θ = π Wang, Kivelson, Lee Need a J-Q model to study this transition!
Sign Problem with S>1/2 Heisenberg model X S i · S j − S 2 � � H J = J ij A few other examples, for S=1: X ( S i · S j ) 2 H bq = − h ij i Harada & Kawashima General Strategy? sign free rotationally symmetric spin- S models
Split-Spin Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004) a = 2 X s a S i = i a = 1 a i h e − β H ( S ) i h i e − β H ( s ) P Z = Tr S = Tr s
Split-Spin idea Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004) a = 2 X s a S i = i a = 1 a i h e − β H ( S ) i h i e − β H ( s ) P Z = Tr S = Tr s (1 e.g. S=1 4 − s · s ) (1 H ij X 4 − s a i · s b 2 = S i · S j − 1 = − j ) a,b
Split-Spin idea Todo & Kato (2001); Kawashima & Gubernatis (1994); Kawashima & Harada (2004) a = 2 X s a S i = i a = 1 a i h e − β H ( S ) i h i e − β H ( s ) P Z = Tr S = Tr s (1 e.g. S=1 4 − s · s ) (1 H ij X 4 − s a i · s b 2 = S i · S j − 1 = − j ) a,b H ij bq = 1 − ( S i · S j ) 2
S=1 designer Desai & Kaul, PRL (2019)
S=1 Phase Diagram Desai & Kaul, PRL (2019) X H ij X H p H = 2 + g 3 ⇥ 3 h ij i p
Order Parameters Desai & Kaul, PRL (2019) B i ( r ) ≡ J S r · S r + e i X e i ( π , 0) · r B x ( r ) /N s φ = r X ψ = ( B x ( r ) − B y ( r )) /N s r 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
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