New routes to topological order:
Toric code in Rydberg atoms and Fractional Chern insulators in moire materials.
Ashvin Vishwanath
Ultra-Quantum Matter Acknowledgements: Giulia Semeghini Harry Levin Sepehr Ebadi Alex Keesling Hannes Pichler
Philip Kim & Amir Yacoby Ruben Verresen Misha Lukin Harvard Harvard Groups
arxiv 2011.12310
Patrick Ledwith & Grisha Tarnopolsky, Eslam Khalaf Harvard (i) MeasureSymmetry order parameter Breaking experimentally States to diagnose - Landau orders a phase: Eg. X-rays on a Crystalline solid (Density(Q)) Landau Order Parameter: Bose Einstein Condensates⟨φ⟩ ≠– JILA0 `95
(i) Measure order parameter experimentally to diagnose a phase: Eg. X-rays on a Crystalline solid (Density(Q)) Bose Einstein Condensates – JILA `95
(ii) Allows for a Classification of possible states Eg. 230 space groups of crystalline solids All are realized in nature!
(ii) Allows for a Classification of possible states (iii) ManyEg. 230 prac cal space applica ons groups of crystalline solids (eg. Magne cAll memories): are realized in nature!
NIST 1 ( ¯ - ¯ ) 2 Neel order Valence Bond Crystal Resonating Valence Bonds Condensates of Loops - Topological Order
States represented by closed loops
Ω = | ⟩ + | ⟩ + … Deconfined phase of an
Reference State Emergent Gauge theory. ∇ ⋅ E = 0 (mod 2) Par cle Condensates - Symmetry Breaking Orders
Landau Order Parameter: Beyond Landau Orders:
⟨φ⟩ ≠ 0 Topological Order
Condensate of particles: Condensate of closed loops:
Ω = + + Ω = | ⟩ + | ⟩ + …
BEC JILA `95 XG Wen Science `19 Realization of Topological Order? Topological Order (Gapped) Quantum spin liquids? Fractional Quantum Hall Loop condensates Promising candidates - but no clearcut example
Triangular Lattice
Organic n=1$
n=2$ n=3$
Kagome Lattice
Gapped states with anyon Kitaev- excitations - Honeycomb+ Self statistics -unlike boson/ B field • fermion • Mutual statistics between From: Broholm et al. Science 2020 excitations Classical Orders versus Topological orders
(i) Measure order parameter experimentally to diagnose Landaua phase: Order: Topological Order: Eg. X-rays on a Crystalline solid⟨φ (Density(Q⟩ ≠ 0 )) Bose Einstein Condensates – JILA `95 (i) Measure order parameter experimentally to diagnose • How to measure? a phase: Eg. X-rays on a Crystalline solid (Density(Q)) (beyond Quantum Hall - edge states Bose Einstein Condensates – JILA `95 and quantized conductance)
(ii) Allows for a Classification of possible states • New Platforms for realization? Eg. 230 space groups of crystalline solids All are realized in nature!
(ii) Allows for a Classification of possible states Eg. 230 space groups of crystalline solids All are realized in nature! (iii) Many prac cal applica ons • quantum memories/computing (eg. Magne c memories):
NIST Explicit Realization of Gauge Fields
• Seek NEW platforms and models where gauge fields are explicit microscopic degrees of freedom eg. dimer model/ toric code
Kitaev `97
1 ( ¯ - ¯ ) 2
Toric code - With 4 body interactions Wegner `71; Fradkin&Shenker `79 Rokhsar and Kivelson `88; Read&Sachdev `90; Sachdev `92; Wen `91
Balents Fisher Girvin `01; Motrunich Senthil `02 Realization in Rydberg atom Array? Rydberg atom array
Ebadi et al. arXiv:2012.12281 Scholl et al. arXiv:2012.12268 Samajdar et al. PRL `20 Samajdar et al. PNAS `21 Ruben Verresen, Lukin, AV arxiv Quantum number n~70 of Rb atom - 2011.12310 large van-der-Waals interactions Rydberg Hamiltonian Rydberg Hamiltonian Dimer Model From Blockade?
Within radius Rb Dimer Model From Blockade
Links of the Kagome
RUBY Lattice
Other 2D lattices: Samajdar et al. PRL `20. Samajdar et al. PNAS `21 Z2 topological order?
• Hilbert space - Z2 gauge theory
• Dynamics - deconfinement?
Confined - Deconfined - Valence Bond Crystal Topological order Dimer resonance
Exact Solution: 32 resonance terms with equal weight Misguich, Serben, Pasquier ‘01 Zeng & Elser ‘95 Pure Dimer Model
Ω → 0 δ
… Crystal δ ↓ Introduce vacancies - Topological order?
12 unitcells Phase Diagram of Blockade Model
δ/Ω
Ruben Verresen, Lukin, AV arxiv 2011.12310
1.Topological Entanglement Entropy
Trivial Phase Topo. Phase Line Operators (BFFM)
m
m
z ∏σ
“ x ” ∏σ Line Operators (BFFM)
m m
K. Gregor et al. `10 Diagnosing Phases
m m
P
Q Modular Matrices
(XC-8)
Yi Zhang et al. `12 Cincio & Vidal `12
(YC-8)
Dimer density signature
topo. order First order
Quantum critical! Ebadi et al. arXiv:2012.12281 Ruben Verresen, Lukin, AV arxiv 2011.12310 Measuring String Operators?
Quench Spin liquid with van der Waals Interactions
Retain: r1, r2, r3, r4,…r16 Each atom couples to 44 other sites Spin liquid with van der Waals Interactions
Ground States
Modular BFFM Matrix
Ruben Verresen, Lukin, AV arxiv 2011.12310
Towards Topological Quantum Bit
Alexei Kitaev Tune boundaries Two-fold degenerate
Boundary phase transition On changing detuning δ
TwistedMonolayerBilayer Bilayer Continuum model • Larger unit cell→ smaller BZone • Bistrizer-Macdonald (BM) model (2011)
• Lattice relaxation: AB stacking favored to AA stacking (Carr et al. ’19, Nam, Koshino `17)
35 2 derived, if it is known at one point in the Brillouin TABLE I. Comparison of magic angles in continuum model Zone. An interesting mathematical aspect here is that with w0 = 0 (first row) and with w0 = w1 (second row). Only the wave function ratios are constructed from meromor- the first magic angles correspond. 2 phic doubly-periodic functions that are ratios of theta functions, similar to those appearing in the Quantum derived, if it is known at one point in the Brillouin ↵ ↵ ↵ ↵ ↵ Hall E↵ect on the torus [44]. The CS-CM has a single TABLE I. Comparison1 of magic2 angles3 in continuum4 5 model Zone. An interesting mathematical aspect here is that withCS-CMw = (here) 0 (first row)0.586 and2.221 with w 3.75= w 5.28(second6.80 row). Only coupling constant ↵ = w1/(2v0kD sin(✓/2)) where v0 0 0 1 the wave function ratios are constructed from meromor- the first magic angles correspond. and kD arephic the doubly-periodic bare velocity functions and crystal that are momentum ratios of theta BM-CM (Ref. [35]) 0.606 1.27 1.82 2.65 3.18 of graphene’sfunctions, Dirac similar fermions. to those We appearing show that in the pertur- Quantum ↵ ↵ ↵ ↵ ↵ bation theoryHall E to↵ect high on ordersthe torus (up [44]. to ↵ The8) matches CS-CM has with a single 1 2 3 4 5 CS-CM (here) 0.586 2.221 3.75 5.28 6.80 numericalcoupling results constant very accurately↵ = w1 near/(2v0k theD sin( first✓/2)) magic where vsiders0 two layers of graphene described by an e↵ective and k are the bare velocity and crystal momentum BM-CM (Ref. [35]) 0.606 1.27 1.82 2.65 3.18 angle. The sequenceD of magic angles that we find ↵ = Dirac fields near K, K points of the moire (mini) Bril- of graphene’s Dirac fermions. We show that pertur- 0 0.586, 2.221, 3.751, 5.276, 6.795, 8.313, 9.829, 11.345,... louin Zone, each rotated by an angle ✓/2, and coupled bation theory to high orders (up to ↵8) matches with reveals a remarkable asymptotic quasi-periodicity of through a Moir´epotential T (r) [3, 14,± 35, 39]: ↵ 3/numerical2. Comparing results with very the accurately reported near magic the angles first magic siders two layers of graphene described by an e↵ective ' angle. The sequence of magic angles that we find ↵ = for the BM-CM, we see significant di↵erences except for Dirac fields near K, K0 points of the moire (mini) Bril- iv0 ✓/2 T (r) the first magic0.586, angle,2.221, 3 see.751 Table, 5.276 I., We6.795 finally, 8.313 turn, 9.829 on, 11 the.345,... louinH Zone,= each rotatedr by an angle ✓/2,, and coupled(1) reveals a remarkable asymptotic quasi-periodicity of T (r) iv ± AA-coupling and study numerically how the bandwidth through a Moir´epotential† T (r0) [3,✓/ 14,2 35,! 39]: ↵ 3/2. Comparing with the reported magic angles r and gap evolves' and discuss the possibility of studying i✓ i✓ for the BM-CM, we see significant di↵erences except forwhere = e 4 z ( , )e 4 z , =(@ , @ ) and ✓/2 xiv0y ✓/2 T (r)x y the secondthe magic first magic angle angle, in experiments. see Table I. We finally turn on the H = r r , (1) T †(r) iv0 ✓/2 ! ContinuumAA-coupling model for and Twisted study numerically Bilayer Graphene. how the bandwidthThe 3 r iqj r continuum model describing a single valley of TBG con- T (ri)=✓ T e i✓ (2) and gap evolves and discuss the possibility of studying z j z where = e 4 ( x, y)e 4 , =(@x, @y) and the second magic angle in experiments. ✓/2 j=1 X r Continuum model for Twisted Bilayer Graphene. The 3 1 a. ↵1 =0.586 b. ↵2 =2.221 c. ↵3 =3.751 p iqj r continuum model describing a single valley of TBG con-with q1 = k✓(0, 1), q2T,3(r=)=k✓( Tj3e/ 2, 1/2) and (2) 1.5 ± 0.4 0.08 j=1 Tj+1 = w0 0 + w1 cos( Xj) x +sin( j) y , (3) 1.0 0.3 0.06 I. MAGIC ANGLES a. ↵1 =0.586 b. ↵2 =2.221 c. ↵3 =3.751 with q = k (0, 1), q = k ( p3/2, 1/2) and 0.2 1 ✓ 2,3 ✓ 1.5 0.04 ± 0.5 0.4 where =2⇡/3, k✓ =2kD sin(✓/2) is the Moir´emod- 0.1 0.02 0.08 ulation vectorTj+1 and= wk0D 0↵=4+1 w⇡1 /↵cos((32 a0 )↵j)3 is x the+sin(↵4 magnitude ↵j5) y , of(3) 1.0 0.3 0.06 0 0 0 the Dirac wave vector, where a is the lattice constant 0.2 0.04 where =2⇡/3, k =2 k 0sin(✓/2) is the Moir´emod- Energy 0.5 -0.1 -0.02 CS-CM 0.586✓ 2.221D 3.75 5.28 6.80 -0.5 0.1 0.02 of monolayer graphene. The Hamiltonian (1) acts on the -0.2 -0.04 ulation vector and kD =4T ⇡/(3a0) is the magnitude of 0 0 0 spinorthe ( Diracr)=(BM-CM wave 1, 1 vector,,0.606 2, 2 where1.27) and1.82a theis2.65 the indices lattice3.18 1, 2 constant rep- -1.0 -0.3 -0.06 0 Energy -0.1 -0.02 resent the graphene layer. Here w0 is responsible for the -0.5 -0.08 of monolayer graphene. The Hamiltonian (1) acts on the -1.5 -0.4 -0.2 -0.04 T AA couplingspinor and(r)=(w1 is1, for 1, AB 2, and2) andAB thecouplings. indices 1, 2 rep- A B C-1.0 D A A B C-0.3 D A A B-0.C06 D A Chiral Model Chirallyresent the symmetric graphene continuum layer. Here model.w0 is responsibleIn this Letter, for the -0.08 iq1r i iq2r i iq3r e.-0.42.0 U(r)=e +Tarnopolski,e e Kruchkov,+ AVe e (1) -1.5 Bandwidth we studyAA coupling a model andobtainedw1 is from for ABPRL Hamiltonian 2019and AB couplings. (1) by set- A B C D A A1.6B C D A A B C D A d. Moir´eBZ ↵1 ↵2 ↵3 ting wWe#consider#a#continuum#model#0 Chirally= 0 [45]. symmetric Note, one continuum can rotate model. theIn spinors this Letter, to 1.2 e. 2.0 exact we study a model obtained from Hamiltonian (1) by set- zerosBandwidtheliminatefor#bilayer#graphene#with## theSwitch rotation off inAA the coupling. kinetic termsOnly AB coupling✓/2 in K 0.8 1.6 ting w = 0 [45]. Note, one can rotate± the spinors! to Energy ↵1 ↵2 ↵3 the absence0 of the w term. The dimensionless Hamilto- A d. Moir´eBZ 0 0.4 1.2 exact eliminate the rotation inw theU (r kinetic) w U terms(r) T in D zeros nian now acts on a spinor 0(r)=(0 1,1 2, 1, 2✓)/2, and B K K 0 0.8 T (r)= ± ! (2)
Energy the absence of the w term. The dimensionless Hamilto- A 0 1 2 3 4 5 6 ↵ can be compactly written00 in the form 1 0.4 w1U ⇤( r) w0U0(r) T D nian now acts on a spinor (r)=( 1, 2, 1, 2) , and B K f. 0 Band gap C 0.5 0 1 2 3 4 5 6 ↵ Dimensionless#Hamiltonian#can#be#represented#in#a#can be compactly written@ in the form A 0.4 f. ¯ C 0.3 0.5 Band gap convenient#form0 ⇤( r) 2i@↵U(r) 0.4 = D , (r)= , 0.2 H (r)0 D ↵U( r) ¯ 2i@¯ Chiral Symmetry
Energy 0.3 0 ⇤(!r) 2i@↵U(r!) 0.1 D= D , (r)= ,z 1, =0 0.2 H (r)0 D ↵U( r) 2i@¯(4) { ⌦ H}
0 Energy ! ! 0 1 02.1 3 4 5 6 ↵ D (4) 0 where @¯ = 1 (@ + i@ ) and U(r)=e iq1r + ei e iq2r + sublattice 0 1 2 3 4 5 6 ↵ 2 x y
Many special properties: ψ (x, y) (i) Chern number+ sublattice polarized K ψA(x, y) = z − z0 f(x + iy) θ1( |ω) a1
Valley K’ K C= +1 A C2� B C= -1 A C= +1 A C= -1 C2� B B C2� C= +1 C= -1 Chern =±1; Sublattice polarized �
Ledwith, Tarnopolsky, Khalaf, AV `20 Wang, Zheng, Millis, Cano `20 Becker, Embree, Wittsten, Zworski `20 Valley K’ K C= -1 A C= +1 A
C2� B B C2� C= +1 C= -1 �
Aligned h-BN substrate. Break A-B symmetry
A. Sharpe…D. G-Gordon (2019) M. Serlin… Young (2019)
Spontaneous Integer Hall at ν = 3 - Fractional? Chiral Limit Wavefunctions
Many special properties: Ledwith, Tarnopolsky, Khalaf, AV `20 (i) Bloch wavefunctions - analytic in k=kx+iky
(ii) nearly uniform Berry curvature Berry Curvature: κ = 0 (iii) Isotropic ideal droplet condition -
κ = 0.8
Ideal for Fractional Chern insulators quantum metric:
Roy Phys. Rev. B 90, 165139 (2014) Claassen et. al. Phys. Rev. Lett. 114, 236802 (2015) ⌘(k)= ⌦(k) Jackson, Möller, Roy Nat. Comm. 6 8629 (2015). Ledwith, Tarnopolsky, Khalaf, AV `20 AA/BB Tunneling => non-ideal band geometry
2 Ω • Berry curvature variation: F = − C ⟨( 2π ) ⟩ BZ
• Violation of ideal droplet condition: T¯ = tr g − |Ω| ⟨ ⟩BZ
κ = 0.7
• FCI states often still appear in exact diagonalization
Abouelkomsan, Liu, Bergholtz Phys. Rev. Lett. 124 106803 (2020) Repellin, Senthil Phys. Rev. Research. 2 023238 (2020) Wilhelm, Lang, Läuchli arXiv:2012.09829 (2020) Conclusions:
Topological order, emergent gauge fields, non-local quantum entanglement could be more widespread than previously thought.
Need to explore new platforms and detection schemes.
Rydbergs: Magic angle graphene: Low symmetry Remarkable (and mysterious) similarity Long range interactions to lowest Landau level Tunability
Bernien et al.