A new perspective on long range SU(N) spin models

Transcription

1 A new perspective on long range SU(N) spin models Thomas Quella University of Cologne Workshop on Lie Theory and Mathematical Physics Centre de Recherches Mathématiques (CRM), Montreal Based on work with Roberto Bondesan (arxiv: ) See also related work by Tu, Nielsen and Sierra (arxiv: ) Funding: SFB TR12 Collaborative Research Center Symmetries and Universality in Mesoscopic Systems (SFB TR12) Center of Excellence Quantum Matter and Materials (QM2)

2 Mathematical setup and examples Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 2/34

3 Mathematical setup 1 2 L Spin operators: Si g (rep matrices of g on H i ) Ingredients Symmetry (here: a simple Lie algebra g) Hilbert space H = i H i (a unitary rep of g) Hamiltonian H End(H) (hermitean, commuting with action of g) Quantities of interest The spectrum of H The properties as L (thermodynamic limit) Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 3/34

4 Examples Some famous Hamiltonians Heisenberg model Gaudin model Haldane-Shastry model H = kl S k S l H = k,l S k S l z k z l H = k,l S k S l z k z l 2 special conditions apply Relevant features Short/long ranged Two-spin couplings Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 4/34

5 The Gaudin model Hamiltonian and arrangement S i S j H = i<j z i z j Quantum integrability H i = j( i) S i S j z i z j [H i, H j ] = 0 Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 5/34

6 The Gaudin model Hamiltonian and arrangement S i S j H = i<j z i z j Specific features There is a close relation to the Kac-Wakimoto/Feigin-Frenkel construction for representations of affine Lie algebras [Feigin,Frenkel,Reshetikhin]... and to the geometric Langlands program [Frenkel] Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 5/34

7 The Haldane-Shastry model Hamiltonian and arrangement H = i<j S i S j z i z j 2 1D Specific features Quantum integrability Yangian symmetry Finitized characters of affine Lie algebras A new type of exclusion statistics: Haldane statistics [Haldane,Ha,Talstra,Bernard,Pasquier] [Bouwknegt,Schoutens] [Haldane] Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 6/34

8 This talk Objective Construct long ranged spin models in 1D and 2D from the structure of Verma modules over an affine Lie algebra ĝ Features Arbitrary positions of spins Arbitrary representations Two- and three-spin couplings (could also be higher) Interesting physical applications Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 7/34

9 Physical motivation Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 8/34

10 Motivation 2D FQH liquid Edge current Anyonic excitations Fractional quantum Hall liquids Strongly correlated, gapped, topological phase of matter Anyonic excitations Chiral edge currents Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 9/34

11 FQH states from 2D conformal field theory (CFT) 2D FQH liquid Edge current Anyonic excitations Paradigm All important aspects of a FQH state are captured by 2D CFT: The groundstate wave function = chiral CFT correlator The braiding and fusion of anyonic excitations The chiral edge theory [Moore,Read] [Wen] Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 10/34

12 The goal of the talk Strategy 1 Start from a CFT/vertex operator algebra 2 Construct a quantum spin model (in either 1D or 2D) 3 Try to solve it in the thermodynamic limit 4 Investigate its relation to the original CFT Concrete choice of CFT here: WZW models Based on affine Lie algebra Natural realization of g-symmetry Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 11/34

13 From CFT to long range spin models Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 12/34

14 Outline of the idea Philosophy (based on a given CFT) [Cirac,Sierra] [Nielsen,Cirac,Sierra] CFT correlation function ψ(z 1,..., z L ) State ψ (wave function) of a quantum mechanical lattice model Hamiltonian with ψ as a (unique) groundstate Sketch Conformal field theory C z j ψ i (z i ) Field insertions Quantum spin model zj Spin locations S i Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 13/34

15 Outline of the idea Philosophy (based on a given CFT) [Cirac,Sierra] [Nielsen,Cirac,Sierra] Correlator ψ m1 m L (z 1,..., z L ) = ψ m1 (z 1 ) ψ ml (z L ) State ψ = {m i } ψ m 1 m L (z 1,..., z L ) m 1 m L Hamiltonian H 0 with H ψ = 0 Sketch Conformal field theory C z j ψ i (z i ) Field insertions Quantum spin model zj Spin locations S i Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 13/34

16 Features and natural questions Features Interpretation as an Infinite Matrix Product State (MPS) Freedom: Type and position of field insertions can be chosen at will Examples random 2D square triangular honeycomb 1D chain Questions Why a lattice model? Potential cold atom implementation Thermodynamic limit: What is the relation to the original CFT? Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 14/34

17 Some examples Existing implementations [Cirac,Sierra] [Nielsen,Cirac,Sierra] [Tu] [Tu,Nielsen,Cirac,Sierra] WZW model (CFT) Groundstate wave function 1D Hamiltonian U(1) k Jastrow/Laughlin SU(2) 1 Jastrow/Laughlin mod Haldane-Shastry SO(N) 1 Projected BCS The SU(2) Haldane-Shastry model [Haldane] [Shastry] [Haldane,Ha,Talstra,Bernard,Pasquier] S k z k z l S l H k<l S k S l z k z l 2 Exactly solvable: Yangian symmetry Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 15/34

18 Application to SU(N) spin chains Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 16/34

19 The SU(N) WZW model Starting point: The SU(N) 1 WZW model Based on affine Lie algebra extending su(n) It has N 1 basic fields with abelian fusion We will only work with the fields ψ(z) and ψ(z) (fundamental/conjugate rep) Affine Lie algebra (written as current algebra) J a (z) J b (w) = if ab c J c (w) (z w) 2 + kκab z w + Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 17/34

20 Some notation Assignment of objects to sites V ψ(z) V ψ(z) Spin model Alternating Uniform WZW model Formalization Set of sites L = {1, 2,..., L} = S S and a parity map d : L Z 2 ( ) ( ) 1, ψ(z), V, k S dk, ψ k (z), H k = ( 0, ψ(z), V ), k S Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 18/34

21 Construction of the Hamiltonian Strategy Find operators P k that annihilate ψ 1 (z 1 ) ψ L (z L ) Define H = k P k P k By construction: H is hermitean H 0 ψ is a groundstate ψ l (z l ) P k k Relevant result The operators P k arise from null vectors in Verma modules over the affine Lie algebra su(n) after making use of WZW Ward identities Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 19/34

22 From null fields to the Hamiltonian Null fields ψ(z) J(w) J(w) ψ 1 (z) ψ 2 (z) χ(z) Energy Analogy: Representations of SU(2) 2 J 1 J 0 J 1 J z Construction of the null operators P k ψ χ a l (z l ) k (z k) ψ l (z l ) ψ k (z k ) = (P k ) a b Jb (w) = 0 dw/w 2πi z k }{{} Projects J b (w)ψ k (z k ) onto χ a k (z k) Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 20/34

23 From null fields to the Hamiltonian Null fields ψ(z) J(w) J(w) ψ 1 (z) ψ 2 (z) χ(z) Energy Analogy: Representations of SU(2) 2 J 1 J 0 J 1 J z Construction of the null operators P k ψ χ a l (z l ) k (z k) ψ l (z l ) ψ k (z k ) = 0 = (P k ) a b S l b z k z l l( k) Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 20/34

24 From null fields to the Hamiltonian Null fields ψ(z) J(w) J(w) ψ 1 (z) ψ 2 (z) χ(z) Energy Analogy: Representations of SU(2) 2 J 1 J 0 J 1 J z Construction of the null operators P k ( P k {zl } ) ψ(z 1 ) ψ(z L ) def = l( k) (P k ) a b S b l z k z l ψ(z1 ) ψ(z L ) = 0 Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 20/34

25 Null vectors in affine Verma modules Decomposition of a Verma module over su(n) into reps of SU(N) L 0 J a 1. J J V J V = V Λ N V H J a (z) = Jn a z n 1 n Z Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 21/34

26 Construction of the projector Objective Construct projector onto the subspaces N and N in J V = V N Λ and J V = V N Λ t Sk Concrete expression P k = (C k C V )(C k C Λ ) [ 1 (C N C V )(C N C Λ ) = 2(N+1) ( Sk t) 2 + (N + 1) S k t + N ] with quadratic Casimir operator C k = ( t + S k ) 2 Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 22/34

27 Simplification of the projector The algebra of spin matrices The matrices S k and I exhaust the space of N N matrices, therefore S a k S b k = i 2 f ab c S c k 1 2 ( 1)d k d ab c S c k + 1 N κab I Final result for projector onto N J H i [Bondesan,TQ] [Tu,Nielsen,Sierra] (P k ) a b = i N+2 4 N+1 f a bc Sk c N( 1)d k 4(N+1) d a bc Sk c + N+2 2(N+1) δa b I Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 23/34

28 Modification of the null operator Generalizing the parameter dependence The aspired groundstate ψ satisfies (P k ) a b S b l ψ = 0 z k z l l( k) (P k ) a b S l b ψ = 0 l( k) w kl (P k ) a b S l b ψ = 0 l( k) with w kl = f (z k) z k z l + g(z k ) Two convenient choices a) w kl = 1 z k z l b) w kl = z k + z l z k z l Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 24/34

29 Evaluation of the Hamiltonian Modified null operator Pk a ( ) (P k ) a {z} = b S l b C a ( ) k {w} = w kl (P k ) a b z k z S l b l l( k) l( k) Corresponding Hamiltonian H = C a ) k( {w} κab C b ( ) k {w} = k k = k i,j k m,n( k) [Bondesan,TQ] [Tu,Nielsen,Sierra] w km w kn S a m P k,ab S b n w ki w kj { i N+2 4 N+1 f abc Si a Sj b Sk c N( 1)d k 4(N+1) d abc Si a Sj b Sk c + N+2 2(N+1) S i S j } Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 25/34

30 The general Hamiltonian Simplified Hamiltonian (up to a constant) { [N H = N+2 4(N+1) + ( 1) d i +d j (N 2) ] w ij } w ki w Si kj S j i j + 1 4(N+1) i<j<k k i,j { } (N + 2) Ω A kij f abc Si a Sj b Sk c N ΩT kij d abc Si a Sj b Sk c Auxiliary tensors Ω T ijk = 2 Re [ ( 1) d i w ij w ik + ( 1) d j w jk w ji + ( 1) dk w ki w kj ] Ω A ijk = 2 Im [ w ij w ik + w jk w ji + w ki w kj ] Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 26/34

31 Absence of higher spin couplings Absence of the f -coupling This vanishes if the w kl are either all real or all imaginary: [ ] Ω A ijk = 2 Im w ij w ik + w jk w ji + w ki w kj = 0 Absence of the d-coupling This decouples from the local dynamics if the tensor is constant [ ] Ω T ijk = 2 Re ( 1) d i w ij w ik + ( 1) d j w jk w ji + ( 1) dk w ki w kj = const Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 27/34

32 Discussion Thermodynamic limit Expectation: Gapless Same CFT (actually not true...) Generic 1D Generic 2D For the general case an analytic solution is beyond reach One analytic result Expectation/hope: Gapped spin liquid 1D edge CFT Anyonic excitations [Tu,Nielsen,Sierra] [Bondesan,TQ] [Tu,Nielsen,Sierra] The exact groundstate is defined in terms of ψ1, q1 (z 1 ) ψ L, ql (z L ) = δ q,0 e if ({ q l }) }{{} (z i z j ) q i, q j known i<j where q l are quantum numbers (weights) with respect to SU(N) Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 28/34

33 More on the WZW correlators Free field realization The SU(N) 1 WZW model is equivalent to a theory of N 1 free bosonic fields compactified on the root lattice of SU(N) Correlation functions WZW correlation functions follow from those of bosonic vertex operators e iµ 1 ϕ(z 1) e iµ L ϕ(z L ) = δ µ,0 (z i z j ) µ i,µ j after the inclusion of certain cocycles i<j Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 29/34

34 The uniform chain Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 30/34

35 The Hamiltonian for the uniform chain S i S i Useful quantity: S j S k S j w ij = z i + z j z i z j Two spin interaction Three spin interaction The Hamiltonian { H = N+2 2(N+1) (N 1) w ij 2 + } ( w ki w Si kj Sj + C N {wij } ) i j k } + 1 4(N+1) w ki w kj { i(n + 2)f abc Si a Sj b Sk c + Nd abcsi a Sj b Sk c i j k }{{} cancels out for special choice of positions z i [Bondesan,TQ] [Tu,Nielsen,Sierra] Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 31/34

36 SU(N) spins on the unit circle arbitrary uniform (ξ i = 0) Useful quantity: ξ i = j( i) w ji Drastic simplification { } H = C 1 1 w ij (ξ i ξ j ) Si z i z j 2 4(N+1) Sj + C 2 S 2 + C 3 d abc S a S b S c +C i j }{{}}{{} 4 uniform: 0 coupling to total spin S [Bondesan,TQ] [Tu,Nielsen,Sierra] The uniform case Reduction to Haldane-Shastry model plus coupling to total spin Exact solution despite absence of Yangian symmetry Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 32/34

37 Summary and Outlook Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 33/34

38 Summary and Outlook Summary [Bondesan,TQ] [Tu,Nielsen,Sierra] Long range SU(N) spin models on arbitrary lattices in 1D or 2D can be constructed based on the null vectors in the SU(N) 1 WZW model Concrete results The 1D uniform case can be reduced to the Haldane-Shastry model All eigenstates and their energies are known explicitly [Bondesan,TQ] [Tu,Nielsen,Sierra] Outlook The 1D alternating case leads to an hitherto unidentified CFT Exploration of various 2D setups Relation to Gaudin Hamiltonians? Application to other symmetry groups [Tu,Nielsen,Cirac] see [Tu] [Bondesan,Peschutter,TQ] [TQ,Tu] for SO(N), GL(M N) and SP(N) Thomas Quella (University of Cologne) A new perspective on long range SU(N) spin models 34/34