An ingenious mapping between integrable supersymmetric chains

Transcription

1 An ingenious mapping between integrable supersymmetric chains Jan de Gier University of Melbourne Bernard Nienhuis 65th birthday, Amsterdam 2017 György Fehér Sasha Garbaly Kareljan Schoutens Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

2 Gefeliciteerd Bernard! A walk with Bernard in 2009 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

3 Eye staring contest 2011 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

4 Ik ben in het ziekenhuis maar het is niet erg I am in hospital but it s alright Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

5 A brief history A brief history of my recent interactions with Bernard Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

6 O(n = 1) loops, boundaries of percolation clusters Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

7 Build configurations layer for layer using transfer matrix (or Hamiltonian for continuous time) States described by connectivity. Here () () (()) () () () (), or Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

8 Temperley-Lieb loops with n = 1 The Hamiltonian can be defined in terms of Temperley-Lieb generators: e 2 i = e i, 1 i L 1 e i e i±1 e i = e i Pictorial representation of e i Algebraic relations: i i + 1 loop weight =1 only connectivities matter Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

9 Temperley-Lieb loops with n = 1 Example L = 6: state space: Action of e 1 on fourth state: Hamiltonian: L 1 H = (e j 1) j=1 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

10 Stochastic process H = Groundstate: H(11, 5, 5, 4, 1) = 0 Razumov-Stroganov combinatorics (Batchelor, dg, Nienhuis) The Hamiltonian defines a stochastic process, i.e. H ij = 0, H ij 0 (i j) i Stationary state: H ψ = 0 d ψ(t) = H ψ(t) dt Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

11 The Raise and Peel Stochastic Process Raise and Peel Stochastic Process (dg, Nienhuis, Pearce, Rittenberg) Raise: Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

12 The Raise and Peel Stochastic Process Peel: O(n = 1) loops are raise and peel contour lines Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

13 XXZ quantum spin chain and six-vertex model bond percolation n = 1 loops raise and peel stochastic process The XXZ quantum spin chain (and six vertex model) H = 1 L ( σ x 2 j σ x j+1 + σ y j σy j+1 + σz j σj+1) z j=1 has a zero groundstate energy at = 1/2 (and appropriate boundary conditions) Similar for XYZ chain (Baxter, Fendley) Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

14 XXZ quantum spin chain and six-vertex model bond percolation n = 1 loops raise and peel stochastic process XXZ chain and six vertex model at = 1/2 Next: supersymmetry on the lattice Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

15 Fermions Definition (Fermions on a lattice) Fermionic creation and annihilation operators c j and c j are defined by, {c i, c j} = δ ij, {c i, c j } = {c i, c j } = 0 These operators act in a fermionic Fock space spanned by ket vectors of the form τ = L i=1 ( c i) τi, where the product is ordered and is the vacuum state defined by c i = 0. Example with L = 8: = c 2 c 5 = c 5 c 2 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

16 Hardcore fermions The fermionic number and hole operators are defined as N i = c i c i, P i = 1 N i. Hardcore fermions: d i = P i 1c i P i+1 d i = P i 1 c i P i+1 Example with L = 8: d 1 = 0 d 7 = d 7 = Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

17 Supersymmetry SUSY charges: and Hamiltonian Q = L L d i, Q = d i Q 2 = 0. i=1 H := {Q, Q} = L i=1 i=1 ( ) d i d i + d i d i + d i d i+1 + d i+1 d i. H is an N = 2 supersymmetry lattice realisation (de Boer, Fendley, Schoutens) Example states:, Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

18 Supersymmetry Alternative form for H: H = L 2f + L i=1 ( ) d i d i+1 + d i+1 d i + d i d id i+2 d i+2, The Hamiltonian as a chemical potential of 2 per fermion, a hopping term and a repulsive potential. Supersymmetry implies that E 0 = 0 and nonzero eigenvalues are positive. All singlets have E = 0 and all other states are (at least) doublets. Mapping to XXZ spin chain with = 1/2 (Nienhuis, Fendley, Schoutens). Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

19 Mapping to XXZ chain Consider the mapping: +, This maps states with length L to length N = L f: is equivalent to H = L 2f + L i=1 ( ) d i d i+1 + d i+1 d i + d i d id i+2 d i+2, H = 1 2 N j=1 with = 1/2 (Note different lenghts N vs L). [ σ x j σ y j+1 + σy j σy j+1 ] σz jσ z j+1 + 3N/4, Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

20 Summary Lattice realisation of N = 2 SUSY quantum mechanics Conserved fermion number Yang-Baxter integrable Spectrum determined via Bethe ansatz Groundstate special combinatorial properties Generalisations: M k models, only k fermions are allowed on consecutive sites (Fendley, Nienhuis, Schoutens) Zig-zag ladder of semionic particles (Fendley, Schoutens) Particle-hole symmetric without fermion conservation (Feher, dg, Nienhuis, Ruzaczonek) Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

21 Particle-hole symmetry Definition (Hole operators) Include particle-hole symmetry d i e i with e i = N i 1 c i N i+1 Example e 3 = SUSY charges (Nienhuis): L Q = (d i + e i), Q = and Hamiltonian H := {Q, Q} = L i=1 i=1 L (d i + e i ), Q2 = 0. i=1 ( ) ( ) d i d i + d i d i + d i d i+1 + d i+1 d i + e i e i + e i e i + e i e i+1 + e i+1 e i ) + (e i d i+1 + d i+1e i + e i+1 d i + d ie i+1. Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

22 Domain walls Cross terms do not conserve fermions: e i d i d i e i But domain walls are conserved Hopping terms: d i+1 d i e i+1 e i domain walls hop with two steps 01 domain walls hop with a minus sign nearby domain walls interact Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

23 Large degeneracies in spectrum Figure 2: Comparison of spectra for system sizes L =7 10 for periodic and antiperiodic boundary conditions. Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

24 Symmetries Supersymmetry: [H, Q] = 0, [H, Q ] = 0. Particle-hole symmetry: Γ = L i=1 γ i, γ i = c i + c i, [Γ, H] = 0. Domain wall non-domain wall L/2 E = (c 2i c 2i ), [E, H] = 0 i=1 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

25 Symmetries Shift symmetry S = L i=1 n i 1 γ i p i+1 + p i 1 γ i n i+1, γ i = c i + c i, [S, H] = 0. Reflection symmetry with consequence n 1 M = δ 4i+1 δ 4i+2, δ j = i (c j c j ) i=0 M(L H) = HM, These are not enough to explain large degeneracy. Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

26 Integrability This model turns out to be integrable (unintentionally!) Spectrum can be analysed by Bethe ansatz on domain wall dynamics Conservation of all domain walls, but also of odd and even domain walls separately Nested Bethe ansatz: Ψ(m; k) = ψ(x 1,..., x m ; p 1,..., p k ) x 1,..., x m ; p 1,..., p k, {xi} {p j } Derive conditions on ψ ɛ such that H Ψ(m; k) = Λ Ψ(m; k). Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

27 Bethe ansatz Bethe ansatz works and the spectrum is described by: Λ = L + 2n i=1 where the numbers z i satisfy the following equations z L j = ±i L/2 m 1 = 2n j=1 k=1 (z 2 i + z 2 i 2). u k (z j 1/z j ) 2 u k + (z j 1/z j ) 2, j = 1,..., 2n u k (z j 1/z j ) 2, k = 1,..., m. u k + (z j 1/z j ) 2 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

28 Zero energy Cooper pairs If z L j = 1, then z L j = 1 = u 1 (z j 1/z j ) 2 u 2 (z j 1/z j ) 2 u 1 + (z j 1/z j ) 2 u 2 + (z j 1/z j ) 2 1 = 2n j=1 u 1 (z j 1/z j ) 2 2n u 1 + (z j 1/z j ) = u 2 (z j 1/z j ) 2 2 u 2 + (z j 1/z j ) 2 j=1 Pair u 2 = u 1 are solutions that do not contribute to Λ = L + 2n i=1 (z 2 i + z 2 i 2). Zero energy Cooper pairs can explain large degeneracy in spectrum Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

29 A supersymmetric zig-zag chain There is another model with similar properties (workshops in Beijing 2016 and research institute MATRIX (Australia) 2017: Supersymmetric Integrable Zero energy Cooper pairs The Fendley-Schoutens zig-zag chain: i 1 1 i 1 i 1 1 i 1 The hopping amplitudes depend on the occupation of the other chain Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

30 FS model Supercharge where Hamiltonian: Q FS = c 2 c 1 + L/2 1 k=1 ( e i π 2 α 2k 2 c 2k 1 + π ) ei 2 α 2k c 2k+1 c 2k, α k = k ( 1) j n j. j=1 H FS = {Q FS, Q FS } L 1 = (c j+1 p jc j 1 + c j 1 p jc j+1 + ic j+1 n jc j 1 ic j 1 n jc j 1 ) j=1 L 1 2 n j n j+1 + 2F 1 + 2F 2 + F 0, j=1 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

31 FS model H FS = {Q FS, Q FS } L 1 = (c j+1 p jc j 1 + c j 1 p jc j+1 + ic j+1 n jc j 1 ic j 1 n jc j 1 ) j=1 L 1 2 n j n j+1 + 2F 1 + 2F 2 + F 0, j=1 This model describes a one-dimensional model where the fermions are hopping on two chains coupled in a zig-zag fashion. The F 1 and F 2 operators are counting the fermions on the two chains. The interaction between the chains is statistical: the hopping amplitude picks up an extra i or i factor, if the fermion "hops over" another fermion on the other chain. There is a further attractive interaction between the two chains. Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

32 Theorem There exists a mapping between the FGNR and FS model: where Q FS = TQ FGNR T, T = PΓM, M turns creation and annihilation of domain walls into hopping of domain walls Γ turns domains walls into particles P fixes phase factors Explicit operators give (L 1)/4 M = i=0 L 1 (c 4i+1 c 4i+1 )(c 4i+2 c 4i+2 ), Γ = Q := ΓMQ FGNR M Γ = c 1 c 2e iπ L/2 1 j=1 n 2j+1 + L/2 1 i=1 i=1 ( ) p i + n i (c i+1 + c i+1), (c 2i+1 c 2ie iπ ) 2i 1 j=1 n j c 2i+1 c 2i+2 e iπ L/2 1 j=i+1 n 2j+1. Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

33 Examples empty ladder : FS FGNR single FS semion : FS FGNR single FS pair : FS FGNR lower leg filled : FS FGNR single FGNR particle : FS FGNR upper leg filled : FS FGNR upper leg plus semion : FS FGNR filled ladder : FS FGNR Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

34 The supercharge Q agrees with Q FS up to phase factors. Still need to find P such that where x i are the positions of the fermions. Theorem (Phase factors) p(x 1,..., x m ) = Q FS = P Q P, P = e i π 2 p(x 1,x 2,...,x m), m x k k=1 i=k+1 Generalized Jordan Wigner transform ( 1 + ( 1) i k ( 1) j + j=1 ) m (1 ( 1) x j ). j=k+1 Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35

35 Conclusions Particle-hole symmetric FGNR model = FS zig-zag chain! Generalised Jordan-Wigner phase factors Integrability Two very different physical pictures High degeneracy in all eigenvalues, zero energy Cooper pairs Jan de Gier (University of Melbourne) An ingenious mapping between integrable supersymmetric chains September 28, / 35