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Integrable Systems and Quantum Deformations Integrable Systems and Quantum Deformations Peter Koroteev University of Minnesota In collaboration with N. Beisert, F. Spill, A. Rej 0802.0777, work in progress MG12, July 13th 2009 Outline Introduction. Integrability and Symmetry What is Integrability Coordinate Bethe Ansatz AdS/CFT Correspondence Quantum Deformation of 1D Hubbard Model 2 Uq(su(2j2) n R ) algebra Hubbard-like Models Universal R-matrix Yangian Universal R-matrix I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. We are focused on QFTs and spin chains What is Integrability Exists an infinite set of independent commuting charges fQαg; [Qα; Qβ] = 0 I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. We are focused on QFTs and spin chains What is Integrability Exists an infinite set of independent commuting charges fQαg; [Qα; Qβ] = 0 I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution We are focused on QFTs and spin chains What is Integrability Exists an infinite set of independent commuting charges fQαg; [Qα; Qβ] = 0 I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. What is Integrability Exists an infinite set of independent commuting charges fQαg; [Qα; Qβ] = 0 I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. We are focused on QFTs and spin chains Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions Some methods I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Coordinate Bethe Ansatz. Mostly for spin chains I Algebraic integrability Signatures of Integrability Explicitly find all commuting charges Some methods I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Coordinate Bethe Ansatz. Mostly for spin chains I Algebraic integrability Signatures of Integrability Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions I Coordinate Bethe Ansatz. Mostly for spin chains I Algebraic integrability Signatures of Integrability Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions Some methods I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Algebraic integrability Signatures of Integrability Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions Some methods I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Coordinate Bethe Ansatz. Mostly for spin chains Signatures of Integrability Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions Some methods I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Coordinate Bethe Ansatz. Mostly for spin chains I Algebraic integrability Hamiltonian L X H = Hk;k+1; k=1 1 Hk;k+1 = Ik;k+1 − Pk;k+1 = 2 1 − ~σk ⊗ ~σk+1 Symmetry su(2) interchanges j"i and j#i. H commutes with all generators What is the spectrum of the linear operator H? Brute force: list all states with given n"; n#, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) XXX Spin Chain Hilbert space j i = j"##" :::i Symmetry su(2) interchanges j"i and j#i. H commutes with all generators What is the spectrum of the linear operator H? Brute force: list all states with given n"; n#, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) XXX Spin Chain Hilbert space j i = j"##" :::i Hamiltonian L X H = Hk;k+1; k=1 1 Hk;k+1 = Ik;k+1 − Pk;k+1 = 2 1 − ~σk ⊗ ~σk+1 What is the spectrum of the linear operator H? Brute force: list all states with given n"; n#, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) XXX Spin Chain Hilbert space j i = j"##" :::i Hamiltonian L X H = Hk;k+1; k=1 1 Hk;k+1 = Ik;k+1 − Pk;k+1 = 2 1 − ~σk ⊗ ~σk+1 Symmetry su(2) interchanges j"i and j#i. H commutes with all generators Brute force: list all states with given n"; n#, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) XXX Spin Chain Hilbert space j i = j"##" :::i Hamiltonian L X H = Hk;k+1; k=1 1 Hk;k+1 = Ik;k+1 − Pk;k+1 = 2 1 − ~σk ⊗ ~σk+1 Symmetry su(2) interchanges j"i and j#i. H commutes with all generators What is the spectrum of the linear operator H? XXX Spin Chain Hilbert space j i = j"##" :::i Hamiltonian L X H = Hk;k+1; k=1 1 Hk;k+1 = Ik;k+1 − Pk;k+1 = 2 1 − ~σk ⊗ ~σk+1 Symmetry su(2) interchanges j"i and j#i. H commutes with all generators What is the spectrum of the linear operator H? Brute force: list all states with given n"; n#, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) Find H12j##i = 0 hence Hj0i = 0 Spin flip jki = j###"k ## :::i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X jpi = eipk jki Act with H to find eigenvalue and dispersion relation +1 X Hjpi = eipk (jki − jk − 1i + jki − jk + 1i) −∞ = 2(1 − cos p)jpi =: e(p)jpi Bethe Ansatz Consider infinite chain. Vacuum (ferromagnetic) j0i = j##### :::i Spin flip jki = j###"k ## :::i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X jpi = eipk jki Act with H to find eigenvalue and dispersion relation +1 X Hjpi = eipk (jki − jk − 1i + jki − jk + 1i) −∞ = 2(1 − cos p)jpi =: e(p)jpi Bethe Ansatz Consider infinite chain. Vacuum (ferromagnetic) j0i = j##### :::i Find H12j##i = 0 hence Hj0i = 0 Momentum eigenstate X jpi = eipk jki Act with H to find eigenvalue and dispersion relation +1 X Hjpi = eipk (jki − jk − 1i + jki − jk + 1i) −∞ = 2(1 − cos p)jpi =: e(p)jpi Bethe Ansatz Consider infinite chain. Vacuum (ferromagnetic) j0i = j##### :::i Find H12j##i = 0 hence Hj0i = 0 Spin flip jki = j###"k ## :::i Hamiltonian homogeneous, eigenvectors are plane waves Act with H to find eigenvalue and dispersion relation +1 X Hjpi = eipk (jki − jk − 1i + jki − jk + 1i) −∞ = 2(1 − cos p)jpi =: e(p)jpi Bethe Ansatz Consider infinite chain. Vacuum (ferromagnetic) j0i = j##### :::i Find H12j##i = 0 hence Hj0i = 0 Spin flip jki = j###"k ## :::i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X jpi = eipk jki Bethe Ansatz Consider infinite chain. Vacuum (ferromagnetic) j0i = j##### :::i Find H12j##i = 0 hence Hj0i = 0 Spin flip jki = j###"k ## :::i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X jpi = eipk jki Act with H to find eigenvalue and dispersion relation +1 X Hjpi = eipk (jki − jk − 1i + jki − jk + 1i) −∞ = 2(1 − cos p)jpi =: e(p)jpi Two excitations P ipk+iql Position State jp < qi = e j::: "k ::: "l :::i k<l \Almost" an eigenstate (spin flips far from each other) Contact term X i(p+q)k ip+iq iq Hjp < qi = (e(p) + e(q))jp < qi + e (e − 2e + 1)j"k "k+1i; k X i(p+q)k ip+iq ip Hjq < pi = (e(p) + e(q))jq < pi + e (e − 2e + 1)j"k "k+1i k Construct eigenstate jp; qi = jp < qi + Sjq < pi with scattering phase eip+iq − 2eiq + 1 S = − = e2iφ(p;q) eip+iq − 2eip + 1 Eigenvalue Hjp; qi = (e(p) + e(q))jp; qi Scattering 6=3! asymptotic regions Match up regions at contact terms, find eigenstate jp1; p2; p3i = jp1 < p2 < p3i + S12jp2 < p1 < p3i + S23jp1 < p3 < p2i; +S13S12jp2 < p3 < p1i + S13S23jp3 < p1 < p2i + S12S13S23jp3 < p2 < p1i eigenvalue e(p1) + e(p2) + e(p3) Integrability: Scattering factorizes for any number of particles Two particles scattering phase enough to construct any eigenstate on infinite chain Move one ip L excitation pk past L sites e k of the chain and k − 1 other Q particles Skj . Should end up with the same state Bethe equations k Y eipk +ipj − 2eipk + 1 1 = e−ipk L − eipk +ipj − 2eipj + 1 j=1 Reparametrise pk = 2arccot2uk via rapidity k uk − i=2L Y uk − uj + i 1 = − uk + i=2 uk − uj + 1 j=1 Total energy k k k X X 2 X i i E = e(pj ) = 4 sin pj =2 = 4 − uj + i=2 uj − i=2 j=1 j=1 j=1 u +i=2 Total momentum eip = Q eipj = Q j uj −i=2 Bethe Equations We have: infinite chain.
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