BEYOND THE TOMONAGA-LUTTINGER MODEL: Dynamical critical behaviour of the xxz spin chain with RG Pereira , J Sirker, JS Caux, JM Maillet, R Hagemans and SR White 1
1 ST LECTURE-small q • Introduction • Band curvature effects at small q • Effective Hamiltonian – exact coupling constants • Exact results for high energy tail of S(q, ω ) • Behaviour near peak of S(q, ω ) • Bethe ansatz and Density Matrix Renormalization group (DMRG) results 2
OUTLINE-2 nd LECTURE • “X-ray edge” field theory methods • Finite size spectrum and Bethe Ansatz • Long time behaviour of self-correlation function • Comparison to Dynamical Density Matrix Renormalization Group results • Open questions 3
Introduction N ∑ = ⋅ + ⋅ + Δ ⋅ − x x y y z z z [ ], H S S S S S S hS + + + j j 1 j j 1 j j 1 j = 1 j r r Δ < < = | | 1 , , h h S S + S N 1 1 N 1 ∞ ∑ ∫ − − ω ω = < > iq ( j j ' ) i t z z ( , ) 0 | ( ) ( 0 ) | 0 S q e dt e S t S ' j j − ∞ N = , ' 1 j j ω Field theory works only here, at low ω ?? q 4 2 π 2k F
Jordan-Wigner transformation: = ∑ N + − + + Δ 1 [ ( . .) ] H c c h c n n + + 1 1 j j j j 2 = 1 j − ≈ ψ + ψ ik j ik j ( ) ( ) c e j e j F F j R L ε k F k ε F Low energy q ≅ 2k F excitation 5
Linearizing the fermion dispersion relation and bosonizing gives: − ≈ ψ + ψ + ψ + ψ + ψ + ψ + ψ + ψ 2 2 z ik j ik j ( ) ( ) S e e F F j L L R R L R R L K ≈ ∂ φ + ⋅ π φ + const cos [ 4 2 ] K k j π x F ψ ∝ − π θ ± π φ exp [ ( / )] i K K / R L With an effective free boson Hamiltonian density: 1 v = Π + ∂ φ 2 2 ( ) H x 2 2 v •The velocity, v, and Luttinger parameter, K, can be determined from Bethe ansatz results •We can obtain low energy S(q, ω ) and fermion Green’s functions in terms of free boson Green’s functions 6
BAND CURVATURE EFFECTS • At small q, Luttinger liquid theory predicts S zz (q, ω )=(K/ π )< ∂ x φ ∂ x φ >=Kq δ ( ω -v|q|) • In free fermion case this simple form is a result of linear dispersion: ω q All particle-hole excitations of wave-vector q have same energy, ω =v|q| 7
Including band curvature gives finite width: ω q width, Δω ~q 2 Δ =0 <S z >= so LL theory is almost correct as q ⇒ 0 8
•What happens if we include both band curvature and interactions ( Δ≠ 0)? •Does width still scale as q 2 ? •What is line shape? •Could steps turn into power law divergences? •Can we study these questions with field theory techniques? •What information can we extract about these questions from Bethe ansatz solution? •What can we learn from numerical techniques (Density Matrix Renormalization Group)? 9
EFFECTIVE HAMILTONIAN: EXACT COUPLING CONSTANTS • Band curvature gives interactions upon bosonizing even for Δ =0! π 2 ( ) + ψ ∂ ψ ≈ ∂ φ 3 2 L L x L 3 •Interactions in lattice model given another cubic term. Including both: •These are most general dimension 3 interactions allowed by parity •They destroy Lorentz invariance and particle-hole symmetry ( ϕ→ - ϕ ) 10
•We may derive exact identities relating η ± to derivatives of Luttinger parameter and velocity with respect to field, h: •v(h) and K(h) can be determined to high accuracy from numerical solution of thermodynamic Bethe ansatz equations •Thus coupling constants, η ± can be determined essentially exactly 11
BEHAVIOUR NEAR TAIL OF S • Lowest order perturbation theory in η ± gives retarded Green’s function: θ ω − 5 2 4 ( | |) Kq d Kq v q = δ ω − + η δ ω − + η + 2 2 K ( | |) ( | |) S Kq v q v q − + ω ω − 2 2 2 2 24 18 ( ) d v v q •Note that, while both corrections are suppressed by 2 extra power of q or ω , they diverge “on mass shell” near ω ~v|q| •We can only trust this perturbation theory far from mass shell: ω -v|q|>> q 2 η ± 12
2 term predicts a “high energy tail” for S • η + •We compared this to Bethe ansatz: •Up to 2000 form factors were calculated: •<0|S z (q)|n>, 2-particle, … up to 8-particle for finite length chains up to length N=600 •We extend the field theory result to finite N by the usual conformal transformation •Field theory predicts states at discrete energies: ω =2 π vn/N, n=1,2,3,… whereas Bethe ansatz states are scattered •We bin BA states to make comparison: 13
14 Bethe ansatz squared form factors
•Red dots are BA results, line is field theory •Note that we have no adjustable parameters (all fixed from thermodynamic BA) and we are at strong coupling 15
BEHAVIOUR NEAR PEAK • We need to somehow sum series in η ± to get sensible result near peak • We explicitly calculated series in η - to 4 th order: 16
Here w ≡ i ω -vq and the 4 th order term gets contributions from all 3 diagrams: 1/[5x2 4 ]=1/144 + 1/504 + 1/280 •Tells us that no simple partial resummation (RPA, SCBA,…) will be sufficient! 17
•Series in η - should sum up to give free fermion result, including band curvature at small q Implying: but we have no idea how to sum η + terms This crude ( η - only) approximation, seems to give correct line-width, ~q 2 and height but misses line shape and power law singularities 18
Δ =.25 σ =-.1 N=6000 •Blue dots are BA results (2 particle states only) •Dashed line is field theory result ignoring η + 19
• We can’t get enough resolution from finite size BA, with limited number of states kept, to study singularities • We don’t know how to sum series in η + to make field theory predictions for singularities • Singularities may also exist at q of O(1) – we would like to study those also q 20 2 π 2k F
X-Ray Edge Field Theory Methods We can calculate S(q, ω ) near ω min (q) for any q using other field theory techniques first developed for study of X-ray edge singularities – Pustilnik, Glazman, … ω ω min (q) q q 2 π 2k F 21
•Consider excitations with one “deep hole” at k 1 and many particles and holes near +k F •These appear to give lower threshold for hole Green’s function at momentum k or of S(q, ω ) at q= ± k F -k 1 k 1 k F k ε F q 22
To calculate G F (k 1 , ω ) and S(+k F -k, ω ), we consider a “low energy” effective Hamiltonian, H k , containing only Fourier modes of fermions near k and + k F for the particular value of k=k 1 of interest Λ k F k 1 k ε F ε 1 23
− ≈ ψ + ψ + ik j ik j ik j ( ) ( ) ( ) c e j e j e d j 1 F F j R L ~ ~ ≈ + + ε − ∂ + κ ψ + ψ + κ ψ + ψ + ( ) ( ) H H d iu d d d 1 1 TLL x L L L R R R •N.B. we haven’t included any terms that would allow the “heavy hole” to decay •Validity of this is complicated and not completely understood •Now looks like model for a single “heavy” hole interacting with a Tomonaga-Luttinger Liquid (see Tsukamoto, Fujii, Kawakami, PRL 100, 126403, 1998) •Now we bosonize the fermions near the Fermi surface only: 1 v + = Π + ∂ φ + ε − ∂ 2 2 ( ) ( ) H d iu d 1 1 x x 2 2 v 1 + + κ ∂ φ + κ ∂ φ ( ) d d π L x L R x R 2 K 24
•All parameters: ε 1 , u 1 , κ L , κ R , depend on our arbitrary choice of heavy hole momentum, k 1 , and are renormalized by the interactions, as we integrate out all wave-vectors except for the 3 narrow bands near k 1 , +k F •Nonetheless, we determine them exactly using Bethe Ansatz •Once these parameters are known, we can calculate the edge exponents by making a unitary transformation: ⎧ ⎫ ∞ [ ] i ∫ + d = − γ φ + γ φ ⎨ ⎬ exp U dx d π R R L L ⎩ 2 ⎭ K − ∞ •The parameters γ L/R are chosen to eliminate the marginal interactions ~ ∂ x φ L/R d + d, leaving only irrelevant ones with 2 or more derivatives = κ γ / L R ± / L R v u 1 25
The transformed Hamiltonian is just free fields: + ε 1 ~ ~ ~ ~ v = Π + ∂ φ + − ∂ 2 2 ( ) ( ) H d iu d 1 1 x x 2 2 v But the fermion d-field now contains a boson factor: ⎡ ⎤ ~ ~ ~ i = − γ φ + γ φ exp ( ) d d ⎢ ⎥ π R R L L ⎣ ⎦ 2 K ~ ~ ~ + ψ − πν φ + πν φ + and thus: p exp [ 2 2 ] d i i d + − R R L ⎡ γ ⎤ ⎛ − ⎞ 1 1 ν = ± ⎜ ⎟ / where: R L 1 ⎢ K ⎥ ± π ⎝ ⎠ ⎣ ⎦ 4 K and similarly for d + ψ L 26
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