Mobile Impurity Models and Spin-Charge Separation Fabian Essler - - PowerPoint PPT Presentation
Mobile Impurity Models and Spin-Charge Separation Fabian Essler (Oxford) Amsterdam, June 2015 A. Perturbed Luttinger Liquids large distances low energies Lattice model LL +... Example: spinless fermions (= spin-1/2 XXZ chain) L ( c
Amsterdam, June 2015
H = −t
L
(c†
jcj+1 + H.c.) + V
njnj+1.
Lattice model LL +... large distances low energies Example: spinless fermions (= spin-1/2 XXZ chain)
H =
term,
HLL(x) = v 16π
K (∂x)2
Luttinger model. The velocity and the Luttinger
free, compact boson irrelevant perturbations
Projection to low-energy degrees of freedom:
lattice spacing
Haldane ’82
Ground state correlation functions for XXZ: Perturbation theory in Sirr[Φ] gives excellent agreement with the lattice model result (even for ℓ=1) !
Lukyanov ’97
(analogous expressions for spin operators in XXZ)
e.g. single-particle spectral function: Map this to a LL for ω≈0 (⇒k≈±kF): LL gives power-law threshold singularities CFT predictions for μ± are wrong !
PT in Sirr[Φ] fails. Why does this happen? Can be understood already for free lattice fermions: Close to kF: LL result infrared singularity Bad news: must sum PT to all orders, singularities don’ t simply exponentiate. There is a neat trick for doing this ....
Pustilnik, Khodas, Kamenev &Glazman ’06
Bethe Ansatz gives full spectrum of our lattice model: thresholds Kinematics: just above the threshold at mtm k only states with a single high energy excitation with mtm ≈k contribute ⇒ augment LL by a single “mobile impurity” with mtm ≈k
A<(ω,k) = 2π
|ψn|ck|ψ0|2δ(ω + En − E0),
impurity field LL lattice Impurity has the same quantum numbers as cj (smallΔ, Bethe Ansatz) Project onto LL & impurity degrees of freedom:
Gret,<(ω,k) ≈ −i ∞ dt eiωt ∞
−∞
dxψ0|χ(0,0)χ†(t,x)|ψ0.
Close to the threshold (at ω<0) we have
The mobile impurity model is solved by a unitary transformation: U = e−i
∞
−∞ dx [γ ϕ(x)+ ¯
γ ¯ ϕ(x)]χ†(x)χ(x).
impurity decouples in new basis!
A<(ω,k) ∼ θ(ǫ(k) − ω)|ǫ(k) − ω|−1+2(γ 2+ ¯
γ 2).
Threshold singularity at ω<0:
How to fix ???
Hamiltonian in presence of a single high-energy excitation.
using mode expansion.
Pereira, Affleck & White ’09 Kitanine et al ’12
confirmed by direct BA calculation
spinful fermions: Hubbard model large distances low energies
Single-particle spectral function: Probes all excitations with quantum numbers S=1/2, Q=±e Bethe Ansatz: excitations at all energies composed of holons (Q=±e, S=0), spinons (Q=0, S=±1/2) (and their bound states).
cf Essler&Korepin ’94
/2
Phs
1 2
Ehs u=1
n=0.5 threshold corresponds to a high-energy spinon + low-energy holon Kinematics: just above the threshold only states with a single high energy excitation contribute ⇒ single mobile spinon impurity!
Projection onto low-energy and impurity d.o.f. now is known
Schmidt, Imambekov &Glazman ’10
Scheme, in which the impurity is weakly interacting and has fractional quantum numbers at variance with Bethe Ansatz
Essler, Pereira &Schneider ’15
☞
bosonized formulation Refermionize in terms of fields carrying only spin/charge: in the Hubbard model these fermions are strongly interacting
cf Coleman ’75
massive Thirring
Projection of lattice fermion operators: U(1) charges fractional JW strings Can add interactions in lattice model to make spin/charge fermions weakly interacting (break SU(2)!!!)
“high-energy” spinon impurity low energy spinons Bosonize low-energy fermions Mobile impurity model:
Projection onto low-energy and impurity d.o.f. becomes Conjecture: for strongly interacting spin/charge fermions the same expressions apply, only the parameters need to be adjusted. Finally
to Bethe Ansatz calculation ⇒ exact results for threshold exponents
G(t, k) ∼ X
α
Aαeiωαt+φαt−γα,
G(t,k)
k=0 k=π/8 U=5
(a) (b)
10 20 30 40 50 60 1.0 0.5 0.0 0.5 1.0 1.0
t
10 20 30 40 50 60 1.0
Translate results for threshold singularities to time domain exact results t-DMRG results
Seabra et al ’14
use as fit parameters
Our construction raises an interesting question: Is there a lattice model of strongly interacting electrons, that maps exactly to free fermionic spinons and holons?
RG analysis of vicinity of LE point is quite interesting.
even at low energies.
calculate dynamical properties at finite frequencies.