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Edge exponent in the dynamic spin
structure factor of the Yang-Gaudin
model
Mikhail Zvonarev Harvard University
Thierry Giamarchi Geneva University
Vadim Cheianov Lancaster University
PRB 80, 201102(R) (2009)
Mobile impurity / itinerant ferromagnet
Mobile impurity
Spin-down particle
Host particles
Spin-up particles
Equivalent problem: dynamics of quantum mobile ferromagnet
One-dimensional itinerant ferromagnets
Bosons in 1D carrying spin (1/2 for simplicity)
Theorem: if the interaction term is rotationally invariant than the ground
state is ferromagnetic
Excitations: longitudinal spin waves (linear dispersion = plasmons)
Excitations: transverse spin waves (quadratic dispersion = magnons)
Effective mass
One-dimensional itinerant ferromagnets – Yang-Gaudin
(iso)spin 1/2 bosons,
spin-independent interaction:
Bethe-Ansatz solvable when
- Yang-Gaudin model
Wave functions, spectrum
C. N. Yang, PRL 19, 1312 (1967);
and thermodynamics are known. M. Gaudin Phys. Lett. A 24, 55 (1967)
Dimensionless coupling:
Dispersion of spinless boson gas:
Dispersion of bosons with spin:
Effective mass in the Yang-Gaudin model (
)
Bogoliubov
Tonks-Girardeau
J. N. Fuchs et. al., PRL 95, 150402 (2005)
Tonks-Girardeau (TG) limit: short-range potential is infinitely strong
Effective mass diverges in the TG limit:
(it costs no energy to flip a spin when
)
Dynamical structure factor for
Exited states are made by one magnon and arbitrary number of plasmons
(particle-hole pairs or density fluctuations)
Similar to a polaron problem
Effective model: magnon carries momentum , its dispersion has a
minimum around . Plasmons have a linear dispersion.
Effective Hamiltonian:
Free plasmons = Luttinger Liquid
Plasmons linearly coupled
to the magnon
Effective theory (indegrability not needed)
Talk by Leonid Glazman
Free plasmons = Luttinger Liquid:
magnon
Plasmons linearly coupled
to the magnon
depend on !!!
Momentum is arbitrary !!!
Dynamical structure factor and integrability
Integrable models:
(a) Infinitely many mutually commuting integrals of motion
(b) Eigenfunctions and spectrum can be found explicitly
We replace
Replace with
with
Integrals of motion
Different
The same
Edge exponents in integrable models
Excitation spectrum near the
edge: all particle-hole pairs
except one are near Fermi
surface.
We replace
with
We require spectrum of
We require
Integrals of motion
be gapless at momentum
Edge exponents in the Yang-Gaudin model
We compare spectrum of
with the spectrum of
and get
through the solution of the integral equation
Edge exponents in the Yang-Gaudin model (continued)
Luttinger parameter K
Small momentum:
Limit of strong repulsion: logarithmic diffusion
Infinite repulsion: spin-down (red) particle cannot exchange the position with its
neighbors
Results:
as
Here
or
“logarithmic
diffusion”
Luttinger parameter
Logarithmic diffusion vs. Lutinger Liquid
Intensity plot of
White dotted line:
crossover from
trapped to open
regime:
Dashed green lines:
lightcone
corresponding to
density excitations
of spinless bosons
in the
plane at
Remarks
Edge exponents were calculated for the Yang-Gaudin model using a
combination of the Bethe Ansatz and effective theory approach
(so far there is only one – Calodgero-Sutherland – model, whose edge exponents are
calculated from the Bethe Ansatz exclusively)
The effective theory is Luttinger Liquid plus a mobile impurity, not just a
Luttinger Liquid, since the dispersion is quadratic at small momentum
There is a possibility to extract the edge exponents from the analysis
(experiment) in the real space & time.