1. with Masaki Oshikawa (UBC-Tokyo Institute of Technology)
Solitons and Breathers in Cu Benzoate and other Antiferromagnetic Chain Compounds
with Masaki Oshikawa
(UBC-Tokyo Institute of Technology)

2. OUTLINE Motivation Antiferromagnets in staggered fields
Sine-Gordon Field Theory
Specific heat
Neutron Scattering
Electron Spin Resonance

3. Motivation -find new theoretical methods to deal with
low dimensional systems where quantum
fluctuations invalidate conventional approaches
-one dimensional systems can be studied by
powerful and well-developed field theory methods

4. ANTIFERROMAGNETS IN STAGGERED FIELDS
This effective Hamiltonian arises in low symmetry
crystals: Cu benzoate, pyridine Cu dinitrate, Yb4As3
(2 Cu’s per unit cell) due to:
-alternating term in g-tensor
-Dzyaloshinskii-Moriya interaction
h = cH - always call applied field direction z
-c depends strongly on actual applied field direction

5. local environment
of Cu2+ ions
alternates
along chains Cu

6. Dzyaloshinski-Moriya Interactions
-in 1 dimension we can always get rid of
DM interactions by a “gauge transformation”
in zero field
-not true in >1 dimensions, in general
-specifically, in 1D we can transform a DM
interaction into a small anisotropy in symmetric
exchange interactions

7. -choose co-ordinates so D points in z-direction
and suppose it is staggered:
Gauge transformation:
Reduces H to standard form:
with:

8. classically, spins would order in xy plane:
(at arbitrary angle to axes)
S2i+1
p-a x
S2i
-however, in 1D quantum fluctuations prevent
true long range order

9. -however, a (uniform) magnetic field transforms
into a uniform plus staggered field:
-this determines a unique classical configuration
so spin fluctuations are gapped and spins order y H x

10. semi-classically, there is a unique ground state
and a gap to all excitations due to staggered field: z x
-semi-classically we may think of excitations as
small vibrations (spin waves) and topological solitons
where spins rotate by 2p in x-y plane as we move
along chain axis y x

11. SINE-GORDON FIELD THEORY
-we can study strong correlation effects
in one dimension using bosonization:
 and  are dual fields:

12. -here the parameter R=(2p)-1/2
-bosonized Hamiltonian density:
H can be eliminated by a redefinition of :
-this only shifts critical wave-vector (for Sz)

13. we are left with the sine-Gordon field theory
for the  field
-a more careful analysis, using Bethe ansatz,
shows that parameters R and vs actually vary
continuously with H in an exactly calculated way
-many things are known exactly
about relativistic sine-Gordon quantum field theory
-behavior depends strongly on R parameter
which determines renormalization group
scaling dimension of interaction term
-spectrum consists of stable single particle states:
solitons and breathers

14. highly accurate numerical solution of Bethe
ansatz equations

15. highly accurate numerical solution of Bethe
ansatz equations

16. q
soliton
1/R x
breather q
1/R x

17. semi-classical quantization of breathers
gives exact spectrum!
-breather masses all lie below 2Ms where
Ms is the soliton (and anti-soliton) mass
-they can be well below, even below Ms,
meaning that they are very strongly bound
-breather masses (and number of breathers)
depend on R
-exact free energy of SG model is known:
determines C and Ms (h)
-exact form factors also known:
<0|cos(2pRq)|si>, <0|sin(2pRq)|si>

18. SPECIFIC HEAT -a standard renormalization group scaling
argument implies that all masses (gaps) scale
as M hn, with n=1/(2-pR2)2/3
-detailed form of C(T) is non-trivial
-crosses over from linear at T>>M to Boltzmann
at T<<M
-Bethe ansatz result agrees quite well with
experiments on Cu Benzoate (Essler)

19. H2/3

20. (F. Essler)

21. NEUTRON SCATTERING -Cu Benzoate (Dender et al.)
-in zero field there are no single particle peaks- just
power law singularities
-at finite H we see expected momentum shift
p pH
-we also see resolution limited single particle peaks
-different mass gaps at wave-vector p, corresponding
to Sx ,Sy Green’s function (breather) and
pH corresponding to Sz Green’s function (soliton)

22. breather
soliton
soliton

23. MB/MS=.79 for pR2=.41, predicted by S-G model
-this value of R is expected for Heisenberg model
for gmBH/J=.52 (H = 7 Tesla, J=1.57 meV)
from Bethe ansatz
-other breathers?
-SG model predicts 2 higher energy breathers
at this value of R with M2=1.45Ms, M3=1.87Ms
-relative intensities down by ½ and 1/6 compared
to 1st breather from SG “form factor”

24. -however, we must also take into account
predicted polarization of modes and factors
of [k2-(ka)] in Saa (k,w)
-1st and 3rd breather are polarized along y
but 2nd breather is polarized along x (staggered
field direction)
-soliton and anti-soliton are polarized along z
-polarization of 1st soliton was determined
experimentally- this determines staggered
field direction within our theory
-k in experiments was nearly along x direction
-so predicted intensity is very small for 2nd breather

25. -we can make a less accurate theoretical estimate
of relative intensity of solitons to 1st breather
-when R=(2p)-1/2 (i.e. at very small H)
1st breather, soliton and anti-soliton form a
degenerate triplet and intensity of breather
is twice that of soliton – experimentally
breather is more intense by about 2.8?
-we interpret other feature at k=p as being
soliton observed at k=H:

26. Electron Spin Resonance
-adsorption intensity of microwave radiation
in a finite static field - from Zeeman coupling
-a k=0 probe: S(k=0,w)
-we can map this into Green’s function of
at k=H

27. -ignoring staggered field (and any other anisotropy
apart from field)
-taking imaginary part gives a d-function
in I(w) at Zeeman energy
-interactions produce a self-energy
-peak now has width =Im P/2H

28. -staggered field term is a relevant perturbation
-we can treat it perturbatively at high enough T
T>D(h)h2/3
-now interaction has dimension ½
-width is finite at H=0
so scaling implies actual result:
h  h2/T2 (for H<<T)
-note this grows with decreasing T

29. we have included in fit a linear term in
width which comes from exchange anisotropy

30. at lower T perturbation theory in h breaks down
-h defines a characteristic energy gap scale, Dh2/3
-when T is of order D or smaller
perturbation theory in h diverges
-fortunately, we know many things
about T=0 and low T behavior from integrability
of sine-Gordon model
-main contribution to  Green’s function
is from breather intermediate state
(with momentum H)
-this gives a d-function peak at T=0
-low T width is given by Boltzmann
factor, exp[-D/T]

31. thus we expect “re-entrant” behavior of width
with T with cross-over temperature  h2/3
-we also predict an abrupt shift of peak frequency:
as nature of excitation changes from a broadened
massless free boson to a massive breather
-confirmed by recent experiments
of Asano et al. (PRL84, 5880, 2000)

35. Conclusions -S=1/2 antiferromagnets with staggered fields
contain gapped soliton and breather excitations
-Dh2/3
-integrability of sine-Gordon field theory leads
to a variety of exact results on heat capacity
and neutron scattering cross-section
-when combined with renormalization group
concepts a fairly complete understanding of
T-dependent electron spin resonance results