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)

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

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

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

local environment of Cu2+ ions alternates along chains Cu

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

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

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

-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

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

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

-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)

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

highly accurate numerical solution of Bethe ansatz equations

highly accurate numerical solution of Bethe ansatz equations

q soliton 1/R x breather q 1/R x

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>

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)

H2/3

(F. Essler)

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)

breather soliton soliton

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”

-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

-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:

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

-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

-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

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

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]

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)

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