NONMAGNETIC IMPURITIES IN QUASI1D S12 ANTIFERROMAGNETS

1
NON-MAGNETIC IMPURITIES IN QUASI-1D S1/2
ANTIFERROMAGNETS

• (with S. Eggert, M. Horton)
• Outline
• doping in 2D antiferromagnets
• 1D case bosonization
• Neel order in pure quasi-1D AFs
• doped 1D AFs finite chains
• Neel temperature in doped quasi-1D case

2
1.Doping in 2D AFs
-a 2D Heisenberg antiferromagnet orders at T0
only (at best) -quantum fluctuations (controlled
by 1/S) reduce ordered moment ?.3 mB for
S1/2 -a high enough density of non-magnetic
impurities (eg. Cu?Zn) must kill the T0
order -surprisingly, this apparently only happens
at the percolation threshold (A. Sandvik)
3
-at percolation threshold (50 doping on a
square lattice) there are no infinite clusters
of connected spins -so, ignoring longer range
interactions, all spins belong to finite
islands-these cant order -surprisingly, the
order seems to persist right up to 50 doping
even for S1/2 case
4
2. 1D case Bosonization
-all low energy, long distance properties are
described by non-interacting massless
phase-boson field theory -Lagrangian is simple
but representation of spin operators is a bit
complicated
5
from free boson correlation function we get spin
correlations

• -N.B. not Néel order- only quasi-long-range
• staggered part of Sz has renormalization
• group scaling dimension ½

6
simple scaling arguments (which agree with
detailed calculations) then determine finite-T
staggered susceptibility
(for L??)
7
similarly, we can determine staggered moment
resulting from staggered field
since L must scale as (length)-2 and staggered
spin density scales as (length)-1/2, h must
scale as (length)-3/2 and therefore
8
Néel order in Quasi-1D AFs
a standard approximation for weakly coupled
chains is to treat inter-chain coupling (J) in
a mean field approximation but intra-chain
coupling exactly (I.A., Singh, Gelfand,
Halperin) -becomes exact at large z
(co-ordination number) -each spin feels a mean
field (-1)iJzm from its z nearest neighbors
where m is staggered moment per spin
(non-frustrated lattice) -we now calculate
staggered moment resulting from staggered field
hJzm and impose self-consistency condition in
purely 1D theory
9
m1D(Jzm,T)m
-here m1D(h,T) is staggered moment in 1D
chain resulting from staggered field h at
temperature T -behavior of TN is determined from
h?0 limit m1D(h) ?c(T)h so zJ c(TN)1, ?TN
?zJ -zero field moment is determined from T0
behavior of function m1D(h,T) ?h1/3
m?(zJ/J)1/2 ?(TN/J)1/2
10
-this seems to agree with mSR experiments -much
better than other theories
11
Néel Order in Doped quasi-1D AFs
-as a first pass at this problem we apply same
approach of calculating exact 1D quantities and
treating inter-chain coupling in mean field
theory (Imry et al. Buyers et al. for Ising
case ) -at large z we can (?) average over
moments on neighboring chains, which depend on
distance to nearest impurity along chain, to
get mean staggered field
12
-if each chain has many neighbouring chains it
makes sense to replace spatially varying
staggered field by a constant staggered field ?h
13
-first we must solve 1D problem with impurities
and a staggered field -in 1D impurities just
break chain up into disconnected segments
(ignoring longer range interactions) -we need
response of a finite length chain to a staggered
field -we can determined exact linear response
for large L, small T, using bosonization -we
use mode expansion of boson field on a finite
line
14
-imposing boundary conditions, we get
oscillators modes, an with momentum pn/L as well
as a zero-momentum soliton mode whose
eigenvalues are the total (ferromagnetic)
magnetization, Sz
we get finite L thermal Greens function of ?
by Boltzmann averaging over occupation number of
each oscillator mode and over Sz Sz is integer
for even L, ½-integer for odd L
15
simplest thing to calculate is groundstate
staggered magnetization density, mj(-1)jltSjgt
of single chain of length L -for even L this
must be zero since groundstate is a spin singlet
-for odd L there are two groundstates with
total Sz?1/2 and mj has opposite sign in
each -in Néel state, mim0 (constant) for long
chain
and total staggered magnetization ? L
16
-in a dimerized phase mj would be zero (or
exponentially small) everywhere except near the
centre
total staggered moment is O(1) -quasi-long-range
ordered phase is intermediate between these 2
extremes
17
we can calculate mj in groundstate from mode
expansion -we get contribution from oscillator
modes and from soliton mode -putting oscillators
in groundstates but allowing arbitrary soliton
number Sz we find
-even L groundstate has Sz0 hence mj0 -odd L
groundstates have Sz ?1/2 and hence
18
-this agrees fairly well with Lanczos
calculations -log corrections lead to slow
convergence to field theory predictions
19
-the maximum staggered moment is near chain
centre and scales as 1/L1/2 -the total staggered
moment goes as L1/2 -note broad distribution of
moments -to get TN we will need staggered
susceptibility of a finite chain at finite T,
c(T,L)
20
-this Greens function can be expressed in terms
of Gamma functions -it has different expressions
for even or odd L -in general, it has a scaling
form ce/o(1/T)fe/o(LT) -we know that
fe/o(x)?constant at x ?? to get correct bulk
behavior -fo(x) ?constant at x ?0 since it is
then dominated by groundstate contribution
-same form at LT/vltlt1 and LT/vgtgt1 !
21
-on the other hand, fe(x) ?x so that ce is finite
at T ?0 -we have calculated c(L,T) by Monte
Carlo for L up to 120 and T down to .2J to test
bosonization prediction -expected logarithmic
corrections slightly spoil scaling forms but
they work fairly well at low T, large L
replacing constant C by
-here a is a fitting parameter (?23) and
exact amplitude of leading log is known exactly
22
NB- for L odd, c drops by about ½ -for L even it
goes to 0
23
-now we can calculate TN for a random system
using exact 1D results and mean field
treatment of inter-chain couplings
24
z4
25
(No Transcript)