Ground-state selection from anharmonic zero-point … selection from anharmonic zero-point energy in...
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Ground-state selection from anharmonic zero-point energy in the pyrochlore antiferromagnet Uzi Hizi and C.L. Henley, Cornell University March 22, 2004 1
Transcript of Ground-state selection from anharmonic zero-point … selection from anharmonic zero-point energy in...
Ground-state selection from
anharmonic zero-point energy
in the pyrochlore antiferromagnet
Uzi Hizi and C.L. Henley,
Cornell University
March 22, 2004
1
Heisenberg Model on the Pyrochlore lattice
H = J∑
〈ij〉
~Si · ~Sj =∑
α
∣
∣
∣
∣
∣
∑
i∈α
~Si
∣
∣
∣
∣
∣
2
+ const.
• Classically, all states with zero sum in
each tetrahedron are degenerate.
• Thermal fluctuations do not break the degeneracy (Reimers
1992, Moessner and Chalker, 1998).
• Quantum fluctuations choose only some of the
collinear ground states (Henley, APS March
Meeting 2001). Ground state is characterized
by Ising variables ηi ∈ {±1},∑
i∈α ηi = 0
• Does the quantum model have a distinct ground state?
2
Large S expansion
• Holstein Primakoff transformation:
– Change to local coordinates: x = ηix,y = y,z = ηiz.
– Express spin components in terms of bosons {ai}, {a†i}.
– Spin deviations σxi ≡ηi
√
S2 (ai+a
†i ), σ
yi ≡−i
√
S2 (ai−a
†i ),
with commutation [σxi , σ
yj ] = iηiSδij.
• Hamiltonian:
H = −JN
(
S +1
2
)2
+J
(
1 +1
2S
)
∑
i
|~σi|2 + J
(
1 +1
4S
)2∑
〈ij〉
~σi · ~σj
+J
4S2
∑
〈ij〉
(
ηiηj |~σi|2|~σj |
2 −1
2~σi · ~σj
(
|~σi|2 + | ~σj |
2)
)
+ O(S−3) .
3
Harmonic Hamiltonian
H = (σx)THσ
x + (σy)THσ
y .
• Equation of motion:
σ̇x = −i
h̄[σx,H] =
2S
h̄ηHσ
y ,
σ̇y = −i
h̄[σy,H] = −
2S
h̄ηHσ
x .
η ≡
η1 0 0 · · · 0
0 η2 0 · · · 0
0 0 η3 · · · 0...
......
. . ....
0 0 0 · · · ηN
• Harmonic modes {vp} are eigenvectors of (ηH)2, with
eigenvalues ω2p.
• If ηH is diagonalizable, {vp} are also eigenvector of ηH and
the zero-point fluctuations are 〈σxi σx
j 〉 =∑
pS
2(vp,ηvp)vipv
jp.
• Otherwise, fluctuations diverge.
• In order to treat quartic terms in Hamiltonian perturbatively,
use a harmonic variational Hamiltonian to removes divergence.
4
Comparison to Kagome lattice
Kagome Pyrochlore
Coplanar spins Collinear spins
H2 doesn’t break degeneracy H2 breaks some of degeneracy
Asymmetry between in-plane Symmetry between
and out-of-plane deviations x and y spin components
Branch of divergent modes Divergences along lines in ~q space
Power law dependence of Logarithmic dependence of
correlations on S correlations on S
Cubic order in H selects from Cubic terms vanish. Quartic
degenerate states (Chubukov terms should select between
it 1992, Henley & Chan 1995) harmonic degenerate states
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Unperturbed Harmonic Hamiltonian
H =J∗
2W
TW, Wαi ≡ δi∈α .
• Harmonic zero point energy depends only on frequencies. All
non-zero frequencies are for modes that have non-zero sums on
tetrahedra. To find non-zero frequencies, only the total
tetrahedron spin ~τα =∑
i∈α ~σi matters (Henley 01).
• A gauge transformation ~τα → ±~τα does not change frequency,
i.e. zero-point energy is invariant.
• This “gauge” transformation must be done is a way that
conserves the zero sum on each tetrahedron.
• Energy can be expressed as an infinite sum over all loops in
lattice. Each term in the sum is a product of spins around the
loop. Harmonic ground states are all (gauge equivalent) states
with negative products of spins around hexagons.
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Variational Hamiltonian
• The most general local harmonic Hamiltonian that conserves
the lattice and Ising symmetries is Hvar = H2 + δηH2η + ε � .
• Spin rotation symmetry requires ε + 4δ = 0.
• Calculate (Gaussian) fluctuations from this Hamiltonian, and
plug into decoupled Hamiltonian. Minimize the total energy to
find the self consistent energy.
• We considered three periodic Harmonic ground states, shown
here in (0,0,1) projection:
↓
↑
↓
↑
↓
↑
↑
↓
↑
↓
↑
↓
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Results
• Scaling:
– ε scales, to leading order as S−0.9. While the expansion is
only valid for large S, the numerical accuracy reduces as S
increases, because contribution to correlations comes from
narrower regions in ~q space.
– E2 ∼ ε1.2 ∼ S−1.0.
– E4 ∼ −0.2 ln ε ∼ 0.17 lnS.
• Ground state selection: 10−3
10−2
E2/S
101
102
103
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SE
4
– Although the leading order scaling is the same for all three
states, one of them has lower energy for all S values: the
one with 32 spins in its unit cell.
8
Conclusions
•
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