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Theory of the Helical Spin CrystalA Proposal for the `Partially Ordered’
State of MnSi
Ashvin VishwanathUC Berkeley
In collaboration with:
Benedikt Binz (UC Berkeley)
Vivek Aji (UC Riverside)
•Phys. Rev. Lett. 96, 207202 (2006).•cond-mat/0608128
MnSi: Experimental Facts
• Elastic Neutron Scattering
This Talk:Focus on the Partial Order StateProposal: Helical Spin Crystal as
intermediate scale structure.
1. Theory of Helical Spin Crystals2. Destroying the Crystal – Disorder
OR Thermal/quantum fluctuations?
Static vs. dynamic? New phase or crossover ?
Dzyaloshinskii-Moriya and the Spiral State
• Origin of DM: Spin orbit interaction – Need to break inversion symmetry.
Consider 2 spins in an insulator
Ferromagnetism
21 SSH
J
J/|| 12d
DM Term
21 SSHDM
12d
12d
1 2
Leads to long wavelength spiral if d<<J.
12d2
J 2E
Landau Theory of a Spiral States 1
• Continuum magnetization in a crystalline itinerant magnet
Note, free energy is rotationally invariant – locking to lattice from higher order terms.
M)(MD
MMM
DM
ferro
aaF
urJF
422)(
M
M)(MDM DF
r<0→ ferromagnet
DM rotates M
DM for the B20 structure
Landau theory of Spiral States II
Single Spiral
const.|M(r)|
)sin()cos((r)M
rqrq qq 21 ˆˆ
)ˆˆ(
;
21
.
qqq
riq
im
em
(r)M
General State: A Superposition M
At quadratic order, for r=0, any superposition of helices is degenerate.
Selected by interactions
Landau Theory of Spiral States III
• The quartic interaction picks the combination of spirals. For uniform U, this is the single mode state – only a single point on the wavevector sphere.
{Reason: is minimized by single mode since it has }
• Finally, Crystal Anisotropy Term for MnSi crystal structure:
drU4
(r)ME4
.const(r)M
λ>0 Implies spiral locked along <111>
not <100> directions.
Why Partial Order state is not a single spiral state
• Obvious anisotropy terms allowed by the crystal field orient spots along (111) or (100) – unnatural to have them along (110).
• `Math’ argument – For a real function on the sphere (i.e. The Anisotropy energy of the single mode state):
(#Maxima) + (#Minima) – (#Saddle-pts) = 2
If all critical points are cubic points: 8 of (111), 6 of (100) and 12 of (110)
Only solution 8+6-12 = 2.(110) Is a saddle point and NOT a Minimum for single
mode states. Unnatural to expect minima at (110).
(111)
(110)
(001)
Proposal: Multi Mode StateHelical Spin Crystal
• Energetics: Stabilizing a Multi-mode (Spiral Solid) state.
• Description of Multi-Mode state
• Phenomenology:– Effect of anisotropy– Effect of magnetic field– Effect of disorder– NMR and muSR– Magnetotransport
• Classical and Quantum Transitions
Simultaneous condensation of spirals at multiple wave-vectors
Energy Scales
•Ferro
•DM
•Interactions U
•Anisotropy
Chaikin and Lubensky, pg. 189
Aside 1: Analogy with Solids
• Order parameter of a solid- density at wavevector q.
• Single mode state – CDW; • Multi-mode state – solid
• Landau theory (weak crystallization) of freezing
• Favours triangles of Bragg spots – triangular lattice in 2D; BCC in 3D.
• Transition first order in mean field theory
Cubic term
Aside 2:Differences from Solids
• Important differences from the problem of crystallization:– M is a vector; no cubic term in free energy.
Freezing transition in mean field can be continuous.
– Spiral state is special: |M(r)|=const. unlike a CDW.
– Simple energetics gives BCC for solids (maximize triangles) – no simple arguments for spirals.
– Coincidentally(?).
MnSi Max intensity in high pressure state→BCC
Stabilizing a Multi Mode Spiral State
• Uniform quartic term gives rise to single mode state – need more structure to stabilize multi-mode state.
iqr
qqem
V
1(r)M
]][[),,(V
1)(321
,,34 3214321
321
qqqqqqqqqq
mmmmqqqUF
Parameterize Quartic interaction
φ/2
θ
1 2
4
3
Choose
Stabilizing a Multi Mode State IIExpanding the interaction in harmonics
Determine energetics for arbitrary combination of 13 modes [(110);(100);(111)] and upto 4 arbitrary spirals.
Phase Diagram: U20=0, U0=W
Relation to other work: (Rossler et al., I. Fischer and A. Rosch) have the term:
24 )]([' 24 MM WWF
220
20
'3
4
'3
4
QWU
QWWU
Which here is:
Stabilizing a Multi Mode State III
Phase Diagram: U20=0, U0=W
Energetics dominated by 1 and 2 mode interactions.
BCC stabilized since reciprocal FCC lattice is close packed.
Can construct toy interactions with BCC as ground state
Landau Theory for BCC state
• Allow for arbitrary amplitudes and phases of 6 modes relative to a reference state.
• Identify quartic invariants under translations, point group. ),...,( 621
BCC state – condensation in all 6 modes. λ>0 BCC1 AND λ<0 BCC2
λ
Analogy to Cholesteric Blue Phases
• Chiral nematics – rod like molecules form spiral states.
• (A) Blue Phase - periodic array of defect lines permeates structure.
• Nematic order parameter naturally has line defects – these then arrange themselves into an array.
• Here – ferromagnetic order parameter that spirals. No line defects.
• But point defects – expect lattice of hedgehogs (?)
The BCC1 state
Sections through the state
END
BCC 1
Generic Cut – Merons and anti-Merons; and vortices
Zeros of the Magnetization -- and Meron Centers --
BCC 1 – Symmetry Properties
Zeros of the Magnetization -- and Meron Centers --
Adding modes does NOT erase line zeros.
Protected by symmetry (not topology).
BCC1 defined by symmetry property: Rotation by 90º about black lines x,y or z, followed by Time Reversal (τ: M → –M) is a symmetry.
•Implies Nodes (along black lines)
•AND implies M around node has anti-vortex form (lowest winding).
•Magnetization directions along red lines as shown.
•M has ‘meron’ form near this line
BCC2 Phase
Magnetization zero at points – but no hedgehogs.
Symmetry 90º Rotn.+ τ+Translation
END
Sections through the state
Nature of Symmetry Breaking ofBCC States
• BCC States:– break continuous Translation symmetry (Tx,Ty,Tz) and Time
reversal symmetry. {Derived from Landau Theory}
– 3 Goldstone modes + 2 types of domains (M→ -M).– Time reversal symmetry breaking without a net magnetization.
• Single Spiral State: only one Goldstone mode (with crystal anisotropy) and does not
break Time reversal symmetry (M→ -M can be achieved by translation). Domains arise from breaking lattice point group symmetry.
BCC1 vs BCC2 – Magnetization Distribution
M
BCC1
BCC2
Histogram of magnetization –
))((1 3 rMMrdV
y
Single mode
If static, should be observable by NMR and μSR
NMR on MnSi
• Zero field NMR on MnSi In the helical spin crystal – Static magnetism above pc
– Broad line shapes
– BUT, drop in intensity.– No signal in muSR
– Time fluctuating BCC order pinned at surfaces?
W. Yu et al. PRl (2004)
Phenomenology 1. Effect of Crystal Anisotropy
From the single mode state orientation [111], we know the sign of the crystalline anisotropy term:
Crystalline anisotropy also locks the orientation of the BCC states.
With the above sign of the anisotropy, we find that the 6 mode state is always oriented along the (110) directions both for BCC1 and BCC2.
Problematic for other theories of the partial order state:proximity to multi-critical point – (Turlakov and Schmalian PRL 04); magnetic liquid-gas transition (Tewari, Belitz, Kirkpatrick Phys. Rev.
Lett. 96, 047207 (2006).)
Would prefer the (111) states.
2444 )(|| qzyxc MqqqF
Phenomenology 2: Effect of a Field
• Applying a Magnetic Field:
– Single Mode State• Anisotropic Susceptibility –
likes to orient q along h.• If q//h, spins can cant towards
field.
)(rMhf
0m
020
2q//
200 mmmmm hUrf ][
U
rJDUr
/21//
Phenomenology 2: Effect of a Field
• Applying a Magnetic Field:– BCC state
• Isotropic Susceptibility – independent of field direction by cubic symmetry. No reorientation transition expected.
• Susceptibility expected to be lower than an oriented single mode state.
Oriented spiral
Polarized
BCC
00q20
2q//
200 mmm|m|m|mm hUUrf
2|][
12
1 /]
3
2[
SpiralBCC U
rJDUUr //
Phenomenology 2: Effect of a Field
Oriented spiral
Polarized
BCC
q2
Effect of a Field on BCC States
• Actually, response of BCC state is more complicated.
))((
|][
0
2
mmmm
mmm|m|m|mm
321 qqq
00q20
2q//
200
hUUrf
q1q3
0
adjusting phases adjusting rel. amplitudes
• State adjusts in a field – susceptibility smaller than single spiral if coupling μ is weak.
Effect of a Field on BCC States
• Effect of Magnetic Field on Bragg Spot Intensities. Starting with a particular BCC1 state (breaks Time reversal)– Applying a field along +[111] enhances
spots (1,3,5) but reduces (2,4,6).
– Applying field along [001] enhances (1) and reduces (2).
– Could be tested by neutron scattering.
Signatures of BCC State in Magneto-Transport
• BCC states break Time reversal symmetry (S=±1) unlike single spiral state. BUT no spontaneous magnetization.– Hence NO Anomalous (zero field) Hall Effect,
• BUT in a single domain crystal:– Anomalous (linear in field) Magneto-resistance
– Quadratic Hall Effect
cBabcsab S Eg. Field along z,
sample along (110) and (1-10).
BS 0
edH BBcdeabc
Hab S '
x
y
B
x
y
BJ
EEg. Hall current parallel to B Field along (110).
Destroying the Order: Effect of Disorder
• Although clean from resistivity viewpoint ( ) disorder may be important for large, soft structures.
• Disorder expected to have a much stronger effect on multi-mode state than single spiral state. – Single spiral state only couples to magnetic disorder since |M(r)|
=const. – But Multi mode state couples to non-magnetic disorder.
– Disorder expected to destroy Goldstone modes and Bragg peaks of BCC in d=3;
– But, T breaking domains survive – finite temperature phase transition expected.
23 )()( rMrVrdf dis
Al 000,1
Effect of Disorder
• Expect maximum smearing of intensity along the softest directions. – Smallest energy scale is crystal locking: Even smaller
for BCC1 state as compared to single spiral – Can be extracted from ratio of critical fields for single
mode state.
5.4/KK effBCC
satur
orient//
H
H
q
q
RMS
RMS ][
][
20
][
][
RMS
RMS
q
q
//
Quantum Phases and Transitions
• Assume crystal anisotropy is irrelevant:
then all modes with wavevector |q|=q0 need to be included (“Bose Sphere”).
2
//2
0//2 ),,())(( tqqMrqqS
T
p
BCC]][[),,(1
)(321,,
34 3214321
321
qqqqqqqqqq
mmmmqqqUS
V
Non-trivial critical point (?) at T=0; requires a “Bose Surface” RG
Eg. Turlakov and Schmalian, z=3-ε expansion
Eg. Magnetic Crystal to Paramagnet
• New phases from low energy wave-vector sphere (to explain the Non-Fermi liquid)?
Conclusions
• multi mode BCC spiral phase proposed as a useful starting point for High Pressure Phase. Naturally captures:– Neutron scattering intensity maximum– Evolution in an applied magnetic field– Predictions for magneto-transport and elastic neutron scattering
in a field.• If structure destroyed by coupling to disorder:
– Enhanced coupling to disorder natural– Expect finite T transition and static magnetism– Why weak signature in resistivity?
• Future Work– Thermal/Quantum fluctuation mechanism destroying order?– Transport anomalies (NFL) at high pressure– Classical and Quantum phases and transitions with a “Bose
surface” of excitations?
Thermal and Quantum Phase Transitions
• Assume crystal anisotropy is relevant:
then we can reduce the number of variables to the modes along (110) [6 complex fields].
Fluctuation induced first order at T>0
Mean field at T=0; could be continuous – expect intervening (111) though.
222 ),()( krkF i
P
BCC
T
(111)
{For a quartic form of the anisotropy:
BCC1 is oriented along (110) but BCC2 is not.}
)(' 444zyxc MMMF