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

qq

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