1. Theory of the Helical Spin Crystal A Proposal for the `Partially Ordered’ State of MnSi Ashvin Vishwanath UC Berkeley In collaboration with: Benedikt Binz (UC Berkeley) Vivek Aji (UC Riverside) Phys. Rev. Lett. 96, 207202 (2006). cond-mat/0608128

2. MnSi: Experimental Facts Elastic Neutron Scattering This Talk: Focus on the Partial Order State Proposal: Helical Spin Crystal as intermediate scale structure. 1.Theory of Helical Spin Crystals 2.Destroying the Crystal – Disorder OR Thermal/quantum fluctuations? Static vs. dynamic? New phase or crossover ?

3. Dzyaloshinskii-Moriya and the Spiral State Origin of DM: Spin orbit interaction – Need to break inversion symmetry. Consider 2 spins in an insulator Ferromagnetism DM Term 1 2 Leads to long wavelength spiral if d<<J.

4. 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. r<0→ ferromagnet DM rotates M DM for the B20 structure

5. Landau theory of Spiral States II Single Spiral General State: A Superposition M At quadratic order, for r=0, any superposition of helices is degenerate. Selected by interactions

6. 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: λ>0 Implies spiral locked along not directions.

7. 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)

8. Proposal: Multi Mode State Helical 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

9. 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

10. 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

11. Stabilizing a Multi Mode Spiral State Uniform quartic term gives rise to single mode state – need more structure to stabilize multi- mode state. Parameterize Quartic interaction φ/2 θ 1 2 4 3 Choose

12. Stabilizing a Multi Mode State II Expanding the interaction in harmonics Determine energetics for arbitrary combination of 13 modes [(110);(100);(111)] and upto 4 arbitrary spirals. Phase Diagram: U 20 =0, U 0 =W Relation to other work: (Rossler et al., I. Fischer and A. Rosch) have the term: Which here is:

13. Stabilizing a Multi Mode State III Phase Diagram: U 20 =0, U 0 =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

14. 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. BCC state – condensation in all 6 modes. λ>0 BCC1 AND λ<0 BCC2 λ

15. 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 (?)

16. The BCC1 state Sections through the state END

17. BCC 1 Generic Cut – Merons and anti-Merons; and vortices Zeros of the Magnetization -- and Meron Centers --

18. 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 90 º 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

19. BCC2 Phase Magnetization zero at points – but no hedgehogs. Symmetry 90º Rotn.+ τ+Translation END Sections through the state

20. Nature of Symmetry Breaking of BCC States BCC States: – break continuous Translation symmetry (T x,T y,T z ) 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.

21. BCC1 vs BCC2 – Magnetization Distribution M BCC1 BCC2 Histogram of magnetization – Single mode If static, should be observable by NMR and μSR

22. NMR on MnSi Zero field NMR on MnSi In the helical spin crystal –Static magnetism above p c –Broad line shapes –BUT, drop in intensity. –No signal in muSR –Time fluctuating BCC order pinned at surfaces? W. Yu et al. PRl (2004)

23. 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.

24. 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.

25. 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

26. Phenomenology 2: Effect of a Field Oriented spiral Polarized BCC

27. q2q2 Effect of a Field on BCC States Actually, response of BCC state is more complicated. q1q1 q3q3 0 adjusting phasesadjusting rel. amplitudes State adjusts in a field – susceptibility smaller than single spiral if coupling μ is weak.

28. 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.

29. 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 Eg. Field along z, sample along (110) and (1-10). x y B x y B J E Eg. Hall current parallel to B Field along (110).

30. 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.

31. 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.

32. Quantum Phases and Transitions Assume crystal anisotropy is irrelevant: then all modes with wavevector |q|=q 0 need to be included (“Bose Sphere”). T p BCC 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)?

33. 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?

35. 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. P BCC T (111)

37. {For a quartic form of the anisotropy: BCC1 is oriented along (110) but BCC2 is not.}