Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi Dietrich Belitz, University of Oregon with Ted Kirkpatrick, Achim Rosch, Thomas Vojta, et al. Ferromagnets and Helimagnets II. Phenomenology of MnSi Theory 1. Phase diagram 2. Disordered phase 3. Ordered phase

I. Ferromagnets versus Helimagnets 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) August 2006 Lorentz Center

I. Ferromagnets versus Helimagnets 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) Helimagnets: (Dzyaloshinski 1958, Moriya 1960) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix August 2006 Lorentz Center

I. Ferromagnets versus Helimagnets 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) Helimagnets: (Dzyaloshinski 1958, Moriya 1960) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix HHM invariant under rotations, but not under x → - x Crystal-field effects ultimately pin helix (very weak) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) (Pfleiderer et al 1997) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p (Pfleiderer et al 1997) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane !) TCP (Pfleiderer et al 1997) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) In an external field B there are “tricritical wings” TCP (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) In an external field B there are “tricritical wings” Quantum critical point at B ≠ 0 TCP (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) In an external field B there are “tricritical wings” Quantum critical point at B ≠ 0 Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) d d direction TCP (Pfleiderer et al 1997) (Pfleiderer et al 2004) (Pfleiderer, Julian, Lonzarich 2001) August 2006 Lorentz Center

II. Phenomenology of MnSi 1. Phase diagram magnetic transition at Tc ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane !) In an external field B there are “tricritical wings” Quantum critical point at B ≠ 0 Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) d d direction Cubic unit cell lacks inversion symmetry (in agreement with DM) TCP (Pfleiderer et al 1997) (Carbone et al 2005) (Pfleiderer et al 2004) (Pfleiderer, Julian, Lonzarich 2001) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above (Pfleiderer et al 2004) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) (Pfleiderer et al 2004) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) (Pfleiderer et al 2004) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) No detectable helical order for T > T0 (p) (Pfleiderer et al 2004) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) No detectable helical order for T > T0 (p) T0 (p) originates close to TCP (Pfleiderer et al 2004) August 2006 Lorentz Center

2. Neutron Scattering Ordered phase shows helical order, see above Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) No detectable helical order for T > T0 (p) T0 (p) originates close to TCP So far only three data points for T0 (p) (Pfleiderer et al 2004) August 2006 Lorentz Center

3. Transport Properties Non-Fermi-liquid behavior of the resistivity: p = 14.8kbar > pc ρ(μΩcm) T(K) ρ(μΩcm) Resistivity ρ ~ T 1.5 o over a huge range in parameter space T1.5(K1.5) ρ(μΩcm) T1.5(K1.5) August 2006 Lorentz Center

III. Theory 1. Nature of the Phase Diagram Basic features can be understood by approximating the system as a FM August 2006 Lorentz Center

III. Theory 1. Nature of the Phase Diagram Basic features can be understood by approximating the system as a FM Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) August 2006 Lorentz Center

III. Theory 1. Nature of the Phase Diagram Basic features can be understood by approximating the system as a FM Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) NB: TCP can also follow from material-specific band-structure effects (Schofield et al), but the many-body mechanism is generic August 2006 Lorentz Center

III. Theory 1. Nature of the Phase Diagram Basic features can be understood by approximating the system as a FM Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) NB: TCP can also follow from material-specific band-structure effects (Schofield et al), but the many-body mechanism is generic Wings follow from existence of tricritical point DB, T.R. Kirkpatrick, J. Rollbühler, PRL 94, 247205 (2005) Critical behavior at QCP determined exactly! (Hertz theory is valid due to B > 0) August 2006 Lorentz Center

Example of a more general principle: Hertz theory is valid if the field conjugate to the order parameter does not change the soft-mode structure (DB, T.R. Kirkpatrick, T. Vojta, Phys. Rev. B 65, 165112 (2002)) Here, B field already breaks a symmetry no additional symmetry breaking by the conjugate field mean-field critical behavior with corrections due to DIVs in particular, d m (pc,Hc,T) ~ -T 4/9 August 2006 Lorentz Center

2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) August 2006 Lorentz Center

2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: Chirality parameter c acts as external field conjugate to chiral OP August 2006 Lorentz Center

2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: Chirality parameter c acts as external field conjugate to chiral OP Perturbation theory Attractive interaction between OP fluctuations! Condensation of chiral fluctuations is possible August 2006 Lorentz Center

2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: Chirality parameter c acts as external field conjugate to chiral OP Perturbation theory Attractive interaction between OP fluctuations! Condensation of chiral fluctuations is possible Prediction: Feature characteristic of 1st order transition (e.g., discontinuity in the spin susceptibility) should be observable across T0 August 2006 Lorentz Center

Proposed phase diagram : August 2006 Lorentz Center

Proposed phase diagram : August 2006 Lorentz Center

Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) August 2006 Lorentz Center

Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) (Lubensky & Stark 1996) August 2006 Lorentz Center

Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) (Lubensky & Stark 1996) (Anisimov et al 1998) August 2006 Lorentz Center

Other proposals: Superposition of spin spirals with different wave vectors (Binz et al 2006), see following talk. Spontaneous skyrmion ground state (Roessler et al 2006) Stabilization of analogs to crystalline blue phases (Fischer & Rosch 2006, see poster) (NB: All of these proposals are also related to blue-phase physics) August 2006 Lorentz Center

3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode August 2006 Lorentz Center

3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? August 2006 Lorentz Center

3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? NO! rotation (0,0,q) (a1,a2,q) cannot cost energy, yet corresponds to f(x) = a1x + a2y H fluct > 0 cannot depend on August 2006 Lorentz Center

3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? NO! rotation (0,0,q) (a1,a2,q) cannot cost energy, yet corresponds to f(x) = a1x + a2y H fluct > 0 cannot depend on August 2006 Lorentz Center

anisotropic! August 2006 Lorentz Center

anisotropic! anisotropic dispersion relation (as in chiral liquid crystals) “helimagnon” August 2006 Lorentz Center

anisotropic! anisotropic dispersion relation (as in chiral liquid crystals) “helimagnon” Compare with ferromagnets w(k) ~ k2 antiferromagnets (k) ~ |k| August 2006 Lorentz Center

4. Ordered Phase: Specific heat Internal energy density: Specific heat: helimagnon contribution total low-T specific heat August 2006 Lorentz Center

4. Ordered Phase: Specific heat Internal energy density: Specific heat: helimagnon contribution total low-T specific heat Experiment: (E. Fawcett 1970, C. Pfleiderer unpublished) Caveat: Looks encouraging, but there is a quantitative problem, observed T2 may be accidental August 2006 Lorentz Center

5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) August 2006 Lorentz Center

5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) August 2006 Lorentz Center

5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) r(T) = r2 T 2 + r5/2 T 5/2 total low-T resistivity August 2006 Lorentz Center

5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) r(T) = r2 T 2 + r5/2 T 5/2 total low-T resistivity Experiment: r (T→ 0) ~ T 2 (more analysis needed) August 2006 Lorentz Center

6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) Use TDGL theory to study magnetization dynamics: August 2006 Lorentz Center

6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) Use TDGL theory to study magnetization dynamics: Bloch term damping Langevin force August 2006 Lorentz Center

6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) Use TDGL theory to study magnetization dynamics: Bare magnetic response function: helimagnon frequency damping coefficient Fluctuation-dissipation theorem: One-loop correction to c : c F August 2006 Lorentz Center

Strictly speaking, helimagnetic order is not stable at T > 0 The elastic coefficients and , and the transport coefficients and all acquire singular corrections at one-loop order due to mode-mode coupling effects: Strictly speaking, helimagnetic order is not stable at T > 0 In practice, cz is predicted to change linearly with T, by ~10% from T=0 to T=10K Analogous to situation in smectic liquid crystals (Mazenko, Ramaswamy, Toner 1983) What happens to these singularities at T = 0 ? Special case of a more general problem: As T -> 0, classical mode-mode coupling effects die (how?), whereas new quantum effects appear (e.g., weak localization and related effects) coth in FD theorem 1-loop integral more singular at T > 0 than at T = 0 ! All renormalizations are finite at T = 0 ! August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase Helimagnons predicted in ordered phase; lead to T2 term in specific heat August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase Helimagnons predicted in ordered phase; lead to T2 term in specific heat NFL quasi-particle relaxation time predicted in ordered phase August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase Helimagnons predicted in ordered phase; lead to T2 term in specific heat NFL quasi-particle relaxation time predicted in ordered phase Resistivity in ordered phase is FL-like with T5/2 correction August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase Helimagnons predicted in ordered phase; lead to T2 term in specific heat NFL quasi-particle relaxation time predicted in ordered phase Resistivity in ordered phase is FL-like with T5/2 correction Hydrodynamic description of ordered phase breaks down August 2006 Lorentz Center

IV. Summary Basic T-p-h phase diagram is understood Possible additional 1st order transition in disordered phase Helimagnons predicted in ordered phase; lead to T2 term in specific heat NFL quasi-particle relaxation time predicted in ordered phase Resistivity in ordered phase is FL-like with T5/2 correction Hydrodynamic description of ordered phase breaks down Main open question: Origin of T3/2 resistivity in disordered phase? August 2006 Lorentz Center

Acknowledgments Ted Kirkpatrick Rajesh Narayanan Jörg Rollbühler Achim Rosch Sumanta Tewari John Toner Thomas Vojta Peter Böni Christian Pfleiderer Aspen Center for Physics KITP at UCSB Lorentz Center Leiden National Science Foundation August 2006 Lorentz Center