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First Order vs Second Order Transitions in Quantum Magnets Dietrich Belitz, University of Oregon Ted Kirkpatrick, University of Maryland I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory 2. Renormalized mean-field theory 3. Effects of flucuations III. Other Transitions I. Quantum Ferromagnetic Transitions: Experiments Sep 2008 Quantum Criticality Workshop Toronto 2 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: Sep 2008 Quantum Criticality Workshop Toronto 3 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) Sep 2008 (clean, pressure tuned) Quantum Criticality Workshop Toronto 4 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: Sep 2008 Quantum Criticality Workshop Toronto 5 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 (Pfleiderer & Huxley 2002) Sep 2008 Quantum Criticality Workshop Toronto 6 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 (Pfleiderer & Huxley 2002) Sep 2008 ZrZn2 (Uhlarz et al 2004) Quantum Criticality Workshop Toronto 7 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 (Pfleiderer & Huxley 2002) ZrZn2 MnSi (Uhlarz et al 2004) (Pfleiderer et al 1997) Sep 2008 Quantum Criticality Workshop Toronto 8 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 (Pfleiderer & Huxley 2002) ZrZn2 MnSi (Uhlarz et al 2004) (Pfleiderer et al 1997) ○ Additional evidence: μSR (Uemura et al 2007) Sep 2008 Quantum Criticality Workshop Toronto 9 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ T=0 1st order transition persists in a B-field, ends at quantum critical point. Sep 2008 Quantum Criticality Workshop Toronto 10 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ T=0 1st order transition persists in a B-field, ends at quantum critical point. Schematic phase diagram: Sep 2008 Quantum Criticality Workshop Toronto 11 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 Sep 2008 (disordered, concentration tuned) Quantum Criticality Workshop Toronto 12 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 (disordered, concentration tuned) ○ Disordered material shows a 2nd order transition down to T=0: Bauer et al (2005) Butch & Maple (2008) Sep 2008 Quantum Criticality Workshop Toronto 13 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 (disordered, concentration tuned) ○ Disordered material shows a 2nd order transition down to T=0: Bauer et al (2005) Butch & Maple (2008) ○ Observed exponents are not mean-field like (see below) Sep 2008 Quantum Criticality Workshop Toronto 14 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. Sep 2008 Quantum Criticality Workshop Toronto 15 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … Sep 2008 Quantum Criticality Workshop Toronto 16 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 Sep 2008 } for both clean and dirty systems Quantum Criticality Workshop Toronto 17 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 } for both clean and dirty systems ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 Sep 2008 Quantum Criticality Workshop Toronto u<0 18 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 } for both clean and dirty systems ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 ■ Problems: u<0 ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like Sep 2008 Quantum Criticality Workshop Toronto 19 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 } for both clean and dirty systems ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 ■ Problems: u<0 ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like ■ Conclusion: Conventional theory not viable Sep 2008 Quantum Criticality Workshop Toronto 20 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) Sep 2008 Quantum Criticality Workshop Toronto 21 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) Sep 2008 Quantum Criticality Workshop Toronto 22 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: Sep 2008 Quantum Criticality Workshop Toronto 23 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: Sep 2008 Quantum Criticality Workshop Toronto 24 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0) Sep 2008 Quantum Criticality Workshop Toronto 25 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ In general, Hertz theory misses effects of soft modes (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0) ● v>0 Sep 2008 Transition is generically 1st order! (TRK, T Vojta, DB 1999) Quantum Criticality Workshop Toronto 26 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass Sep 2008 Quantum Criticality Workshop Toronto ln m -> ln (m+T) tricritical point 27 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes Sep 2008 Quantum Criticality Workshop Toronto 28 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ● Quenched disorder G changes ○ fermion dispersion relation Sep 2008 ln m -> ln (m+T) tricritical point md -> md/2 Quantum Criticality Workshop Toronto 29 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ● Quenched disorder G changes ○ fermion dispersion relation ○ sign of the coefficient Sep 2008 ln m -> ln (m+T) tricritical point md -> md/2 Quantum Criticality Workshop Toronto 30 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ● Quenched disorder G changes ○ fermion dispersion relation ○ sign of the coefficient ln m -> ln (m+T) tricritical point md -> md/2 Renormalized mean-field equation of state: (disordered, d=3, T=0) Sep 2008 Quantum Criticality Workshop Toronto 31 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ● Quenched disorder G changes ○ fermion dispersion relation ○ sign of the coefficient ln m -> ln (m+T) tricritical point md -> md/2 Renormalized mean-field equation of state: (disordered, d=3, T=0) ● v>0 Sep 2008 Transition is 2nd order with non-mean-field (and non-classical) exponents: β=2, δ=3/2, etc. Quantum Criticality Workshop Toronto 32 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● Phase diagrams: G=0 Sep 2008 Quantum Criticality Workshop Toronto 33 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● Phase diagrams: G=0 Sep 2008 T=0 Quantum Criticality Workshop Toronto 34 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) Sep 2008 Quantum Criticality Workshop Toronto 35 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Sep 2008 Quantum Criticality Workshop Toronto 36 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δmc ~ -T4/9 Sep 2008 Quantum Criticality Workshop Toronto 37 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δmc ~ -T4/9 (Pfleiderer, Julian, Lonzarich 2001) ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: Sep 2008 Quantum Criticality Workshop Toronto 38 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δmc ~ -T4/9 (Pfleiderer, Julian, Lonzarich 2001) ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) Sep 2008 Quantum Criticality Workshop Toronto 39 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δmc ~ -T4/9 (Pfleiderer, Julian, Lonzarich 2001) ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) ● So far no OP fluctuations have been considered Sep 2008 Quantum Criticality Workshop Toronto 40 II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δmc ~ -T4/9 (Pfleiderer, Julian, Lonzarich 2001) ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) ● So far no OP fluctuations have been considered ● More generally, Hertz theory works if field conjugate the OP does not change the soft-mode spectrum (DB, TRK, T Vojta 2002) Sep 2008 Quantum Criticality Workshop Toronto 41 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: Sep 2008 Quantum Criticality Workshop Toronto 42 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations Sep 2008 Quantum Criticality Workshop Toronto 43 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) Sep 2008 Quantum Criticality Workshop Toronto 44 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) Sep 2008 Quantum Criticality Workshop Toronto 45 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels: Sep 2008 Quantum Criticality Workshop Toronto 46 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels: ○ Gaussian approx Hertz theory (FP unstable with respect to m q2 term in effective action) Sep 2008 Quantum Criticality Workshop Toronto 47 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels: ○ Gaussian approx Hertz theory (FP unstable with respect to m q2 term in effective action) ○ mean-field approx for OP + Gaussian approx for fermions renormalized mean-field theory (FP marginally unstable) Sep 2008 Quantum Criticality Workshop Toronto 48 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 is marginal for all 0<d<4, and is the only marginal term Sep 2008 Quantum Criticality Workshop Toronto 49 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length Sep 2008 Quantum Criticality Workshop Toronto 50 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length ○ 4-ε expansion does not work! Flow eqs depend singularly on the subdominant time scale: where w = ratio of time scales Sep 2008 Quantum Criticality Workshop Toronto 51 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length ○ 4-ε expansion does not work! Flow eqs depend singularly on the subdominant time scale: where w = ratio of time scales NB: One-loop (or any finite-loop) order yields misleading results Infinite resummation logs Sep 2008 Quantum Criticality Workshop Toronto 52 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ Comparison with experiments: Butch & Maple (2008) Sep 2008 Quantum Criticality Workshop Toronto 53 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) Butch & Maple (2008) Sep 2008 Quantum Criticality Workshop Toronto 54 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1st order?? (Should go the other way: 1st to 2nd !) Butch & Maple (2008) Sep 2008 Quantum Criticality Workshop Toronto 55 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1st order?? (Should go the other way: 1st to 2nd !) ▫ β ≈ 0.8 with no x-dependence, ?? Butch & Maple (2008) Sep 2008 Quantum Criticality Workshop Toronto 56 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1st order?? (Should go the other way: 1st to 2nd !) ▫ β ≈ 0.8 with no x-dependence, ?? Butch & Maple (2008) ○ Needed: ▫ Analysis of width of asymptotic region ▫ Analysis of x-overs to pre-asymptotic region, and to clean behavior Sep 2008 Quantum Criticality Workshop Toronto 57 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for clean case upper critical dimension is d=3 Sep 2008 Quantum Criticality Workshop Toronto 58 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for clean case upper critical dimension is d=3 ○ 3-ε expansion to 1-loop order suggests 2nd order transition is possible in certain parameter regimes (fluctuation-induced 2nd order: u driven negative is counteracted by couplings at loop level). Sep 2008 Quantum Criticality Workshop Toronto 59 II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for clean case upper critical dimension is d=3 ○ 3-ε expansion to 1-loop order suggests 2nd order transition is possible in certain parameter regimes (fluctuation-induced 2nd order: u driven negative is counteracted by couplings at loop level). This analysis is suspect due to the problems with the ε-expansion! More work is needed. Sep 2008 Quantum Criticality Workshop Toronto 60 II. Quantum Ferromagnetic Transitions: Theory 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1st vs 2nd order). . Sep 2008 Quantum Criticality Workshop Toronto 61 II. Quantum Ferromagnetic Transitions: Theory 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1st vs 2nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works! Sep 2008 Quantum Criticality Workshop Toronto 62 II. Quantum Ferromagnetic Transitions: Theory 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1st vs 2nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works! ■ For disordered systems, exotic critical behavior is predicted. Experiments are now available, analysis is needed! Sep 2008 Quantum Criticality Workshop Toronto 63 II. Quantum Ferromagnetic Transitions: Theory 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1st vs 2nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works! ■ For disordered systems, exotic critical behavior is predicted. Experiments are now available, analysis is needed! ■ Role of fluctuations in clean systems needs to be investigated. Sep 2008 Quantum Criticality Workshop Toronto 64 III. Some Other Transitions 1. Metamagnetic transitions . ■ Some quantum FMs show metamagnetic transitions: ● UGe2 (Pfleiderer & Huxley 2002) Sep 2008 Quantum Criticality Workshop Toronto 65 III. Some Other Transitions 1. Metamagnetic transitions . ■ Some quantum FMs show metamagnetic transitions: ● UGe2 (Pfleiderer & Huxley 2002) ● Sr3Ru2O7 (e.g., Grigera et al 2004) (“hidden order”) Possibly a Pomeranchuk instability (Ho & Schofield 2008) Sep 2008 Quantum Criticality Workshop Toronto 66 III. Some Other Transitions 1. Metamagnetic transitions . ■ Some quantum FMs show metamagnetic transitions: ● UGe2 (Pfleiderer & Huxley 2002) ● Sr3Ru2O7 (e.g., Grigera et al 2004) (“hidden order”) Possibly a Pomeranchuk instability (Ho & Schofield 2008) ■ Another example of a restored ferromagnetic QCP: Critical behavior at a ○ metamagnetic end point. Is Hertz theory valid? (magnons!) Sep 2008 Quantum Criticality Workshop Toronto 67 III. Some Other Transitions 2. Partial order transition in MnSi . ■ MnSi is a weak helimagnet with a complicated phase diagram Sep 2008 Quantum Criticality Workshop Toronto 68 III. Some Other Transitions 2. Partial order transition in MnSi . ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase (Pfleiderer et al 2006, Uemura et al 2007): • Magnetic state is a helimagnet with 2π/q ≈ 180 Ǻ, pinning in (111) direction Sep 2008 Quantum Criticality Workshop Toronto 69 III. Some Other Transitions 2. Partial order transition in MnSi . ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase: • 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. Slow dynamics. Sep 2008 Quantum Criticality Workshop Toronto 70 III. Some Other Transitions 2. Partial order transition in MnSi . ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase: •No detectable helical order for T > T0 (p) Sep 2008 Quantum Criticality Workshop Toronto 71 III. Some Other Transitions 2. Partial order transition in MnSi . ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals Sep 2008 Quantum Criticality Workshop Toronto 72 III. Some Other Transitions 2. Partial order transition in MnSi . ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals 1st order transition from a chiral gas (PM phase) to a chiral liquid (partial order phase, “blue quantum fog”) (S. Tewari, DB, TRK 2006) Sep 2008 Quantum Criticality Workshop Toronto 73 III. Some Other Transitions 2. Partial order transition in MnSi . ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals 1st order transition from a chiral gas (PM phase) to a chiral liquid (partial order phase, “blue quantum fog”) (S. Tewari, DB, TRK 2006) ■ Alternative explanations: Analogies to crystalline blue phases (Binz et al 2006, Fischer, Shah, Rosch 2008) Sep 2008 Quantum Criticality Workshop Toronto 74 III. Some Other Transitions 3. Quantum critical point in an inhomogeneous ferromagnet . ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) Sep 2008 Quantum Criticality Workshop Toronto 75 III. Some Other Transitions 3. Quantum critical point in an inhomogeneous ferromagnet . ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): Sep 2008 Quantum Criticality Workshop Toronto 76 III. Some Other Transitions 3. Quantum critical point in an inhomogeneous ferromagnet . ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): Sep 2008 Quantum Criticality Workshop Toronto 77 III. Some Other Transitions 3. Quantum critical point in an inhomogeneous ferromagnet . ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): ■ NB: Mean-field exponents (another example where Hertz theory works!) Sep 2008 Quantum Criticality Workshop Toronto 78 III. Some Other Transitions 3. Quantum critical point in an inhomogeneous ferromagnet . ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): ■ NB: Mean-field exponents (another example where Hertz theory works!) ■ Open problem: Non-equilibrium behavior Sep 2008 Quantum Criticality Workshop Toronto 79 Acknowledgments • • • • • • • • Ted Kirkpatrick Maria-Teresa Mercaldo Rajesh Narayanan Jörg Rollbühler Achim Rosch Ronojoy Saha Sharon Sessions Sumanta Tewari • John Toner • Thomas Vojta • Peter Böni • Christian Pfleiderer • Aspen Center for Physics • KITP at UCSB • Lorentz Center Leiden National Science Foundation Sep 2008 Quantum Criticality Workshop Toronto 80