first order vs second order transitions in quantum magnets i. quantum ferromagnetic transitions:...
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Quantum Criticality Workshop Toronto 3 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero:TRANSCRIPT
![Page 1: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/1.jpg)
First Order vs Second Order First Order vs Second Order Transitions in Quantum Transitions in Quantum
MagnetsMagnets
I. Quantum Ferromagnetic Transitions: Experiments
II. Theory 1. Conventional (mean-field) theory 2. Renormalized mean-field theory 3. Effects of flucuations
III. Other Transitions
Dietrich Belitz, University of Oregon
Ted Kirkpatrick, University of Maryland
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Quantum Criticality Workshop Toronto 2Sep 2008
I. Quantum Ferromagnetic Transitions: Experiments
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Quantum Criticality Workshop Toronto 3Sep 2008
I. Quantum Ferromagnetic Transitions: Experiments
■ Itinerant ferromagnets whose Tc can be tuned to zero:
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Quantum Criticality Workshop Toronto 4Sep 2008
I. Quantum Ferromagnetic Transitions: Experiments
■ Itinerant ferromagnets whose Tc can be tuned to zero:
● UGe2, ZrZn2, (MnSi) (clean, pressure tuned)
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Quantum Criticality Workshop Toronto 5Sep 2008
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:
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Quantum Criticality Workshop Toronto 6Sep 2008
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:
(Pfleiderer & Huxley 2002)
UGe2
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Quantum Criticality Workshop Toronto 7Sep 2008
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:
(Pfleiderer & Huxley 2002)
UGe2 ZrZn2
(Uhlarz et al 2004)
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Quantum Criticality Workshop Toronto 8Sep 2008
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:
(Pfleiderer & Huxley 2002)
UGe2 ZrZn2 MnSi
(Pfleiderer et al 1997)
(Uhlarz et al 2004)
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Quantum Criticality Workshop Toronto 9Sep 2008
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:
○ Additional evidence: μSR (Uemura et al 2007)
(Pfleiderer & Huxley 2002)
UGe2 ZrZn2 MnSi
(Pfleiderer et al 1997)
(Uhlarz et al 2004)
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Quantum Criticality Workshop Toronto 10Sep 2008
I. Quantum Ferromagnetic Transitions: ExperimentsI. Quantum Ferromagnetic Transitions: Experiments
■■ Itinerant ferromagnets whose TItinerant ferromagnets whose Tc c can be tuned to zero:can be tuned to zero:
● ● UGeUGe22, ZrZn, ZrZn22, (MnSi) (clean, pressure tuned), (MnSi) (clean, pressure tuned)
○ T=0 1st order transition persists in a B-field, ends at quantum critical point.
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Quantum Criticality Workshop Toronto 11Sep 2008
I. Quantum Ferromagnetic Transitions: ExperimentsI. Quantum Ferromagnetic Transitions: Experiments
■■ Itinerant ferromagnets whose TItinerant ferromagnets whose Tc c can be tuned to zero:can be tuned to zero:
● ● UGeUGe22, ZrZn, ZrZn22, (MnSi) (clean, pressure tuned), (MnSi) (clean, pressure tuned)
○ T=0 1st order transition persists in a B-field, ends at quantum critical point.
Schematic phase diagram:
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Quantum Criticality Workshop Toronto 12Sep 2008
I. Quantum Ferromagnetic Transitions: ExperimentsI. Quantum Ferromagnetic Transitions: Experiments
■■ Itinerant ferromagnets whose TItinerant ferromagnets whose Tc c can be tuned to zero:can be tuned to zero:
● URu2-xRexSi2 (disordered, concentration tuned)
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Quantum Criticality Workshop Toronto 13Sep 2008
I. Quantum Ferromagnetic Transitions: ExperimentsI. Quantum Ferromagnetic Transitions: Experiments
■■ Itinerant ferromagnets whose TItinerant ferromagnets whose Tc c can be tuned to zero: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)
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Quantum Criticality Workshop Toronto 14Sep 2008
I. Quantum Ferromagnetic Transitions: ExperimentsI. Quantum Ferromagnetic Transitions: Experiments
■■ Itinerant ferromagnets whose TItinerant ferromagnets whose Tc c can be tuned to zero: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)
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Quantum Criticality Workshop Toronto 15Sep 2008
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.
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Quantum Criticality Workshop Toronto 16Sep 2008
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 + …
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Quantum Criticality Workshop Toronto 17Sep 2008
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 anddirty systems
![Page 18: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/18.jpg)
Quantum Criticality Workshop Toronto 18Sep 2008
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
■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 u<0
} for both clean anddirty systems
![Page 19: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/19.jpg)
Quantum Criticality Workshop Toronto 19Sep 2008
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
■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 u<0
■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like
} for both clean anddirty systems
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Quantum Criticality Workshop Toronto 20Sep 2008
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
■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 u<0
■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like
■ Conclusion: Conventional theory not viable
} for both clean anddirty systems
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Quantum Criticality Workshop Toronto 21Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff)
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Quantum Criticality Workshop Toronto 22Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
![Page 23: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/23.jpg)
Quantum Criticality Workshop Toronto 23Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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:
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Quantum Criticality Workshop Toronto 24Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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:
![Page 25: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/25.jpg)
Quantum Criticality Workshop Toronto 25Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
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Quantum Criticality Workshop Toronto 26Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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 Transition is generically 1st order! (TRK, T Vojta, DB 1999)
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Quantum Criticality Workshop Toronto 27Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point
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Quantum Criticality Workshop Toronto 28Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes
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Quantum Criticality Workshop Toronto 29Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2
![Page 30: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/30.jpg)
Quantum Criticality Workshop Toronto 30Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2
○ sign of the coefficient
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Quantum Criticality Workshop Toronto 31Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2
○ sign of the coefficient
Renormalized mean-field equation of state:
(disordered, d=3, T=0)
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Quantum Criticality Workshop Toronto 32Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2
○ sign of the coefficient
Renormalized mean-field equation of state:
(disordered, d=3, T=0)
● v>0 Transition is 2nd order with non-mean-field (and non-classical) exponents: β=2, δ=3/2, etc.
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Quantum Criticality Workshop Toronto 33Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● Phase diagrams:
G=0
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Quantum Criticality Workshop Toronto 34Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory ● Phase diagrams:
G=0 T=0
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Quantum Criticality Workshop Toronto 35Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. Renormalized mean-field theory● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005)
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Quantum Criticality Workshop Toronto 36Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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!
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Quantum Criticality Workshop Toronto 37Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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
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Quantum Criticality Workshop Toronto 38Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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
■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram:
(Pfleiderer, Julian,
Lonzarich 2001)
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Quantum Criticality Workshop Toronto 39Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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
■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970)
(Pfleiderer, Julian,
Lonzarich 2001)
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Quantum Criticality Workshop Toronto 40Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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
■ 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
(Pfleiderer, Julian,
Lonzarich 2001)
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Quantum Criticality Workshop Toronto 41Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory2. 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
■ 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)
(Pfleiderer, Julian,
Lonzarich 2001)
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Quantum Criticality Workshop Toronto 42Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations■ Complete theory: Keep all soft modes explicitly:
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Quantum Criticality Workshop Toronto 43Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations
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Quantum Criticality Workshop Toronto 44Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
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Quantum Criticality Workshop Toronto 45Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
![Page 46: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/46.jpg)
Quantum Criticality Workshop Toronto 46Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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:
![Page 47: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/47.jpg)
Quantum Criticality Workshop Toronto 47Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
![Page 48: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/48.jpg)
Quantum Criticality Workshop Toronto 48Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)
![Page 49: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/49.jpg)
Quantum Criticality Workshop Toronto 49Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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
![Page 50: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/50.jpg)
Quantum Criticality Workshop Toronto 50Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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
![Page 51: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/51.jpg)
Quantum Criticality Workshop Toronto 51Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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
![Page 52: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/52.jpg)
Quantum Criticality Workshop Toronto 52Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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
![Page 53: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/53.jpg)
Quantum Criticality Workshop Toronto 53Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. Order-parameter fluctuations○ Comparison with experiments:
Butch & Maple (2008)
![Page 54: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/54.jpg)
Quantum Criticality Workshop Toronto 54Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. Order-parameter fluctuations○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs)
Butch & Maple (2008)
![Page 55: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/55.jpg)
Quantum Criticality Workshop Toronto 55Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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)
![Page 56: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/56.jpg)
Quantum Criticality Workshop Toronto 56Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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)
![Page 57: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/57.jpg)
Quantum Criticality Workshop Toronto 57Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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, ??
○ Needed: ▫ Analysis of width of asymptotic region ▫ Analysis of x-overs to pre-asymptotic region, and to clean behavior
Butch & Maple (2008)
![Page 58: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/58.jpg)
Quantum Criticality Workshop Toronto 58Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. Order-parameter fluctuations○ RG analysis for clean case upper critical dimension is d=3
![Page 59: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/59.jpg)
Quantum Criticality Workshop Toronto 59Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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).
![Page 60: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/60.jpg)
Quantum Criticality Workshop Toronto 60Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations3. 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.
![Page 61: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/61.jpg)
Quantum Criticality Workshop Toronto 61Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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)..
![Page 62: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/62.jpg)
Quantum Criticality Workshop Toronto 62Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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!
![Page 63: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/63.jpg)
Quantum Criticality Workshop Toronto 63Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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!
![Page 64: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/64.jpg)
Quantum Criticality Workshop Toronto 64Sep 2008
II. Quantum Ferromagnetic Transitions: TheoryII. 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.
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Quantum Criticality Workshop Toronto 65Sep 2008
III. Some Other Transitions1. Metamagnetic transitions
.■ Some quantum FMs show metamagnetic transitions:
● UGe2 (Pfleiderer & Huxley 2002)
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Quantum Criticality Workshop Toronto 66Sep 2008
III. Some Other Transitions1. 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)
![Page 67: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/67.jpg)
Quantum Criticality Workshop Toronto 67Sep 2008
III. Some Other Transitions1. 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!)
![Page 68: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/68.jpg)
Quantum Criticality Workshop Toronto 68Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. Partial order transition in MnSi
.■ MnSi is a weak helimagnet with a complicated phase diagram
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Quantum Criticality Workshop Toronto 69Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. 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
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Quantum Criticality Workshop Toronto 70Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. 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.
![Page 71: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/71.jpg)
Quantum Criticality Workshop Toronto 71Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. 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)
![Page 72: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/72.jpg)
Quantum Criticality Workshop Toronto 72Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. Partial order transition in MnSi
.■ Theory: Chiral OP
in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals
![Page 73: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/73.jpg)
Quantum Criticality Workshop Toronto 73Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. 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)
![Page 74: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/74.jpg)
Quantum Criticality Workshop Toronto 74Sep 2008
III. Some Other TransitionsIII. Some Other Transitions2. 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)
![Page 75: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/75.jpg)
Quantum Criticality Workshop Toronto 75Sep 2008
III. Some Other TransitionsIII. Some Other Transitions3. 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)
![Page 76: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/76.jpg)
Quantum Criticality Workshop Toronto 76Sep 2008
III. Some Other TransitionsIII. Some Other Transitions3. 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):
![Page 77: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/77.jpg)
Quantum Criticality Workshop Toronto 77Sep 2008
III. Some Other TransitionsIII. Some Other Transitions3. 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):
![Page 78: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/78.jpg)
Quantum Criticality Workshop Toronto 78Sep 2008
III. Some Other TransitionsIII. Some Other Transitions3. 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!)
![Page 79: First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory](https://reader035.vdocument.in/reader035/viewer/2022062504/5a4d1b947f8b9ab0599c2a10/html5/thumbnails/79.jpg)
Quantum Criticality Workshop Toronto 79Sep 2008
III. Some Other TransitionsIII. Some Other Transitions3. 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
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Quantum Criticality Workshop Toronto 80Sep 2008
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