Download Breakdown of the Standard Model

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
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I. Quantum Ferromagnetic Transitions: Experiments
■ Itinerant ferromagnets whose Tc can be tuned to zero:
Sep 2008
Quantum Criticality Workshop Toronto
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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II. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory
● Phase diagrams:
G=0
Sep 2008
Quantum Criticality Workshop Toronto
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II. Quantum Ferromagnetic Transitions: Theory
2. Renormalized mean-field theory
● Phase diagrams:
G=0
Sep 2008
T=0
Quantum Criticality Workshop Toronto
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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
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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
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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
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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
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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
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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
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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
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II. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations
■ Complete theory: Keep all soft modes explicitly:
Sep 2008
Quantum Criticality Workshop Toronto
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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
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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
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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
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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
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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
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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
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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
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II. Quantum Ferromagnetic Transitions: Theory
3. Order-parameter fluctuations
○ Comparison with experiments:
Butch & Maple (2008)
Sep 2008
Quantum Criticality Workshop Toronto
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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III. Some Other Transitions
1. Metamagnetic transitions
.
■ Some quantum FMs show metamagnetic transitions:
● UGe2 (Pfleiderer & Huxley 2002)
Sep 2008
Quantum Criticality Workshop Toronto
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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
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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
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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
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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
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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
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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
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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
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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)
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Quantum Criticality Workshop Toronto
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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)
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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
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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):
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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
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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
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