First Order vs Second Order Transitions in Quantum Magnets

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

Quantum Criticality Workshop Toronto 2Sep 2008




■ 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








UGe2 ZrZn2

(Uhlarz et al 2004)








UGe2 ZrZn2 MnSi

(Pfleiderer et al 1997)

(Uhlarz et al 2004)







○ Additional evidence: μSR (Uemura et al 2007)


UGe2 ZrZn2 MnSi

(Pfleiderer et al 1997)

(Uhlarz et al 2004)


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.




● ● 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:




● URu2-xRexSi2 (disordered, concentration tuned)





○ Disordered material shows a 2nd order transition down to T=0:

Bauer et al (2005)

Butch & Maple (2008)





○ Disordered material shows a 2nd order transition down to T=0:

Bauer et al (2005)


○ Observed exponents are not mean-field like (see below)


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








■ 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










■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like

■ Conclusion: Conventional theory not viable



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)



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)



2. Renormalized mean-field theory■ In general, Hertz theory misses effects of soft modes (TRK & DB 1996 ff) ● Soft modes (clean case)



● Renormalized mean-field equation of state:

(clean, d=3, T=0)

● v>0 Transition is generically 1st order! (TRK, T Vojta, DB 1999)



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



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



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)





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.



2. Renormalized mean-field theory2. Renormalized mean-field theory ● Phase diagrams:

G=0



2. Renormalized mean-field theory2. Renormalized mean-field theory ● Phase diagrams:

G=0 T=0



2. Renormalized mean-field theory2. Renormalized mean-field theory● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005)



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!



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)





■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970)


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


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)


Lonzarich 2001)



3. Order-parameter fluctuations■ Complete theory: Keep all soft modes explicitly:



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:







● Analysis at various levels: ○ Gaussian approx Hertz theory (FP unstable with respect to m q2 term in effective action)







● 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)



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



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




○ 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



3. Order-parameter fluctuations3. Order-parameter fluctuations○ Comparison with experiments:




3. Order-parameter fluctuations3. Order-parameter fluctuations○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs)




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 !)




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, ??




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




3. Order-parameter fluctuations3. Order-parameter fluctuations○ RG analysis for clean case upper critical dimension is d=3



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).



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.



4. Summary of quantum ferromagnetic transitions■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1st vs 2nd order)..



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!



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!



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.


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)





● 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!)


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




■ 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.




■ 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)



.■ 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)





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)


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!)





■ NB: Mean-field exponents (another example where Hertz theory works!)

■ Open problem: Non-equilibrium behavior


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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First Order vs Second Order Transitions in Quantum Magnets

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