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Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Stoner Ferromagnetic instability of a Fermi Liquid
Shivam Ghosh
Statistical Physics Class Presentation
Department of Physics
Cornell University
December 2, 2010
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Outline
1 Magnetic Instabilities:Stoner
Stoner Ferromagnetism
QPT at T=0
2 RG Transformation and �ows
Landau-Ginzburg-Wilson functional for interacting
Paramagnons
Carrying out RG steps
3 Quantum-classical crossover
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Outline
1 Magnetic Instabilities:Stoner
Stoner Ferromagnetism
QPT at T=0
2 RG Transformation and �ows
Landau-Ginzburg-Wilson functional for interacting
Paramagnons
Carrying out RG steps
3 Quantum-classical crossover
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Stoner instability
A magnetic phase transition of a Fermi liquid with net non
zero magnetization
There can be other magnetically ordered states with no net
magnetization (eg. AFM) . General class of magnetic
transitions is speci�ed by ordering wave vectors−→Q . Also
inludes Spin Density Waves
Will focus here on Ferromagnetic Instability. A simple
illustration of QPT
Phase transition occurs on varying a system parameter
(Coulombic repulsion U in Stoner case)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Stoner instability
A magnetic phase transition of a Fermi liquid with net non
zero magnetization
There can be other magnetically ordered states with no net
magnetization (eg. AFM) . General class of magnetic
transitions is speci�ed by ordering wave vectors−→Q . Also
inludes Spin Density Waves
Will focus here on Ferromagnetic Instability. A simple
illustration of QPT
Phase transition occurs on varying a system parameter
(Coulombic repulsion U in Stoner case)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Stoner instability
A magnetic phase transition of a Fermi liquid with net non
zero magnetization
There can be other magnetically ordered states with no net
magnetization (eg. AFM) . General class of magnetic
transitions is speci�ed by ordering wave vectors−→Q . Also
inludes Spin Density Waves
Will focus here on Ferromagnetic Instability. A simple
illustration of QPT
Phase transition occurs on varying a system parameter
(Coulombic repulsion U in Stoner case)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Stoner instability
A magnetic phase transition of a Fermi liquid with net non
zero magnetization
There can be other magnetically ordered states with no net
magnetization (eg. AFM) . General class of magnetic
transitions is speci�ed by ordering wave vectors−→Q . Also
inludes Spin Density Waves
Will focus here on Ferromagnetic Instability. A simple
illustration of QPT
Phase transition occurs on varying a system parameter
(Coulombic repulsion U in Stoner case)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hubbard Model
Hubbard model Hhubbard =−t ∑<ij>
c†i cj +U∑
ini↑ni↓
Electrons reduce U by favoring magnetic ordering
Cost: Gain in Kinetic energy
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hubbard Model
Hubbard model Hhubbard =−t ∑<ij>
c†i cj +U∑
ini↑ni↓
Electrons reduce U by favoring magnetic ordering
Cost: Gain in Kinetic energy
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hubbard Model
Hubbard model Hhubbard =−t ∑<ij>
c†i cj +U∑
ini↑ni↓
Electrons reduce U by favoring magnetic ordering
Cost: Gain in Kinetic energy
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Phase digram and Instability Criterion
Instability occurs when UN(EF ) > 1 (Mean �eld), System has
ferromagnetic ordering
E�ect of adding �uctuations can be seen in the divergence of
the Spin response function χ0(q,ω) which diverges for
q,ω → 0 for FM
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Phase digram and Instability Criterion
Instability occurs when UN(EF ) > 1 (Mean �eld), System has
ferromagnetic ordering
E�ect of adding �uctuations can be seen in the divergence of
the Spin response function χ0(q,ω) which diverges for
q,ω → 0 for FM
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Outline
1 Magnetic Instabilities:Stoner
Stoner Ferromagnetism
QPT at T=0
2 RG Transformation and �ows
Landau-Ginzburg-Wilson functional for interacting
Paramagnons
Carrying out RG steps
3 Quantum-classical crossover
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hertz-1976
Want to study Stoner transition both at T = 0 and extend to
�nite T
At low T (compared to EF ) Quantum �uctuations bring out
new physics
Change of e�ective dimensionality: Dynamic exponent z
Quantum to Classical crossover exponent zν
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hertz-1976
Want to study Stoner transition both at T = 0 and extend to
�nite T
At low T (compared to EF ) Quantum �uctuations bring out
new physics
Change of e�ective dimensionality: Dynamic exponent z
Quantum to Classical crossover exponent zν
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hertz-1976
Want to study Stoner transition both at T = 0 and extend to
�nite T
At low T (compared to EF ) Quantum �uctuations bring out
new physics
Change of e�ective dimensionality: Dynamic exponent z
Quantum to Classical crossover exponent zν
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Hertz-1976
Want to study Stoner transition both at T = 0 and extend to
�nite T
At low T (compared to EF ) Quantum �uctuations bring out
new physics
Change of e�ective dimensionality: Dynamic exponent z
Quantum to Classical crossover exponent zν
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Basic ProblemQPT at T=0
Phase Diagram with Quantum, Classical Exponents
Aim to understand Phase diagram using Renormalization
Group
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Outline
1 Magnetic Instabilities:Stoner
Stoner Ferromagnetism
QPT at T=0
2 RG Transformation and �ows
Landau-Ginzburg-Wilson functional for interacting
Paramagnons
Carrying out RG steps
3 Quantum-classical crossover
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Order Parameter φ(~x ,τ)
We want to �nd an order parameter which distinguishes the
para-ferro phase
Obvious choice: magnetization density φ . Also need an
'e�ective' theory forS [φ ]- describing dynamics of φ
φ = φ(~x ,τ) is a �uctuating (in imaginary time τ ∼ β ) �eld.
Quantum �uctuations arise from dependance on τ!
About rc⇒ Classical statistics in a critical fan.
Quantum statistics elsewhere
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Order Parameter φ(~x ,τ)
We want to �nd an order parameter which distinguishes the
para-ferro phase
Obvious choice: magnetization density φ . Also need an
'e�ective' theory forS [φ ]- describing dynamics of φ
φ = φ(~x ,τ) is a �uctuating (in imaginary time τ ∼ β ) �eld.
Quantum �uctuations arise from dependance on τ!
About rc⇒ Classical statistics in a critical fan.
Quantum statistics elsewhere
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Order Parameter φ(~x ,τ)
We want to �nd an order parameter which distinguishes the
para-ferro phase
Obvious choice: magnetization density φ . Also need an
'e�ective' theory forS [φ ]- describing dynamics of φ
φ = φ(~x ,τ) is a �uctuating (in imaginary time τ ∼ β ) �eld.
Quantum �uctuations arise from dependance on τ!
About rc⇒ Classical statistics in a critical fan.
Quantum statistics elsewhere
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Order Parameter φ(~x ,τ)
We want to �nd an order parameter which distinguishes the
para-ferro phase
Obvious choice: magnetization density φ . Also need an
'e�ective' theory forS [φ ]- describing dynamics of φ
φ = φ(~x ,τ) is a �uctuating (in imaginary time τ ∼ β ) �eld.
Quantum �uctuations arise from dependance on τ!
About rc⇒ Classical statistics in a critical fan.
Quantum statistics elsewhere
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
φ in mode space
In Fourier space φ = φ(q,ω). ω captures the dynamics of QF
Compare to φ4 theory where φ = φ(q). No ω dependence. No
QF!
Statics and dynamics intricately mixed up in Quantum
statistics
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
φ in mode space
In Fourier space φ = φ(q,ω). ω captures the dynamics of QF
Compare to φ4 theory where φ = φ(q). No ω dependence. No
QF!
Statics and dynamics intricately mixed up in Quantum
statistics
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
φ in mode space
In Fourier space φ = φ(q,ω). ω captures the dynamics of QF
Compare to φ4 theory where φ = φ(q). No ω dependence. No
QF!
Statics and dynamics intricately mixed up in Quantum
statistics
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Quantum Classical←→Classical d ←→ d +1
L
Lτ
ξ
ξτ
0
β
Coarse Grain
Close analogy with a '�nite' classical system. A box with
in�nite d dimensions and a �nite ′d +1′th dimension.
Quantum←→ Classical ⇐⇒ d ←→ d +1(ξ ∼ Ld+1)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Paramagnons
Fluctuations of φ(q,ω) quasiparticles wave vector q frequency
ω . Called Paramagnons! (Recall Mihir's lecture, running
horses etc.)
Mass r0 of paramagnons gets renormalized by scattering o�
electrons, holes. We will soon �nd how r0 �ows under RG.Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Paramagnons
Fluctuations of φ(q,ω) quasiparticles wave vector q frequency
ω . Called Paramagnons! (Recall Mihir's lecture, running
horses etc.)
Mass r0 of paramagnons gets renormalized by scattering o�
electrons, holes. We will soon �nd how r0 �ows under RG.Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Building S [φ ]
Begin with Hhubbard which describes hopping fermions ψfermion
interacting through a short ranged repulsive potential U
'Integrate out' ψfermion close to Fermi surface to get an
e�ective model for φ
S [φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2 +
u04Nβ
∑qi ,ωi
φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Building S [φ ]
Begin with Hhubbard which describes hopping fermions ψfermion
interacting through a short ranged repulsive potential U
'Integrate out' ψfermion close to Fermi surface to get an
e�ective model for φ
S [φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2 +
u04Nβ
∑qi ,ωi
φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Building S [φ ]
Begin with Hhubbard which describes hopping fermions ψfermion
interacting through a short ranged repulsive potential U
'Integrate out' ψfermion close to Fermi surface to get an
e�ective model for φ
S [φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2 +
u04Nβ
∑qi ,ωi
φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Outline
1 Magnetic Instabilities:Stoner
Stoner Ferromagnetism
QPT at T=0
2 RG Transformation and �ows
Landau-Ginzburg-Wilson functional for interacting
Paramagnons
Carrying out RG steps
3 Quantum-classical crossover
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
E�ective Action S [φ ]
S [φ ] = S (2)[φ ] +S (4)[φ ](A quadratic and a quartic part)
S (2)[φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2
S (4)[φ ] =u04Nβ
∑qi ,ωi
φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑
3i=1ωi )
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
E�ective Action S [φ ]
S [φ ] = S (2)[φ ] +S (4)[φ ](A quadratic and a quartic part)
S (2)[φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2
S (4)[φ ] =u04Nβ
∑qi ,ωi
φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑
3i=1ωi )
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
E�ective Action S [φ ]
S [φ ] = S (2)[φ ] +S (4)[φ ](A quadratic and a quartic part)
S (2)[φ ] = 12 ∑q,ω
(r0 +q2 + |ω|
q
)|φ(q,ω)|2
S (4)[φ ] =u04Nβ
∑qi ,ωi
φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑
3i=1ωi )
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Interaction parameters in S [φ ]
Seen r0 and q2 before.
F [m] =∫dd r
[r0m
2 + |∇m|2 +u0m4 + ...
]r0 = 1−UN(EF )∼ [U−Uc(T )]/Uc(T ) measures distance
from criticality
Ferromagnetic instability occurs at r0 = 0⇒ UN(EF ) = 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Interaction parameters in S [φ ]
Seen r0 and q2 before.
F [m] =∫dd r
[r0m
2 + |∇m|2 +u0m4 + ...
]r0 = 1−UN(EF )∼ [U−Uc(T )]/Uc(T ) measures distance
from criticality
Ferromagnetic instability occurs at r0 = 0⇒ UN(EF ) = 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Interaction parameters in S [φ ]
Seen r0 and q2 before.
F [m] =∫dd r
[r0m
2 + |∇m|2 +u0m4 + ...
]r0 = 1−UN(EF )∼ [U−Uc(T )]/Uc(T ) measures distance
from criticality
Ferromagnetic instability occurs at r0 = 0⇒ UN(EF ) = 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
RG Transformation
Step 1: Separate out modes into 'slow' and 'fast' and
integrate out 'fast' modes
Step 2: Rescale q and ω such that they lie in the original k
space interval
Step 3: Rescale �elds φ to absorb change in rede�nition of
couplings
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
RG Transformation
Step 1: Separate out modes into 'slow' and 'fast' and
integrate out 'fast' modes
Step 2: Rescale q and ω such that they lie in the original k
space interval
Step 3: Rescale �elds φ to absorb change in rede�nition of
couplings
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
RG Transformation
Step 1: Separate out modes into 'slow' and 'fast' and
integrate out 'fast' modes
Step 2: Rescale q and ω such that they lie in the original k
space interval
Step 3: Rescale �elds φ to absorb change in rede�nition of
couplings
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
RG Transformation
Step 1: Separate out modes into 'slow' and 'fast' and
integrate out 'fast' modes
Step 2: Rescale q and ω such that they lie in the original k
space interval
Step 3: Rescale �elds φ to absorb change in rede�nition of
couplings
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Shaving o� modes
The cut o� in summations for the momentum and energy are
Λ∼ 1/a and EF
Rescale q by q/Λ and integrate out modes lying in a thin shell
[Λe−`,Λ]
Similarly, rescale ω by EF and integrate out modes lying
within [Λe−`,Λ]
e−`⇐⇒ s−1 for Kyungmin's MSRG . It parametrizes RG steps
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Shaving o� modes
The cut o� in summations for the momentum and energy are
Λ∼ 1/a and EF
Rescale q by q/Λ and integrate out modes lying in a thin shell
[Λe−`,Λ]
Similarly, rescale ω by EF and integrate out modes lying
within [Λe−`,Λ]
e−`⇐⇒ s−1 for Kyungmin's MSRG . It parametrizes RG steps
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Shaving o� modes
The cut o� in summations for the momentum and energy are
Λ∼ 1/a and EF
Rescale q by q/Λ and integrate out modes lying in a thin shell
[Λe−`,Λ]
Similarly, rescale ω by EF and integrate out modes lying
within [Λe−`,Λ]
e−`⇐⇒ s−1 for Kyungmin's MSRG . It parametrizes RG steps
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Shaving o� modes
The cut o� in summations for the momentum and energy are
Λ∼ 1/a and EF
Rescale q by q/Λ and integrate out modes lying in a thin shell
[Λe−`,Λ]
Similarly, rescale ω by EF and integrate out modes lying
within [Λe−`,Λ]
e−`⇐⇒ s−1 for Kyungmin's MSRG . It parametrizes RG steps
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 2
After RG step 1
S (2)[φ ]→ S ′(2)[φ ] = 12βN
∫ e−`
0ddqdω
(2π)d+1
(r ′0 +q2 + |ω|
q
)|φ(q,ω)|2
Note: r0→ r ′0 under step 1 of RG (postponed till later)
To restore cuto� de�ne: q′ = qe`,ω ′ = ωez`(RG Step 2)
Notice the di�erent RG step sizes along q and ω .
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 2
After RG step 1
S (2)[φ ]→ S ′(2)[φ ] = 12βN
∫ e−`
0ddqdω
(2π)d+1
(r ′0 +q2 + |ω|
q
)|φ(q,ω)|2
Note: r0→ r ′0 under step 1 of RG (postponed till later)
To restore cuto� de�ne: q′ = qe`,ω ′ = ωez`(RG Step 2)
Notice the di�erent RG step sizes along q and ω .
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 2
After RG step 1
S (2)[φ ]→ S ′(2)[φ ] = 12βN
∫ e−`
0ddqdω
(2π)d+1
(r ′0 +q2 + |ω|
q
)|φ(q,ω)|2
Note: r0→ r ′0 under step 1 of RG (postponed till later)
To restore cuto� de�ne: q′ = qe`,ω ′ = ωez`(RG Step 2)
Notice the di�erent RG step sizes along q and ω .
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 2
After RG step 1
S (2)[φ ]→ S ′(2)[φ ] = 12βN
∫ e−`
0ddqdω
(2π)d+1
(r ′0 +q2 + |ω|
q
)|φ(q,ω)|2
Note: r0→ r ′0 under step 1 of RG (postponed till later)
To restore cuto� de�ne: q′ = qe`,ω ′ = ωez`(RG Step 2)
Notice the di�erent RG step sizes along q and ω .
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Dynamic exponent z!
Choosing q′ = qe−`,ω ′ = ωe−z`
S ′(2)[φ ] =12βNe−(d+z)`
∫ 10 D
′(r ′0 +q′2e−2` + |ω ′|e−z`
q′e−`
)|φ(q′e−`,ω ′e−z`)|2
If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.
Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)
S ′(2)[φ ] = 12βN
∫ 10 D
′(r ′0e
2` +q′2 + |ω ′|q′
)|φ ′(q′,ω ′)|2
Under RG r0→ r(`) = r ′0e2`
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
S (4)[φ ] term
Carry out same steps for quartic term to �nd
u0→ u(`) = u′0e[4−(d+z)]` = u′0e
ε`
ε = 4− (d + z)
To test stability of Gaussian �xed point : du(`)d` = εu(`)
Stable ε < 0⇒ d > 4− z .Expect Mean�eld exponents for d > 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
S (4)[φ ] term
Carry out same steps for quartic term to �nd
u0→ u(`) = u′0e[4−(d+z)]` = u′0e
ε`
ε = 4− (d + z)
To test stability of Gaussian �xed point : du(`)d` = εu(`)
Stable ε < 0⇒ d > 4− z .Expect Mean�eld exponents for d > 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
S (4)[φ ] term
Carry out same steps for quartic term to �nd
u0→ u(`) = u′0e[4−(d+z)]` = u′0e
ε`
ε = 4− (d + z)
To test stability of Gaussian �xed point : du(`)d` = εu(`)
Stable ε < 0⇒ d > 4− z .Expect Mean�eld exponents for d > 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
S (4)[φ ] term
Carry out same steps for quartic term to �nd
u0→ u(`) = u′0e[4−(d+z)]` = u′0e
ε`
ε = 4− (d + z)
To test stability of Gaussian �xed point : du(`)d` = εu(`)
Stable ε < 0⇒ d > 4− z .Expect Mean�eld exponents for d > 1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 1 & Flow equations
To get the �ow equations we need to carry out Step 1 carefully
Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1
De�ne fast φ>(q) and slow φ<(q) modes with q in outer and
inner shells
S (4) = u04βN ∑
qi ,ωi
∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )
Only few terms out of 16 contribute!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 1 & Flow equations
To get the �ow equations we need to carry out Step 1 carefully
Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1
De�ne fast φ>(q) and slow φ<(q) modes with q in outer and
inner shells
S (4) = u04βN ∑
qi ,ωi
∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )
Only few terms out of 16 contribute!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 1 & Flow equations
To get the �ow equations we need to carry out Step 1 carefully
Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1
De�ne fast φ>(q) and slow φ<(q) modes with q in outer and
inner shells
S (4) = u04βN ∑
qi ,ωi
∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )
Only few terms out of 16 contribute!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 1 & Flow equations
To get the �ow equations we need to carry out Step 1 carefully
Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1
De�ne fast φ>(q) and slow φ<(q) modes with q in outer and
inner shells
S (4) = u04βN ∑
qi ,ωi
∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )
Only few terms out of 16 contribute!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Step 1 & Flow equations
To get the �ow equations we need to carry out Step 1 carefully
Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1
De�ne fast φ>(q) and slow φ<(q) modes with q in outer and
inner shells
S (4) = u04βN ∑
qi ,ωi
∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )
Only few terms out of 16 contribute!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Important contributions
2 kinds of terms contribute: φ<,1φ<,2φ<,3φ<,4 and
φ<,1φ<,2φ>,3φ>,4
S (4) =u0
4Nβ∑
qi<e−l
4
∏i
φ<,i (qi ,ωi )δ
(∑i
qi
)δ
(∑i
ωi
)+
+3u02Nβ
∑qi<e−l
|φ<(q,ω)|2 ∑1>qi>e−l
|φ>(q,ω)|2
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Integrating out fast modes
Z> =∫Dφ>exp
[−1
2
(r0 +q2 + |ω|
q+ 3u0
βN ∑ |φ<(q,ω)|2)|φ>(q,ω)|2
]Z> =
(r0 +q2 + |ω|
q+ 3u0
βN ∑ |φ<|2)−1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Integrating out fast modes
Z> =∫Dφ>exp
[−1
2
(r0 +q2 + |ω|
q+ 3u0
βN ∑ |φ<(q,ω)|2)|φ>(q,ω)|2
]Z> =
(r0 +q2 + |ω|
q+ 3u0
βN ∑ |φ<|2)−1
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Partition Function Z
Z =∫Dφ<
(S<,quadraticS<,quarticS>
)S<,quadratic = exp
(−1
2 ∑q<e−`
(r0 +q2 + |ω|
q
)|φ<|2
)
S<,quartic = exp
(−1
4u0βN ∑
q<e−`∏
4i φ<(qi ,ωi )δ
(∑iqi
)δ
(∑iωi
))
S> = exp
(−1
2 ∑q>e−`
ln
(r0 +q2 + |ω|
q+ 3u0
βN ∑q<e−`
|φ<|2))
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Partition Function Z
Z =∫Dφ<
(S<,quadraticS<,quarticS>
)S<,quadratic = exp
(−1
2 ∑q<e−`
(r0 +q2 + |ω|
q
)|φ<|2
)
S<,quartic = exp
(−1
4u0βN ∑
q<e−`∏
4i φ<(qi ,ωi )δ
(∑iqi
)δ
(∑iωi
))
S> = exp
(−1
2 ∑q>e−`
ln
(r0 +q2 + |ω|
q+ 3u0
βN ∑q<e−`
|φ<|2))
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Partition Function Z
Z =∫Dφ<
(S<,quadraticS<,quarticS>
)S<,quadratic = exp
(−1
2 ∑q<e−`
(r0 +q2 + |ω|
q
)|φ<|2
)
S<,quartic = exp
(−1
4u0βN ∑
q<e−`∏
4i φ<(qi ,ωi )δ
(∑iqi
)δ
(∑iωi
))
S> = exp
(−1
2 ∑q>e−`
ln
(r0 +q2 + |ω|
q+ 3u0
βN ∑q<e−`
|φ<|2))
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Partition Function Z
Z =∫Dφ<
(S<,quadraticS<,quarticS>
)S<,quadratic = exp
(−1
2 ∑q<e−`
(r0 +q2 + |ω|
q
)|φ<|2
)
S<,quartic = exp
(−1
4u0βN ∑
q<e−`∏
4i φ<(qi ,ωi )δ
(∑iqi
)δ
(∑iωi
))
S> = exp
(−1
2 ∑q>e−`
ln
(r0 +q2 + |ω|
q+ 3u0
βN ∑q<e−`
|φ<|2))
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Renormalized couplings
Expanding log in S>to 4th order in φ<will give renormalized
couplings
r0→ r ′0 = r0 + 3u0βN ∑
q>e−`∑
ω>e−zl
(r0 +q2 + |ω|
q
)−1u0→ u′0 = u0− 9u20
βN ∑q>e−`
∑ω>e−zl
(r0 +q2 + |ω|
q
)−2
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Renormalized couplings
Expanding log in S>to 4th order in φ<will give renormalized
couplings
r0→ r ′0 = r0 + 3u0βN ∑
q>e−`∑
ω>e−zl
(r0 +q2 + |ω|
q
)−1u0→ u′0 = u0− 9u20
βN ∑q>e−`
∑ω>e−zl
(r0 +q2 + |ω|
q
)−2
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Renormalized couplings
Expanding log in S>to 4th order in φ<will give renormalized
couplings
r0→ r ′0 = r0 + 3u0βN ∑
q>e−`∑
ω>e−zl
(r0 +q2 + |ω|
q
)−1u0→ u′0 = u0− 9u20
βN ∑q>e−`
∑ω>e−zl
(r0 +q2 + |ω|
q
)−2
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Flow equations and comparison toφ4
Flow equations:
dr(`)
d`= 2r(`) +A
′u(`)
du(`)
d`= (4− (d + z))u(`)−B
′u(`)2
Complex scalar φ4 theory
dr(`)
d`= 2r(`) +Au(`)
du(`)
d`= (4−d)u(`)−Bu(`)2
Notice appearance of z due to quantum �uctuations!
Draw same conclusions as for φ4 but due to quantum
statistics!Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Flow equations and comparison toφ4
Flow equations:
dr(`)
d`= 2r(`) +A
′u(`)
du(`)
d`= (4− (d + z))u(`)−B
′u(`)2
Complex scalar φ4 theory
dr(`)
d`= 2r(`) +Au(`)
du(`)
d`= (4−d)u(`)−Bu(`)2
Notice appearance of z due to quantum �uctuations!
Draw same conclusions as for φ4 but due to quantum
statistics!Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Flow equations and comparison toφ4
Flow equations:
dr(`)
d`= 2r(`) +A
′u(`)
du(`)
d`= (4− (d + z))u(`)−B
′u(`)2
Complex scalar φ4 theory
dr(`)
d`= 2r(`) +Au(`)
du(`)
d`= (4−d)u(`)−Bu(`)2
Notice appearance of z due to quantum �uctuations!
Draw same conclusions as for φ4 but due to quantum
statistics!Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps
Flow equations and comparison toφ4
Flow equations:
dr(`)
d`= 2r(`) +A
′u(`)
du(`)
d`= (4− (d + z))u(`)−B
′u(`)2
Complex scalar φ4 theory
dr(`)
d`= 2r(`) +Au(`)
du(`)
d`= (4−d)u(`)−Bu(`)2
Notice appearance of z due to quantum �uctuations!
Draw same conclusions as for φ4 but due to quantum
statistics!Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Classical d ←→ d +1 transition
In analogy with the d ←→ d +1 transition in a classical
system living in a box in�nite in ′d ′ dimensions and �nite along′d +1′th dimension
Criticality occurs when correlation length along the �nite
dimension diverges!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Classical d ←→ d +1 transition
In analogy with the d ←→ d +1 transition in a classical
system living in a box in�nite in ′d ′ dimensions and �nite along′d +1′th dimension
Criticality occurs when correlation length along the �nite
dimension diverges!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Analogy to Quantum-Classical crossover
L
Lτ
ξ
ξτ
0
β
Coarse Grain
Similarly, our system also reaches criticality (with classical
statistics existing in a 'critical fan' about the criticalpoint)
when correlation length along imaginary time τ direction
diverges ξτ ∼ β
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Analogy to Quantum-Classical crossover
L
Lτ
ξ
ξτ
0
β
Coarse Grain
Similarly, our system also reaches criticality (with classical
statistics existing in a 'critical fan' about the criticalpoint)
when correlation length along imaginary time τ direction
diverges ξτ ∼ β
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
The cross over exponent
Begin with a small ξτ (quantum regime) and transition to
classical as ξτ ∼ β
ξτ ∼ ξ z and since ξ ∼ r−ν
0 with ν = 1/2 we get ξτ ∼ r−zν
0
Cross over condition:
ξτ = β = r−zν
0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2
zν is a new cross over exponent!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
The cross over exponent
Begin with a small ξτ (quantum regime) and transition to
classical as ξτ ∼ β
ξτ ∼ ξ z and since ξ ∼ r−ν
0 with ν = 1/2 we get ξτ ∼ r−zν
0
Cross over condition:
ξτ = β = r−zν
0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2
zν is a new cross over exponent!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
The cross over exponent
Begin with a small ξτ (quantum regime) and transition to
classical as ξτ ∼ β
ξτ ∼ ξ z and since ξ ∼ r−ν
0 with ν = 1/2 we get ξτ ∼ r−zν
0
Cross over condition:
ξτ = β = r−zν
0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2
zν is a new cross over exponent!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
The cross over exponent
Begin with a small ξτ (quantum regime) and transition to
classical as ξτ ∼ β
ξτ ∼ ξ z and since ξ ∼ r−ν
0 with ν = 1/2 we get ξτ ∼ r−zν
0
Cross over condition:
ξτ = β = r−zν
0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2
zν is a new cross over exponent!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Crossover exponent
Boundary between classical and quantum statistics given by
T = r zν0
Classical statistics for T > r zν0 and
Quantum statistics for T < r zν0
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Summary
Looked at QPT in stoner ferromagnetism
Quantum dynamics changes condition for stability of Gaussian
�xed point
z enters as a dyanmic exponent, treats spatial and time
correlations di�erently
Also see a new quantum to classical classover exponent!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover
Summary
Acknowledgements
All of us would like to thank Prof. Sethna for giving us an
opportunity to work on this very interesting problem and for
many helpful discussions
Shivam would like to thank Kyungmin for the wonderful
�gures!
Have a good break!
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid
Appendix
References
Hertz J.A. (1976) Phys.Rev .B 14, 1165
Chapter 12 'Phase Transitions of Fermi Liquids' Quantum
Phase Transitions, Subir Sachdev
Millis, A. J. (1993) Phys.Rev .B 48, 7183
'Renormalization-group approach to interacting fermions' R.
Shankar, Rev. Mod. Physics, Vol. 66, No. 1, January 1994
Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid