Nonperturbative functional
renormalization groupfor disordered systems:
The case of therandom field Ising model
G. Tarjus and M. Tissier(LPTMC, CNRS-Univ. PARIS 6)

Physics of〝disordered systems″• Systems in the presence of quenched disorder (due toimpurities, dislocations, random environment, etc, frozen on
the relevant time scale) pose new challenges to statisticalphysics:
new phases and phase transitions (spin glass, glassy phases, Griffiths phases,...)✴ new phenomena (localization, pinning,...)✴ slow relaxation, aging and hysteresis.
✴

• One often needs new theoretical tools=> Nonperturbative functional RG (NP-FRG)• Here: focus on the equilibrium behavior of classical systems.

Random field model• Prototypical model in theory of ‶disordered systems″
In the field-theoretical description (RFIM) :Sh [φ] = SB [φ] −�h(x)φ(x); SB =
x

�

�

1
τ
u
2
2
d x (∂µ φ(x)) + φ(x) + φ(x)4
2
2
4!
d

with a quenched random field drawn
from a given probability distribution(e.g., Gaussian)�h(x) = 0,h(x)h(y) = ∆B δ(d)(x − y)• Physical realizations in soft and hard condensed matter:✴ Near critical fluids in disordered porous materials✴ Dilute antiferromagnets in a uniform magnetic fluid✴ Hysteresis in dirty magnets✴ Vortex phases in disordered type-II superconductors

Generic difficulties ofdisordered systems• Due to quenched disorder (h), one loses translational invariance.
Way out: average over disorder, but what ?, how ?

• Presence of many low-energy (low-action) “metastable states”.
• Possible influence of rare events, rare spatial regions or rare
samples.

Average over the disorder[‶self-averaging″, ‶replica trick″, etc.]• RFIM equilibrium partition function in a given random-fieldsample h :Zh [J] = eWh [J] =

�

Dφ e

�
−Sh [φ]+ x J(x)φ(x)• Wh [J] is a random functional of the source =>✴ in principle, one needs its whole probability distribution✴ or equivalently, the infinite set of its disorder-averaged

cumulants:
W1 [J] = Wh [J], W2 [J1 , J2 ] = Wh [J1 ]Wh [J2 ]|c , · · ·Known results about the RFIM
• Existence of a Z2 symmetry breaking transition for d>2 for the Ising

version [transition for d<4 for the O(N>2) version]. The upper criticaldimension is dUC=6.

• The critical behavior is associated with a zero-temperature fixed
point (thermal fluctuations are formally irrelevant) and one candirectly work at T=0.

• For a given realization of the disorder h(x), the ground state is unique[except for rare values/configurations of the external source J(x)].

reduction in the number of degrees of freedom by a factor b . The transformation generates aflow in the space of the naive scaling fields T /J, h/J, and H/J, which eventually terminates
in one of the fixed points of the system. The existence of three fixed point will be assumed(Fig. 4), in addition to the trivial, high temperature fixed point:(i) A totally unstable “thermal” fixed point C at T = Tc , H = h = 0 (the random field is arelevant perturbation, see our discussion in 3.2) .
(ii) A fixed point R at T = H = 0 and h = hR which is unstable in two, but stable in one
directions and is therefore a critical point.
(iii) A totally stable fixed point F at T = h = H = 0, which corresponds to the lowtemperature phase for d > dl .

Known results about the RFIM (contd.)
Zero-temperature fixed point and its consequences
Disorderh/JstrengthhR

0R

Schematic phase (and RGflow) diagram of the RFIM
for d>2F
C

Temperature

T/J• Additional exponent for the temperature flow: θ > 0
H/J

Figure 4: Schematic renormalization group flow of the random field Ising model

pairthecorrelation
functions:
In general
RNG procedure generates
also new terms in the Hamiltonian. We will
• Two distinct

assume, that these terms are irrelevant in the RNG–sense and can therefore be neglected.
In order to calculate the critical behavior we have to linearize the RNG flow close to
the fixed point �R. The eigenvalues and eigenvectors of the linearized RNG–transformation
� d−4+η̄
deliver the critical exponents and scaling
fields. Phenomenological arguments concerning
the RNG flow suggest
1
T
H
T
,
τ
=
(h
−
h
)
+
c
and
(23)
R
�
�
J
J
J
J
� d−2+η
as the scaling fields. Close to the fixed point R J, τ and H transform under the RNG
coarse graining as
J → J ! = J b yJθ
τ → τ ! = τ b yτ
(24)
!
y
H
H →H = Hb .

1
< φ(x) >< φ(x ) > ∼|x − x |, with θ = 2 + η − η̄,T< φ(x)φ(x ) > − < φ(x) >< φ(x ) > ∼|x − x |• For T>0: very slow “activated” critical dynamics, τ ∼ exp(c ξ ) ,with ξ the correlation length (that10diverges at the critical point).

Metastable states• At zero temperature, the equilibrium behavior of the RFIM isdetermined by the ground state configuration [absolute minimumof Sh = SB − (h + J)φ], which is solution of the stochastic field
equation:δSB [φ]
=
h(x)
+
J(x)
δφ(x)

• However, for low disorder strength and in the region of interest(near the critical point), the equation has many solutions =>many minima of the bare action (“metastable states”).

• What is their effect on the long-distance properties ?

[Also known to go with slow relaxation, hysteresis and ‶glassiness″]

Parisi-Sourlas supersymmetric approachof the RFIM
• At T=0, generating functional of the correlation functions:ˆ =Zh [J, J]
✴

�

� 2
� �
� δSB [φ]�� δ SB [φ] �ˆ
J(x)φ(x)��xDφ δ−h−J �e�δφ
δφδφ

If there is a unique solution of the stochastic field equation, usual manipulations:Introduce auxiliary fields φ̂(x), ψ(x), ψ(x),average over disorder h (Gaussian probability distribution),
introduce a superspace with 2 Grassmann coordinates x = (x, θ, θ)422dθ̄dθ ,and supermetric dx = dx +∆B2a superLaplacian ∆SS = ∂µ + ∆B ∂θ ∂θ ,a superfield Φ(x) = φ(x) + θψ(x) + ψ(x)θ + θθ φ̂(x) , super-etc...

Parisi-Sourlas supersymmetric approachof the RFIM (contd.)• The generating functional Zh can then be obtained from asuperfield theory with action:
SSU SY [Φ] =

� �
x

1
τ
u
2
− Φ(x)∆SS Φ(x) + Φ(x) + Φ(x)4
2
2
4!

�

• Invariant under SUSY (super-rotations in superspace)=> leads to ‶dimensional reduction″: RFIM in d dimensions
is equivalent
to the pure theory in
d-2.
��
�
��
�
4
d xdθdθ̄ f (x +θθ̄) =
∆B
d

2

4π
∆B

Beautiful, but wrong!!
Problem with multiple solutions!!

dd−2 x f (x2 )Rare events: toy model (d=0 RFIM)SBδSB (φ)= J + h with
Stochastic equation:
δφϕ• For zero temperature T=0, select the ground state:ϕGS,hThe pair correlation function for slightly different sources,
φGS,h (J + δJ)φGS,h (J − δJ) =2

φGS,h (J)2 + A(J)|δJ| + O(δJ )“avalanche”
-h

0

J

has a nonanalytic behavior (a “cusp”) when J->0 due to the “avalanches”.

• For a small T>0, Boltzmann weighting of the minima ( e−SB (φ)−(J+h)φT

The cusp is rounded in a “thermal boundary layer”< φ(J + δJ + h) >< φ(J − δJ + h) > = < φ(J + h) >2 + T f (J,• ISSUE: Does this persist at long distance when d>0 ?

|δJ|) + ···T):

Long-standing puzzles concerning
random-field systems• Critical behavior: what is the way out of dimensional
reduction?

• What is the phase diagram of the d-dimensionalrandom-field O(N) model in the whole (N,d) plane?

Why does one need anonperturbative functional RG ?
• RG, because one is interested in the long-distance properties near tothe critical point; in particular, the “metastable states” of potential
relevance are not those of the bare action but those of a scaledependent renormalized functional;• Functional, because the influence of the rare events (avalanches anddroplets) can only be described through a singular dependence of the
cumulants of the renormalized disorder on their arguments;• Nonperturbative, because standard perturbation theory completely fails(dimensional reduction), relevant dimensions are far from d=6, disorder
grows strong under coarse graining.

Program for RG study of RFIM
[Search for the proper T=0 IR (critical) fixed point]• Select with high probability the ground state at the running IR scale kamong the solutions if several of them and ensure that only theground state is considered when k -> 0.

• Describe full functional dependence of cumulants of renormalized
disorder and allow for nonanalytical dependence on theirarguments.

• Start the RG flow with a ‶regularized″ stochastic field equation
having a unique solution.

• Use a nonperturbative truncation and be able to recover dimensional
reduction if it has a range of validity.

=> NP-FRG in a superfield setting

Superfield formalism for the RFIM
• Several copies + a weighting factor => Generating functional:
� 2�� δSB [φa ]�δ SB [φa ](β)ˆ
Zh [{Ja , Ja }] =Dφa δ− h − Ja det
δφaδφa δφaa����ˆa (x)φa (x) −β SB [φa ]− [h(x)+Ja (x)]φa (x)
J
x
×e x
e
��

Average over disorder generates cumulants with full functional dependence:Zh [{Ja , Jˆa }] =

�

ˆa ]
W
[J
,
J
a
h
e
=e

a

�

�
1
ˆ
ˆ
ˆ
a Wh [Ja ,Ja ]+ 2
ab Wh [Ja ,Ja ]Wh [Jb ,Jb ]|c +···

Grassmannian space• Introduce superfields and a ‶curved″�� �Φ(θ) = φ + θψ + ψθ + θθφ̂ ;=dθdθ(1 + βθθ)

θ

��

� �
�
1

(β)
S
[{Φa }] =S1 [Φa (θ)] +S2 [Φa (θ1 ), Φa (θ2 )]
=>
2
θ
θ1 θ2
a
ab
�
� �
�
�2
1�
S1 =
∂µ Φa (θ, x) + UB (Φa (θ, x) ; S2 =∆B Φa (θ1 , x)Φb (θ2 , x)x 2xSuperfield formalism (contd.)• Add coupling to supersources

��
a

θ,x

Ja (θ, x)Φa (θ, x) -> W (β) [{Ja }](β)Γ[{Φa }]+ Legendre transform -> Effective action• The action is invariant under a large group of symmetries andsupersymmetries (Sn between copies, global Z2 and Euclidean translations+ rotations, isometries of the curved Grassmann subspace copy by copy).

• The expansion in increasing number of sums over copies generates the‶cumulant expansion″ of the 1PI generating functional (effective action):(β)Γ[{Φa }] =

�
a1

1
(β)
Γ1 [Φa1 ] −�

2 a ,a12(β)Γ2 [Φa1 , Φa2 ] + · · ·NP-FRG in superfield formalism• Add an IR regulator to the action:(β)∆Sk [{Φa }] =

�
�
1
2

ab

x1

�

x2

Φa (x1 )Rk,ab (x1 , x2 )Φb (x2 )�k (q 2 ) + R
�k (q 2 ) : suppresses fluctuations of ϕ field and random field
Rk,ab (x1 , x2 ) = δθ1 ,θ2 R• ERGE for the effective average action at scale k:(β)∂k Γk [{Φa }] =

•

�
�
1
2

ab

x1

�

x2

(2)
∂k Rk,ab (x1 , x2 )(Γk [{Φa }] + Rk )−1(b,x2 )(a,x1 )(β)
Through the expansion of Γk [{Φa }] in increasing number of copies:

Hierarchy of coupled ERGE’s for the cumulants (functionals of thesuperfields):(β)(β)∂k Γk,1 [Φ1 ] = · · · , ∂k Γk,2 [Φ1 , Φ2 ] = · · · , etc‶Grassmannian ultralocality″
and superrotational invariance• Property of the generating functionals when a unique solution of thestochastic equation is included:
�(β)
W‶Grassmannian ultralocality″: h [J ] = W [J (θ)]θ• When β → ∞ , ‶ultralocality″ (UL) becomes exact, with the pth
cumulant of the effective average action given by (more later!)(β)Γk,p [Φa1 , ..., Φap ] =

�

...
θa

1

�

θ ap

�

(U L)
Γk,p [Φa1 (θa1 ), ..., Φap (θap )] + N U L corrections• When ‶Grassm. UL″, β drops out of the FRG equations.

�Then, for supersources that reduce the theory to a 1-copy problem,
the theory is invariant under superrotations (SUSY)=> Ward-Takahashi (WT) identities.

NP-FRG and SUSY breaking• Grassm. ultralocality => hierarchy of ERGE’s for the cumulants withphysical field arguments (Φ ≡ φ) :�� (2)
�−1 � (11)��1��k∂t Γk1 [φ] = ∂t T r Γk1 [φ] + R̂kΓk2 [φ, φ] − R2∂t Γk2 [φ1 , φ2 ] = · · ·[t = ln(k/Λ)]!!! Recall: The auxiliary parameter β then drops out of the ERGE’s !!!

• As a result, superrotational invariance for 1 copy is a priori preserved

along the RG flow: From the WT identities, one can show that it leads(nonperturbatively) to dimensional reduction.

• What can go wrong ?
✴ Spontaneous breaking of superrotation invariance:some 1PI vertex blows up when copy fields become equal.
✴ Dimension reduction is broken when a cusp
(11)
(11)
Γk,2 (ϕ1 , ϕ2 ) − Γk,2 (ϕ1 , ϕ1 ) ∼ |ϕ2 − ϕ1 | as ϕ2 → ϕ1appears at a finite scale kL.

SUSY-compatible approximation and RG flow
• Ansatz for effective average action (under ‶Grassm. ultralocality″):��12
Γk1 [φ] =Uk (φ(x)) + Zk (φ(x))(∂µ φ(x))
2x ��Γk2 [φ1 , φ2 ] =Vk (φ1 (x), φ2 (x)),Γk,p>2 = 0x+ Regulators: R̂k = Zk k 2 r(q 2 /k 2 ), R̃k = −(∆k /Zk )∂q2 R̂(q 2 )[ SUSY WT identity: Δk=ΔB Zk ]• Introduce scaling dimensions for T=0 fixed point (critical). Then,∂t u�k (ϕ) = · · ·∂t zk (ϕ) = · · ·(11)∂t δk (ϕ1 , ϕ2 ) = ∂t vk (ϕ1 , ϕ2 ) = · · ·ηk = −∂t Zk

η̄k = 2ηk + ∂t ∆k• If no linear cusp in δk (ϕ1 , ϕ2 ), then ∂t δk (ϕ, ϕ) = ∂t zk (ϕ) (WT id.)•and exact dim. reduction follows: found for d > dDR ≃ 5.1• Numerical resolution on a grid.

11
SUSY is broken at a finite IR scale along the RG flow
and a breakdown of dimensional reduction takes place.
By using the procedure (1) detailed in the above subsection, we have numerically located this critical dimension
as dDR ! 5.1 ± 0.1 (the precise value has a residual dependence on the chosen cutoff function). Note that the
DR with thevalue dDR ! 5.1 obtained here is consistentvalue found in our previous, and somewhat cruder, NPFRG approach of the RFO(N )M when extrapolatingtheDRtransition line dDR (N ) down to N = 1 (see Figure 4 ofpaper II2 ).
For initial conditions of the RG flow at or near the critical point, the second derivative δk,2 (ϕ) = ∂y2 δk (ϕ + y, ϕ −A blows
secondδk blowsy)|y=0up atordera finitederivative
RG “time” tof
L = log(kL /Λ)for d <
stays time
finite up
the fixed
point
updDR
in, whereasa finiteit RG
fortod<dDR (redfor d > dDR . We illustrate the difference of behavior
forin d>d
betweencurve),these twonot
cases
Fig. 1DRfor(bluea fieldcurve)
configuration ϕ = 0. The divergence
the full function δk,2 (ϕ)=> SUSYof breaking

Results
0

!500
!1000
δk,2(ρ) !1500!2000

Above d ≃ 5.1: no cusp in δk (ϕ1 , ϕ2 ) .
Below d : cusp in δk (ϕ1 , ϕ2 ) and SUSY breaking
ρin a finite RG time.
|t|
1

2

3

4

FIG. 2: NP-FRG flow of δk,2 (ρ) = ∂y2 δ(ρ, y)|y=0 for d = 5 <dDR . The initial conditions at k = Λ (i.e., t = 0) for u!k (ρ) andzk (ρ) are taken at the fixed-point solution, u!∗ (ρ) and z∗ (ρ),
and those for δk,0 (ρ) and δk,2 (ρ) are chosen as δk=Λ,0 (ρ) =Flowsecondz∗ (ρ) and δk=Λ,2(ρ) =of0. theOne dimensionlessobserves that the divergence
first takes place for small values of ρ.
cumulant δk in d=4

PRL 107, 041601 (2011)0

PHYSICAL REVI

!40

d=5.2

δ%,2(ρ)

!500

d=5.0

δk,2(0)

|tL |

!1500

0

1

2

3

!80
!120
!160

!1000

!2000

0

0
0.002
0.004
0.006
0.008
5 0.01

4

5

6

7

|t|
FIG. 1: NP-FRG flow of δk,2 (0) in the regime where SUSY isvalid. The initial conditions at k = Λ (i.e., t = 0) for u!k (ρ)
and zk (ρ) = δk,0 (ρ), with ρ = ϕ2 /2, are taken at the fixedpoint solution, u!∗ (ρ) and z∗ (ρ) [∂t u!k (ρ)|∗ = ∂t zk (ρ)|∗ = 0],10.95 00.9 0.0020.004
0.006
0.85 ρ
5.17 5.18 5.19
5.16
5.15
0.008 5.13 5.14
0.8
d
0.75

5.2

0
FIG. 3: Fixed-point solution δ∗,2
0.4√(ρ) for dimensions d > dDR .
Note the square-root behavior d − dDR as a function of di0.8
1.2
mension.
−0.08 −0.04
1.6
0
0.04
0.08
associated WT identities. This is shown for the fixed21

Results: Critical exponents η and η̄Breakdown of dimensional reduction appears continuously indimension d(pure,d−2)
η̄
=
η
[=
η
]
•Dimensional reduction:
week ending
E W L E T T E R S •Below dDR: η̄ > η22 JULY 2011
1.2

Very good agreementwith ‶best estimates″:
In d=3,
η =0.57 ± 0.05
[0.51±0.04]
η̄ =1.08 ± 0.05
[1.02-1.10]

1
0.8
0.6
0.4
0.2
0

3

3.5

4

4.5

5

5.5

6

22

Role of the (auxiliary) temperature• If one adds the first ‶non-ultralocal″ corrections, one finds that they
go to finite fixed-point values and that the flow of the ultralocal
quantities is generically of the form
� θ�
k
(U L)
(N U L)β δk(ϕ1 , ϕ2 )∂t δk (ϕ1 , ϕ2 ) = βδk (ϕ1 , ϕ2 ) +β• The second term drops out when (1/β)=0 (zero auxiliary temperature)and one is back to the purely ‶ultralocal″ contributions.

• When β finite and k -> 0, nonuniform convergence to the ‶ultralocal″
cuspy fixed point: cusp in δk (ϕ1 , ϕ2 ) is rounded in a thermal
θ
|ϕ
−
ϕ
|/k
boundary layer in 1.
2

• The boundary layer is related to the presence of rare ‶droplet″
excitations.

Results: N-d phase diagram of the RFO(N)M
20
18
16

IV

14
12

III

N 10

NDR (d)

8

I

6

II

4

Ising

NC

2
1.5

2

2.5

3

3.5

4

d
Region IV:

4.5

5

5.5

6

dDR ≈ 5.1

Weak non-analyticity (at fixed pt.); dim. red. predictions O.K.

Regions I and II:

Spontaneous SUSY breaking at finite RG scale;cusp in renormalized second cumulant; breakdown of dim. red. (II: QLRO)Region III:No phase transition

Conclusion• The description of the long-distance physics of systems with

quenched disorder requires special theoretical tools to accountfor loss of translational invariance/average over disorder, rare
events, metastable states, etc...

• NP-FRG in a superfield setting (with introduction of many

copies and of a weighting factor to select the proper solution)
solves the 30-year-old pending problems concerning the criticalbehavior in random field systems.

• It could be a useful formalism for other problems described bya stochastic field equation with multiple solutions (metastablestates in glasses, shocks in fluid turbulence, Gribov copies in
non-Abelian gauge theories,...).

Results (contd)
Below dDR ≃ 5.1: cuspy fixed point forδk (ϕ1 , ϕ2 )Dimensionless second cumulant at the fixed point in d=3δ∗ (ρ, y) 1.1
1

0.9
0.8
0.7
0.6
0.5
0.4

ρ=

�

0.07
0.06
0.05
0.04
2
0.03
0.02
0.01

|ϕ1 ϕ |
2

0 0.2

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

ϕ1 − ϕ2
y=
2
26

0.75

0
0.4
0.8
1.2
|t|
1.6

2

Results (contd)

!0.08

!0.04

0

0.04

0.08

y

0
!2

0

ϕ2for
) &d =Below: cuspflow
in ofδkδ(ϕspontaneous
SUSY breaking
1 ,y)FIG. 4:dDR
NP-FRG(0,
4 < dDR . The initialkconditions at k = Λ (i.e., t = 0) for u!k (ρ) and zk (ρ) are taken
FIG. 6: Fixed-pointat the fixed-point solution, u!∗ (ρ) and z∗ (ρ), and that for δkfrom 3 (steepest) tois chosen as δk=Λ (ρ, y) = 1. One observes that a linear cusp
been rescaled byappears at a finite RG time. By construction δk (0, 0) = 1
Breakdown of the (SUSY) WT identity at the fixed pointhas
ind=3
represented on the samalong the flow.
2.5
3

2
2

1.5

z! (ρ)z! (ρ), δ!(ρ, 0)

1

1

0.5
0

0

0.02

0.04

0.06

0.08

ρ

FIG. 5: Fixed-point solution in d = 3 for z (ρ) (solid line)

027

Results (contd.)
Above NDR =18: no cusp and dimensional reduction
Below NDR=18: cusp and breakdown of dimensional reduction
Anomalous dimensions in d=4+ε (1-loop exact FRG)
MATTHIEU TISSIER AND GILLES TARJUS

PHYSICAL REVIEW B 74, 2144

FIG. 2. Ratios *DR / * !upper cu¯*DR / ¯* !lower curve" vs N for the RFO
at first order in ! = d − 4 $the dim
reduction value for the exponents is= ! / !N − 2"%. The critical value of N at w
* and ¯* diverge is Nc = 2.8347. . ..

At the critical value Nc, the beta function for R!!z" !unscaled by !" in exactly d = 4 has a !cuspy" fixed-point solution R*!!z" for any arbitrary value of the renormalized disorder strength R*!!1". We have noted in Ref. 21 that thesituation bears some similarity with the pure O!N" model&%!z" = %!z"„R̃*!!z"z3 + 2R̃*!!z"z − R̃*!!z"z + NR̃*!!1" −!

+ !4z2 + N − 3"R̃*"!z" − 1… + %!1"„!N − 3"R̃*!!

28− !N + 1"zR̃"!z" − !z − 1"R̃!!z"… + %!!z"„R̃!
2

Results: Critical exponents η and ῆ(contd.)Optimization of the cut-off (to ensure a better stability of the results)
0.7

2.

Regularization of the numerical instabil

0.68
0.66
0.64
0.62

η

0.6
0.58
0.56
0.54
0.52
0.5
1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

a
1.2
1.18
1.16

Illustrate in Fig. ????? the appearance of nu
instabilities in the physically unimportant region
fields.

1.14
1.12

η̄

1.1
1.08
1.06
1.04
1.02
1
1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

a
FIG. 10: Dependence of the anomalous dimensions η (upper
panel) and η̄ (lower panel) on the parameter a of the IR cutoff29