Shear Induced Rigidity in Athermal Materials: Supplementary Information
Sumantra Sarkar and Bulbul Chakraborty
Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454, USA
MEANFIELD CALCULATION
The Hamiltonian defining our model is given by:
X
X
X
H = −J
Si Sj −
hi Si − H
Si
<i,j>
+ ∆(H)
X
i
Si2
i
.
(S1)
i
∆ is a chemical potential that controls the number of
zero spins. The quenched random field hi is Gaussian
with zero mean and standard deviation R. The two order
parameters are:
1 X 2
S = P (Si = 1) + P (Si = −1)
N i i
1 X
Si = P (Si = 1) − P (Si = −1) (S2)
M =
N i
Using these probabilities, and the definition of M and X
(Eq. S2), we obtain:
 h
1
√

erf ∆(H)+H+M

2
2R

i

√
∆>0
−erf ∆(H)−H−M
(S7)
M =
2R



erf H+M
√
∆≤0
2R
 h
∆(H)+H+M
1

√

2R
 2 erf i
∆(H)−H−M
X =
(S8)
√
∆>0
+erf
2R



0
∆≤0
1−X =
where P (Si = x) measures the probability that the ith
spin takes the value x (±1 or 0).
In the meanfield (MF) approximation, the energy of a
spin Si is given by:
E(Si ) = −JM Si − HSi − hi Si + ∆Si2
(S3)
Therefore,
E(Si = 1) ≡ E1 =
− (JM
+ H + hi ) + ∆
E(Si = −1) ≡ E−1 = (JM + H + hi ) + ∆
E(Si = 0) = 0
(S4)
Special disorders for α > 1 trajectories
The meanfield equations for α > 1, and ∆0 < ∆c admit three lines of transitions which end at a critical point
(Rc , ∆c ). These lines are defined by Rt (∆0 ), RDST (∆0 ),
and Rm (∆0 ) in descending order of magnitude (Fig.1(a)
in main text). The line Rt (∆0 ), marks the transition
from a single solution, M (H = 0) = 0, for R > Rt (∆0 )
to multiple solutions for M (H = 0), whereas the line
Rm (∆0 ) marks the transition from multiple solutions for
M (H = 0) (R > Rm (∆0 )) to a single stable solution
M (H = 0) = 0. The transition at Rt (∆0 ) is continuous, whereas the transition at Rm (∆0 ) is accompanied
by discontinuous changes of both order parameters at
H = 0. Rt (∆0 ) and Rm (∆0 ) can be calculated analytically
s from the mean field equations
sand yields: Rt (∆0 ) =
−∆20
In our zero-temperature dynamics, a spin Si will be in
the +1 state if E1 < 0, and E1 < E−1 . This condition is
satisfied if
(
∆ − JM − H if ∆ > 0
hi >
−JM − H
if ∆ ≤ 0
whence

 1 erf c ∆−H−JM
√
if ∆ > 0
2
2R P (Si = 1) ≡ P (1) =
 1 erf c −H−JM
√
if ∆ ≤ 0
2
2R
W
π∆2
0,− 2 0
,
−∆20
and Rm (∆0 ) =
W
−1,−
π∆2
0
2
.
Here
W (k, x) is the product log function, also known as Lambert’s W function. The transition at RDST (∆0 ) is a
unique feature of our model and marks the onset of
system-size avalanches in which spins flip from ±1 to
0. RDST (∆0 ) is difficult to calculate analytically, and
the line shown in Fig. 1(a) of the main text has been
obtained numerically. Apart from ∆0 very close to ∆c ,
RDST (∆0 ) ≈ 0.4. Fig. S1 illustrates the behavior of
the system near these special disorders by comparing the
M -hysteresis.
(S5)
A similar calculation yields:

 1 erf c ∆+H+JM
√
2
2R
P (Si = −1) ≡ P (−1) =
 1 erf c H+JM
√
2
2R
α ≤ 1, ∆ > ∆c TRAJECTORIES
if ∆ > 0
if ∆ ≤ 0
(S6)
The dynamics of the system depends crucially on the
functional dependence of ∆ on H. As mentioned in the
main text, we use a trajectory of the form ∆ = α|H| +
2
(a)
(b)
(c)
(d)
(e)
(f)
M
1
0
M
−1
1
0
−1
−0.2
0
H
0.2 −0.2
0
H
0.2 −0.2
0
H
0.2
Fig. S1: Hysteresis of M : meanfield calculations for ∆0 = 0.2 for different R. (a) R = 0.09 (just below Rm ), (b) R = 0.1 (just
above Rm ), (c) R = 0.3 (below RDST ), (d) R = 0.4 (RDST ), (e) R = 0.6 (RDST < R < Rt ), and (f) R = 0.9 (R > Rt ).
∆0 . For α < 1, as the field is increased, M increases
and X decreases, indication that Si = ±1 proliferate
(Fig.S2 (a)). This trajectory is, therefore, not relevant
to shear induced rigidity where grains with three or more
contacts (Si = 0) proliferate, as the system is driven
towards jamming.
Lying between α < 1 and α > 1 trajectories, α = 1
trajectories exhibit an interesting dynamics (Fig.S2(b)).
Since the chemical potential, ∆(H) changes at the same
rate as H, the applied field, the production of ±1 spins
favored by H competes equally with the production of 0
spins favored by ∆. For α > 1 trajectories, the magnetization M (H) starts to decrease with increasing H for
H > Hpeak (R), as depicted in Fig. 2 of the main text.
In contrast, for α = 1, we observe that both the magnetization M , and the fraction of zero spins X asymptote to a disorder-dependent values Msat and Xsat for
H >> Hpeak . As R increases, Msat increases while Xsat
decreases as shown in Fig. S3.
Simulations of the model (Eq. S1) using zero temperature Monte Carlo dynamics show that these asymptotic states have a non-trivial spatial distribution of
spins. As shown in Fig. S5, there is micro-phase separation between ±1 and 0 spins. This spatial structure
is reminiscent of shear bands observed in shear jamming
experiments[1].
SIMULATION RESULTS
Solving the model in the meanfield approximation provides the insight necessary to understand its qualitative features. However, it is not possible to obtain
the spatial configuration of spins from the MF equations. Also, as mentioned in the main text, hysteresis
appears in MF solution if and only if there are system
size avalanches[2]. To understand the features of the
model not captured by MF solutions, we performed zero
temperature Monte Carlo simulations on a square lattice
3
1
(a)
1
(b)
M
X
0
0
0
20 0
10
H
(c)
0.5
0.5
0.5
1
2
0
0
4
H
10
5
H
Fig. S2: Comparison of trajectories with different α, ∆0 > ∆c . The asymptotic (H >> Hpeak ) dynamics is governed by α. M
monotonically increases to 1 while X decreases to zero for α < 1 trajectories (a). Exactly opposite trend is observed for α > 1
trajectories (b). For α = 1, both M and X increases monotonically and saturate to a value less than 1 (c). The saturation
value depends on disorder.
0.7
1
H
1
1.5
2
2
(b)
regime, however numerical results for hysteresis do exhibit qualitatively similar features. We will undertake a
careful numerical study of the DST regime in the near
future.
0.1
0
0
1
2
H
Fig. S3: X (a) and M (b) as a function of field H for α = 1
(∆0 = 2) trajectories at different R (legend) obtained from
meanfield. Both order parameters increase monotonically,
and saturate to a value less than 1. The saturation value
depends on R. For M , the saturation value increases with R
while for X it decreases.
with periodic boundary conditions. We should emphasize, that the model is not restricted to two dimensions,
and one can generalize these numerical calculations easily to three dimensions. The initial spin configuration at
any field H is obtained by slowly annealing the system
from a large value of the external field (|H| >> Hpeak ),
where the spin configuration is known. In order to identify the SJ and DST regimes in simulations, we compare
the meanfield and numerical phase diagrams in Fig. S4
for α > 1. The color map, based on the peak value
of M (H) shows that the same qualitative features are
present in both phase diagrams but there are quantitative differences. In the current work, we have not focused
on using numerical simulations to probe the fate of the
meanfield transitions and the critical point in the DST
2
0
1
−5
0
0
1
R
2
−10
∆0
X
0.8
M
(a)
0.9
0
0.2
∆0
1
4
0
2
−5
0
0
2
4
−10
R
Fig. S4: Phase diagram based on the value of Mpeak : MF
(left) compared to the phase diagram obtained from simulations (right). The colorbar represents the decade in which
Mpeak lies. The simulations were performed on 642 spins, and
averaged over 20 different configurations for each (R, ∆0 ).
Movies from simulation
The dynamics of the system depends on four parameters: H, α, ∆0 , and R. In this section, we show three
videos (Figs. S5, S6, S7 ) that elucidate how the dynamics depends on these parameters. We have chosen trajectories that best illustrate the differences, and, therefore,
the parameter sets that appear in the videos are different in different videos. In all three videos black, red, and
white pixels denote +1, 0, and -1 spins respectively. The
top panel shows the evolution of M (red circle) as H is
4
increased while the bottom panel shows the evolution of
the spin configurations. The videos are constructed from
results of a zero temperature MC simulation of 2562 spins
on a square lattice with periodic boundary conditions,
and initial spin configurations were obtained following
the procedure described in the previous section.
[1] J. Zhang, T. S. Majmudar, A. Tordesillas, and R. P.
Behringer, Granular Matter 12, 159 (2010).
[2] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl,
B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70,
3347 (1993).
5
α<1
α=1
M
1
α>1
0.5
0.1
0.05
0
2
0
0.5
0
0
1
H
2
0
0
1
H
1
H
2
Fig. S5: : alphaComparison.avi Compares dynamics for different α at constant ∆0 > ∆c , and R = 2. The asymptotic
(H >> Hpeak ) dynamics depends crucially on the value of α. The asymptotic spin configuration consists of only zero spins for
α > 1, only +1 spins for α < 1, and a frozen mixture of +1 and 0 spins for α = 1. Color code: black = +1, red = 0, and white
= -1.
∆ <∆
M
0
∆0 > ∆c
c
1
0.02
0.5
0.01
0
0
0.5
H
1
0
0
0.5
H
1
Fig. S6: deltaComparison.avi Compares dynamics for trajectories with ∆0 > ∆c and ∆ < ∆c for α > 1 and R = 0.6 < Rt (∆0 ).
For ∆0 < ∆c , we see large 1 → 0 avalanches, whereas for ∆0 > ∆c , the avalanches are rare. Color code: black = +1, red = 0,
and white = -1.
6
M
R = 0.35
R = 0.6
1
1
0.5
0.5
0
0
0.5
H
1
0
0
R=2
0.5
0.5
H
1
0
0
0.5
H
1
Fig. S7: disorderComparison.avi Compares dynamics for trajectories with different disorders, for ∆0 < ∆c , α > 1. R = 0.35
is less than RDST , and the system relaxes through system-size avalanches. R = 0.6 is slightly larger than RDST , and we
observe avalanches of all sizes. Preliminary data on avalanche statistics shows that the avalanche size distribution approaches
a power law as the disorder is decreased towards RDST (not shown). R = 2 >> RDST and consequently we observe only small
avalanches. Color code: black = +1, red = 0, and white = -1.