Impurity Effects in Highly Frustrated Diamond Lattice Antiferromagnets

Abstract

We consider the effects of local impurities in highly frustrated diamond
lattice antiferromagnets, which exhibit large but non-extensive ground state
degeneracies. Such models are appropriate to many A-site magnetic spinels. We
argue very generally that sufficiently dilute impurities induce an ordered
magnetic ground state, and provide a mechanism of degeneracy breaking. The
states which are selected can be determined by a "swiss cheese model" analysis,
which we demonstrate numerically for a particular impurity model in this case.
Moreover, we present criteria for estimating the stability of the resulting
ordered phase to a competing frozen (spin glass) one. The results may explain
the contrasting finding of frozen and ordered ground states in CoAl2O4 and
MnSc2S4, respectively.

Full text

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Impurity Effects in Highly Frustrated Diamond Lattice Antiferromagnets
Lucile Savary
Ecole Normale Sup´ erieure de Lyon, 46, all´ ee d’Italie, 69364 Lyon Cedex 07
Emanuel Gull
Department of Physics, Columbia University, New York, NY 10027
Simon Trebst
Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106
Jason Alicea
Department of Physics and Astronomy, University of California, Irvine, CA 92697
Doron Bergman
Department of Physics, California Institute of Technology, Pasadena, CA 91125
Leon Balents
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106
(Dated: March 28, 2011)
Weconsidertheeffectsoflocalimpuritiesinhighlyfrustrateddiamondlatticeantiferromagnets, whichexhibit
largebutnon-extensivegroundstatedegeneracies. SuchmodelsareappropriatetomanyA-sitemagneticspinels.
We argue very generally that sufficiently dilute impurities induce an ordered magnetic ground state, and provide
a mechanism of degeneracy breaking. The states which are selected can be determined by a “swiss cheese
model” analysis, which we demonstrate numerically for a particular impurity model in this case. Moreover, we
present criteria for estimating the stability of the resulting ordered phase to a competing frozen (spin glass) one.
The results may explain the contrasting finding of frozen and ordered ground states in CoAl2O4and MnSc2S4,
respectively.
PACS numbers: 75.10.Jm, 75.10.Pq
I.INTRODUCTION
A common feature of highly frustrated magnets is the exis-
tence of a large (classical) ground state degeneracy in model
Hamiltonians.1Although this degeneracy is accidental, in the
sense that the multitude of ground states are generally not
symmetry-related, it nevertheless yields striking physical con-
sequences. For instance, over a broad temperature range the
system resides in a “cooperative paramagnetic” or “classical
spin liquid” regime, where the spins avoid long-range order
but fluctuate predominantly within the ground state manifold.
The ultimate fate of such highly frustrated spins at the lowest
temperatures poses an interesting and experimentally impor-
tant problem. Typically, at very low temperatures entropic or
quantum fluctuations alone are sufficient to lift the degeneracy
and produce an ordering transition via “order by disorder”.2,3
However, additional weak effects which would otherwise be
negligible in unfrustrated systems—such as small further-
neighbor exchange4,5, spin-lattice coupling4,6, and dipolar
interactions7—can also provide a degeneracy-lifting mecha-
nism, which indeed often dominates over fluctuation effects.
In this paper we discuss degeneracy breaking by quenched
random impurities. Generally even a non-magnetic defect (i.e.
one which does not break spin-rotational symmetry) such as
a random bond, an interstitial spin, or a vacancy, will locally
distinguish the various degenerate states of the pristine sys-
tem. This brings up a number of issues. First, can impurities
consequently lead to ordering, i.e. “order by quenched disor-
der”? Or, by virtue of their randomness, do they lead instead
to a glassy disordered state? Do these impurities influence the
spins in their vicinity independently from one another? Or are
their effects rendered highly coordinated by the correlated na-
ture of fluctuations in the cooperative paramagnetic regime?
The answers to these questions probably depend in detail
upon the nature of the magnetic system under consideration,
particularly the degree of frustration. Generally, with increas-
ingfrustrationcomesincreasinggroundstatedegeneracy. One
often useful characterization scheme for frustration involves
counting the distinct magnetic ordering wavevectors which
are possible within the classical ground state manifold. In
mildly frustrated magnets, such as the nearest-neighbor tri-
angular antiferromagnet, this wavevector is unique. In the
nearest-neighbor fcc antiferromagnet, the ordering wavevec-
tors form continuous one-dimensional lines.8The much more
frustrated nearest-neighbor kagome and pyrochlore antiferro-
magnets, by contrast, have ordering wavevectors that fill all of
reciprocal space.9,10
In the latter kagome and pyrochlore cases, the degeneracy
is local—i.e. the ground state entropy is extensive, and states
within the ground state manifold are related by modifications
of only a small number of spins. An impurity can then fix
the spin configuration in its neighborhood, while constrain-
ing the spins outside of its vicinity very little.11Since each
random impurity fixes a spin configuration in its neighbor-
arXiv:1103.4985v1 [cond-mat.str-el] 25 Mar 2011

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hood, roughly independently of the others, one may expect
as a result a globally random ground state, i.e. a spin glass.
In fact, the T > 0 dynamics of such defective pyrochlore
and kagome systems is rather subtle, and the actual spin glass
freezing temperature can sometimes be highly suppressed as a
result.11Nevertheless, spin glass behavior is very commonly
observed in highly frustrated magnets,12even when the disor-
der is nominally very weak.
For the other classes of frustrated systems noted above, in
which the ground state ordering wavevectors occupy a smaller
subset of reciprocal space, the degeneracy is sub-extensive.
An infinite number of spins must then be varied in order to
transform one ground state to another. Thus, different impuri-
ties cannot independently determine their local environments.
In this paper, we develop a formalism for dealing with their
effects, focusing for concreteness on the most degenerate case
(of which we are aware) of a sub-extensive degeneracy: frus-
trated diamond lattice antiferromagnets. In a J1− J2model
on the diamond lattice, the ordering wavevectors (for antifer-
romagnetic J2 > |J1|/8) form a 2d surface within the 3d
momentum space.5This example is of particular recent in-
terest due to its relevance to the A-site magnetic spinel ma-
terials, with chemical formula AB2X4, in which magnetic
A sites form a diamond sublattice with non-magnetic B and
X atoms.13–15Because it represents an extreme case of sub-
extensive degeneracy, we expect that the conclusions obtained
for this case apply fairly generally to other less degenerate
frustrated magnets.
Our conclusion is that, for this class of systems, despite the
large ground state degeneracy, long-range magnetic order is
stabilized—and indeed a specific ground state is selected—at
sufficiently low impurity concentrations. Each impurity in-
duces a small, finite region around it in which the spins are
deformed from an ideal spiral pattern, like holes in “swiss
cheese” (Emmentaler).The swiss cheese model allows a
calculation of the global ground state wavevector, based on
certain properties of an individual defect. We calculate this
wavevector for the A-site spinel case, with a specific impurity
model. We show how the same theoretical framework deter-
mines other physical properties such as the ordered moment
observed in neutron scattering, and the transition temperature.
The swiss cheese model also signals its own demise, in one
of two ways. First, if the holes in the cheese strongly over-
lap, the assumption of their independence fails. Second, even
when the holes do not overlap, if the underlying “stiffness”
of the bulk spiral is too small, then the impurities may induce
strong fluctuations. In either case, the long-range order is ex-
pected to give way to a disordered spin glass ground state.
These two possibilities provide criteria, whereby the stability
of the ordered spiral state can be quantitatively estimated. In
the case of the A-site spinels, we suggest that this method con-
sistently explains the contrasting glassy and ordered ground
states found in CoAl2O414,16and MnSc2S413,15,17, respec-
tively.
The remainder of the paper is organized as follows. We
consider a single impurity in Sec. II. Using a non-linear sigma
model, it is shown quite generally that, on long length scales,
a single defect can generate only small deformations away
from a uniform spiral ground state of the clean system. We
then demonstrate via Monte Carlo simulations that the classi-
cal degeneracy is indeed lifted by the impurity, which favors
specific wavevectors along the spiral surface, thereby provid-
ing a mechanism of “order by quenched disorder”. A single
impurity is further characterized by a length scale ξ (the size
of the hole) outside of which the spins are well-described by
a uniform spiral. In Sec. III, we extend this analysis to the
case of multiple impurities. There, we discuss the interplay
between impurity and entropic effects, and make quantitative,
verifiable predictions for how Tcvaries with impurity con-
centration. We conclude in Sec. IV with a discussion of our
results in the context of experiments and impurity effects in
other models.
II.SINGLE IMPURITY
In this section, we discuss the physics of a single impu-
rity. First, we will consider the possibility that the impurity
induces a slow variation of the spins extending over infinite
distances. By analyzing the energy as a function of order pa-
rameter variations, we show that this is not the case. Instead,
the deformation of the spins by each impurity is local, and
decays to a uniform spiral as the distance from the defect in-
creases. We then show that the impurity physics can be char-
acterized by an impurity energy function, Ea(q), which gives
the difference between the ground state energies of the sys-
tem with and without a single impurity of type a, under the
constraint that far from the impurity the spins adopt a spiral
configuration with wavevector q. Employing extensive Monte
Carlo simulations we calculate this function numerically for a
specific impurity model. In order to check the validity of the
swiss-cheese model, we characterize the local region of defor-
mation around an impurity: we compute locally the q-vector
from Monte Carlo realizations of spin configurations. We find
that in our simulations, variations of q are extremely local, and
most changes happen within one unit cell.
A.General considerations
Consider an arbitrary local defect, for which the Hamilto-
nian of the system can only be modified in a finite vicinity of
the impurity (involving only a finite number of spins). Also,
for simplicity, we will assume the defect is “non-magnetic”,
meaning it preserves the spin-rotational invariance of the
Hamiltonian.
The energy of the system in the presence of the impurity
then consists of a contribution in the region where the defect
has modified the Hamiltonian, and a contribution from the re-
mainder of the system. For any spin configuration, the former
is finite and the latter contains a leading term proportional to
V and subdominant corrections. By choosing the spin config-
uration equal to that of one of the ground states in the absence
of the defect, we can make the energy density E/V = ?0
in the large V → ∞ limit equal to that of the pure system,
and therefore the ground states in the presence of the impurity

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FIG. 1: Cubic cell of an AB2X4 spinel. The sublattice of A sites
(blue spheres) is a diamond lattice, while the sublattice of B sites
(red spheres) is a pyrochlore lattice.
must also achieve this same energy density ?0. This implies
that spins far from the impurity must locally resemble one of
the ground states of the pure system.
1.Spiral order parameter
To make our discussion more concrete, we now special-
ize to the case of the frustrated diamond lattice antiferromag-
net with first and second nearest neighbor interactions. The
ground states of this system were determined in Ref. 5. For
J2/|J1| > 1/8, which is the parameter regime we focus on
hereafter, they consist of coplanar spirals whose propagation
wavevector q lies anywhere on a continuous “spiral surface”
in reciprocal space. The configuration of the spiral is de-
scribed by
??deiq·r+iγ(q;r)?
where the phase γ(q;r) = ±γ(q) when r is on the I or II
diamond sublattice, respectively. (The parametrization is such
that the I diamond sublattice contains the site at (0,0,0) and
the II sublattice that at1
plane of the spiral in spin space and its phase. It takes the form
?S(r) = Re
,
(1)
8(1,1,1)). The vector?d specifies the
?d = ˆ e1+ iˆ e2,
(2)
where ˆ e1, ˆ e2are orthogonal unit vectors and |?d| is fixed at
√2. The spiral surface itself (i.e. the locus of allowed q)
deforms smoothly with J2/|J1| (except at the isolated value
of J2/|J1| = 1/4 where it changes topology).
To specify a ground state, one must therefore specify both
?d and the wavevector q, constrained to the spiral surface. One
can then regard (?d,q) as the order parameter. Far from the
impurity, the spin configuration must locally take the ground
state form of Eq. (1), but we must consider the possibility
that these parameters may vary slowly (relative to the largest
micro-scale of the spiral, the wavelength 2π/|q|) in space. We
will now argue that such variations are insignificant: far from
the impurity, the spiral wavevector and the?d vector are uni-
form in the ground state (and indeed all finite energy states).
To do so, we consider the energy of a slowly-varying order
parameter that is macroscopically non-uniform and show that
it is divergent. Encoding the slow variations na¨ ıvely requires
5 continuous real functions: three angles to specify?d, and two
more to specify the position of q on the surface. However,
the actual number of degrees of freedom is smaller due to an
additional gauge symmetry: To see this we note that a change
in the wavevector, q → q + δq can be compensated by the
shift?d →?de−iδq·r−iδγwithnochangetothespins(hereδγ =
γ(q + δq) − γ(q)). Therefore there is a “gauge” redundancy
in these variables. We can “fix” the gauge in a variety of ways.
A simple choice is to allow only for spatial variations in d and
not in q, i.e. we write:
??d(r)eiq0·r+iγ(q;r)?
where?d(r) is assumed to be slowly varying in space, and q0
is a constant “reference” wavevector. We emphasize that this
still allows the physical wavevector to be different from q0.
For instance, if?d(r) =?d0eiδq·rwith constant?d0, the physical
wavevector is q = q0+ δq. In general, we can define the
physical wavevector as
?S(r) = Re,
(3)
qµ= qµ
0+1
2Im
??d∗· ∂µ?d
?
.
(4)
Note that for Eq. (3) to correspond locally to a proper mini-
mum energy spiralground state, the first argumentq ofγ must
be the physical wavevector given by Eq. (4), not q0.
2. Energy of weakly deformed spirals
It is sufficient to consider just small spatial variations of?d,
since we will find that these are already prohibitively costly at
long distances. Let
?d(r) =?d0+ δ?d(r).
(5)
To preserve the unit vector constraint of the spins?S2= 1
in Eq. (3), a small δ?d(r) must be of the form
δ?d(r) = iφ(r)?d0+ ψ(r)ˆ e3,
(6)
where φ and ψ are arbitrary small real and complex fields,
respectively, and
ˆ e3= ˆ e1× ˆ e2= −1
2Im
??d ×?d∗?
.
(7)
φ describes the rotation of the vector?d within the spiral
plane (spanned by ˆ e1, ˆ e2), and includes simple variations in
the physical wavevector, while ψ describes variations outside

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the spiral plane. To this linearized order, we have simply
q = q0+ ∇φ.
Now consider the energy density as a function of φ,ψ
and their gradients. First, the energy must be unchanged for
constant values of these functions, since these correspond to
global O(3) spin rotations. The first non-trivial terms in a
Taylor expansion can arise at quadratic order in these fields,
and from the above reasoning, must include only spatial gra-
dients so that they vanish for constant configurations. Finally,
this quadratic form must be positive semi-definite, because the
un-deformed configuration obtains the minimal energy.
An additional constraint is given by frustration: the energy
must also be unchanged for deformations corresponding to
changes of the wavevector within the spiral surface. Such a
deformation is of the form φ(r) = δq·r, where δq is an arbi-
trary (small) vector in the plane tangent to the spiral surface at
q0. This constraint is highly restrictive. Consider the structure
of allowed quadratic terms in φ with two gradients:
Eφ=1
2cµν∂µφ∂νφ,
(8)
where a sum over µ,ν is implied, and cµνis an arbitrary real
symmetric matrix. This energy density should vanish for a de-
formation corresponding to a constant spiral with a wavevec-
tor shifted slightly within the spiral surface, which implies
cµνδqµδqν= 0,
(9)
for δq in the tangent plane. Eq. (9) reduces cµνto a single
undetermined coefficient c, such that cµν= cˆ nµˆ nν, where ˆ n
is the unit normal vector to the spiral surface. The energy cost
to deform φ in the directions parallel to the spiral surface is
thus higher order in derivatives. Along the same lines one may
deduce the most general allowed energy density quadratic in
the φ,ψ fields with the minimal number of gradients to ensure
stability:
E =
c
2(∇⊥φ)2+ c?∇⊥φ∇2
+d∇⊥ψ∗∇⊥ψ + d?∇?ψ∗· ∇?ψ,
where ∇⊥ ≡ ˆ n · ∇, ∇? = ∇ − ˆ n∇⊥, and c,c?,c??,d,d?
are undetermined coefficients. For the energy to be bounded
by the ground state value, one needs c,c??,d,d?> 0, and
(c?)2≤ cc??. To simplify Eq. (10), we have actually assumed
at least a three-fold rotational symmetry about the axis of the
ordering wavevector q0. In the most general case, the terms
involving ∇?should be replaced by less isotropic forms, e.g.
∇2
However, such changes do not alter the results of the analysis
at the scaling level we consider in this paper.
Now we estimate the energy cost of a deformation. Con-
sider first ψ, whose energy is determined by the last two terms
in Eq. (10). The scaling is fully isotropic (k⊥∼ k?) as usual
for an ordinary Goldstone mode (phonon or magnon) in three
dimensions. This leads to the conventional estimate of the
energy cost for a “twist” in the order parameter: if ψ varies
by some finite amount δψ over a region of size L, the energy
?φ +c??
2(∇2
?φ)2
(10)
?→ gµν∂µ∂ν, with µ,ν spanning the tangent directions.
density is increased by an amount of order |δψ|2/L2, which
integrates to a total energy of order |δψ|2×L over the volume
of size L3. Since this grows unboundedly with L, such order
one distortions of ψ cost infinite energy in the thermodynamic
limit, and cannot be compensated by any local energy gain.
The energy for twists of φ (which includes wavevector vari-
ations) is less conventional. Here the scaling is anisotropic: if
φ is distorted by an amount δφ over a distance L?in a direc-
tion parallel to the spiral surface, it will typically relax over a
larger distance of order L⊥∼ L2
lar to the surface. This is seen simply by comparing the pow-
ers of derivatives in the first three terms of Eq. (10). The en-
ergy density for such a deformation is then (δφ)2/L4
should be integrated over the volume L⊥L2
total energy which does not scale with length. Thus deforma-
tions of the phase might occur with O(1) disorder contribu-
tions, but there could be subtleties involving thermal fluctua-
tions and anharmonic elasticity.18
In fact, the preference for uniform wavevectors at large dis-
tances is stronger than the above estimate might lead one to
believe. The reason is that since δq = q − q0 = ∇φ, a
wavevector shift δq (in the spiral surface) over a region of
size L?leads already to a large (not O(1)) deformation of
φ: δφ ∼ L?× (δq). Following the prior arguments, one sees
that a variation of the wavevector of δq over a region of size
L?costs an energy ∼ (δq)2L2
fects could allow for large scale variations of φ (see Eq.(33)
and the corresponding discussion), large scale twists of q are
certainly energetically forbidden in the ground state.
?in the direction perpendicu-
?, which
?to obtain a
?∼ L4
?. Thus while more subtle ef-
B.Characterization of single-impurity effects
The preceding discussion implies quite generally that a sin-
gle impurity can induce order-one deviations from a uniform
spiral only locally. Nevertheless, such corrections are im-
portant to quantify as they break the large spiral degeneracy
present in the pure system (at zero temperature), leading to
rich physics. In the following we explain this degeneracy
breaking and characterize the resulting ground states.
1. Single-impurity quantities
To characterize a single impurity, we examine its effect on
the spiral ground states of the pure system. The simplest and
most important quantity is the minimum energy of the system
in the presence of the impurity E(q), relative to the minimum
energy without the impurity, given that infinitely far from the
impurity the spins are in a spiral configuration with wavevec-
tor q. Formally, for an impurity a, Ea(q) is
Ea(q) = energy(q;with impurity)
−energy(q;without impurity).
We only need to consider wavevectors q on the spiral surface,
in which case the locality arguments above imply that E(q) is
(11)

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finite in the infinite volume limit. This energy quantifies the
splitting of the degenerate spiral states by such impurities.
One may also examine the spatial range of the impurity-
induced deformation. To this end we can locally calculate, at
each site ri, a local spiral wavevector qifrom the surrounding
spin configuration and then consider the deviation cos(δq) =
qi· q/(|qi||q|) from the wavevector q taken at infinity. The
localmeasurementofthespiralwavevectorqiisperformedby
considering a set of neighboring spins on the same sublattice
and fitting to
?Si×?Sj= sin(q · rji)i
2
?d ×?d∗= sin(q · rij)ˆ e3,
(12)
where rji= rj−riand ˆ e3defines the spin axis perpendicular
to the spiral plane as given in Eq. (7).
With the calculated qiwe can then define the wavevector
deformation length ξqas the radius outside which the angle
between qiand q at infinity is less than some angle θ0. A
cautionary remark is in order. The finiteness of these lengths
does not mean that the deformation around an impurity de-
cays exponentially away from it. Rather it means only that
the deformation decays toward a uniform spiral, reaching a
“good” approximation of it within length ξq. However, the
approach to the uniform spiral is expected to be in the form of
a power-law rather than exponential, since there is no gap in
the spectrum of normal modes of the spiral state.
Despite this non-exponential decay, the lengths are signif-
icant because the larger they are, the less local the impurity
effects become, and the more sensitive the system is to dis-
order. Specifically, we can no longer regard the impurities
as dilute when their concentration is larger than ξ−3
the above general scaling arguments, we would expect ξqto
be typically of the order of a few lattice spacings, though it
might grow larger near special points in the phase diagram.
To check for this possibility, we consider explicitly the size of
the impurity deformation region in a specific impurity model
below, and find that it remains small throughout the parameter
range of interest.
q . From
2.Specific impurity model
In the following we investigate in detail one particular type
of impurity relevant for the spinels, which brings out the gen-
eral features of the problem. Specifically, we consider the
effect of a magnetic ion on a B site of the spinel structure
AB2X4. Each B-site atom has six nearest-neighbor A sites,
and the distance in this case is smaller than the A-A nearest-
neighbor distance. Thus, the dominant effect of this impurity
is to generate an exchange coupling Jimpbetween the mag-
netic B-site and its six nearest-neighbor A-sites (see Figure 2),
which isexpected to be muchstronger than theA-A exchange,
i.e. Jimp? J1,J2. We therefore model a single B-site impu-
rity by adding to the Hamiltonian the term
δH = Jimp
?
?a,i?
Sa· Si,
(13)
FIG. 2: Impurity model: A non-magnetic impurity resides on a B-
site indicated by the black sphere. The six nearest-neighbor A-sites
form a distorted hexagon around the impurity.
where the sum is over the six A-site nearest neighbors i to
the B-site impurity labeled by a. Since we expect Jimp ?
J1,J2, the natural, simplest approximation is to take Jimp→
∞, in which case the impurity spin Sacan be eliminated and
Eq. (13) reduces to a boundary condition that the six spins in
the vicinity of the impurity are aligned.
It is noteworthy that the B-site does not have the full point
group symmetry of the lattice. Instead there are four distinct
B sites, which transform into one another under the full set of
cubic operations (see Fig. 2). Therefore we must distinguish
the four impurity positions within the unit cell, which we label
a = 1,2,3,4 in the energy function Ea(q), as these will favor
different ordered states.
3.Numerical results
We have simulated the B-site impurity model numerically
by employing extensive classical Monte Carlo simulations.
We set up our simulations such that the impurity is embedded
into systems of N = 8 × L3spins with system sizes ranging
up to N = 8 × 93= 5832 spins. In order to define the spi-
ral state at large distances from the impurity site we employ
fixed boundary conditions by embedding the simulation cube
of length L into a cube of extent L+1, where the spins in the
boundary layer are aligned to form a uniform spiral of a given
wavevector q. In the vicinity of the impurity we consider the
Jimp→ ∞ limit and force the six nearest-neighbor spins of
the impurity to be aligned and point in the same direction at
all times in the simulations. We explore the zero temperature
physics of this impurity model by setting the simulation tem-
perature much lower than all energy scales in the problem,
thereby mimicking a steepest descent energy minimization.
We checked the convergence of this procedure by simulating
systems with different initial spin configurations and obtained
indistinguishable results when starting from random spin con-
figurations or unperturbed spiral states, pointing to the exis-

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FIG. 3: ‘Spiral surfaces’ comprising the degenerate spiral ground-state vectors for varying coupling strengths J2/J1. The surfaces are color-
coded according to the energies E1(q) and E(q), respectively. The colors indicate high values as blue, low values as red, and green being the
absolute minima. The top row shows results for a single impurity E1(q), while the bottom row shows results averaged over the four possible
impurity sites E(q).
tence of a unique (and well accessible) energy minimum.
Since the four distinct impurity sites within the diamond
lattice unit cell are related by simple rotations, we have calcu-
lated the energy Ea(q) only for the impurity at one of these
4 sites. For a given value of interactions J2/|J1| we have run
simulations for a set of 1,000 distinct spiral wavevectors q
on the ‘spiral surface’ appropriate for the value of couplings.
A summary of our numerical results for a medium sized sys-
tem of N = 512 = 8 × 43spins is plotted in the top row of
Fig. 3. The impurity energies E1(q) are found to vary on the
spiral surfaces and clearly reflect the reduced symmetry of the
single B site impurity problem. For instance, in the coupling
range 1/8 ≤ J2/|J1| ≤ 1/4, where the spiral surface is a dis-
torted sphere, the minimum energy wavevectors for E1(q) are
q1points which are along the 1¯11 direction, while the energy
for wavevectors in the¯111 direction (and others in the ?111?
octet) is not an energy minimum.
For J2/|J1| > 1/4, the spiral surface develops ‘holes’ cen-
tered around the 111 directions and we find that E1(q) devel-
ops three energy minima located symmetrically on the spiral
surface along the 111∗direction for all couplings J2/|J1| >
1/4 as indicated in the top row of Fig. 3.
Our numerical simulations also allow us to probe the spiral
deformations in the vicinity of the impurity. In particular, we
measurethelocalspiralwavevectorqiasdescribedindetailin
section IIB1 using Eq. (12). Since we are mostly interested in
estimating the deviation of this local wavevector qifrom the
wavevector q for a given spin spiral configuration fixed at the
boundary, wecalculatethedeviationδq ofthelocalspiralstate
defined as the angle between the spiral wavevectors qiand q,
e.g. cos(δq) = qi· q/(|q||qi|). Our results for the so-defined
spiral deformation for various couplings and boundary spiral
states are summarized in Fig. 4.
We find that the local rearrangement of spins in the vicin-
ity of the impurity gives rise to a significant deviation of the
(angle of the) local spiral wavevector of O(1), while spins
being separated from the impurity by about one unit cell spac-
ing rearrange themselves in a spiral state which differs only
marginally from the one fixed at the boundary. This short-
range behavior of the spiral deviations is found to be quite
insensitive to the size of the system and the distance from the
fixed boundary configuration; for a more detailed discussion
of finite-size effects see Appendix C.
We further analyze how the pattern of local wavevector
deviations changes as we vary the couplings in the range
1/8 < J2/|J1| < 1/4. This is shown in the two panels of
Fig. 4 for fixed boundary spirals pointing in the 1¯11 and 100
directions, respectively. We see that the region of significant
deformation of the spiral is in all cases restricted to the very
close vicinity of the impurity, and varies only slightly with
varying J2/J1.
III.DILUTE IMPURITIES
A. Ground state with many impurities
1.swiss cheese model
We turn now to the case of many impurities. We have seen
that a single impurity already breaks the ground state degen-
eracy of the pure system, as well as the cubic symmetry of
the crystal, thus favoring a unique state. However, the (four)
different impurity positions within the unit cell break the cu-
bic symmetry of the crystal in a different way, and hence each
favors a different ordered state. For example, an impurity at
one B-site will favors a spiral with wavevector along the (111)
direction, while another favors a wavevector along the (111)

Page 7

7
01234
56
distance from impurity
0
0.4
0.8
1.2
0
0.4
0.8
1.2
deviation δq / π
0
0.4
0.8
1.2
0
0.4
0.8
1.2
J2 = 0.13
q = 0.65π
J2 = 0.18
q = 1.96π
J2 = 0.20
q = 2.25π
J2 = 0.24
q = 2.80π
a) q in 111 direction
01234
56
distance from impurity
0
0.4
0.8
1.2
0
0.4
0.8
1.2
q deviation / π
0
0.4
0.8
1.2
0
0.4
0.8
1.2
J2 = 0.13
q = 0.35π
J2 = 0.18
q = 1.02π
J2 = 0.20
q = 1.14π
J2 = 0.24
q = 1.30π
b) q in 100 direction
FIG. 4: Deviation of the local spiral wavevector qi from the one of the spiral state at the boundary q. The left panel is for boundary spiral
states with q pointing along the 1¯11 direction in the coupling range J2/|J1| < 1/4. The right panel is for boundary spiral states with q
pointing along the 100 direction in the coupling range J2/|J1| < 1/4. The impurity is embedded into a system of N = 2,744 = 8×73spins.
The symbols correspond to two different sets of neighboring spins, P (circles) and Q (boxes), used to calculate the local spiral wavevectors,
details are given in the text.
direction. In the physical system, equal densities of each type
of impurity should be simultaneously considered.
Given that the four impurity types favor incompatible or-
ders, what is the nature of the ground state that emerges here?
We will address this question in the dilute limit, by which we
mean that the impurity density nimpis assumed to be much
smaller than ξ−3. One naive candidate ground state in this
limit consists of domains such that around each defect the
spins are close to a spiral with wavevector favored by that im-
purity type. However, this possibility can be dismissed since
such a configuration would necessitate large scale deviations
in wavevector between domains, which we have seen in Sec.
II cost a prohibitively large energy. A more plausible outcome
is that the ground state consists of a uniform spiral deformed
locally around the defects, whose wavevector reflects a com-
promise between the different impurity types. Putting it more
colloquially, the system looks like a “swiss cheese” (Emmen-
taler) with the bulk consisting of an ordered spiral and a set
of holes in which the spins are strongly deformed about each
impurity. In this case, since the energy is the sum of the en-
ergy shifts due to an equal number of each type of impurities,
the ground state wavevector for the many-impurity case mini-
mizes
E(q) =1
4
4
?
a=1
Ea(q).
(14)
Eq. (14) constitutes a large simplification, justified by the im-
purity diluteness—the many-impurity ground state is deter-
mined from an average over single-impurity quantities. In this
sense, the impurities in this limit act independently.
The above discussion asserts that the ground state away
from the impurities is essentially undeformed on scales com-
parable to the impurity separation and somewhat larger. This
is indeed a consequence of the assumption of dilute impuri-
ties and the locality arguments of Sec. IIA2. However, this
does not rule out the possibility that small deformations of the
spiral on the scale of the impurity separation could add up on
much longer distances to a larger deviation from long-range
spiral order. We consider this carefully below. We find that
the wavevector of the spiral indeed remains macroscopically
uniform for dilute impurities, even on the longest scales, with
small fluctuations. This is sufficient to guarantee the correct-
ness of the energy estimate in Eq. (14), and hence correctly
predictthewavevectorfavoredbydiluteimpurities. Thephase
of the spiral, however, fluctuates considerably more, and our
arguments suggest that there may be considerable reduction
of the long-range ordered moment of the spiral by this mech-
anism.
To see this, we will construct a “coarse grained” energy
function for the system containing many impurities, and con-
sider the stability against perturbations to a macroscopically
uniform spiral. We use the parametrization of an arbitrary
slowly-varying deviation from a spiral state with wavevector
q0from Sec. IIA, in terms of the fields φ and ψ. The energy
cost in the clean system for such a deviation is described by
Eq. (10). We must add to this the impurity energy density,
Eimp(q(r),r) =
?
a
Ea(q(r))na(r),
(15)
where the impurity density is
na(r) =
?
Ra
δ(r − Ra),
(16)
and Raare the impurity positions. The impurity density is
a random function. For long-wavelength properties, the cen-
tral limit theorem implies that it is well-characterized by its
first few moments. Taking the impurities to be uniformly and

Page 8

8
independently distributed over the system volume with a to-
tal average density x (or x/4 per impurity type), we find the
mean and two-point correlation
na(r) = x/4,
(17)
na(r)nb(r?) − na(r) nb(r?) =
x
4δ(r − r?)δab,
(18)
in the infinite volume limit. From this, we can evaluate the
mean and second cumulant of the impurity energy density.
The mean is
Eimp(q,r) = xE(q).
(19)
This is precisely the energy in Eq. (14), and is, as expected,
linearly proportional to the impurity concentration x.
As a consequence, the impurity-averaged energy is mini-
mized by the spiral wavevectors that minimize E(q). To as-
certain the stability of these minima in the impurity distribu-
tion we now turn to analyze fluctuations about the minima of
E(q), parametrized as q = q0+ ∇φ (this is the same slowly
varying φ field from Section IIA2). Consider fluctuations in
the impurity energy
δEimp(q(r),r) = Eimp(q(r),r) − Eimp(q(r),r),
and expand to linear order in φ:
(20)
δEimp(q(r),r) ≈ [Eimp(q0,r) − E(q0)] − fimp(r) · ∇φ.
The first term in the brackets is φ-independent and can
be neglected. The second term represents a “random force”,
given by
?
Since E(q) has a minimum at q0, it has vanishing first or-
der derivatives at this point. This also implies that fimp(r) ∼
∇qE(q0) = 0. The second cumulant of the force is however
non-zero:
(21)
fimp(r) = −
a
na(r)∇qEa(q0).
(22)
fµ
imp(r)fν
imp(r?) = x∆µν(q0)δ(r − r?),
(23)
with
∆µν(q0) =1
4
?
a
∂Ea(q0)
∂qµ
∂Ea(q0)
∂qν
.
(24)
∆µνis generally non-zero and positive unless q0is a saddle
point for all impurity types. This is not the case for our prob-
lem, but even if it were, it would only further strengthen the
tendency of the system to order.
We are now in a position to consider the full energy func-
tion. Since ψ does not couple to the impurities, we can ne-
glect it. The energy density involving φ then combines the
first terms in Eq. (10), the mean impurity contribution near a
minimum of E(q) to quadratic order in φ
Eimp(q(r),r) ≈ xE(q0) +x
2
∂2E(q0)
∂qµ∂qν
∂µφ∂νφ,
(25)
and the random force from Eq. (21). Up to an unimportant
additive constant, we find
E =
c
2(∇⊥φ)2+ c?∇⊥φ∇2
+x
2∂qµ∂qν
?φ +c??
2(∇2
?φ)2
∂2E(q0)
∂µφ∂νφ − fimp(r) · ∇φ.
(26)
To proceed, we note that for dilute impurities (small x), the
fourth term in Eq. (26) is much smaller than the first two
except when considering the energy cost for gradients ∇?φ
parallel to the spiral surface, and therefore keep only these
components. For simplicity, we will approximate these com-
ponents as isotropic, and replace
∂2E(q0)
∂qµ∂qν
∂µφ∂νφ → cimp(∇?φ)2.
(27)
It is now straightforward to minimize the energy in Eq. (26)
in Fourier space:
φ(k) =
−ik ·˜fimp(k)
?+ c??k4
ck2
⊥+ c?k⊥k2
?+ xcimpk2
?
.
(28)
Finally, we can evaluate the local variance of the wavevector
δq = ∇φ:
?
= x∆µν
k
(ck2
δq(r)2=
k
k2φ(k)φ(−k)
?
(29)
k2kµkν
⊥+ c??k4
?+ xcimpk2
?)2− (c?)2k2
⊥k4
?
.
To estimate the integral for small x, we note the denominator
of the integrand vanishes more rapidly with k?than with k⊥,
and hence the largest terms will be those in which the mo-
menta in the numerator are taken in the k?directions. Hence,
up to angular factors which do not affect the scaling with x,
we estimate
?
|δq(r)|2∼ x|∆|
d2k?dk⊥
k4
?
(ck2
⊥+ c??k4
?+ xcimpk2
?)2− (c?)2k2
(30)
⊥k4
?
.
The integral over k⊥can be performed directly to obtain
?Λ
|δq(r)|2∼ x|∆|1
√c
0
dk?
k2
?
(˜ c
??k2
?+ xcimp)1/2(c
??k2
?+ xcimp),
(31)
where ˜ c??= c??−(c?)2/(4c), and we have introduced the radial
momentum coordinate k?and introduced a high momentum
(short distance) cut-off Λ. The integral is readily seen to be
logarithmically divergent for small x, hence
|δq(r)|2∼|∆|x
√c
ln(1/x).
(32)
In the limit x → 0 the fluctuations of the wavevector van-
ish, and therefore fluctuations never diverge. The wavevector
is indeed expected to remain uniform over the entire system,
with only small fluctuations for small x.

Page 9

9
A more subtle question concerns the deformation of the
phase φ rather than the wavevector, because two well-
separated regions of the sample can become arbitrarily out of
phase as small deformations of the spiral accumulate between
them. A similar analysis to above gives
|φ(r)|2=
?
k
φ(k)φ(−k)
(33)
∼ x|∆|1
√c
?Λ
0
dk?
1
(˜ c
??k2
?+ xcimp)1/2(c
??k2
?+ xcimp).
The integral in this case is much more singular. For small x it
is dominated by small k?and independent of Λ. By rescaling,
one finds it is proportional to 1/x, canceling the x dependence
of the prefactor:
|φ(r)|2 >
∼
|∆|
√cc??.
cimp
(34)
Because Eq. (34) is independent of x, there is no particular
reduction of the spatial variations of the spiral phase for di-
lute impurities. This is symptomatic of the “softness” of the
degenerate spiral manifold.
Inspecting both Eqs. (34,32), we see that, although fluctua-
tions do not become large for small x, they do become large
for small c. Since c vanishes on approaching the Lifshitz point
J2/J1 = 1/8, we expect that the spiral ordering should be-
come unstable to impurity deformations in the neighborhood
of this part of the phase diagram. We return to this point in the
Discussion.
2. Numerical results
Wehavearguedabovethatdiluteimpuritiesbasicallyactin-
dependentlyofeachotherandthattheyfavorauniqueground-
state wavevector which minimizes the energy E(q). As a
consequence, it is straightforward to estimate the impurity av-
erage E(q) from our numerical calculations of Ea(q) for a
single impurity in section IIB3. Our results for the impu-
rity averaged energies E(q) are summarized in the bottom
row of Fig. 3. Note that while Ea(q) does not have the full
point group symmetry of the lattice, cubic symmetry is re-
stored when calculating the average E(q).
In particular, our numerical results allow us to determine
the direction of the long-distance spiral wavevector favored
by an ensemble of dilute impurities. For couplings 1/8 <
J2/|J1| < 1/4, multiple defects favor a long-distance spi-
ral wavevector residing on the spiral surface along one of the
100 directions. For couplings J2/|J1| > 1/4 where the spiral
surface develops ‘holes’ centered around the 111 directions,
we find that also the long-distance spiral wavevector favored
by an ensemble of dilute impurities first jumps to the 1¯11∗
direction for 1/4 < J2/|J1|<
moves to the 100∗directions for J2/|J1|>
in Fig. 3.
∼0.30, and then continuously
∼0.30 as illustrated
B.Interplay between impurity and entropic effects
We have argued that at zero temperature, dilute impurities
lift the spiral degeneracy inherent in the pure system, gener-
ating “order by quenched disorder”. As discussed above and
in Ref. 5, entropy provides another degeneracy lifting mech-
anism at finite temperature via “thermal order by disorder”.
The interplay between these mechanisms leads to interesting
physics as we will now discuss. In particular, over a wide
range of J2/J1, disorder and thermal fluctuations favor decid-
edly different ordered states; e.g., for J2/J1 = 0.2, thermal
fluctuations favor the 111 directions while impurities prefer
the 100 directions. In such cases, since entropic corrections
giving rise to thermal order by disorder vanish as T → 0, the
system is expected to exhibit multi-stage ordering, from an
impurity-driven phase at the lowest T to an entropically stabi-
lized phase at moderate T to a disordered paramagnet at still
higher T. As an aside, we note that other interactions beyond
those considered in our model and/or quantum fluctuations
can compete with impurity effects at low T, but may similarly
lead to multiple phase transitions. If the energetic corrections
coming from impurity or other effects are too large, however,
then the entropically stabilized phase will be removed, leaving
a single ordered state.
Another interesting effect arising from the interplay be-
tween entropy and disorder, which can be probed experimen-
tally, pertains to the shift in transition temperature Tcat which
the system first orders. Roughly, should entropy and disorder
favor the same state, then Tcis expected to be enhanced rel-
ative to the pure system; otherwise a reduction is anticipated.
To estimate this shift, we note that the transition is first-order
and that at Tcthe free energies for the paramagnet and the
ordered phase must equal,
fsp(Tc) + xδFsp(Tc) = fPM(Tc) + xδFPM(Tc).
(35)
Here, x is the impurity concentration, fspand fPM are the
free energies for a clean system in the spiral phase and para-
magnet, respectively, and xδFspand xδFPM are the corre-
sponding changes in free energy due to the impurities. For a
well defined thermal order-by-disorder phase, an approximate
derivation (see Appendix D) yields the following result
Tc− T∗
c= T∗
cx[E(q)]S− E(q0)
l∗
,
(36)
where T∗
system, l∗is the latent heat density to go from the ordered
phase to the paramagnetic one in the clean crystal, S is the
degeneracy (spiral) surface and q0is the momentum favored
by thermal order-by-disorder in the clean system. We defined
the surface average
?
cis the (upper) ordering temperature for the clean
[E(q)]S=
q∈SdqE(q)
?
q∈Sdq
.
(37)
The quantities determining the Tcshift in Eq. (36) can be
extracted from numerics. We find that the latent heat l∗is
roughly independent of J2/J1. However, there is significant

Page 10

10
0.140.160.180.200.220.24
?0.05
0.00
0.05
0.10
0.15
J2?J1
?E?q??S?E?q0?
FIG. 5: Plot of [E(q)]S−E(q0) versus frustration J2/J1. The shift
of Tcfor the order-by-disorder phase per impurity is proportional to
this quantity (see Eq. (36)).
dependence of the numerator in Eq. (36) on this ratio. This
is plotted in Fig. 5. The Tctracks this quantity, and is thus
sensitive to the degree of frustration.
IV.DISCUSSION
In this manuscript we have explored the effect of dilute im-
purities on the J1−J2model on the diamond lattice. General
considerations led us to conjecture that impurities may pro-
vide a mechanism for ground state degeneracy breaking. We
established that, under rather general conditions, even highly
frustrated magnets are induced to order by low concentrations
of impurities. Moreover, the mechanism and energetics of this
ordering was explained in terms of a simple “swiss cheese”
picture. To expose the mechanism in more detail, we consid-
ered a very specific impurity model, namely B site magnetic
ions being added to the system, and confirmed the general
structure of the impurity-induced ordering by numerical and
analytical means in this situation.
Let us briefly discuss this picture in relation to CoAl2O4
and MnSc2S4, the two A-site magnetic spinels exhibiting the
largest frustration parameters without the complications of or-
bital degeneracy. Disorder in the form of inversion – A and B
site atoms interchanging with one another – is prevalent in
many spinels, including these, at the level of at least a few
percent. One intriguing feature of the measurements on these
materialsistheobservationofglassyfreezinginCoAl2O4, but
not in MnSc2S4, despite comparable levels of inversion in the
two materials. This suggests that CoAl2O4is more sensitive
to defects than MnSc2S4, and our results corroborate this hy-
pothesis. Theoretically, we argued that, generically, the effect
of sufficiently dilute impurities is to induce order, not a spin
glass. Fortheorderedstatetobestable, wearguedthat: (1)the
impurity “halos” should not overlap, and (2) the fluctuations
in wavevector induced by the randomness of the impurity po-
sitions should be small. In Sec. IIIA1, we saw that the second
criteria is highly sensitive to the magnitude of the stiffness c,
thefluctuationsbecominglargeascdecreases. Inthediamond
lattice antiferromagnets, the stiffness c actually vanishes on
approaching the Lifshitz point J2/J1 = 1/8. Prior investi-
gations concluded that in fact CoAl2O4has exchange param-
eters close to this point, while in MnSc2S4, J2/J1≈ 0.85,5
where c is not small. Thus we suggest that the freezing behav-
ior in CoAl2O4may be understood as arising from proximity
to the Lifshitz point. It may be interesting to directly study
disorder physics in this region by field theoretic methods in
the future.
We would like to emphasize the generality of this argu-
ment. The only assumption is that the impurity positions are
not strongly correlated, but otherwise this conclusion is inde-
pendent of the type of defects. Indeed, we do not maintain
any direct relevance of the specific impurity modeled studied
in the numerical portions of this paper to the A-site spinels.
For CoAl2O4, the existence of magnetic ions on the B sites
is probably suspect, as inverted Co+3on the B sites would
be expected to have a non-magnetic ground state. However,
the expected spin “vacancies” induced by Al atoms on the A
sites would lead to the same general conclusions. What would
requireamoreappropriatemicroscopicmodelwouldbeanes-
timate of the size of the region of deformed spins around an
impurity.
Very recent experiments have greatly clarified the situation
in CoAl2O4. Through a careful study of elastic and inelas-
tic neutron scattering high quality single crystal, MacDougall
et al.16have argued that the freezing transition in CoAl2O4
signals an “arrested” first order transition in which the sam-
ple breaks up into antiferromagnetic domains. These domains
are evidenced by a substantial Lorentzian-squared component
to the elastic scattering. Moreover, below the freezing tem-
perature spin-wave excitations were observed, a fit of which
determined J2/J1≈ 0.1. This parameter ratio takes CoAl2O4
close to the Lifshitz point but 0.1 < 1/8, so the commensu-
rateN´ eelstatewouldbeexpectedatlowtemperature. Thefirst
order nature of the transition is consistent with theoretical ex-
pectations based on the order-by-disorder mechanism.5Given
these exchange parameters, the detailed analysis of this paper
does not directly apply, since we have assumed J2/J1> 1/8
and focused on spiral ground states. However, arguments very
similar to those we applied here to show a strong sensitivity to
impurities close to the Lifshitz point on the spiral side also im-
ply a similar sensitivity close to the Lifshitz point on the N´ eel
side. Thus the findings are quite consistent with the general
reasoning espoused here.
We conclude by describing an interesting feature of our nu-
merical simulations, which might be of interest in future the-
oretical and experimental studies. We found that, while the
ground states of the pure system are coplanar, the spin con-
figuration around the impurity might acquire a sizable out-of-
plane spin component, i.e. a spin component orthogonal to
the plane in which the spin spiral state lies at long distances
away from the impurity. This is in particular true for those
spirals with wavevectors q such that their energies E1(q) (see
Eq.(B11)) are far away from the overall minimum. It is pos-
sible that, collectively, impurities might therefore induce non-
coplanar spin ordering. Such non-coplanar order is relatively
rare, and interesting insofar as it can induce non-trivial Berry
phases, related to anomalous Hall effects in conducting sys-
tems.

Page 11

11
Acknowledgments
We would like to thank Leo Radzihovsky for extensive dis-
cussions during the prehistory of this project, and apologize
for his wasted time. Our numerical simulations were based on
the classical Monte Carlo code of the ALPS libraries19. L.B.
was supported by the Packard Foundation and National Sci-
ence Foundation through grants DMR-0804564 and PHY05-
51164. J.A. acknowledges support from the National Science
Foundation through grant DMR-1055522.
Appendix A: Definition of local wavevector
Here we describe in detail how the local spiral wavevec-
tor is defined on the lattice, as used in Sec. IIB3 and
Figs. 4 and 6. Depending on the unperturbed spiral
wavevector q taken at infinity we consider distinct sets
of three neighboring sites out of the 12 second-neighbor
sites which are nearest neighbors on the identical (fcc)
sublattice. In particular, for q pointing in the 111 di-
rection we consider two sets of vectors {rij}, namely
P=
{(1/2,−1/2,0);(1/2,0,1/2);(0,−1/2,1/2)} and
Q = {(−1/2,−1/2,0);(1/2,0,−1/2);(0,−1/2,−1/2)}.
For
q
pointing inthe
sider two alternative sets of vectors {rij},
P?
=
{(1/2,1/2,0);(1/2,−1/2,0);(1/2,0,1/2)} and
Q?
= {(1/2,1/2,0);(1/2,0,1/2);(1/2,0,−1/2)}.
place the local wavevector qiat position ri−1
which is always located inside the (convex) manifold spanned
by the four spins.
100
direction wecon-
namely
We
4
?3
j=1rij,
Appendix B: Symmetries
In this appendix, we give explicit expressions for the sym-
metry transformations and their effects, within our conven-
tions for the spinel lattice. The space group is generated by
the following operations:
1. A three-fold rotation about the (1,1,1) axis:
T1: (x,y,z) −→ (z,x,y).
(B1)
2. A two-fold rotation about the (0,0,1) axis:
T2: (x,y,z) −→ (−x,−y,z).
(B2)
3. Reflection through a (1,−1,0) plane:
T3: (x,y,z) −→ (y,x,z).
(B3)
4. Inversion:
T4: (x,y,z) −→ (1
4− x,1
4− y,1
4− z).
(B4)
We define the following four impurity positions, ua(a =
1,2,3,4) modulo Bravais lattice transformations:
u1 = (3/8,5/8,3/8)
u3 = (5/8,3/8,3/8)
These positions are mapped into one another by the four
spacegroupgenerators. Correspondingtoeachofthesegener-
atorsisanassociatedlineartransformationinreciprocalspace.
This transformation of wavevectors is identical to the trans-
formation of real space coordinates except that translational
components of the transformation are dropped. That is, if the
coordinates transform according to r → Or+a (O is an O(3)
matrix), then the corresponding momentum transformation is
just q → Oq.
As a consequence, any given impurity position may be
mapped to the other three by such an O(3) operation. One
finds that (up to Bravais lattice vectors), the impurity positions
transform according to
u2 = (3/8,3/8,5/8), (B5)
u4 = (5/8,5/8,5/8).
Taub= uc(b,a),
(B6)
where c(a,b) can be represented as the matrix
c(a,b) =



2 3 3 1
3 4 2 2
1 1 1 3
4 2 4 4

,

(B7)
where a and b specify the row and column of the matrix, re-
spectively. We see from this that, for instance, an impurity on
position 4 retains the symmetries generated by T1, T3and T4,
but not T2.
Moreover, we observe that each impurity position can be
mapped to position 1 in the following way:
u1 = T1◦ T1u2,
u1 = T1u3,
u1 = T1◦ T1◦ T2u4.
(B8)
(B9)
(B10)
This allows one to calculate the energies Ea(q) with a =
2,3,4 from E1(q?) with an appropriate q?. Specifically
E2(qx,qy,qz) = E1(qy,qz,qx),
E3(qx,qy,qz) = E1(qz,qx,qy),
E4(qx,qy,qz) = E1(qy,−qz,qx).
Therefore, the average energy can be written as
?
+E1(qz,qx,qy) + E1(−qy,qz,−qx)
(B11)
E(qx,qy,qz) =
1
4
E1(qx,qy,qz) + E1(qy,qz,qx)
?
.
(B12)
We note that, taking into account the subgroup of the full
space group which leaves position 1 invariant, the first im-
purity energy obeys
E1(−qy,−qz,qx) = E1(qz,qy,qx) = E1(qy,qz,−qx)
= E1(−qz,−qy,−qx) = E1(qx,−qz,−qy) = E1(qz,−qx,−qy)
= E1(−qx,qz,qy) = E1(−qz,qx,qy) = E1(qx,qy,qz)
= E1(−qy,−qx,qz) = E1(−qx,−qy,−qz) = E1(qy,qx,−qz).
(B13)

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12
The average energy, by construction, has the full cubic space
group symmetry, i.e.
E(qx,qy,qz) = E(saqa,sbqb,scqc),
(B14)
where sa,sb,sc= ±1 and (qa,qb,qc) is an arbitrary permuta-
tion of qx,qy,qz. As a consequence, 1/48th of the solid angle
in q space is enough to recover the full function E(q). We
therefore carry out numerical simulations only for such a sec-
tion, which we choose, arbitrarily, to be the one defined by:
(qx> 0) ∧ (qy> 0) ∧ (qz> 0) ∧ (qx> qy) ∧ (qx< qz).
(B15)
All points defined by Eq. (B15) are inequivalent to one an-
other, and conversely, can used to generate E(q) for an arbi-
trary point using Eq. (B14).
Appendix C: Locality of spin deformation and finite-size effects
Our numerical simulations of the B-site impurity model in-
dicate that the deformation of the spin spiral state in the vicin-
ity of the impurity is limited to a small numbers of spins in
the unit cell around the impurity. Our numerical calculations
are performed in an L × L × L simulation cube embedded in
a larger cube of extent L + 1, where the spins in the ‘bound-
ary cube’ are fixed to a particular spin spiral state. One might
thus wonder whether the locality of the spin spiral deforma-
tion originates from the impurity physics, as opposed to arti-
facts due to the fixed boundary conditions. To exclude the lat-
ter we have calculated the spiral deviation for different system
sizes L and positioning of the impurity site, as summarized in
Fig. 6 where we fix the boundary spiral wavevector to the 1¯11
direction, which minimizes E1(q) for the chosen ratio of cou-
plings J2/|J1| = 0.2. We find that the deviation is insensitive
to varying the system size as shown in the various panels. Fur-
ther, we also do not find a striking change of our results when
embedding the impurity into a system of even extent L = 4
(second panel from top in Fig. 6), which places the impurity
site rather asymmetrically with respect to the fixed boundary
spiral.
Appendix D: Derivation of the transition temperature shift
Inthisappendixweconsiderthetransitionbetweenthehigh
temperature paramagnetic phase, and an ordered phase where
the spins order in a spiral configuration with a wavevector se-
lected by entropy. We explore how adding impurities shifts
the transition temperature, assuming that the impurities them-
selvesdonotchangethenatureoftheorderedstate. Quantities
in the clean limit are denoted by a star.
For a first-order phase transition, at the transition tempera-
ture:
fsp(Tc) + xδFsp(Tc) = fPM(Tc) + xδFPM(Tc)
fsp(T∗
c) = fPM(T∗
c),
(D1)
01234
56
7
8
distance from impurity
0
0.4
0.8
1.2
0
0.4
0.8
1.2
deviation δq / π
0
0.4
0.8
1.2
0
0.4
0.8
1.2
L = 3
L = 4
L = 7
L = 9
FIG. 6: Finite size effects: Deviation of the local spiral wavevector
qi from the one of the spiral state at the boundary q for systems of
varying size N = 8 × L3. For the chosen couplings J2/|J1| = 0.2
the spiral wavevector q points along the 1¯11 direction with length
|q| = 0.71π. The symbols correspond to two different sets of neigh-
boring spins, P (circles) and Q (boxes), used to calculate the local
spiral wavevectors, details are given in the text.
where fsp,PMare the free energy densities of the spiral phase
and paramagnetic phase, respectively, and similarly δFsp,PM
refer to the free energy density corrections when impuri-
ties are included. A small impurity concentration will only
slightly shift the transition temperature, and so we expand the
free energy density to first order in the temperature shift
f(Tc) ≈f(T∗
=f(T∗
c) +∂f(T∗
c)
∂T
c)(Tc− T∗
(Tc− T∗
c)
c) + s(T∗
c)
?Tc
=f(T∗
c) + (f(T∗
c) − ?)
T∗
c
− 1
?
,
(D2)
where s(T) is entropy, and ? is the energy density. From this
we find
?Tc
Now we turn to estimate the free energy densities δF,
which can be estimated from F = −T logTr?e−βH?, when
a small change in the free energy
?
≈ −T log[1 − β?δH?] ≈ +?δH? ,
where the angle brackets denote a thermal average. Each im-
purity will contribute a term of the form of (13) to δH. Next
we estimate the energy thermal average in each phase. In
the ordered phase, the system remains mostly in the ground
state configuration, and so we estimate δFsp ≈ E(q0),
(?PM− ?sp)
T∗
c
− 1
?
= x(δFPM(Tc) − δFsp(Tc)) .
(D3)
varying the Hamiltonian by a small term xδH. This will yield
δF = −T log
Tr?e−β(H+δH)?
Tr[e−βH]
?
(D4)

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13
where E(q) is the same as in (15), and q0 is the spiral
wavevector. In the paramagnetic phase, close to the tran-
sition temperature, the system thermally fluctuates mostly
amongst the different spiral states (this has been shown ex-
plicitly for the clean system in Ref. 5) and so we estimate
δFPM≈?
?Tc
q∈SdqE(q)/(?
q∈Sdq) where S is the spiral sur-
face. We find therefore
(?PM− ?sp)
T∗
c
− 1
?
= x
??
q∈SdqE(q)
(?
q∈Sdq)
− E(q0)
?
(D5)
,
and finally
Tc− T∗
c=T∗
cx
l∗
??
q∈SdqE(q)
(?
q∈Sdq)
− E(q0)
?
,
(D6)
where l∗= (?PM− ?sp) is the latent heat density to go from
the order-by-disorder phase to the paramagnetic phase in a
clean system.
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