Hydrogenic states of monopoles in diluted quantum spin ice

0.82 MB·English

by  Olga Petrova

Hydrogenic states of monopoles in diluted quantum spin ice
Olga Petrova,1 Roderich Moessner,1 and S. L. Sondhi1,2
1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
2Department of Physics, Princeton University, Princeton, NJ 08544, USA
We consider the effect of adding quantum dynamics to a classical topological spin liquid, with
particular view to how best to detect its presence in experiment. For the Coulomb phase of spin
ice, we find quantum effects to be most visible in the gauge-charged monopole excitations. In the
presence of weak dilution with nonmagnetic ions we find a particularly crisp phenomenon, namely
theemergenceofhydrogenicexcitedstatesinwhichamagneticmonopoleisboundtoavacancyat
various distances. Via a mapping to an analytically tractable single particle problem on the Bethe
lattice, we obtain an approximate expression for the dynamic neutron scattering structure factor.
5
1
0 Thequestforspinliquidsisanimportantenterprisein tice induced mixing—they involve a magnetic monopole
2
strongly correlated many body physics in an era when bound to the impurity site into an infinite set of levels.

n a huge amount of theoretical interest has focused on In the presence of a dilute set of such impurities, these
a
forms of order outside the canonical broken symmetry states give rise to a characteristic signature in neutron
J
paradigm[1–4]. Thesearchinvolvesidentifyingrelatively scattering at low temperatures which we discuss. Read-
1
simple Hamiltonians that host spin liquids and finding ers may note the family resemblance of these hydrogenic
1
experimental systems and signatures—the latter being monopole states to hydrogenic states in doped semicon-

] moreelusivethaninLandauorderedsystems. Indeed,at ductors [21, and references therein], although we caution
l
e thispointthelistofexperimentalsystemswherethereis that the details have crucial differences. We also note
- strong evidence of spin liquid behavior is small. Among thattheresponseofspinliquidstoimpuritiesisofbroad
r
t them is the celebrated spin ice system, arising in some interest as a diagnostic of their internal dynamics: what
s
. rare earth pyrochlore magnets, which exhibits a U(1) happens when you dope a spin liquid is the fundamental
t
a spin liquid and excitations which are condensed matter – and to date largely unresolved – question of the RVB
m
analogsofCoulombicallyinteractingmagneticmonopoles theory of high temperature superconductivity [22].
- [5]. Spin ice is truly special at this point in hosting a In the balance of the paper we begin by briefly re-
d
three dimensional spin liquid, but, owing to large mag- viewing how the dynamics of quantum spin ice can be
n
o neticmoments,islimitedtotheclassicalregime,inwhich formulatedasthequantummechanicsofmonopoles. We
c coherent quantum dynamics appears to play little role. then concentrate our attention on the problem of a va-
[ cancyspinanddescribehowitcanbemappedtoagood
Logically, much recent interest has focused on looking
1 forquantumgeneralizationsofspinice. Thereareseveral approximationtoamonopoleonaBethelatticeinteract-
v ingwithafixedCoulombiccharge. Thismodelleadstoa
candidate materials for quantum spin ice behavior, such
5 familyofhydrogenicboundstatesofthemonopolealong
asTb Ti O [6–9],Yb Ti O [10–12]andPr Zr O [13],
4 2 2 7 2 2 7 2 2 7
withacontinuumband. Inthetechnicalheartofthepa-
4 but an unambiguous experimental signature of quantum
perwesolvethisproblemandobtainanexactclosedform
2 spin ice has been lacking. Logically, much recent theo-
0 retical work has focused on looking for quantum gener- solution for the onsite Green’s functions. We use these
. results to obtain the signature of the hydrogenic states
1 alizations of spin ice in which quantum fluctuations can
in the structure factor of spin ice containing a dilute set
0 lead to a fully quantum U(1) spin liquid [14–20] .
5 of vacancy spins. We conclude with some comments and
Here we investigate the addition of quantum fluctua-
1 pointers to future work.
: tions to spin ice but in a different limit which is, plau- Quantum dipolar spin ice: Our model Hamiltonian
v
sibly, of relevance to existing materials. Fundamentally,
i (cid:88)
X wewishtounderstandtheleadingordereffectsofadding H =H + t·S (1)
QDSI DSI i
r quantum dynamics about the classical spin ice limit. As i
a
we will detail below, this has a parametrically larger ef-
consists, firstly, of the classical dipolar spin ice Hamil-
fect on monopole motion than on monopole-free ground
tonian, defined for Ising spins S living on the sites of
states so the leading manifestations of quantum fluctua- i
pyrochlore lattice and pointing along the local easy axis
tions appear when monopoles are present.
joiningcentersofneighboringtetrahedra[23,24,andref-
We begin this program by studying the simplest man- erences therein]:
ifestation of the quantum mechanics of monopoles—a
(cid:34) (cid:35)
striking effect that appears in the response of quantum H = µ0µ2 (cid:88) Si·Sj − 3(Si·rij)(Sj ·rij) . (2)
spin ice to the introduction of a vacancy or missing spin. DSI 4π r3 r5
i<j ij ij
Wefindthatthelowestlyingexcitedstatesinthevicinity
of the vacancy resemble those of hydrogen modulo lat- The second term in Eq. (1) is the transverse field, ori-
2
from a classical spin ice state (Fig. 1a) leaves behind
two monopoles of charges ±q [26] (Fig. 1b). A bulk
-1 m
monopole with charge ±2q can be ‘emitted’ by the va-
m
cancyviaflippingoneofthetwomajorityspinsateachof
+1 the tetrahedra adjacent to the ghost spin (Fig. 1c). The
bulkmonopoleisthenfreetomovearoundinthesystem.
Wedefineaquantitythatcanbethoughtofastheioniza-
a b
tionenergyofthevacancy: I = µ0µ2+2v µ2. Thisisthe
2πa2 0a2
d d
total energy cost of an emitted monopole moved out to
-1 infinity. Emitting another monopole into the bulk would
- 2 costadditionalenergyoftheorderof2v0µa22,sothelowest
d
+ 2 -1 + 2 energy charged excitation in the presence of an isolated
vacancy is a single monopole with charge ±2q . Once
m
the monopole is emitted into the bulk, it is free to hop
c d
throughfluctuation-inducedspinflips,whileanetcharge
of opposite sign ∓2q remains at the vacancy (Fig. 1d).
FIG. 1. Spin ice projected onto a plane, with each vertex m
of the resulting square lattice in (a) corresponding to a py- Adding quantum dynamics via the transverse field in
rochlore tetrahedron. (b) A missing spin gives rise to a +1 Eq. (1) has only a weak effect on the ground states, as
and a −1 charges; (c) flipping one of the majority spins ad- connecting two of them requires flipping spins in closed
jacenttothevacancycreatesabulkcharge+2,invertingthe
loops, minimally six of them on a hexagon of the py-
sign of one of the vacancy charges; (d) the bulk charge prop-
rochlore lattice. Near the classical limit, v (cid:29) t, such
agates in the system through further spin flips, while the net 0
processes come with a prohibitively small energy scale,
charge of vacancy is approximated as a single −2 monopole.
∼t6/v5. By contrast, for a state containing a monopole,
0
the lowest order effect – a monopole hopping onto a
ented perpendicular to the local easy axis, which adds neighboring tetrahedron by flipping a majority spin – is
thesimplestquantumdynamicsintheformofsinglespin parametrically stronger: linear in t!
flips. Thissimpleformisconvenientforafirsttheoretical Thus, in experiment, the most promising place to
analysis,foramorecompletesymmetry-basedanalysisof see quantum effects in spin ice is in the gauge-charged
quantum terms in the Hamiltonian, see [17, 25]. monopole excitations, rather than its gauge-neutral gap-
Ghost spins and the Bethe lattice: At this point we less emergent photons. Analogous considerations apply
switch from a spin description to that referred to as the intheproximityofavacancy,wherewefocusonthecase
dumbbell model [5] which we quickly review. Each spin of a monopole emitted into the bulk (Fig. 1d), also with
is replaced by a pair of magnetic charges ±qm = µ/ad low-order signatures. For this reason, here we perform a
of opposite sign, where (cid:126)ad is a vector pointing between quantum calculation for the monopole states, and do a
the centres of neighboring tetrahedra. By summing up thermal sum over the nearly degenerate spin ice configu-
the net charge at the center of each tetrahedron (Qα ≡ rations.
(cid:80)
q = 0,±2q ,±4q ), we can replace the dipolar
i∈α i m m We treat the problem as that of two Coulombic
piece of the spin Hamiltonian (2) by
charges, one of which is stationary. As the charge prop-
agates through the bulk, it changes the spin ice back-
H = µ0 (cid:88) QαQβ + v0 (cid:88)Q2 , (3) ground. This process is difficult to capture exactly, but
4π r 2 α
α<β αβ α fortunately it is possible to make considerable progress
via an effective model that we describe next. From this,
Coulomb interactions between charges on the diamond
we are able to extract the bound states in considerable
(cid:16) (cid:113) (cid:17)
lattice, with v = µ0 2 1+ 2 the cost of creating a detail, followed by a continuum band, much as we would
0 4πa 3
monopole, which can be shifted by a nearest-neighbor expect for the Hydrogen atom.
exchange term. In spin ice ground states, Q ≡ 0. In order to investigate the problem of an isolated va-
α
Flipping a spin in a ground states yields a pair of mag- cancy that has emitted a free monopole into the bulk,
neticmonopolesofcharges±2q onadjoiningtetrahedra weswitchtothestate lattice description[27]. First, con-
m
which can then move apart via further spin flips at finite sider a new basis of the following (classical) states: a
costinenergy. Eachchargehasthreemajority spinsthat spin ice state with a vacancy, which we label |0(cid:105), and
areallpointinginoroutofthetetrahedron, andasingle states with an emitted monopole in the bulk, connected
minority spin pointing in the opposite direction. to |0(cid:105) through single spin flips. Next, each site of the
With this in hand let us discuss the energetics of sub- state lattice represents one of the basis states |n(cid:105); while
stituting a magnetic ion on the pyrochlore lattice by a bondsconnectthosesiteswhosecorrespondingstatesare
non-magnetic impurity, see Fig. 1. Removing a spin connected by single spin flips. Apart from site 0 (repre-
3
potential (C = −µ0µ2) between the two charges. In
π a2
d
conventional spin ice, the cost of having a monopole is
larger than the magnitude of the Coulomb interaction
between two charges, I > |C|. For concreteness, we use
C = −I/3; t = I/10 in the following. Since we restrict
ourselves to a particular starting spin ice configuration
andomitotherdegenerateicestatesfromthediscussion,
the mapping from Eq. (1) to (4) is accurate to O(t5/v4).
FIG. 2. The Bethe lattice describing state space. Its root, 0
Our final approximation concerns the distance between
n=0, corresponds to the unionized vacancy state |0(cid:105).
monopoles. Since the four sites at the first generation of
the Bethe lattice correspond to the bulk monopole being
one spin flip away from the vacancy, it is natural to ap-
senting|0(cid:105)),thestatelatticeistrivalent. Itcanbeshown
proximate d in Eq. (4) by the generation of the Bethe
thatthesmallestclosedcycleinthestatelatticeofdisor- n
latticen. Thisdefinitionfailstobeexactalreadybeyond
deredpyrochlorespinicehaslength20[28]. Wetherefore
O(t2), but should work sufficiently well in the I,|C|(cid:29)t
approximatethestatelatticebyacycle-freeinfiniteCay-
regime, when the bulk monopole prefers not to move too
ley tree (the Bethe lattice) rooted at site 0 (Fig. 2). The
far. In return for these approximations, we are able to
monopolepropagatinginrealspacecorrespondstoasin-
solve exactly our idealized model, that of a single par-
gle particle hopping on this lattice in the presence of the
ticle hopping on the Bethe lattice in the presence of a
Coulomb potential:
Coulomb potential I+C/n for n>0.
The Bethe lattice problem: We calculate the diago-
4 (cid:18) (cid:19) 3
H|0(cid:105)=−t (cid:88) |m(cid:105); H|n(cid:105)= I+ C |n(cid:105)−t (cid:88) |m(cid:105) nalelementsGii(ω)ofthelatticeGreen’sfunctiontoinfi-
m=1 dn m=1 nite order in t [29, 30]. We find [28] that each Gii(ω) can
(4) be written down in terms of a finite number of GF(ω),
k
where the sums run over states reached by flipping ma- infinite sums involving particle hopping from a site at
jority spins of the monopole. d denotes the distance generation k to sites at generations g > k. The latter
n
between the vacancy and the monopole in the bulk in have a closed form expression in terms of the Gauss hy-
units of a , such that C/d is the attractive Coulomb pergeometric functions F2(a,b,c,z) [31]:
d n 1
(cid:16) √ (cid:17)
GF(ω)= √ 2k/ω 1 F12 1− √C1+/ωx2,k+1,k+1− √C1+/ωx2,11−+√11++xx22 (5)
k 1+x2+1k− √C/ω F2(cid:16)1− √C/ω ,k,k− √C/ω ,1−√√1+x2(cid:17)
1+x2 1 1+x2 1+x2 1+ 1+x2
where x2 = −8t2. This yields the exact expression for states, of width linear in t, is confined to the region
ω2 √ √
any of the diagonal elements of the Green’s function; for I − 8t2 < ω < I + 8t2. (Introducing closed cycles
instance at the root site into the lattice has the effect of adding band tails, ex-
tending beyond these edges.)
G00(ω)=(cid:0)ω−4t2G1F(ω−I)(cid:1)−1. Signatures of monopoles in neutron scattering:
Oneofourcentralresultsisthedynamicstructurefactor,
The full Green’s function yields the energy levels via its
defined as
poles and the local densities of states for each Bethe lat-
tice generation, proportional to its imaginary part. The S((cid:126)q,∆ω)=(cid:88)δ(E −E −∆ω)|(cid:88)(cid:104)f|S+|i(cid:105)eiq(cid:126)·R(cid:126)|2. (6)
local density of states at site 0 in Fig. 3 indicates that f i R(cid:126)
f R(cid:126)
indeed there are bound states followed by the contin-
uum energy band. While the classical ground state (a In order to extract the information that is most relevant
spin ice state with a vacancy) would have zero energy, to spin ice experiments from the Bethe lattice model,
the ground state energy of the quantum problem ω is we calculate a one dimensional version of Eq. (6), aver-
0
lowered due to the hopping t. Low-lying excited states aged over all directions of (cid:126)q. Such a quantity, S(q,∆ω),
are separated from the ground state by a gap, which is can be measured directly in a powder averaged neutron
also decreased from the classical value I through hop- scattering experiment. The details of our calculation,
ping and Coulomb attraction. They accumulate below carried out in the limit of dilute nonmagnetic impuri-
√
theedgeofthecontinuumband,locatedatL=I− 8t2. ties, are given in the Supplemental Material [28]. The
In the Bethe lattice problem, the band of the extended dynamic structure factor S(q,∆ω), plotted in Fig. 4(a),
4
thepropertiesofmagneticmonopolesindipolarquantum
spinice. Wehavedemonstratedthatthesearetheprime
indicators of the presence of quantum dynamics. In the
presence of nonmagnetic impurities we have found both
4 sharphydrogenicboundstatesaswellasabroadcontin-
uum energy band. While we believe these results to be
robust, there is clearly much scope for further, presum-
ablynumerical,modelingtakingintoaccountthedetailed
L I
0 1 2 lattice structure, as well as any material specific single-
ion physics and terms in the quantum Hamiltonian.
FIG.3. Localdensityofstatesatsite0oftheCoulombprob-
lem on the Bethe lattice, for C = −I/3; t = I/10. Bound Thequantumdynamicsofapairofmonopolespresents
states(red)appearassharppeaks;theloweredgeofthecon- a more difficult problem due to the pair’s center of mass
tinuum (blue) is labeled L, the classical ground state energy motion. We are planning to address this issue, as well
by 0 and the ionization energy by I. as clarify the detailed character of the continuum band
of states in the vacancy problem, in future work. Addi-
Δω S(q,Δω)/S(0,Δω) tionally, despite neutron scattering being the method of
continuum↥ n n choiceforinvestigatingmagneticmaterials,localdisorder
E2 −Ei is an attractive subject for other types of experimental
E −E probes, such as nuclear magnetic resonance. While such
1 i
techniques are beyond the scope of this work, our the-
oretical model can also be employed for calculating real
space quantities accessible by the local probes.
6
Acknowledgments: The authors thank Subhro Bhat-
tacharjee, Claudio Castelnovo, Radu Coldea, Siddhardh
4 Morampudi, Satoru Nakatsuji, and Oleg Tchernyshyov
forusefuldiscussions,andYenTingLinandNikosBagis
forpointingouttherelevanceofRamanujan’sworktothe
2 0 Green’s function calculation. The authors acknowledge
1
support of the Helmholtz Virtual Institute New States
5
0 of Matter and their Excitations,theUSNationalScience
00 π/2 πq 0 10 πq Foundation under Grant No. DMR-1311781 (SS), the
Alexander von Humboldt Foundation (OP and SS), and
(a) (b)
theGermanScienceFoundation(DFG)viatheGottfried
Wilhelm Leibniz Prize Programme.
FIG. 4. Left: dynamic structure factor S(q,∆ω) for pow-
der averaged neutron scattering. Each line is multiplied
by Exp[0.7|n−1|], where n labels the all-even energy lev-
els with n = 0 the ground state. Right: Lineshapes
S(q,∆ω )/S(0,∆ω ) for n=0,1,5,10. C =−I/3;t=I/10
n n
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SUPPLEMENTARY MATERIAL
Minimal closed cycle in the state graph of pyrochlore spin ice
4
3
5
2
9 8
6
1
7
FIG. 5. A fragment of the real-space diamond lattice that the magnetic monopoles of pyrochlore spin ice reside on. Spins are
depicted as arrows pointing along the lattice edges. Left: the shortest closed cycle in the state graph involves flipping spins
along two hexagons that share two edges in common (red). Right: an example of a spin configuration, that gives rise to the
shortestpossiblecycleonthestatelattice,withtwoflippablehexagonsandamagneticmonopoleatoneofthesiteswherethey
intersect.
One of the arguments for using the Bethe lattice to approximate the state graph of pyrochlore spin ice is that the
closed cycles present in the actual state lattice are long and thus can be neglected. The spins can be thought of as
pointing along the edges of the diamond lattice [5]. Its smallest closed loop has length six, so inverting spins along a
flippable hexagon (such that six spins are arranged to point head-to-tail) brings the charge back to where it started.
The spin configuration is changed by such a process, and one needs to flip the same six spins a second time to bring
the system back to the initial state. However, this process constitutes a self-retracing path on the state graph, rather
than a closed cycle.
Introducing a loop into the state graph requires adding another hexagon into the real-space picture. The shortest
possible loop is obtained by maximizing the overlap between the two hexagons: they share two edges as shown
6
in Fig. 5. A non-trivial closed loop on the state graph involves ten spins on two such overlapping hexagons, and
positioning a magnetic monopole on a site from which it can hop along two different paths [charge −2q on site 2
m
in Fig. 5]. Each spin can be labeled by two numbers that correspond to the diamond lattice sites at the two ends of
the dumbbell. An example of a closed cycle in the state graph, written down as a sequence of spin flips as labeled in
Fig. 5, is then:
(cid:104)23(cid:105)(cid:104)34(cid:105)(cid:104)45(cid:105)(cid:104)56(cid:105)(cid:104)61(cid:105)(cid:104)12(cid:105)(cid:104)29(cid:105)(cid:104)98(cid:105)(cid:104)87(cid:105)(cid:104)76(cid:105)(cid:104)65(cid:105)(cid:104)54(cid:105)(cid:104)43(cid:105)(cid:104)32(cid:105)(cid:104)21(cid:105)(cid:104)16(cid:105)(cid:104)67(cid:105)(cid:104)78(cid:105)(cid:104)89(cid:105)(cid:104)92(cid:105).
This closed cycle has length 20. We would like to emphasize that this is the shortest possible loop in the state lattice
of disordered classical ice states. Other spin configurations have closedcycles thateitherhave the same length, or are
even longer.
Lattice Green’s function: the continued fraction method
In order to solve the Coulomb problem on the Bethe lattice, we employ the continued fraction approach [29, 30].
The strategy is to start with a perturbative treatment (where H is the potential, and nearest neighbor hopping t is
0
the perturbation) and proceed to solve the problem exactly by calculating the self-energy to infinite order in t in the
particle’s lattice Green’s function. Recursively solving Dyson’s equation
G(ω)=G (ω)+G (ω)Σ(ω)G(ω)
0 0
(where Σ(ω) is the self energy) gives rise to continued fractions for the Green’s function. For instance, the diagonal
element of the Green’s function for site 0 is:
1 1
G (ω)= = .
0 ω−Σ (ω) 4t2
0
ω−
2t2
ω−I−C−
ω−I− C −...
2
The self energy Σ (ω) above is given by the sum of all paths on the lattice going away from site 0 and back to it.
0
The elements of the Green’s function for sites at all generations can be defined in terms of a finite number of GF(ω),
k
infinite continued fractions involving hops only to generations higher than k:
1
GF(ω)= .
k 2t2
ω− C −
k 2t2
ω− C −
k+1 ω− C −...
k+2
UsingatheorembyRamanujan[31],wearriveatthefollowingexactclosedformexpressionforthecontinuedfraction
above:
(cid:16) √ (cid:17)
GF(ω)= √ 2k/ω 1 F12 1− √C1+/ωx2,k+1,k+1− √C1+/ωx2,11−+√11++xx22
k 1+x2+1k− √C/ω F2(cid:16)1− √C/ω ,k,k− √C/ω ,1−√√1+x2(cid:17)
1+x2 1 1+x2 1+x2 1+ 1+x2
where x2 =−8t2 and F2(a,b,c,z) is the Gauss hypergeometric function.
ω2 1
Calculation of the dynamic structure factor
Quantum numbers of the Bethe lattice eigenstates
Consider a set of only three sites: site 1 at generation n and sites 2 and 3, connected to it, at generation (n+1).
Exchanging sites 2 and 3 leaves the Hamiltonian (4) invariant. Now examine what happens when Eq. (4) acts on
two states: |Ψ (cid:105) and |Ψ (cid:105), symmetric and antisymmetric combinations of the particle being at sites 2 and 3. The
S A
7
ρ(n)
n
FIG.6. Probabilitiesofthefourlowestenergystatesintheallevensector,plottedasafunctionoftheBethelatticegeneration.
C =−I/3;t=I/10.
diagonal part of the Hamiltonian will be the same for both states, and both states allow for the particle to hop to
higher generation sites. However, hopping back to site 1 is eliminated by |Ψ (cid:105), since the contributions from sites
A
2 and 3 cancel out. Similarly, each Bethe lattice generation n ≥ 1 can be assigned an even/odd quantum number,
corresponding to the symmetries that involve exchanging left and right sites at each generation. If a particle starts
at a state that is odd at kth generation, it can only hop to sites at generations n > k. It follows that the particle’s
wavefunctionhaszeroamplitudeatallgenerationsn<k. Therefore,onlytheallevenstateshaveanonzeroamplitude
at the origin.
Powder averaged dynamic structure factor
The dynamic structure factor is defined as
S((cid:126)q,∆ω)=(cid:88)δ(E −E −∆ω)|(cid:88)(cid:104)f|S+|i(cid:105)eiq(cid:126)·R(cid:126)|2.
f i R(cid:126)
f R(cid:126)
We calculate the exact expression for S((cid:126)q,∆ω) that is averaged over all directions of (cid:126)q for the Bethe lattice problem.
ThisgivesustheapproximatedynamicstructurefactorS(q,∆ω)thatcanbemeasuredinapowderaveragedneutron
scattering experiment on quantum spin ice, in the limit of weak dilution.
The ground state |i(cid:105) of the Coulomb problem on the Bethe lattice can be written down in the following form:
|i(cid:105)=α|0(cid:105)+β|1(cid:105)+γ|2(cid:105)+...
where the Greek letters represent the ground state’s amplitude per lattice generation. In the course of a scattering
experiment, a neutron coming in flips a spin. Since we restrict ourselves to having at most one free monopole with
charge ±2q , the spin flip either nucleates a monopole at the vacancy, eliminates it, or shifts an existing monopole
m
by a lattice spacing:
(cid:88)eiq(cid:126)·R(cid:126)S+|i(cid:105)=α(cid:0)4eiq|1(cid:105)(cid:1)+β(cid:0)eiq|0(cid:105)+2ei2q|2(cid:105)(cid:1)+γ(cid:0)ei2q|1(cid:105)+2ei3q|3(cid:105)(cid:1)+...
R(cid:126)
R(cid:126)
Since the ground state belongs to the all even sector, the signs of the amplitudes for the sites at the same generation
are equal, and the contributions from the spin flips add up (hence the numerical prefactors in the expression above).
Whenwetaketheinnerproductof(cid:80) eiq(cid:126)·R(cid:126)S+|i(cid:105)withexcitedstateswhichcontainoddquantumnumbers, however,
R(cid:126) R(cid:126)
the hops to sites at the same generation cancel out. Therefore, only the all even states contribute to the powder
averaged structure factor. We can see how these states are distributed in real space by plotting their probabilities as
a function of Bethe lattice generations n (Fig. 6). In particular, the ground state is localized near the origin, which
leads to the scattering intensity being high for lower energy states, and low for the states that accumulate near the
continuum band edge, whose wavefunctions near the vacancy differ in amplitude, but not much in shape.