Anomalous Hall metals from strong disorder in class A systems on partite lattices

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by  Eduardo V. Castro

Anomalous Hall metals from strong disorder in class A systems on partite lattices
Eduardo V. Castro,1,2,∗ Raphael de Gail,3 M. Pilar López-Sancho,3 and María A. H. Vozmediano3
1CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2Beijing Computational Science Research Center, Beijing 100084, China
3Instituto de Ciencia de Materiales de Madrid, CSIC,
Sor Juana Inés de la Cruz 3, Cantoblanco, E-28049 Madrid, Spain
(Dated: February 1, 2016)
Topological matter is a trending topic in con- intwodimensions,weexpectthefollowingsituation: All
6 densed matter: From a fundamental point of states will be localized in the orthogonal class AI (time
1 view it has introduced new phenomena and tools, reversal symmetry T with T2 =1 preserved); a mobility
0 and for technological applications, it holds the edge11, i.e., a well defined energy separating a region of
2
promiseofbasicstablequantumcomputing. Sim- extended states from the localized states, is expected in

n ilarly, the physics of localization by disorder, an the symplectic class AII (time reversal symmetry with
a oldparadigmofobvioustechnologicalimportance T2 = −1 preserved). Finally, in the unitary class A (T
J
in the field, continues revealing surprises when broken), extended states can remain at particular ener-

8 newpropertiesofmatterappear. Thisworkdeals gies. OnlyclassesAandAIIsupporttopologicalindices.
2
with the localization behavior of electronic sys- TheprototypicalexamplesinclasseAaresystemsshow-

tems based on partite lattices with special at- ing the integer quantum Hall effect (IQHE) and anoma-
]
n tention to the role of topology. We find an un- lous quantum Hall systems, the later exemplified by the
n expected result from the point of view of local- Haldane model12. Spin Hall systems13,14 belong to class
-
ization properties: A robust topological metallic AII.
s
di state characterized by a non–quantized Hall con- Theinterplayoftopologyandlocalizationwasfirstan-
. ductivity arises from strong disorder in class A alyzed in the context of the robustness under disorder
at (time reversal symmetry broken) insulators. The of the Hall conductivity quantization in the IQHE15–18.
m key issue is the nature of the disorder realiza- This is an example of a Chern insulator that belongs to
- tion: selective disorder in only one sublattice in symmetryclassA(alldiscretesymmetriesarebroken)in
d systems based on bipartite lattices. The gener- the standard classification. The mechanism for localiza-
n ality of the result is based on the partite nature tion in both topological classes A, and AII, is referred
o
of most recent 2D materials as graphene or tran- to as “levitation and annihilation"19. For moderate dis-
c
[ sition metal dichalcogenides, and the possibility order, the states in the edges of the conduction and va-
of the physical realization of the particular disor- lence bands start to localize. As disorder increases, the
1 der demonstrated in1. An anomalous Hall metal gap is totally populated by localized states and the ex-
v
arises also when the original clean insulator is tended states carrying the Chern number shift towards
0
1 topologically trivial. one another and annihilate leading to the topological
9 phase transition. The difference between the two classes
7 isthat,whileinthesymplecticclassAII,afiniteregionof
0 I. INTRODUCTION extendedstateswithawelldefinedmobilityedgeremains
.
1 until the transition takes place, there isno mobility edge
0 After the seminal work of P. Anderson2 it was under- in the unitary class A systems. The extended states car-
6 stood that in a non-interacting two dimensional elec- rying the Chern number are located at particular single
1 tron system at zero temperature in spacial dimension energies.
:
v D ≤ 2 and in the thermodynamic limit, the electronic We use the Haldane model12 as a typical example of
Xi wave function will be localized by disorder. In more class A system based on a bipartite lattice. As it is
realistic situations the scaling theory of localization al- known, depending on the parameter values, the model
r
a lowed a classification of the localization behavior of ma- can represent a Chern or a trivial insulator. The main
terials into universality classes set by symmetry and result of this work is the finding of an extended region
space dimensionality3,4 based on the Altland-Zirnbauer of delocalized states with a well defined mobility edge
sets of random matrices5. The advent of topological for strong disorder in class A systems when disorder is
insulators6–8 provided a new class of delocalized states, selectively distributed in only one sublattice. Why this
theedgestates,robustunderdisorderprovidedsomedis- is surprising is because there is no mobility edge in this
crete symmetries were preserved. The symmetry classes class. Hence our result implies that the standard classi-
werethenadaptedtoincludethetopologicalfeaturesand fication has to be modified. Moreover the final metallic
a “tenfold way" classification was set9,10. state is an anomalous Hall metal even in the case when
Centering the attention on the three non–chiral sym- thecleanstartingsystemisatopologicallytrivialinsula-
metryclassesoftheoriginalWigner–Dysonclassification tor.
2
Potential (Anderson) disorder is implemented by
adding to the Hamiltonian the term (cid:80) ε c†c , with
i∈A,B i i i
a uniform distribution of random local energies, ε ∈
i
[−W/2,W/2]. Wewilldiscusstwocases: disorderequally
or selectively distributed among the two sublattices. For
selective disorder the sum runs only over one sublattice.
TheHaldanemodelbelongstosymmetryclassAwhere
thedifferenttopologicalphasescanbecharacterizedbya
Z–topologicalnumber,theChernnumberν. Intheclean
insulating system it can be computed from the single
particle Bloch states u (k) as:
n
FIG. 1. Phase diagram of the Haldane model as a function 1 (cid:90)
oftheparametersMandφwith|t /t |<1/3. Thecondition ν = Ωn(k)dS, (2)
√ 2 1 n 2π z
|M|=3 3t2sinφsetstheboundarybetweenatrivial(Chern S
number ν=0 ) and a topological insulator ν =±1. where the integral is over the unit cell and Ωn(k) is
z
the z component of the Berry curvature: Ωn(k) =
∇ ∧A (k)definedfromtheBerryconnection: A (k)=
In addition to its fundamental interest, the physics k n n
(cid:104)u (k)|−i∇ |u (k)(cid:105). Thenontrivialtopologyofmetal-
of this work can be realized in actual material systems. n k n
lic states (anomalous Hall systems) is associated to a fi-
Many of the 2D materials relevant for technological or
nite, non quantized Hall conductivity that can be com-
fundamentalphysicsarebasedonbipartitelattices. Most
puted using a Kubo formula. The main technical diffi-
prominent examples like graphene and its siblings sil-
culty in addressing disordered systems is the breakdown
icene, germanene, or stanene; black phosphorus, boron
of translational symmetry which prevents working di-
nitride, or transition metal dichalcogenides MX (M=
2
rectly in momentum space. The subject being very old,
Mo, W and X= S, Se) are based on the Honeycomb
many numerical and analytical tools have been worked
lattice20. The experimental possibility of inducing dis-
out to deal with this oddity. Topological systems share
order selectively in one sublattice only, has been proven
the same problem as most topological indices are natu-
in1.
rallydefinedinkspace. Wehaveusedanumericalrecipe
based on the Kubo formula a to compute the Hall con-
ductivity in the disordered tight binding model similar
II. MODEL AND METHODS
to that described in23.
The localization behavior of the system has been ex-
We use the Haldane model12 as a generic example of
plored with standard tools: Level spacing statistics, and
a Chern topological insulator. The Haldane model tight
inverse participation ratio (IPR)10. A transfer matrix
binding Hamiltonian can be written as
method24 hasbeenalsousedtocomputethelocalization
(cid:88) (cid:88)
H =−t c†c +−t e−iφijc†c (1) length and confirm the presence of a mobility edge.
i j 2 i j
<ij> <<ij>>
(cid:88)
+M ηic†ici+H.c., III. WARMING UP: DISORDER EQUALLY
i DISTRIBUTED IN BOTH SUBLATTICES
wherec =A,B aredefinedinthetwotriangularsublat-
i
tices that form the honeycomb lattice. The first term t We first present the case of Anderson disorder equally
representsastandardrealnearestneighborhoppingthat distributed in the two sublattices which shows the stan-
links the two triangular sublattices. The next term rep- dard behavior of class A systems: extended states car-
resentsacomplexnextnearestneighborhoppingt2e−iφij rying the topological index remain at singular energies,
acting within each triangular sublattice with a phase φ approacheachotherasdisorderincreases(levitation)and
ij
that has opposite signs φ = ±φ in the two sublattices. merge (annihilation of the topological index). Figure 2
ij
This term breaks time–reversal symmetry and opens a shows the spectrum for the Haldane model with Ander-
non–trivial topological gap at the Dirac points. The son disorder equally distributed over the two sublattices
last term represents a staggered potential(η = ±1). It for a disorder strength W = 3t. The dots correspond to
i
breaks inversion symmetry and opens a trivial gap at a given eigenenergy for a given disorder realization in a
the Dirac points. The topological transition occurs at finite lattice with size d = 30. We have performed 1000
√
|M| = 3 3t(cid:48)sinφ as indicated in Fig. 1. We have done disorder realizations. Superimposed to the spectrum we
our calculations for the simplest case φ = π/2 and a show the level spacing variance as a function of energy.
typical value t =0.1t. M has been set to zero except Thevarianceofthelevelspacingvariationcontainsinfor-
2
when analyzing the topologically trivial case. A physi- mationonthelocalizationofthestatesatagivenenergy
calrealizationofthemodelwithopticallatticeshasbeen region (details can be found in the supporting informa-
presented in21(see also22). tion). Gaussian Unitary Ensemble (GUE) and Poisson
3
FIG. 2. Level statistics analysis for the Haldane model
with Anderson disorder W= 3t equally distributed over the
twosublattices. Statesarelocalized(Poissondistribution)all
alongtheenergyrange. Extendedstates(GUE)arefoundat
the two singular energies marked in the figure (green lines).
This result agrees with the analysis done in Ref.25.
FIG.4. LevelstatisticsanalysisfortheHaldanemodelwith
Andersondisorderselectivelydistributedoveronesublattice.
ThereddottedhorizontallinemarksthevarianceoftheGUE
associated to the presence of extended states. As disorder
increases, the singular energies where extended states were
located at moderate disorder strength W/t=20−30, evolve
FIG. 3. Localization transition studied through level statis-
to a full extended region of delocalized states with a well
ticsfortheHaldanemodelwithAndersondisorderequallydis-
defined mobility edge.
tributed over the two sublattices. The horizontal green line
marks the GUE variance. The two extended states present
at W=3t merge around W=5t and annihilate as disorder in-
the extended states are separated from the localized
creases. All states become localized for W >W ∼5t.
c
states at the band edges by a well defined mobility edge.
Figure 4 shows a level spacing statistic analysis of the
system for increasing disorder strength. What we see
(P) ensemble statistics are associated to extended and
in the figure is the statistics associated to two charac-
localized states respectively. It is clear that there are
teristic energies in the spectrum: one at the edge (blue
two extended states, one below the gap and another one
line) where states start to localize first when disorder is
above, where the variance clearly approaches the GUE
introduced, and one at the middle of the band (green
variance 0.178. These results are in perfect agreement
with those presented in Ref.25. In Fig. 3 we show the line) where extended states are expected to persist up to
higher disorder strength. The red dotted horizontal line
level statistics variance and the DOS for three different
marksthevarianceoftheGUEassociatedtothepresence
disorder strengths: W = 4t,5t,6t. Levitation and anni-
ofextendedstates. Weseethat,asdisorderincreases,the
hilation is clearly operative, and the critical disorder for
singular energies where extended states were located at
localization is in good agreement with that obtained in
ref.26 for the topological transition, 4t<W <5t. moderatedisorderstrengthW/t=20−30,evolvetoafull
c
extended region of delocalized states with a well defined
mobility edge. Fig. 5 shows that the extended region of
delocalizedstatesisarobustfeaturethatpersistsuptoa
IV. MAIN RESULTS
disorder strength of W=200t. We have also set up a cal-
culation of the localization length via a transfer matrix
A. Disorder selectively distributed in only one method to confirm the presence of the mobility edge.
sublattice: Topological model Thetopologicalnatureofthemetallicstateisreflected
in the calculation of the Hall conductivity shown in Fig.
The first unexpected result obtained is that, for se- 6. In our previous publication26 we showed that the
lectively distributed disorder in only one sublattice, the Cherninsulatorsufferedatopologicaltransitiontoatriv-
class A system ends up in a robust metallic state where ial state at a critical disorder strength around W ∼50t.
c
4
FIG. 5. Level spacing variation for increasing disorder
strengths W/t in the selectively distributed disorder case.
ThemiddleregionhasthesamevarianceasthatofGUEand
correspondstoextendedstates. Eventhoughthetransitionis
becomingsharper,theregionisnotshrinking. Thisisaclear
evidencefortheexistenceofanextendedregionofdelocalized
states. Amobilityedgeinthecenterofthebandhasemerged
from the singular, isolated energies by increasing disorder.
FIG.7. Levelspacingvariationforthedisorderedtrivialinsu-
lator in the selectively distributed disorder case. The results
are very similar to these in the topological case in Fig. 5.
FIG. 6. Hall conductivity of the resulting metallic state
emerging from the Chern insulator for disorder strengths
above the critical value for the topological transition. The
conductivity is not quantized and depends on the chemical
potential. Despite the big numerical error bars a finite non-
zero value can be granted.
What we see here is the further evolution to an anoma-
lousHallmetalwhendisorderisfurtherincreasedandthe
metallic state is well established. The panels in Fig. 6
showthattheHallconductivitystaysfiniteinthemetal-
FIG.8. AcomparisonoftheHallconductivityoftheresulting
lic region for W > W . The different curves correspond
c metallic state emerging from the topological (left hand side)
todifferentsizesofthesystem. Weseethatforincreasing
andtrivialinsulator(rightandside). AfiniteHallconductiv-
system sizes σ is not decreasing what proofs that we
xy ity is obtained in both cases.
arenotdealingwithafinitesizeeffect. Despitethelarge
numerical error bars (bigger for smaller sizes), a finite
conductivity can be granted.
gapwithzeroChernnumber. Thelocalizationproperties
B. Disorder selectively distributed in only one areshowninfig. 7. Thefinalstateismetallicwithawell
sublattice: Topologically trivial case definedmobilityedge. Thetopologicalnatureofthefinal
state is reflected in the Hall conductivity shown in Fig.
Thesecondunexpectedresultisobtainedwhenanalyz- 8. WecomparethetopologicalM =0casewiththetriv-
√
ingthetrivialcase. Asitwasmentionedwhendescribing ial |M| > 3 3t sinφ for t /t = 0.1eiπ/2. We have used
2 2
the Haldane model, the parameters can be tuned to de- larger system sizes, and higher number of disorder real-
scribeatrivialinsulatorforvaluesofthestaggeredpoten- izations; 105, 105, 104, and 5000 disorder realizations,
√
tial |M|>3 3t sinφ. In order to ascertain the possible respectively for d = 6, 12, 18, and 22. Note that σ is
2 xy
roleoftopologyinthedevelopmentofthemetallicphase, finiteonlyintheenergyregionwherestateshavealarger
we have analyzed the localization behavior of the trivial amplitude in the non-disorder sublattice. These findings
casewithgenericvaluesoftheparameterschosensothat agree well with the results for the level spacing variance
T is still broken but the original insulator has a trivial discussed above.
5
V. UNDERSTANDING WHAT IS GOING ON: propagate to a very long distant site by hopping to the
SIDE QUESTIONS. clean sublattice, propagate there, and hop back. The
probability of the process is suppressed to be of order
This work rises a number of additional questions. We αt2 but it is never zero. This also explains why the final
have addressed some of them, others remain open. metallic state is a topological metal since the anomalous
For simplicity, a purely imaginary value of the t pa- Halleffectisduetotheinterbandmatrixelementsofthe
2
rameter t = i0.1t has been used through the work current operators27.
2
(φ = π/2). This choice induces an accidental particle– The different behavior of IQHE and anomalous Hall
hole symmetry to the system that technically belongs to metals under disorder has been examined in refs.28,29.
classD.Wehaveperformedsomeadditionalcalculations The properties of disordered topological metals arising
with a non–zero real part for t to ensure that we are from clean topological insulators have been analyzed in
2
discussing class A physics and found no qualitative dif- ref. 30. The metallicity of the final state seems to be
ferences. It could be thought that, since disorder affects at odds with the non-linear sigma model results10,31 so
onlyonesublattice,thefinalmetallicstatecoincideswith it would be very interesting to implement the selective
the trivial metal of the triangular lattice. The result of disorder case in this approach.
the Hall conductivity makes obvious that this is not so. The physics described in this work can be realized in
We have performed some calculations of partial IPR and topological materials based on other more complicated
sawthatthewavefunctionofextendedstateshasalways partite lattices32. The results presented in this work are
someweightinthedisorderedsublattice. Itisinteresting conceptually important, although we recognize that to
tonotethatthisissoalsointhecaseofvacancydisorder implement the selectively distributed disorder might be
when the disordered sublattice is depleted. a hard task. To this respect, we note that experiments
Irrespective of the topological character of the clean have been done in graphene where defects are located
system,thefinalstateintheclassAanalyzedisaTbro- selectively in one sublattice to check the magnetic prop-
ken disordered metal with finite Hall conductivity. Our ertiesofthesystem1. Artificial33 oropticallattices21 are
resultsshowthat,whileatopologicalinsulatorcanbebe- other possibilities to realize this physics.
come trivial by an appropriate tuning of the parameters
as happens in the Haldane model, the anomalous Hall
metal is a very robust and a stable phase for T broken ACKNOWLEDGMENTS
metals in the absence of an external magnetic field.
Aswementionedabove,evenwhenforthehighestval- We gratefully acknowledge useful conversations with
ues of disorder, the disordered sublattice is never decou- Alberto Cortijo, Belén Valenzuela, Fernando de Juan,
pled from the ordered one. The physics that we observe Adolfo G. Grushin, and J. A. Vergés. EC acknowl-
all along the work is that of the disordered Honeycomb edges the financial support of FCT-Portugal through
lattice, as proven by the fact that the extended states grant No. EXPL/FIS-NAN/1728/2013. This research
foundintheextremedisordercasehaveanonzeroweight was supported in part by the Spanish MECD grants
in the disordered sublattice. Even though the weight in FIS2014-57432-P, the European Union structural funds
the disordered sublattice is orders of magnitude smaller and the Comunidad de Madrid MAD2D-CM Program
than in the clean sublattice, the two sublattices do not (S2013/MIT-3007),theEuropeanUnionSeventhFrame-
“decouple". This explains the metallic nature of the fi- work Programme under grant agreement no. 604391
nal state: An electron in the disordered sublattice can Graphene Flagship FPA2012-32828.
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