Effective field theory of the disordered Weyl semimetal

0.68 MB·English

by  Alexander Altland

Effective field theory of the disordered Weyl semimetal
Alexander Altland and Dmitry Bagrets1
1Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany
(Dated: January 27, 2015)
In disordered Weyl semimetals, mechanisms of topological origin lead to the protection against
Andersonlocalization,andatthesametimetodifferenttypesoftransverseelectromagneticresponse
– the anomalous Hall, and chiral magnetic effect. We here apply field theory methods to discuss
themanifestationofthesephenomenaatlengthscaleswhicharebeyondthescopeofdiagrammatic
perturbation theory. Specifically we show how an interplay of symmetry breaking and the chiral
5 anomaly leads to a field theory containing two types of topological terms. Generating the uncon-
1 ventionalresponsecoefficientsofthesystem,thesetermsremainlargelyunaffectedbydisorder,i.e.
0 information on the chirality of the system remains visible even at large length scales.
2
PACSnumbers: 75.47.-m,03.65.Vf,73.43.-f
n
a
J
Weyl semimetals are paradigmatic examples of gap- a system comprising two Weyl nodes separated in mo-
6
less topological condensed matter systems. A Weyl mentumspaceshowsananomalousHalleffect(AHE)[9].
2
semimetal comprises an even number of linearly dis- Within the field theoretic framework below, this effect

] persive band touching points embedded in a three- willderivefroma3dextensionofatwo-dimensionaltopo-
l
l dimensional Brillouin zone. The presence of these hot- logical θ-term, familiar from the theory of the quantum
a
spotsimpliesaresponsetoperturbationsthatisinterme- Hall effect.
h
- diate between that of metals and insulators (see Ref. [1] Field theory — In the following, we will derive an effec-
s
for review). This ‘semi-metallicness’ also signifies in the tivefeldtheorydescribingthesestructures. Ourstarting
e
m physics of the disordered system [2–5]: on the one hand, point is the bi-nodal Hamiltonian (cf. Fig. 1)
the vanishing of the nodal density of states weakens dis-
.
at order scattering cross sections; on the other hand, suf- Hˆ =vk/ˆσ3n+(v/b+µ)+V(x), (1)
m ficiently strong disorder will generate a finite band cen-
- ter density of states to eventually overpower the above where k/ˆ ≡ k·σ, σ is a vector of Pauli matrices, kˆ the
d effect. It has been shown [4] that the above competi- vector momentum operator, and v a characteristic ve-
n
tion manifests itself in the presence of a critical disorder locity. The Pauli matrix σn acts in a two-component
o 3
c strength below/above the system flows towards a clean space discriminating between two nodes split by a vec-
[ fixed point/a regime of strong impurity scattering. It is tor 2b≡2be in momentum space and an increment 2µ
3
the purpose of the present paper to derive and discuss in energy. The model is coupled to disorder by a Gaus-
1
v the effective theory describing the latter phase at length siandistributedpotentialV(x)withvarianceγ0. Wedis-
7 scales exceeding the system’s scattering mean free path. criminate between disorder correlated over length scales
3 At large length scales, impurity scattering will render (cid:38)b−1,whichissoftinthesensethatthetwoWeylnodes
5
the motion of individual excitations diffusive which sug- are not coupled by impurity scattering, and the opposite
6
0 geststhatthesystemwillendupintheuniversalityclass case of short range correlated disorder mixing the nodes.
. ofthe3dAndersonmetal(i.e. abovethephasetransition Implicittothemodelisahighmomentumcutoff|k|<Λ
1
0 point separating a 3d metal from an insulator.) This ex- limiting the range of linearizability of an underlying lat-
5 pectation is, in fact, a certainty given that a single Weyl tice model.
1 node may be interpreted as an effective surface theory of To access the transport properties of the system at
v: a bulk 4d topological insulator; finite conduction is pro- energies (cid:15), we introd´uce a supersymmetric generating
i tected by topology. At the same time, topology implies functional Z[a] = ´ D(ψ¯,ψ) exp(−S[ψ¯,ψ]), with ac-
X
a number of differences distinguishing the Weyl system tion S[ψ¯,ψ,a] = −i d3xψ¯((cid:15)+iδτ −Hˆ)ψ. Here ψ =
3
r from a generic metal: First, an individual Weyl node {ψα (x)} is a sixteen-component field of integration
a s,i,n
breaks parity symmetry, and it is known (e.g. from the variables where, n = 1,2 labels the two nodes, i = 1,2
exampleofthed-wavesuperconductingquasiparticlesys- the components of a Weyl spinor, s = ± distinguishes
tem[6])thatthebreakingofdiscretesymmetriesisgener- between advanced and retarded (ar) Green functions,
allyremembered,eveninthepresenceofstrongdisorder. i.e. (τ ) = sδ such that integration over ψ gen-
3 ss(cid:48) ss(cid:48) s=±
Indeed,wewillfindthatthelowenergytheoryofindivid- erates matrix elements of Gˆ± = ((cid:15) ± iδ − Hˆ)−1, and
ual nodes system contains a parity breaking non-abelian α = 1,2 defines a two-dimensional space (bf-space) of
Chern-Simons(CS)term,whichdescribesthesurvivalof complex commuting/Grassmann variables ψ1/2. In the
the so-called chiral magnetic effect (CME) [7, 8] in the commuting sector, ψ¯1 = ψ1†, while the commuting vari-
disordered environment. Second, it has been shown that ablesψ¯2,ψ2 areindependentGrassmannvariables. From
2
Z, transport observables may be computed by introduc-
ing suitably defined source fields a, which are presently
suppressed to keep the notation simple.
To explore the influence of disorder on the system,
we´integrate over V to generate the quartic contribution
γ dx(ψ¯ψ)2.Thefateofthisnonlinearityunderchanges
2
of the cutoff Λ has been studied [5, 10] by evaluating
the results in one- and two-loop renormalized perturba-
tion theory in 2 + (cid:15) dimensions [11] at (cid:15) = 1. It has
been found that for bare amplitudes larger than a crit-
ical value γ∗ = π2v2/Λ the effective disorder strength Figure 1. (Color online) Schematic of two Weyl nodes split
in energy and momentum by 2µ and 2b, resp.
increases under renormalization. To describe the pertur-
batively inaccessible regime beyond the scattering mean
free path, Λ−1 ≡ l ∼ γ/v2, we decouple the nonlinear-
rious anomaly, this operation is invalid, the action needs
ity by a supermatrix field B, and integrate over the then
to be regularized first. Following a strategy previously
quadratic fermions [12]´. As a result, we obtain the effec- applied to the 2d d-wave superconductor, we regularize
tive action S[B] = 1 d3xstrB2 −strln(Gˆ[B]), where
2γ bysubtractionofatermS[Q]≡S0[Q]−Sη[Q],whereSη
Gˆ[B] = ((cid:15)+iδτ −Hˆ −B)−1, Hˆ is the clean Hamil- differs from S by a replacement κ→η (cid:38)0 and setting
3 0 0 0
tonian and str is the supertrace [12]. The difference be- (cid:15) = 0. In the limit η → 0 the Q-dependence of S [Q]
η
tweenthecasesofhardandsoftdisorder,resp.,isthatin drops out so that S [Q] = 0. On the other hand, for
η→0
the former/latter case the two nodes couple to the same large momenta v|k|(cid:29)q and fixed η, the two action con-
(B = B ⊗In)/independent (B = bdiag(B ,B )n) ma- tributions cancel against each other, i.e. S[Q] is UV reg-
1 2
trix fields. For definiteness, we first consider the soft ularized(inagauge-invariantway.) Thesimilaritytrans-
case, for which the two nodes can be discussed sepa- formationmaynowsafelybeappliedtobothS ,sothat
0,η
rately; the effect of impurity mixing can be described we obtain an effective action S[Q] = (S [A]−S [A]) ,
0 η reg
by a locking B = B at any later stage. Focusing on where the subscript ‘reg’ means that only UV finite con-
1 2
node n = 1 and writing B = B for notational simplic- tributions to the subsequent expansion in A are to be
1
ity, we proceed by subjecting the action to a mean field kept. Notice that in the language of the A-fields, the
analysis. A variation of the action yields the equation local gauge invariance is no longer manifest. Rather,
B¯ =! γtrGˆ(x,x;[B¯]),whichissolved[11]bythediagonal Ai → k−1(A + k∂ik−1)k transforms as a non abelian
ansatz, B¯ = −iκτ describing the ‘spontaneous symme- gauge field, and invariance of the action becomes a non-
3
try breaking’ of the infinitesimal δ a finite impurity self trivial consistency condition.
energy κ = κ((cid:15)). Specifically, at (cid:15) = 0 (semimetal) one In the expansion of the action, we keep terms of order
finds κ = (2/π)vΛ(1−γ∗/γ) [13], while far away from two (O(∂A,A2)) and three (O(A3,∂A)) derivatives. To
Weyl node (metal) κ = γπν, where ν = (cid:15)2/2π2v3 is the second order we obtain the result
ˆ
clean density of states. σ1 (cid:88)
S [A]=− xx d3xstr([A ,τ ]2),
Turning to the analysis of fluctuations around the d 8 i 3
mean field, we focus on configurations B ≡ iκQ ≡ iˆ
iκTτ3T−1, whose action vanishes in the limit of slowly S [A]= σx1y(cid:15)3ij d3xstr(τ ∂ A ), (2)
varying fluctuation matrices T(x) → const. Here, T ∈ top 2 3 i j
G/H takesvaluesinaGoldstonemodecosetspacewhere
where the longitudinal and Hall conductivity of node 1,
G = GL(2|2) is the group of 4 × 4 invertible super-
σ1 and σ1 are determined by the microscopic model
matrices and H = GL(2|2) the subgroup of matrices xx xy
parameters as discussed below. We note that the ac-
k commutative with the mean field, [k,τ ] = 0. Since
3 tion (2) affords the man´ifestly gauge invariant refor-
sSt0r[(QQ]2≡) =str0l,n(t(cid:15)h−e vack/t+ioniκQwe).nFoowrlnaeteedrrteofecroenncsied,ewrerenaodtes mσxuyl(cid:15)a3tiijo´ndSxds[tQr(]Q=∂ Q−∂σ8xQx).dT3xhestrfi(r∂sQt 2o)f tahnedseSctoopn[tQri]bu=-
that the action has two continuous symmetries: a global 8 i j
tions has been constructed in Ref. [11] on phenomeno-
symmetry Q→T QT−1, where T ∈G is constant, and
0 0 0 logical grounds within a non-regularized framework. (In
a local gauge symmetry T →Tk, k =k(x)∈H.
ourcurrentapproachregularizationplaysavitalrole;the
Effective field theory — Our goal is to expand the ac- ‘crossterm’Aτ Aτ ofthecommutator[A,τ ]2isweighed
3 3 3
tion in gradients T−1∂iT ≡ Ai. One might be tempted with an UV divergent ‘fermion bubble’. Only after reg-
to start this program with a similarity transformation, ularization it combines with the finite coefficient of A2
S [Q]=? strln(T−1((cid:15)−vk/+iκQ)T)=strln((cid:15)−vk/+ivA/+ to a gauge invariant commutator.) Before discussing the
0
iκτ )≡S[A], which may then be expanded in powers of physics of these expressions, we complete the derivation
3
the‘non-abliangaugefield’A. However,duetothenoto- of the action and consider terms of cubic order in A.
3
ThetermsofO(A3)arethe‘trianglegraphs’pervasive crossover,theconductanceg takesvaluesofO(1),close
xx
inthetheoryof(2+1)or(3+0)dimensionalrelativistic to but larger [18] than the critical value g∗ marking the
gauge theories. On general grounds [14] we expect the 3d Anderson transition.
appearance of a Chern-Simons action at this order. A
The bare coefficient σ1 multiplying the action S
straightforward if lengthy calculation indeed yields the xy top
is the contribution of node 1 to the Hall conductiv-
result
ity of the system at crossover length scales to the dif-
fusive regime. As in the derivation of the field the-
S [A]=SI [A]+SII[A], (3)
CS CS CˆS ory of the quantum Hall effect [19] we obtain σ as
xy
SI [A]= i(cid:15)ijk (cid:88)s d3xstr(A Ps∂ A Ps), a thermodynamic coefficient σxy = V−1∂BN, where
CS 8π i j k V ≡ L L2 is the volume of the sample, and the deriva-
s=± z
ˆ tive probes the sensitivity of the number of states N =
SII[A]= i(cid:15)ijk (cid:88)s d3xstr(A PsA PsA Ps), −1 ´0 d(cid:15)Imtr(G+((cid:15))) below zero energy to the pres-
CS 12π i j k π −∞
s=± ence of an external magnetic field in 3-direction. Within
the framework of the gradient expansion, G+ is the re-
where P± is a projector on advanced/retarded indices.
tarded Weyl node Green function coupled to disorder
Apartfromthepresenceoftheseprojectormatrices,this
in the mean field/self consistent Born approximation.
hasthecharacteristicstructureofanon-abelianCSterm.
Following the strategy of Ref. [9], we evaluate this ex-
(However,(3)doesnotdefinea‘real’CSaction,inasmuch
pression by interpreting the quantized values of the 3-
as A does not describe a field-gauge coupling, but repre- momentum, m ≡ v(b+k ), k = 2πnL−1 as masses
n 3,n 3 z
sents the nonlinear σ-model field itself; the situation is
entering effective 2d layered Dirac Hamiltonians govern-
conceptuallysimilartothatconsideredinRefs.[15,16].)
ing the planes perpendicular to the 3-direction. Within
The CS action does not afford a representation in (cid:80)
this interpretation, σ = (2π/L ) σ is obtained
terms of Q-fields, which reflects the lack of com- xy z n xy,n
by summing over the contribution of the layers, where
plete gauge invariance of this action piece [14]: it is
σ = C /2π is given by the disorder averaged Chern
xy,n n
straightforward to verify that under a gauge transfor- (cid:80)
number, C =(1/2π) arctan[(m +σ(cid:15))/κ]+1/2,
mation by k ≡ bdiag(k ,k )ar ∈ H, and for a ficti- n σ=± n
+ − of the nth layer. Since only changes of C at zero cross-
n
tious system without boundaries, the CS action trans-
ings of the effective masses can be unambiguously de-
forms as SC´S[A] → SCS[A] + Stop[k], where Stop[k] = termined from the linearized theory, we fix the absolute
−i (cid:80) s d3xstr(k−1∂k )∧3. The integral yields a
24π s=± s s value of the sum by the condition σxy = 0 at b = 0,
quantized value, viz. 24π2×n , where n is the is wind-
s s which follows from matching to the band structure of
ing number of a configuration kff ∈ SU(2) in three di-
s the bi-nodal Weyl system. In the limit Lzb (cid:29) 1, this
mensional space. (The non-compact bb sector of the su- gives σ1/2 =b/2π for both nodes, irrespective of the en-
xy
permatrices k is topologically empty.) For ‘large’ gauge
ergy (cid:15) or the disorder strength κ. Both for soft and hard
transformations with non-vanishing winding numbers,
disorder, the two contributions to the Hall conductivity
the CS action changes by a factor iπ(n + n ). The
+ − add, and we obtain σ = b/π, a result known as the
origin of this phenomenon was explained in the clas- xy
anomalous Hall effect (AHE).
sic reference [14], where it was shown that these factors
get canceled by a gauge anomaly of the regulator action The CS contribution to the action accounts for the
S → S +iπ(n +n ), where the appearance of the thermodynamic response of the system to imbalances
η η + −
extra terms is caused by zero-crossings of the regulariz- (b,µ) between the nodes. That this term probes equi-
ing Dirac operator under a large gauge transformation. librium properties follows from the presence of the pro-
Overall the action is gauge invariant. jector matrices Ps which prevent coupling between the
Discussion—TheactionS +S ofthediffusionmodes retarded and advanced sector of the theory (a necessary
d top
A appeared for the first time in connection with the ingredient to any type of dynamic response.) A finite
multi-layer quantum Hall effect [17], a system conceptu- equilibrium response is obtained if we couple the sys-
ally similar to the present one if the 3-direction of Weyl tem to an external field a = {ai}, where ai = B2(cid:15)3ijxj,
node splitting is interpreted as the stacking direction of i = 1,2 represents an external magnetic field Be3, and
a layered system of 2d class A topological insulator lay- a3 = a(x)σ3bf ⊗ τ3 is a source field. The latter is de-
ers [9]. The action S controls the fluctuations of diffu- fined in such a way that differentiation of the parti-
d
sion modes in terms of the dimensionless coupling con- tion function, 4iπδa(x)Z[a] = −π1Im(cid:104)tr[G+(x,x)σ3](cid:105) =
stant g ≡ σ1 Λ−1 Within the framework of our gra- (cid:104)j (x)δ((cid:15)−Hˆ)(cid:105)≡j yieldsthecontributionofstatesat
xx xx 3 3,(cid:15)
dient expansion we find σ = ((cid:15)2 +3κ2)/6πκv, which energy(cid:15)totheequilibriumvalueofthe3-currentdensity
xx
simplifies to σ1 = κ/2πv at the Weyl node and asymp- of node 1. We compute this expression by adding the
xx
totes to the Drude conductivity σ1 = v2/3γ at higher externalfieldtotheinternalone, A→A+a, andsubsti-
xx
energies (cid:15) (cid:29) κ. At the nodes, (cid:15) = 0, and at bare length tuting this configuration into the CS action. In the sim-
scales Λ ∼ l−1 characteristic for the ballistic/diffusive plest approximation A = 0 (for the above ‘equilibrium’
4
choiceofsourcetermsfluctuationcorrectionsaroundthe persymmetricfieldtheorydescriptionofdisorderedWeyl
A = 0 vanish by su´persymmetry anyway), we then ob- semimetals and metals at length scales exceeding the
tain S [a]=−iB dxa(x), and hence j = 1 B. To meanfreepath. Theprofileofthetheoryisessentiallyde-
CS π 3,(cid:15) 4π2
obtain the full response of the system, we need to add terminedbyaninterplayofsymmetryconditionsandthe
the (opposite) contribution of the second node and inte- chiral anomaly. This mechanism stabilizes metallic be-
grate over filled energy states up to some Fermi energy havioratlargelengthscales,alongwithvariousdisorder-
(cid:15) . Taking into account that the existence of a bare lin- insensitive response coefficients of topological origin.
F
earizationcutoffΛimpliesacutoff|(cid:15)|<|(cid:15) ±µ|, (cid:15) ≡vΛ Acknowledgments: We acknowledge discussions with P.
0 0
for the accessible energy states (cf. Fig. 1), this leads to Brouwer, M. Hermanns, and S. Trebst. Work supported
´ ´
j = EF d(cid:15)j − (cid:15)F d(cid:15)j = µB, i.e. an equilib- by SFB/TR 12 of the Deutsche Forschungsgemeinschaft.
ri3um c−u(cid:15)r0r+enµt pr3o,p(cid:15)orti−on(cid:15)0a−lµto an3,(cid:15)exter2nπ2al magnetic field, Inthefinalstagesofthepreparationofthispaperwebe-
the so-called chiral magnetic effect (CME) [7, 8] (for a came aware of arXiv:1412.7678 where an action similar
discussion how the nonvanishing of this expression may to ours is motivated from a different perspective, viz. by
be understood from the perspective of the Fermi-liquid dimensional reduction from a bulk 4d-topological insula-
theory, see Ref. [20]). tor.
Renormalization – What happens if short distance fluc-
tuations in the field theory are integrated out to probe
thephysicsatlengthscalesbeyondtheballistic/diffusive
crossover regime? An answer to this question has been
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(Cambridge University Press, Cambridge, 1997).
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3 B [13] For γ <γ∗, the mean field equations have only a trivial
meansthatwithintheframeworkofourtheorytheCME,
solution κ = 0, which is consistent with the presence of
too, fully protected against renormalization by disorder. a critical point at ∼γ∗.
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[18] Evidence that we start out on the metallic side of the
and renormalized values of the longitudinal conductance transition follows from (i) the topological protection
g : intheformer/lattercase, g isinitiallysmall/large against localization, (ii) matching with the RG flow of
xx xx
to begin with. However, in either case, g increases, the underlying ψ-field theory which is directed towards
xx
andasymptotestoOhmicbehavioratlargelengthscales. metallicbehavior,and(iii)thefactthattheRGanalysis
oftheσ-modelind=2+(cid:15),(cid:15)=1,dimensionspredictsa
While the system then behaves similar to a three di- √
value (cf. e.g. Ref.[17]) g∗ =1/ 3π2.
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Summarizing, we have microscopically derived a su- 181602 (2012).