Composite Dirac liquids: parent states for symmetric surface topological order
David F. Mross,
∗
Andrew Essin,
∗
and Jason Alicea
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
We introduce exotic gapless states—‘composite Dirac liquids’—that can appear at a strongly interacting
surface of a three-dimensional electronic topological insulator. Composite Dirac liquids exhibit a gap to all
charge excitations but nevertheless feature a single massless Dirac cone built from emergent
electrically neutral
fermions. These states thus comprise electrical insulators that, interestingly, retain thermal properties similar
to those of the non-interacting topological insulator surface. A variety of novel fully gapped phases naturally
descend from composite Dirac liquids. Most remarkably, we show that gapping the neutral fermions via Cooper
pairing—which crucially does
not
violate charge conservation—yields symmetric non-Abelian topologically
ordered surface phases captured in several recent works. Other (Abelian) topological orders emerge upon alter-
natively gapping the neutral Dirac cone with magnetism. We establish a hierarchical relationship between these
descendant phases and expose an appealing connection to paired states of composite Fermi liquids arising in
the half-filled Landau level of two-dimensional electron gases. To controllably access these states we exploit a
quasi-1D deformation of the original electronic Dirac cone that enables us to
analytically
address the fate of the
strongly interacting surface. The algorithm we develop applies quite broadly and further allows the construction
of symmetric surface topological orders for recently introduced bosonic topological insulators.
I. INTRODUCTION
Three-dimensional topological insulators (3D TIs)
1–5
pos-
sess an electrically inert interior, yet harbor a wealth of
physics at their surfaces that simply could not exist without
the accompanying bulk. For example, photoemission stud-
ies efficiently identify TI materials by detecting an odd num-
ber of massless surface Dirac cones
6,7
—an impossible band
structure in strictly two-dimensional, time-reversal-invariant
media. Introducing magnetism or superconductivity at the
boundary gaps these Dirac cones but leaves nontrivial physics
behind. A magnetically gapped region of the surface en-
ters an integer quantum Hall state with an anomalous
half-
integer
Hall conductance
8
. Domain walls between oppositely
magnetized regions accordingly bind chiral gapless charge
modes
9,10
. Gapping the surface instead by externally im-
posing Cooper pairing generates topological superconduc-
tivity in which vortices bind widely sought Majorana zero
modes
11
. The induced topological superconducting phase,
while reminiscent of a spinless two-dimensional (2D)
p
+
ip
superconductor
12
, preserves time-reversal symmetry and
hence can not be ‘peeled away’ from the TI bulk.
With perturbatively weak electron-electron interactions the
above discussion summarizes the essence of 3D TI boundary
physics: massless Dirac cones imply preservation of both time
reversal and charge conservation, while conversely a gapped
surface necessitates breaking at least one of these symme-
tries. But does symmetry imply surface metallicity more gen-
erally? That is, with strong interactions is it possible to con-
struct a
symmetry-preserving, fully gapped
surface state for an
electronic TI? Several groups, quite remarkably, recently an-
swered this question in the affirmative
13–16
, overturning con-
ventional wisdom on the subject. An important precursor to
these studies originated from ingenious work on bosonic 3D
topological insulators
17
, which were shown to support gapped
surfaces that either break symmetries
or
exhibit topological
order that is forbidden in strictly 2D systems with the same
symmetry. Soon after symmetric gapped surface topological
orders were identified for electronic TIs as well. Interestingly,
in contrast to the bosonic case these gapped surface states nec-
essarily support non-Abelian anyons, emergent particles that
enable inherently fault-tolerant storage and manipulation of
quantum information
18,19
.
Symmetric topological orders for the electronic TI surface
have so far been accessed using two quite different means.
One approach starts from the broken-symmetry supercon-
ducting surface and then condenses a particular multiplet of
vortices to restore charge conservation
13,14,16
; the other
15
ex-
plicitly constructs the surface phases using so-called Walker-
Wang models
20
. The latter benefits from exact solvability
but sacrifices direct connection to the original electronic de-
grees of freedom. Two distinct, symmetric topological or-
ders emerge from these approaches. Using the language of
Ref. 15, the phenomenological vortex condensation picture
naturally accesses a ‘Pfaffian-antisemion’ phase. The Walker-
Wang construction also captures this state but additionally re-
veals a ‘T-Pfaffian’ phase with fewer quasiparticle types (see
also Ref. 13). Both phases relate closely to the non-Abelian
Moore-Read state
21
—also known as the Pfaffian—expected
to occur at filling factor
ν
= 5
/
2
in clean GaAs quantum
wells. We stress, however, that neither the T-Pfaffian nor the
Pfaffian-antisemion topological order can appear in isolated
2D systems preserving time-reversal and charge-conservation
symmetry.
In this paper we re-examine the strongly interacting TI sur-
face using an alternative approach that both works directly
with the original surface electrons
and
follows a controlled,
Hamiltonian-based formulation. Our main result is the dis-
covery of a new class of symmetric, strongly correlated gap-
less surface states that we christen composite Dirac liquids
(CDLs). Apart from exhibiting numerous striking physical
response properties, these gapless states also constitute ‘par-
ents’ for both kinds of gapped symmetric topological orders
noted above (among other nontrivial phases). We will show
that constructing CDLs allows us to capture the T-Pfaffian and
Pfaffian-antisemion surface phases in a single, unified frame-
arXiv:1410.4201v1 [cond-mat.str-el] 15 Oct 2014
2
Chiral edge states:
Charged semion (
ν
= 1
/
2)
Neutral semion (
ν
= 1
/
2)
Neutral fermion (
ν
= 1)
Majorana
FIG. 1. Executive summary of results. Stripping the electric charge off of the original surface Dirac cone (top left) yields a composite Dirac
liquid (top center) that exhibits a charge gap but features a gapless Dirac cone formed by
electrically neutral
fermions. Analogously stripping a
fictitious ‘pseudocharge’ from the latter yields a nested composite Dirac liquid (top right) with a second-generation neutral Dirac cone. These
composite Dirac liquids serve as parent states for topologically ordered surface phases (bottom row) obtained by gapping the neutral Dirac
cones with pairing or magnetism. Importantly, pairing proceeds
without
breaking electric charge conservation—because the paired fermions
are neutral—and hence produces symmetric surface topological orders captured by previous works
13–16
.
work that, interestingly, reveals a hierarchical relationship be-
tween these topological orders.
The simplest CDL follows by systematically stripping off
the charge from the original surface Dirac electrons—leaving
a
neutral
massless Dirac cone immersed in a gapped, fraction-
alized, bosonic background that encodes the charge physics.
Because the charge sector acquires a gap the CDL is char-
acterized by incompressible, electrically insulating behavior
for the TI surface. The neutral Dirac sea nevertheless yields
a
T
2
specific-heat contribution (
T
denotes temperature) and
underlies metallic longitudinal heat transport. In other words,
the CDL essentially retains the thermal but not charge char-
acteristics of the original surface Dirac cone, thereby sharply
violating the Wiedemann-Franz law.
As our nomenclature suggests, the CDL reflects a surface
analogue of the well studied composite Fermi liquid
22
opera-
tive in the half-filled lowest Landau level for GaAs quantum
wells, but (as usual) cannot live in 2D with the same sym-
metries. Reviewing the latter provides a useful perspective
on our results. To understand the half-filled Landau level
it proves exceedingly useful to decompose the electrons in
terms of composite fermions bound to fictitious flux quanta
that, on average, cancel the applied magnetic field. Despite
the strong Lorentz force acting on the electrons, the effec-
tively neutral composite fermions move in straight lines over
long distances and form a Fermi sea. One can profitably view
the composite fermion’s Fermi sea as a 2D counterpart of the
CDL’s neutral Dirac cone. (We stress however, that compos-
ite Fermi liquids—which are compressible, metallic states—
exhibit rather different response properties, partly for symme-
try reasons.) Composite Fermi liquids serve as mother states
for various interesting incompressible fractional quantum Hall
phases. Notably, the non-Abelian Moore-Read state arises
upon gapping the Fermi sea by ‘weakly pairing’ the com-
posite fermions; forming a ‘strong pairing’ condensate out of
tightly bound composite-fermion pairs yields Abelian descen-
dant quantum Hall states.
12
Like its composite-Fermi-liquid-cousin, the CDL provides
a convenient window for accessing nontrivial proximate
phases for the electronic TI surface. Most interesting, once
we strip off the electric charge, the neutral Dirac cone can
acquire a pairing gap
without spoiling charge conservation
symmetry
. The resulting symmetric, gapped state is precisely
the T-Pfaffian. Conversely, we show that magnetically gap-
ping the neutral Dirac cone instead yields a time-reversal-
breaking Abelian topological order corresponding to the 113
state. These CDL descendants are very similar in spirit to the
weak- and strong-pairing phases
12
that derive from 2D com-
posite Fermi liquids.
Nesting the procedure above yields surface topological or-
ders with richer structure.
In particular, one can form a
new CDL out of the neutral Dirac cone rather than the orig-
inal, charged Dirac electrons. This nested CDL supports
a second-generation neutral Dirac cone that coexists with a
further fractionalized bosonic background. Pairing the neu-
tral fermions generates the Pfaffian-antisemion state captured