REVIEW
A symmetry-respecting topologically-ordered surface phase of 3d electron topological insulators
Strongly Correlated Electrons · Submitted 14 Jun 2013 · 44 references
Pith's verdict: A careful effective-field-theory derivation of a gapped, symmetry-respecting ETI surface with Moore-Read × U(1)_{-2} order; the construction holds together, and the main open questions are microscopic realization and minimality, both acknowledged.
T0 · AI review
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Referee report
The paper
The paper claims
A topological insulator's surface need not be gapless: with strong interactions it can become a gapped, symmetric, topologically ordered phase hosting non-Abelian charge-1/4 anyons.
The reading
What carries the argument
The engine is vortex condensation on the superconducting surface, made precise by weakly gauging the global $U(1)$ so that the Abelian part of flux-tube statistics is well defined. A slab with a superconducting top surface and a $σ_{xy}=1/2$ bottom surface behaves as a 2d $p_x+ip_y$ superconductor, so its flux tubes carry Ising $×$ $U(1)_{-8}$ statistics: odd vorticity gives $σe^{ikθ}$, even vorticity gives $e^{ikθ}$ or $fe^{ikθ}$, with the $U(1)_{-8}$ charge equal to the vorticity. Subtracting the bottom-surface Chern-Simons response leaves the top-surface action, and the smallest Abelian bosonic vortex is the flux $4π$ object $e^{4iθ}$. Condensing flux $8π$ restores particle number, produces a charge-$1/4$ boson $e^{iφ}$ dual to the flux quantum, and yields a 96-anyon phase $T_{96}$; condensing $e^{4iθ}$ then reduces $U(1)_{-8}→U(1)_{-2}$ and leaves Moore-Read $×$ $U(1)_{-2}$. The reverse transition is condensation of the charge-$1/2$ boson $e^{2iφ_ρ}s̄$, an $XY^*$ transition whose $XY$ field has the Cooper pair as its fourth power, driven from the superconducting side by proliferation of flux $4π$ vortices.
Extended reading notes
Core claim
The central claim is that a symmetry-respecting gapped surface termination of an electron topological insulator exists and carries Moore-Read $×$ $U(1)_{-2}$ topological order: 24 anyon types counting the electron, 12 modulo it, with total quantum dimension $√32$ ($4$ modulo the electron). The Moore-Read part has the same anyon content as the $ν=1/2$ Pfaffian state — Ising $×$ $U(1)_8$ restricted to excitations local with respect to the electron — so the surface supports non-Abelian $σ$-type anyons of electric charge $1/4$, while the $U(1)_{-2}$ part is a neutral anti-semion $s̄$. Time reversal acts non-trivially, mapping $σe^{imφ_ρ}↔σe^{imφ_ρ}s̄$ and exchanging $e^{imφ_ρ}↔fe^{imφ_ρ}$ for $m≡2$ mod 4, with Kramers parities tabulated in Eq. (2.2). The authors then argue this order cannot be realized strictly in 2d with both symmetries: a $2π$-flux instanton would have to nucleate an anyon of charge $∓1/2$, impossible at vanishing Hall conductivity, whereas on the ETI surface the same event is a magnetic monopole tunnelling through the surface and picking up the half-integer charge required by the Witten effect. They also prove that a time-reversal-invariant interacting electron insulator with no intrinsic topological order, $θ=π$, and Kramers-singlet electrons ($T^2=+1$) cannot exist.
Load-bearing premise
The entire anyon labelling descends from one identification: that a slab of topological insulator with a superconducting top face and a $σ_{xy}=1/2$ bottom face is equivalent to a two-dimensional $p_x+ip_y$ superconductor, which fixes the flux-tube statistics to Ising $×$ $U(1)_{-8}$ and hence everything later condensed. The paper argues this carefully from the boundary Chern-Simons form of the bulk $θ$-term but does not derive it from a microscopic lattice model.
Editorial extensions
If this is right
- The gapless odd-Dirac-cone surface of an electron topological insulator is a weak-coupling description, not a theorem: sufficiently strong interactions can drive the surface into a fully gapped, symmetric, topologically ordered phase.
- That surface phase supports non-Abelian charge-$1/4$ anyons and a neutral anti-semion without any applied magnetic field or fractional quantum Hall sample, giving an alternative materials route to non-Abelian excitations.
- Because the phase provably cannot exist in 2d with both symmetries intact, observing it on a surface is itself a certificate that the bulk carries the $θ=π$ electromagnetic response of a non-trivial topological insulator.
- The transition between the topological surface phase and the surface superconductor is a direct second-order $XY^*$ transition, with the Cooper pair playing the role of the fourth power of the $XY$ field, giving a concrete critical theory to search for.
- No interacting electron insulator with $θ=π$ can be time-reversal invariant if the electrons are Kramers singlets, sharpening the classification of interacting topological insulators beyond the non-interacting case.
Reading between the lines
- The technical move of gauging the global $U(1)$ to give flux tubes unambiguous braiding phases, then condensing, looks reusable: it turns 'what surface orders are allowed' into a definite anyon-condensation computation for other symmetry-protected surfaces built on $U(1)⋊T$.
- The neutral anti-semion sector carries chirality without electric charge, so a surface thermal Hall measurement (or edge energy transport) could in principle distinguish this phase from an ordinary quantum Hall layer even when $σ_{xy}=0$.
- The no-go linking $θ=π$ to $T^2=-1$ hints at a broader constraint tying the sign of the time-reversal square to which quantized electromagnetic responses are permitted, which could be tested in other symmetry classes with anti-unitary generators.
- The paper leaves open whether Moore-Read $×$ $U(1)_{-2}$ is the smallest possible surface order, and a 12-anyon alternative has been proposed elsewhere; if that alternative is continuously connected to the standard non-interacting topological insulator, a surface phase transition between the two must exist and would be a concrete target.
Original abstract
A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this paper we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic 2d topological order, Moore-Read x anti-semion. The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, anti-semion sector is electrically neutral. Time-reversal symmetry is implemented in this surface phase in a highly non-trivial way. Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle number and time-reversal symmetries. A 1d edge on the ETI surface between the topologically-ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma_H = 1/2, carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode and a left-moving bosonic neutral mode. The topologically-ordered phase is separated from the surface superconductor by a direct second order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically-ordered side, and proliferation of a flux 4\pi-vortex, when approached from the superconducting side. In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a \theta-angle, \theta = \pi, do not exist if the electrons transform as Kramers singlets under time-reversal.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this paper we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic 2d topological order, Moore-Read x anti-semion. The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, anti-semion sector is electrically neutral. Time-reversal symmetry is implemented in this surface phase in a highly non-trivial way. Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle number and time-reversal symmetries. A 1d edge on the ETI surface between the topologically-ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma_H = 1/2, carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode and a left-moving bosonic neutral mode. The topologically-ordered phase is separated from the surface superconductor by a direct second order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically-ordered side, and proliferation of a flux 4\pi-vortex, when approached from the superconducting side. In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a \theta-angle, \theta = \pi, do not exist if the electrons transform as Kramers singlets under time-reversal.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. | background 1.00 | 0.07 | 0.02 | asserted 1.00 |
| s1 | It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. | background 0.90 | 0.03 | 0.02 | asserted 0.77 |
| s2 | A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s3 | In this paper we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic 2d topological order, Moore-Read x anti-semion. | contribution 0.54 | 0.96 | 0.93 | asserted 1.00 |
| s4 | The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, anti-semion sector is electrically neutral. | result 0.88 | 0.76 | 0.06 | asserted 1.00 |
| s5 | Time-reversal symmetry is implemented in this surface phase in a highly non-trivial way. | result 0.74 | 0.79 | 0.05 | asserted 1.00 |
| s6 | Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle number and time-reversal symmetries. | result 0.89 | 0.83 | 0.08 | asserted 1.00 |
| s7 | A 1d edge on the ETI surface between the topologically-ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma_H = 1/2, carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode and a left-moving bosonic neutral mode. | result 0.99 | 0.82 | 0.04 | asserted 1.00 |
| s8 | The topologically-ordered phase is separated from the surface superconductor by a direct second order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically-ordered side, and proliferation of a flux 4\pi-vortex, when approached from the superconducting side. | result 0.97 | 0.85 | 0.06 | asserted 1.00 |
| s9 | In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a \theta-angle, \theta = \pi, do not exist if the electrons transform as Kramers singlets under time-reversal. | result 0.96 | 0.89 | 0.07 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
45 arXiv papers cite this in 86 sentences
Papers citing it · 63 sentences without a confident label
Contests it 1 paper
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Thus the way to gap out the boundary states in Ref. [ref] fails in this situation.
Topological Orders with Global Gauge Anomalies · 2. Example with perturbative anomaly · contest 0.85
Uses its method or tool 2 papers
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Ref. [ref] discussed how to drive the boundary of $3d$ topological insulator to a topological order, starting with the superconductor phase of the boundary which spontaneously breaks the U(1) symmetry.
Topological Orders with Global Gauge Anomalies · 2. Example with perturbative anomaly · 0.85
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Directly studying strong four-fermion interactions is difficult, so we will follow the same logic as in Ref. [ref] : we will first manually break a subgroup of the $1×_2$ symmetry by condensing an order parameter that transforms nontrivially under these symmetries.
Interacting Topological Insulator and Emergent Grand Unified Theory · 2. A toy model · 0.96
Uses its data or values 1 paper
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Fortunately this topological spin $e^iθ_n$ has been computed already in Ref. [ref] , and is given by e^iθ_n=1 if n=±1 (mod 8), e^iθ_n=-1 if n=±3 (mod 8).
Bound states of three fermions forming symmetry-protected topological phases · The 3D equivalence from symmetric surface states · 0.98
Compares with it 3 papers
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This result identifies the pairing-gapped nested CDL with the Pfaffian-antisemion symmetric surface phase obtained by completely different means in Refs. [ref] .
Composite Dirac liquids: parent states for symmetric surface topological order · Pfaffian-antisemion phase · 0.99
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We recognize many of these TQFTs as those found in [ref] .
Gapped Boundary Phases of Topological Insulators via Weak Coupling · Outline · 0.84
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Fisher which obtains the same STO [ref] , and a related work by P.
Gapped Symmetry Preserving Surface-State for the Electron Topological Insulator · Discussion · 0.86
Cites it as background 11 papers
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With interactions though other states are possible [ref] .
Half-filled Landau level, topological insulator surfaces, and three dimensional quantum spin liquids · Correlated surface states of 3d topological insulators · 0.82
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It was demonstrated that under strong interaction the boundary of a $3d$ TI can have various topological orders that cannot be realized in a pure $2d$ system [ref] .
"Self-Dual" Quantum Critical Point on the surface of $3d$ Topological Insulator · Introduction · 0.83
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[ref] A topological order is characterized by a set of anyon types and their topological properties, such as fusion and braiding.
Bulk characterization of topological crystalline insulators: stability under interactions and relations to symmetry enriched U(1) quantum spin liquids · A symmetric surface topological order · 0.93
What it cites · 9 sentences without a confident label
Uses the method or tool of 1 paper
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[ref] Imagine building a Walker-Wang model [ref] based on the “restricted" $Ising×U(1)_-8$ braided tensory category introduced in section [ref].
Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model · Topological - superconducting slab. · 0.83
Cites as background 2 papers
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[ref] Note that the massless Dirac theory is a rare example of a 2d CFT that is well-understood.
Topological Insulators Avoid the Parity Anomaly · Introduction. · 0.90
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[ref] Note that the massless Dirac theory is a rare example of a 2d CFT that is well-understood.
Bosonic topological insulator in three dimensions and the statistical Witten effect · Introduction. · 0.90
Discussion
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