Download The breakdown of the topological classification Z for gapped phases

The breakdown of the topological classification Z
for gapped phases of noninteracting fermions by
quartic interactions
Christopher Mudry1
Takahiro Morimoto2,3
1 Paul
Akira Furusaki2
Scherrer Institut, Switzerland
2 RIKEN,
3 University
Japan
of California at Berkeley
UIUC, October 20 2015
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
1 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Main results
Breakdown of the tenfold way (noninteracting topological classification
of fermions) by quartic contact interactions:
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
T
0
0
+1
+1
0
−1
−1
−1
0
+1
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ5
0
1
0
1
0
1
0
1
0
1
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
d=1
0
Z4
0
Z8 , Z4
Z2
Z2
0
Z2 , Z2
0
0
d=2
Z
0
0
0
Z
Z2
Z2
0
Z
0
d=3
0
Z8
0
0
0
Z16
Z2
Z2
0
Z4
d=4
Z
0
Z
0
0
0
Z
Z2
Z2
0
d=5
0
Z16
0
Z16 , Z8
0
0
0
Z16 , Z16
Z2
Z2
d=6
Z
0
Z2
0
Z
0
0
0
Z
Z2
d=7
0
Z32
Z2
Z2
0
Z32
0
0
0
Z32
d=8
Z
0
Z
Z2
Z2
0
Z
0
0
0
Breakdown of the three-dimensional AII+R topological crystalline
classification Z to Z8 by quartic contact interactions.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
2 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
3 / fermio
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Introduction: What is the tenfold way?
The tenfold way applies to noninteracting fermions that realize a
noninteracting manybody ground state, a Slater determinant, such
that,
(i) it is unique and it is separated from all excitations by a gap, when
space is boundaryless,
(ii) while it supports gapless excitations that are extended along but
bounded away from any boundary of space otherwise.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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93
Comment 1: This definition does not require translation invariance in
the direction parallel to a boundary. In particular, it applies when static
impurities are present.
Comment 2: This definition implies a notion of topology since two
spectral properties (the existence of gapless and extended modes)
depend on the topological distinction between manifolds without (e.g.,
sphere, torus) and with (e.g., cube, cylinder) boundaries.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Introduction: Chronological examples
1
2
3
4
5
Zero modes at domain walls in polyacetylene (Jackiw and Rebbi 1976;
Su, Schrieffer, and Heeger 1979).
Chiral edge states along the boundary of an heterojunction displaying
the IQHE (Laughlin 1981; Halperin 1982).
Chiral Majorana zero modes along the boundary of a two-dimensional
(p + ip) superconductor (Read and Green 2000).
A pair of helical zero modes along the boundary of a two-dimensional Z2
topological insulator (Kane and Mele 2005).
The tenfold way (Schnyder, Ryu, Furusaki, and Ludwig; Kitaev 2008):
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
T
0
0
+1
+1
0
−1
−1
−1
0
+1
C. Mudry (PSI)
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ5
0
1
0
1
0
1
0
1
0
1
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
d=1
0
Z
0
Z
Z2
Z2
0
Z
0
0
d=2
Z
0
0
0
Z
Z2
Z2
0
Z
0
d=3
0
Z
0
0
0
Z
Z2
Z2
0
Z
d=4
Z
0
Z
0
0
0
Z
Z2
Z2
0
d=5
0
Z
0
Z
0
0
0
Z
Z2
Z2
d=6
Z
0
Z2
0
Z
0
0
0
Z
Z2
d=7
0
Z
Z2
Z2
0
Z
0
0
0
Z
d=8
Z
0
Z
Z2
Z2
0
Z
0
0
0
The breakdown of the topological classification Z for gapped phases of noninteracting
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Robustness of the gapless extended boundary states arises iff certain
symmetries hold. The distinct combinations of symmetries produce 10
symmetry classes. These symmetries are local in the table
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
T
0
0
+1
+1
0
−1
−1
−1
0
+1
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ5
0
1
0
1
0
1
0
1
0
1
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
d=1
0
Z
0
Z
Z2
Z2
0
Z
0
0
d=2
Z
0
0
0
Z
Z2
Z2
0
Z
0
d=3
0
Z
0
0
0
Z
Z2
Z2
0
Z
d=4
Z
0
Z
0
0
0
Z
Z2
Z2
0
d=5
0
Z
0
Z
0
0
0
Z
Z2
Z2
d=6
Z
0
Z2
0
Z
0
0
0
Z
Z2
d=7
0
Z
Z2
Z2
0
Z
0
0
0
Z
d=8
Z
0
Z
Z2
Z2
0
Z
0
0
0
but this need not be so. Crystalline topological insulators are
noninteracting fermions that support gapless extended boundary
states iff crystalline symmetries hold. An all inclusive terminology is
Short-ranged entangled (SRE) topological phases of fermionic matter
⇐⇒
Symmetry-protected topological (SPT) phases of fermionic matter
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Examples of diagnostic for SRE topological phases
For band insulators on a torus H = ⊕k ∈BZ H(k ). Single-particle
observables for topological SRE phases can then be chosen to be
Polyacetylene:
H(k ) :=
0
q † (k )
q(k )
0
,
ν =
i
2π
+π
Z
†
dk tr q ∂k q (k ).
−π
IQHE for a Chern band insulator:
H(k ) :=
X
a
a
ab
|u (k )ihu (k )|,
a
Aµ (k ) := hu (k )|
a
∂
∂k µ
b
|u (k )i,
ν =
i
2π
+π
Z
2
d k tr F12 (k ) occupied bds .
−π
Z2 topological insulator in two-dimensional space:
H(k ) :=
X
a
a
|u (k )ihu (k )|,
a
C. Mudry (PSI)
ab
a
b
wµ (k ) := hu (−k )|Θ|u (+k )i,
ν =
Y
Pf [w(K )]|occupied bds .
K =−K ∈BZ
The breakdown of the topological classification Z for gapped phases of noninteracting
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The problem is that such observables are not useful in the presence of
static disorder or in the presence of interactions. Alternative
observables can be constructed in the presence of static disorder.
The only observable that I understand in the presence of static
disorder and interactions is the Hall conductivity averaged over twisted
boundary conditions, Niu and Thouless 1984; Niu, Thouless, and Wu
1985
Z2π Z2π ∂Ψ ∂Ψ
∂Ψ ∂Ψ
i
dφ dϕ
−
C=−
2π
∂φ ∂ϕ
∂ϕ ∂φ
0
0
with Ψ the manybody ground state obeying twisted boundary
conditions.
If C is an integer in the noninteracting limit, and if the interactions are
not too strong (compared to the band gap), then C is unchanged by
interactions. This is so because there is no backscattering allowed
among the chiral edge states
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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z
Laughlin 1981: The Hall conductivity
must be rational and if it is not an
integer, the ground state manifold must
be degenerate and support fractionally
charged excitations
B
y
x
(a)
(b)
Halperin 1982: Chiral edges are immune to backscattering within each
traffic lane
Integer Quantum Hall Effect
C. Mudry (PSI)
Fractional Quantum Hall Effect
The breakdown of the topological classification Z for gapped phases of noninteracting
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This argument (no backscattering) is tied to the boundary being a
one-dimensional manifold.
Question: Are all the topological entries of the tenfold way robust to
manybody interactions?
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Partial answer for strong topological insulators
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
T
0
0
+1
+1
0
−1
−1
−1
0
+1
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ5
0
1
0
1
0
1
0
1
0
1
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
d=1
0
Z4
0
Z8 , Z4
Z2
Z2
0
Z2 , Z2
0
0
d=2
Z
0
0
0
Z
Z2
Z2
0
Z
0
d=3
0
Z8
0
0
0
Z16
Z2
Z2
0
Z4
d=4
Z
0
Z
0
0
0
Z
Z2
Z2
0
d=5
0
Z16
0
Z16 , Z8
0
0
0
Z16 , Z16
Z2
Z2
d=6
Z
0
Z2
0
Z
0
0
0
Z
Z2
d=7
0
Z32
Z2
Z2
0
Z32
0
0
0
Z32
d=8
Z
0
Z
Z2
Z2
0
Z
0
0
0
Fidkowsky and Kitaev 2010; ...
Kitaev 2011 (unpublished); Fidkowski, Chen, and Vishwanath 2014; ...
Tang and Wen 2012
Wang and Senthil 2014
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Partial answer for crystalline topological insulators
There also are examples of crystalline topological insulators that are
unstable to quartic interactions:
Z → Z8 for DIII+R in 2d (Yao and Ryu 2013; Qi 2013)
topological insulators with inversion symmetry (You and Xu 2014).
Z → Z8 for AII+R in 3d (Isobe and Fu 2015; Yoshida and
Furusaki; this work 2015)
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Many different approaches were used to reach these conclusions. Is
there an approach, perhaps a simpler one, that reproduces all known
results and extends readily to all entries of the tenfold way?
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
14 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
15 / fermio
93
Strategy
The strategy that we use to study the robustness of ν boundary modes
to quartic contact fermion-fermion interactions is related to the
approached advocated by You and Xu 2014 and (unpublished) Kitaev
2015.
It consists of three steps.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Step 1
A noninteracting topological phase is represented by the manybody
ground state of a massive Dirac Hamiltonian with a matrix dimension
that depends on ν.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
17 / fermio
93
Step 2
A Hubbard-Stratonovich transformation is used to trade a generic
quartic contact interaction in favor of dynamical Dirac mass-like
bilinears coupled to their conjugate fields (that will be called dynamical
Dirac masses). These dynamical Dirac masses may violate any
symmetry constraint other than the particle-hole symmetry (PHS).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Step 3
The ν boundary modes that are coupled with a suitably chosen subset
of dynamical Dirac masses are integrated over. The resulting dynamical theory on the (d − 1)-dimensional boundary is a bosonic one,
a quantum nonlinear sigma model (QNLSM) in [(d − 1) + 1] space and time,
with a target space that depends on ν.
These QNLSM can have the following phase diagrams:
Without topological term
d ≤ 2 g=0
d > 2 g=0
gc
C. Mudry (PSI)
With topological term
gs
g=0
gs
g=0
★
g
s
gc
★
g
s
The breakdown of the topological classification Z for gapped phases of noninteracting
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Step 4
The reduction pattern is then obtained by identifying the smallest value
of ν for which:
This QNLSM cannot be augmented by a topological term that is
compatible with local equations of motion.
The quantum fluctuations driven by the interactions have restored
all the symmetries broken by the saddle point.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
20 / fermio
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Comment 1: The question that we address in this paper is whether
the topological classification of noninteracting fermions is reduced by
interactions. A complete classification of fermionic SRE phases
(combined with that for the bosonic SRE phases) is beyond the scope
of our approach.
Comment 2: The presence or absence of topological terms in the
relevant QNLSM is determined by the topology of the spaces of
boundary dynamical Dirac masses, i.e., the topology of classifying
spaces. Now, K-theory provides a systematic way to study the
topology of the classifying space.
Comment 3: This is why the same approach that was used to obtain
the tenfold way of noninteracting fermions can be relied on to deduce a
classification of topological short-range entangled (SRE) phases for
interacting fermions.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
21 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
22 / fermio
93
Example: Symmetry class AII in d = 3
Consider the single-particle static massive Dirac Hamiltonian [this is
THE Dirac Hamiltonian (Dirac 1928)]
H(0) (x) := −i∂x X21 − i∂y X11 − i∂z X02 + m(x) X03 ,
(1a)
Xµµ0 := σµ ⊗ τµ0
(1b)
where
for µ, µ0 = 0, 1, 2, 3. Because
H(0) (x) = +T H(0) (x) T −1 ,
T := iX20 K,
(1c)
we interpret this Hamiltonian as realizing a noninteracting topological
insulator in the three-dimensional symmetry class AII.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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The static domain wall in the mass (Jackiw and Rebbi 1976)
m(x, y , z) = m∞ sgn(z)
(2a)
binds a zero mode to the boundary z = 0 that is annihilated by the
boundary single-particle Hamiltonian
(0)
Hbd (x, y ) = − i∂x σ2 − i∂y σ1
(0)
= Tbd Hbd (x, y ) Tbd−1 ,
(2b)
where
Tbd := iσ2 K.
C. Mudry (PSI)
(2c)
The breakdown of the topological classification Z for gapped phases of noninteracting
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The boundary dynamical Dirac Hamiltonian
(dyn)
Hbd
(τ , x, y ) = −i∂x σ2 − i∂y σ1 + M(τ , x, y ) σ3
(3)
belongs to the symmetry class A, as the Dirac mass Mσ3 breaks TRS
unless M(−τ , x, y ) = −M(τ , x, y ).
The space of normalized boundary dynamical Dirac mass matrices
{±σ3 } is homeomorphic to the space of normalized Dirac mass
matrices in the symmetry class A
Vν=1 =
1
[
U(ν)/ [U(k ) × U(ν − k )]
(4)
k =0
when space is two dimensional.
The domain wall in imaginary time (the instanton)
M(τ, x, y ) = M∞ sgn(τ )
(5)
prevents the gapping of the boundary zero mode.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
25 / fermio
93
We conclude that the noninteracting topological classification Z2 of
three-dimensional insulators in the symmetry class AII is robust to the
effects of interactions under the assumption that only
fermion-number-conserving interacting channels are included in the
stability analysis. The logic used to reach this conclusion is
summarized by Table
D
0
1
πD (C0 )
Z
0
ν
1
Topological obstruction
Domain wall
once the line corresponding to ν = 1 has been identified. This line is
fixed by the first integer D that accommodates a non-trivial entry for
the corresponding homotopy group.
Moreover, one verifies by introducing a BdG (Nambu) grading that this
robustness extends to interaction-driven dynamical superconducting
fluctuations.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
26 / fermio
93
Example: Symmetry class AII in d = 3 with reflection
symmetry R
We consider again the bulk, boundary, and dynamical boundary
Hamiltonian defined in Eqs. (1)–(4). We observe that the
single-particle Hamiltonian (1a) has the symmetry
Rx H(0) (−x, y , z) ( Rx )−1 = +H(0) (x, y , z),
(6a)
R2x = −1,
(6b)
where
Rx := iX10 ,
[T , Rx ] = 0,
in addition to the TRS (1c).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
27 / fermio
93
The presence of the additional reflection symmetry allows one to
define a mirror Chern number (n+ ∈ Z) for the sector with the
eigenvalue Rx = +i on the two-dimensional mirror plane (kx = 0) in
the three-dimensional Brillouin zone (Hsieh, Lin, Liu, Duan, Bansil, and
Fu 2012).
Thus, the ν linearly independent zero modes that follow from tensoring
the single-particle Hamiltonian (1a) with the ν × ν unit matrix along the
domain wall (2a) are stable to strong one-body perturbations on the
boundary that preserve the reflection symmetry. (If we forget the
reflection symmetry and keep only the TRS, it is only the parity of ν
that is stable to strong one-body perturbations on the boundary.)
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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If we only consider dynamical masses that preserve the
fermion-number U(1) symmetry, the space of normalized boundary
dynamical Dirac mass matrices after tensoring the boundary
dynamical Dirac Hamiltonian (4) with the unit ν × ν matrix is
homeomorphic to the space of normalized Dirac masses in the
symmetry class A
Vν =
ν
[
U(ν)/[U(k ) × U(ν − k )]
(7)
k =0
when space is two dimensional.
The limit ν → ∞ of these spaces is the classifying space C0 .
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
29 / fermio
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Integrating the boundary Dirac fermions delivers a QNLSM in
(2+1)-dimensional space and time. In order to gap out dynamically the
boundary zero modes without breaking the symmetries, this QNLSM
must be free of topological obstructions. We construct explicitly the
spaces for the relevant normalized boundary dynamical Dirac mass
matrices [M(τ , x, y ) in Eq. (3)] of dimension ν = 2n with n = 0, 1, 2, 3 in
the following.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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93
Case ν = 1:
There is a topological obstruction of the domain-wall type as the target
space is
S 0 = {±1}
(8)
and π0 (S 0 ) 6= 0.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Case ν = 2:
There is a topological obstruction of the monopole type as the target
space is
S 2 = {c1 X1 + c2 X2 + c3 X3 | c12 + c22 + c32 = 1}
(9)
and π2 (S 2 ) = Z.
C. Mudry (PSI)
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Case ν = 4:
There is a topological obstruction of the WZ type as the target space is
5
X 2
ci = 1, ci ∈ R
S = c1 X13 + c2 X23 + c3 X33 + c4 X01 + c5 X02 4
i=1
(10)
and π4 (S 4 ) = Z.
C. Mudry (PSI)
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Case ν = 8:
There is no topological obstruction as one can find more than five
pairwise anticommuting matrices such as the set
{X133 , X233 , X333 , X013 , X023 , X001 , X002 }.
C. Mudry (PSI)
(11)
The breakdown of the topological classification Z for gapped phases of noninteracting
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Reduction from Z to Z8 due to interactions for the topologically equivalent classes of the
three-dimensional topological insulators with time-reversal and reflection symmetries (AII+R).
We denote by Vν the space of ν × ν normalized Dirac mass matrices in boundary (d = 2) Dirac
Hamiltonians belonging to the symmetry class A. The limit ν → ∞ of these spaces is the
classifying space C0 . The second column shows the stable D-th homotopy groups of the
classifying space C0 . The third column gives the number ν of copies of boundary (Dirac)
fermions for which a topological obstruction is permissible. The fourth column gives the type of
topological obstruction that prevents the gapping of the boundary (Dirac) fermions.
D
0
1
2
3
4
5
6
7
πD (C0 )
Z
0
Z
0
Z
0
Z
0
C. Mudry (PSI)
ν
1
Topological obstruction
Domain wall
2
Monopole
4
WZ term
8
None
The breakdown of the topological classification Z for gapped phases of noninteracting
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This Z8 classification is unchanged if all boundary dynamical masses
that break the fermion-number U(1) symmetry are accounted for. The
corresponding target spaces for the boundary dynamical masses and
their topological obstructions are derived by extending the
single-particle Hamiltonian [Eq. (3)] to a BdG Hamiltonian
(dyn)
Hbd
= −i∂x σ2 ⊗ ρ3 − i∂y σ1 ⊗ ρ0 ⊗ Iν×ν + γ 0 (τ, x, y ),
(12a)
where ρ0 and ρµ are unit 2 × 2 and Pauli matrices, respectively, acting
on the particle-hole (Nambu) space, Iν×ν is the unit ν × ν matrix, and
the particle-hole symmetry is given by C = ρ1 K. In this case, the target
spaces of the QNLSM made of normalized boundary dynamical Dirac
mass matrices γ 0 of dimension ν = 2n with n = 0, 1, 2, 3 are modified
as listed in the following with the notation
Xµµ0 µ00 µ000 ... = σµ ⊗ ρµ0 ⊗ τµ00 ⊗ τµ000 . . . .
C. Mudry (PSI)
(12b)
The breakdown of the topological classification Z for gapped phases of noninteracting
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93
The relevant homotopy groups are given in
D
πD (R0 )
ν
Topological obstruction
0
Z
1
Z2
1
Vortex
2
Z2
2
Monopole
3
0
4
Z
4
WZ term
5
0
6
0
7
0
8
Z
8
None
We note that these target spaces are closed under the global U(1)
transformation generated by ρ3 .
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
37 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
38 / fermio
93
Application to the surfaces of SnTe
The crystal SnTe is a three-dimensional topological crystalline
insulator protected by time-reversal and reflection symmetries (AII+R).
SnTe supports four Dirac cones on the [001] surface and six Dirac
cones on the [111] surface.
If strong interaction effects are present, we expect the following.
On the [001] surface, the ν = 4 phase described by a QNLSM with
a WZ term should be realized.
On the [111] surface, the ν = 6 = 4 + 2 phase should be realized,
whereby the effective field theory is that of a QNLSM with a WZ
term for 4 out of the six surface Dirac cones and that of a QNLSM
with a topological term arising from a gas of monopoles for the
remaining two surface Dirac cones.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
39 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
40 / fermio
93
Summary
Breakdown of the tenfold way (noninteracting topological classification
of fermions) by quartic contact interactions:
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
T
0
0
+1
+1
0
−1
−1
−1
0
+1
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ5
0
1
0
1
0
1
0
1
0
1
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
d=1
0
Z4
0
Z8 , Z4
Z2
Z2
0
Z2 , Z2
0
0
d=2
Z
0
0
0
Z
Z2
Z2
0
Z
0
d=3
0
Z8
0
0
0
Z16
Z2
Z2
0
Z4
d=4
Z
0
Z
0
0
0
Z
Z2
Z2
0
d=5
0
Z16
0
Z16 , Z8
0
0
0
Z16 , Z16
Z2
Z2
d=6
Z
0
Z2
0
Z
0
0
0
Z
Z2
d=7
0
Z32
Z2
Z2
0
Z32
0
0
0
Z32
d=8
Z
0
Z
Z2
Z2
0
Z
0
0
0
Breakdown of the three-dimensional AII+R topological crystalline
classification Z to Z8 by quartic contact interactions.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
41 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
42 / fermio
93
The classifying spaces and their homoytopy groups
are
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
V
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
C. Mudry (PSI)
π0 (V )
Z
0
Z
Z2
Z2
0
Z
0
0
0
T
0
0
+1
+1
0
−1
−1
−1
0
+1
π0 (V )
0
Z
0
Z
Z2
Z2
0
Z
0
0
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ
0
1
0
1
0
1
0
1
0
1
π0 (V )
Z
0
0
0
Z
Z2
Z2
0
Z
0
Extension
Cld → Cld+1
Cld+1 → Cld+2
Cl0,d+2 → Cl1,d+2
Cld+1,2 → Cld+1,3
Cld,2 → Cld,3
Cld,3 → Cld,4
Cl2,d → Cl3,d
Cld+3,0 → Cld+3,1
Cld+2,0 → Cld+2,1
Cld+2,1 → Cld+2,2
π0 (V )
0
Z
0
0
0
Z
Z2
Z2
0
Z
π0 (V )
Z
0
Z
0
0
0
Z
Z2
Z2
0
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
π0 (V )
0
Z
0
Z
0
0
0
Z
Z2
Z2
π0 (V )
Z
0
Z2
0
Z
0
0
0
Z
Z2
π0 (V )
0
Z
Z2
Z2
0
Z
0
0
0
Z
The breakdown of the topological classification Z for gapped phases of noninteracting
43 / fermio
93
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
44 / fermio
93
Main results (AZ stands for Altland and Zirnbauer)
Case d = 1
Five AZ symmetry classes
Three AZ symmetry classes
(A)
(B)
g
g
g
Two AZ symmetry classes
N=2 and so on
m
0
Insulator
ν=+1
m
(C)
g
Insulator
ν=0
Insulator Insulator
ν=−1/2 ν=+1/2
ν=−1
Insulator
Insulator
0
(B)
N=1
Insulator
ν=0
m
0
Insulator
ν=+1
m
0
Case d = 3
Five AZ symmetry classes
Three AZ symmetry classes
(A)
(B)
g
g
Metal
(B)
N=1
g
Two AZ symmetry classes
N=2 and so on
(C)
g
Metal
Metal
Metal
ν=0
ν=−1/2 ν=+1/2
Insulator
0
m
C. Mudry (PSI)
0
m
ν=−1
ν=+1
0
ν=0
m
ν=+1
0
m
The breakdown of the topological classification Z for gapped phases of noninteracting
45 / fermio
93
Recent related works
Case d = 1 chiral classes: A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and
H. Schmiedt, Quantum criticality of quasi one-dimensional topological
Anderson insulators, Phys. Rev. Lett. 112, 206602 (2014),
Case d = 1 class D: M.-T. Rieder and P. W. Brouwer, Density of states at
disorder-induced phase transitions in a multichannel Majorana wire,
arXiv:1409.1877; M.-T. Rieder, P. W. Brouwer, and I. Adagideli, Reentrant
topological phase transitions in a disordered spinless superconducting wire,
Phys. Rev. B 88, 060509(R) (2013); and
taken from A. Altland, D. Bagrets, and A. Kamenev, Topology vs. Anderson
localization: non-perturbative solutions in one dimension, arXiv:1411.5992.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
46 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
47 / fermio
93
The Dirac equation: an application of Clifford algebra
Dirac (1928) [taken from P. A. M. Dirac, Principles of Quantum
Mechanics (4th ed.), Clarendon, page 255 (1958)]:
where
{αi , αj } = 2 δij ,
C. Mudry (PSI)
{αi , β} = 0,
i, j = 1, 2, 3,
β 2 = 1.
The breakdown of the topological classification Z for gapped phases of noninteracting
47 / fermio
93
1931: Dirac introduces topology in physics
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
48 / fermio
93
1932-1939: Tamm and Schockley surface states
I. Tamm, “On the possible bound states of electrons on a crystal
surface,” Phys. Z. Soviet Union 1, 733 (1932).
W. Shockley, “On the Surface States Associated with a Periodic
Potential,” Phys. Rev. 56, 317 (1939).
P-Y. Chang, C. Mudry,
and S. Ryu 2014
C. Mudry (PSI)
Direct sum of a
px + ipy and of a
px − ipy BdG
superconductor in a
cylindrical geometry.
The breakdown of the topological classification Z for gapped phases of noninteracting
49 / fermio
93
Anderson localization before 1981
P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,”
Phys. Rev. 109, 1492 (1957): What is the fate of a single-particle Bloch
state when perturbed by a static potential that breaks all space-group
symmetries?
In a noninteracting many-body setting, there are three possible
phases:
The metallic phase:
σ = ∞ at T = 0.
The insulating phase:
σ = 0 at T = 0.
A quantum critical phase:
σ = σc at T = 0.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
50 / fermio
93
1953: Dyson’s exception
F. J. Dyson, “The Dynamics of a Disordered Linear Chain,” Phys. Rev.
92, 1331 (1953).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
51 / fermio
93
1963: The threefold way for random matrices
F. J. Dyson, “The Threefold Way. Algebraic Structure of Symmetry
Groups and Ensembles in Quantum Mechanics,” J. Math. Phys. 3,
1199 (1962):
P(θ1 , · · · , θN ) ∝
Y
β
iθj
e − eiθk ,
β = 1, 2, 4.
1≤j<k ≤N
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
52 / fermio
93
1976: Jackiw and Rebbi introduce Fermion number
fractionalization
R. Jackiw and C. Rebbi, “Solitons with fermion number 1/2,” Phys. Rev.
D 13, 3398 (1976): Dirac equation with a single point defect in a
background field supports a single zero mode that carries the fermion
number 1/2.
W. P. Su, J. R. Schrieffer, and H. J. Heeger, “Soliton excitations in
polyacetylene,” Phys. Rev. B 22, 2099 (1980): Propose polaycetylene
as a realization of fermion fractionalization
00
11
00
11
00
11
ε (k)
k
−π/2
C. Mudry (PSI)
+π/2
The breakdown of the topological classification Z for gapped phases of noninteracting
53 / fermio
93
1981: Nielsen-Ninomiya theorem
H. B. Nielsen and M. Ninomiya, “A no-go theorem for regularizing
chiral fermions,” Phys. Lett. B105, 219 (1981): The Nielsen-Ninomiya
theorem is a no-go theorem that prohibits defining a theory of chiral
fermions on a lattice in odd-dimensional space.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
54 / fermio
93
1981: Jackiw and Rossi introduce a localized
Majorana zero mode in a two-dimensional relativistic
superconductor
R. Jackiw and P. Rossi, “Zero modes of the vortex-fermion system,”
Nucl. Phys. B190, 681 (1981).
L. Fu and C. L. Kane, “Superconducting
Proximity Effect and Majorana Fermions at the
Surface of a Topological Insulator,” Phys. Rev.
Lett. 100, 096407 (2008) have proposed to
realize this localized Majorana zero mode on
the core of a superconducting vortex induced by
proximity effect on the surface of a strong Z2
topological insulator.
C. Mudry (PSI)
s−waveSC
TI
The breakdown of the topological classification Z for gapped phases of noninteracting
55 / fermio
93
Anderson localization after 1981
The discovery of the Integer Quantum Hall effect, Klitzing, Dorda, and
Pepper (1980), lead to the understanding that there are topologically
dictinct insulating phases separated by critical point at which a plateau
transition takes place.
Example: Graphene deposited on SiO2 /Si, T =1.6 K and B=9 T (inset
T =30 mK): ν = ±2, ±6, ±10, · · · = ±2(2n + 1), n ∈ N
after Zhang et al., 2005.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
56 / fermio
93
The Integer Quantum Hall Effect
εm h ω c
Without disorder
εF
εm h ω c
m= 1
m= 1
εF
With disorder
2
but no interactions
6
−2
m= 0
DOS
σxy
m= 0
DOS
−6
m=−1
σxy
[e^2/h]
m=−1
At integer fillings of the Landau levels, the noninteracting ground state
is unique and the screened Coulomb interaction Vint can be treated
perturbatively, as long as transitions between Landau levels or outside
the confining potential Vconf along the magnetic field are suppressed by
the single-particle gaps:
Vint ~ ωc Vconf ,
C. Mudry (PSI)
ωc = e B/(m c).
The breakdown of the topological classification Z for gapped phases of noninteracting
57 / fermio
93
1981-1982: Laughlin and Halperin introduce the
bulk-edge correspondence in the Quantum
Hall Effect
z
B
Laughlin 1981: The Hall conductivity
must be rational and if it is not an
integer, the ground state manifold must
be degenerate and support fractionally
charged excitations
y
x
(a)
(b)
Halperin 1982: Chiral edges are immune to backscattering within each
traffic lane
Integer Quantum Hall Effect
C. Mudry (PSI)
Fractional Quantum Hall Effect
The breakdown of the topological classification Z for gapped phases of noninteracting
58 / fermio
93
1982: TKNN relate linear response to topology in the
Quantum Hall Effect
Thouless, Kohmoto, Nightingale, and den Nijs, (1982),
Avron, Seiler, and Simon (1983); Simon (1983),
Niu and Thouless (1984); Niu, Thouless, and Wu (1985)
The Hall conductance is proportional to the first Chern number
i
C=−
2π
Z2π
Z2π
dφ
0
dϕ
∂Ψ ∂Ψ
∂Ψ ∂Ψ
−
∂φ ∂ϕ
∂ϕ ∂φ
0
with Ψ the many-body ground state obeying twisted boundary
conditions.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
59 / fermio
93
1983-1985: Khmelnitskii and Pruisken introduce the
scaling theory of the Integer Quantum Hall Effect
A topological term modifies the scaling analysis of the gang of four:
σxx
Khmelnitskii 1983
Pruisken 1985
d ln g
d ln L
0
1
2
3
σxy
φ=π
1
0
ln g
−1
d=2
φ=0
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
60 / fermio
93
1983-1984: Haldane introduces the θ term for spin
chains and Witten achieves non-Abelian bosonization
F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin Heisenberg
Antiferromagnets: Semiclassically Quantized Solitons of the
One-Dimensional Easy-Axis Néel State,” Phys. Rev. Lett. 50, 1153
(1983), “Continuum dynamics of the 1-D Heisenberg antiferromagnet:
Identification with the O(3) nonlinear sigma model,” Phys. Lett. A 93,
464 (1983) : Topological θ term modifies the RG flow in the
two-dimensional O(3) non-linear-sigma model.
O(3) NLSM+ θ=π
O(3) NLSM
g
SU(2)1
g
E. Witten, “Nonabelian bosonization in two dimensions,” Commun.
Math. Phys. 92, 455 (1984): Non-Abelian bosonization.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
61 / fermio
93
1988: Haldane model for a Chern band insulator
F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau
Levels: Condensed-Matter Realization of the ‘Parity Anomaly’,” Phys.
Rev. Lett. 61, 2015 (1988).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
62 / fermio
93
1994: Random Dirac fermions in two-dimensional
space
A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein,
“Integer quantum Hall transition: An alternative approach and exact
results,” Phys. Rev. B 50, 7526 (1994): Effects of static disorder on a
single Dirac fermion in two-dimensional space.
AIII
IQHE
D
AII
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
63 / fermio
93
1997: The tenfold way for random matrices
A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in
mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B
55, 1142 (1997).
Cartan label
A (unitary)
AI (orthogonal)
AII (symplectic)
AIII (ch. unit.)
BDI (ch. orth.)
CII (ch. sympl.)
D (BdG)
C (BdG)
DIII (BdG)
CI (BdG)
T
C
S
Hamiltonian
G/H (ferm. NLSM)
0
+1
−1
0
0
0
0
0
0
U(N)
U(N)/O(N)
U(2N)/Sp(2N)
U(2n)/U(n) × U(n)
Sp(2n)/Sp(n) × Sp(n)
O(2n)/O(n) × O(n)
0
+1
−1
0
+1
−1
1
1
1
U(N + M)/U(N) × U(M)
O(N + M)/O(N) × O(M)
Sp(N + M)/Sp(N) × Sp(M)
U(n)
U(2n)/Sp(2n)
U(2n)/O(2n)
0
0
−1
+1
+1
−1
+1
−1
0
0
1
1
SO(2N)
Sp(2N)
SO(2N)/U(N)
Sp(2N)/U(N)
O(2n)/U(n)
Sp(2n)/U(n)
O(2n)
Sp(2n)
The column entitled “Hamiltonian” lists, for each of the ten symmetry classes, the symmetric space of which the quantum
mechanical time-evolution operator exp(it H) is an element. The last column entitled “G/H (ferm. NLσM)” lists the (compact
sectors of the) target space of the NLσM describing Anderson localization physics at long wavelength in this given symmetry
class.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
64 / fermio
93
1998-2000: Brouwer et al. establish that there are five
symmetry class that display quantum criticality in
disordered quasi-one-dimensional wires
The “radial coordinate” of the
transfer matrix M from the Table
below makes a Brownian motion
on an associated symmetric space.
Class
O
U
S
chO
chU
chS
CI
C
DIII
D
TRS
Yes
No
Y
Y
N
Y
Y
N
Y
N
SRS
Y
Y(N)
N
Y
Y(N)
N
Y
Y
N
N
C. Mudry (PSI)
mo
1
2
4
1
2
4
2
4
2
1
ml
1
1
1
0
0
0
2
3
0
0
D
2
2(1)
2
2
2(1)
2
4
4
2
1
M
CI
AIII
DIII
AI
A
AII
C
CII
D
BDI
H
AI
A
AII
BDI
AIII
CII
CI
C
DIII
D
δg
−2/3
0
+1/3
0
0
0
−4/3
−2/3
+2/3
+1/3
h− ln gi
2L/(γ`)
2L/(γ`)
2L/(γ`)
2mo L/(γ`)
2mo L/(γ`)
2mo L/(γ`)
2ml L/(γ`)
2m L/(γ`)
pl
4 L/(2πγ`)
p
4 L/(2πγ`)
ρ(ε) for 0 < ετc 1
ρ0
ρ0
ρ0
ρ0 | ln |ετc ||
πρ0 |ετc ln |ετc ||
(πρ0 /3)|(ετc )3 ln |ετc ||
(πρ0 /2)|ετc |
ρ0 |ετc |2
πρ0 /|ετc ln3 |ετc ||
πρ0 /|ετc ln3 |ετc ||
The breakdown of the topological classification Z for gapped phases of noninteracting
65 / fermio
93
2000: Read and Green introduce the chiral p-wave
topological superconductor
N. Read and D. Green, “Paired states of fermions in two dimensions
with breaking of parity and time-reversal symmetries and the fractional
quantum Hall effect,” Phys. Rev. B 61, 10267 (2000).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
66 / fermio
93
2005: Kane and Mele introduce the strong Z2
topological insulator
The spin-orbit coupling is ignored in the QHE as the breaking of
time-reversal symmetry provides the dominant energy scale.
C. L. Kane and E. J. Mele, “Z2 Topological Order and the Quantum
Spin Hall Effect,” Phys. Rev. Lett. 95, 146802 (2005): Combine a pair
of time-reversed Haldane models with a small Rashba coupling and
find protected helical edge states.
(for a
E
k0
EF
E0
ky
Γ
S
kx
Bix Pb1−x /Ag(111) surface alloy, say)
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
67 / fermio
93
2008: The tenfold way for topological insulators and
superconductors
Schnyder, Ryu, Furusaki, and Ludwig 2008 and 2010; Kitaev 2008
complex case:
Cartan\d
A
AIII
0
Z
0
1
0
Z
2
Z
0
3
0
Z
4
Z
0
5
0
Z
6
Z
0
7
0
Z
8
Z
0
9
0
Z
10
Z
0
11
0
Z
···
···
···
8
Z
Z2
Z2
0
2Z
0
0
0
9
0
Z
Z2
Z2
0
2Z
0
0
10
0
0
Z
Z2
Z2
0
2Z
0
11
0
0
0
Z
Z2
Z2
0
2Z
real case:
Cartan\d
AI
BDI
D
DIII
AII
CII
C
CI
0
Z
Z2
Z2
0
2Z
0
0
0
C. Mudry (PSI)
1
0
Z
Z2
Z2
0
2Z
0
0
2
0
0
Z
Z2
Z2
0
2Z
0
3
0
0
0
Z
Z2
Z2
0
2Z
4
2Z
0
0
0
Z
Z2
Z2
0
5
0
2Z
0
0
0
Z
Z2
Z2
6
Z2
0
2Z
0
0
0
Z
Z2
7
Z2
Z2
0
2Z
0
0
0
Z
···
···
···
···
···
···
···
···
···
The breakdown of the topological classification Z for gapped phases of noninteracting
68 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
69 / fermio
93
Definition and question to be addressed
Consider the random Dirac Hamiltonian in d-dimensional space of
rank r given by
Hd,r =
d
X
i=1
αi
∂
+ M(x) + · · · ,
i∂xi
{αi , αj } = 2 δij ,
i, j = 1, · · · , d,
where
1
tr[M(x) − M] [M(y) − M] =: g2 e−|x−y|/ξdis ,
r
with all higher cumulants vanishing.
Is the single-particle energy eigenstate at ε = 0 delocalized or
localized?
A necessary but not sufficient condition for localization is that a Dirac
mass matrix is premitted i.e., M(x) 6= 0.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
69 / fermio
93
Why “necessary but not sufficient”?
Choose d = 1 and r = 2:
Hd=1,r =2 = τ1 [−i∂ + A1 (x)] + τ0 A0 (x) + τ2 m2 (x) + τ3 m3 (x).
There is a single localized phase.
Choose d = 2 and r = 2:
Hd=2,r =2 = τ1 [−i∂1 + A1 (x)] + τ2 [−i∂2 + A2 (x)] + τ0 A0 (x) + τ3 m(x).
There are two topological distinct phases and a critical line separating them if
m(x) = 0.
Choose d = 3 and r = 2:
Hd=3,r =1 = τ1 [−i∂1 + A1 (x)] + τ2 [−i∂2 + A2 (x)] + τ3 [−i∂3 + A3 (x)] + τ0 A0 (x).
Localization is prohibited.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
70 / fermio
93
A normalized Dirac mass matrix β is defined by the condition
{α, β} = 0,
β 2 = 1.
When d = 1 and r = 2, the normalized Dirac mass matrix is
β(x) = τ2 cos θ(x) + τ3 sin θ(x),
tan θ(x) :=
m3 (x)
.
m2 (x)
As a space, it is the unit circle.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
71 / fermio
93
A normalized Dirac mass matrix β is defined by the condition
β 2 = 1.
{α, β} = 0,
When d = 2 and r = 2, the normalized Dirac mass matrix is
β(x) =
m(x)
τ ,
|m(x)| 3
As a space, it is two points.
If m(x) = m, then τxy = 12 sgn m. The domain wall m(x, y ) = m(x) with
limx→±∞ m(x) = ∓m supports the boundary states
Ψ(x, y ; k ) ∝ e
ik y −iτy
Rx
−∞
dx 0 m(x 0 )
1
−i
with the dispersion
m(x)
y
x
x
ε(k ) = k .
This is the minimal model for the IQHE.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
72 / fermio
93
A normalized Dirac mass matrix β is defined by the condition
{α, β} = 0,
β 2 = 1r .
When d = 3 and r = 2, the normalized Dirac mass matrix is not
allowed.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
73 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
74 / fermio
93
The tenfold way for the classifying spaces when d = 1
Symmetry class A:
H(k ) := τ3 k + τ3 A1 + τ2 M2 + τ1 M1 + τ0 A0 ,
A
1
Vd=1;r
=2 = S .
Symmetry class AII:
H(k ) = +τ2 H∗ (−k ) τ2 ,
AII
Vd=1;r
=2 = ∅.
Symmetry class AI:
H(k ) = +τ1 H∗ (−k ) τ1 ,
AI
1
Vd=1;r
=2 = S .
Symmetry class AIII:
H(k ) = −τ1 H(k ) τ1 ,
AIII
Vd=1;r
=2 = {±τ2 }.
Symmetry class CII: No Dirac Hamiltonian when r = 2.
Symmetry class BDI:
H(k ) = −τ1 H(k ) τ1 = +τ1 H∗ (−k ) τ1 ,
C. Mudry (PSI)
BDI
Vd=1;r
=2 = {±τ2 }.
The breakdown of the topological classification Z for gapped phases of noninteracting
74 / fermio
93
Symmetry class D:
H(k ) = −H∗ (−k ),
D
Vd=1;r
=2 = {±τ1 }.
Symmetry class DIII:
H(k ) = −H∗ (−k ) = +τ2 H∗ (−k ) τ2 ,
DIII
Vd=1;r
=2 = ∅.
Symmetry class C:
H(k ) = −τ2 H∗ (−k ) τ2 = τ3 A1 + τ2 M2 + τ1 M1 .
PHS squaring to minus unity prohibits a kinetic energy in the symmetry
class C.
Symmetry class CI:
H(k ) = −τ2 H∗ (−k ) τ2 = +τ1 H∗ (−k ) τ1 = τ2 M2 + τ1 M1 .
PHS squaring to minus unity prohibits a kinetic energy in the symmetry
class CI.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
75 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
76 / fermio
93
Existence and uniqueness of normalized Dirac
masses
For each symmetry class, there exists a minimum rank
N 3 rmin (d) > 2, for which the d-dimensional Dirac Hamiltonian
supports a mass matrix and below which either no mass matrix or no
Dirac kinetic contribution are allowed by symmetry.
Suppose that the rank of the d-dimensional massive Dirac Hamiltonian
is r = rmin (d) N. When there exists a mass matrix squaring to unity (up
to a sign) that commutes with all other symmetry-allowed mass
matrices, we call it the unique mass matrix.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
76 / fermio
93
Existence and uniqueness of normalized Dirac
masses
There are the following three cases:
(a) π0 (Vd ) = {0}: For any given mass matrix, there exists another
mass matrix that anticommutes with it for N ≥ 1.
(b) π0 (Vd ) = Z: There always is a unique mass matrix for N ≥ 1.
(c) π0 (Vd ) = Z2 : There is a unique mass matrix when N is an odd
integer. However, when N is an even integer, there is no such
unique mass matrix.
When N = 1 and when the Dirac Hamiltonian has a unique mass
matrix, then topologically distinct ground states are realized for
different signs of its mass. When the mass is varied smoothly in space,
domain boundaries where the sign-of-mass changes are accompanied
by massless Dirac fermions, whose low-energy Hamiltonian is of rank
rmin (d)/2. It follows that rmin (d − 1) = rmin (d).
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
77 / fermio
93
Path connectedness of the normalized Dirac masses
Case (a): π0 (V ) = {0}
Trivial phase
Odd N
.....
ν=−1/2
ν=+1/2
.....
Case (b): π0 (V ) = Z
Even N
.....
Case (c): π0 (V ) = Z2
C. Mudry (PSI)
ν=−1
ν=0
ν=0
ν=+1
.....
ν=1
The breakdown of the topological classification Z for gapped phases of noninteracting
78 / fermio
93
Network model and criticality
For each realization of a random potential perturbing the
B
A
massless Dirac Hamiltonian defined in d-dimensional
(domains) of linear size ξdis . In each of these domains,
B
B
the normalized Dirac masses correspond to a unique
B
value of their zeroth homotopy group. At the boundary
between domains differing by the values taken by their
A
A
A
zeroth homotopy group, the Dirac masses must vanish.
Such boundaries support zero-energy boundary states.
B
A
Euclidean space, it may be decomposed into open sets
B
B
When it is possible to classify the elements of the zeroth homotopy group by
the pair of indices A and B, we may assign the letters A or B to any one of
these domains as is illustrated. When the typical “volume” of a domain of type
A equals that of type B, quasi-zero modes undergo quantum percolation
through the sample and thus establish either a critical or a metallic phase of
quantum matter in d-dimensional space.
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
79 / fermio
93
The solution for the classifying spaces is
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
Class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
V
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
C. Mudry (PSI)
π0 (V )
Z
0
Z
Z2
Z2
0
Z
0
0
0
T
0
0
+1
+1
0
−1
−1
−1
0
+1
π0 (V )
0
Z
0
Z
Z2
Z2
0
Z
0
0
C
0
0
0
+1
+1
+1
0
−1
−1
−1
Γ
0
1
0
1
0
1
0
1
0
1
π0 (V )
Z
0
0
0
Z
Z2
Z2
0
Z
0
Extension
Cld → Cld+1
Cld+1 → Cld+2
Cl0,d+2 → Cl1,d+2
Cld+1,2 → Cld+1,3
Cld,2 → Cld,3
Cld,3 → Cld,4
Cl2,d → Cl3,d
Cld+3,0 → Cld+3,1
Cld+2,0 → Cld+2,1
Cld+2,1 → Cld+2,2
π0 (V )
0
Z
0
0
0
Z
Z2
Z2
0
Z
π0 (V )
Z
0
Z
0
0
0
Z
Z2
Z2
0
Vd
C0+d
C1+d
R0−d
R1−d
R2−d
R3−d
R4−d
R5−d
R6−d
R7−d
π0 (V )
0
Z
0
Z
0
0
0
Z
Z2
Z2
π0 (V )
Z
0
Z2
0
Z
0
0
0
Z
Z2
π0 (V )
0
Z
Z2
Z2
0
Z
0
0
0
Z
The breakdown of the topological classification Z for gapped phases of noninteracting
80 / fermio
93
where
Label
C0
C1
R0
R1
R2
R3
R4
R5
R6
R7
Label
C0
C1
R0
R1
R2
R3
R4
R5
R6
R7
C. Mudry (PSI)
π0 (V )
Z
0
Z
Z2
Z2
0
Z
0
0
0
π1 (V )
0
Z
Z2
Z2
0
Z
0
0
0
Z
Classifying space V
∪N
n=0 U(N)/ U(n) × U(N − n)
U(N)
∪N
n=0 O(N)/ O(n) × O(N − n)
O(N)
O(2N)/U(N)
U(2N)/Sp(N)
N ∪n=0 Sp(N)/ Sp(n) × Sp(N − n)
Sp(N)
Sp(N)/U(N)
U(N)/O(N)
π2 (V )
Z
0
Z2
0
Z
0
0
0
Z
Z2
π3 (V )
0
Z
0
Z
0
0
0
Z
Z2
Z2
π4 (V )
Z
0
Z
0
0
0
Z
Z2
Z2
0
π5 (V )
0
Z
0
0
0
Z
Z2
Z2
0
Z
π6 (V )
Z
0
0
0
Z
Z2
Z2
0
Z
0
π7 (V )
0
Z
0
Z
Z2
Z2
0
Z
0
0
The breakdown of the topological classification Z for gapped phases of noninteracting
81 / fermio
93
The parameters for the phase diagram: π0 (V ) = {0}
For any of the five AZ symmetry classes with V = C1 , R3 , R5 , R6 , R7 ,
we can always choose the parameter space (m, g) ∈ R × [0, ∞[ by
selecting
α = αmin ⊗ 1N ,
β0 := βmin ⊗ 1N ,
and
M(x) =: m β0 ,
1 tr [M(x) − m β0 ][M(y) − m β0 ] =: g2 e−|x−y|/ξdis .
r
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
82 / fermio
93
The parameters for the phase diagram: π0 (V ) = Z
For any of the three AZ symmetry classes with V = C0 , R0 , R4 , we can
always choose the parameter space (m, g) ∈ R × [0, ∞[ by selecting
α = αmin ⊗ 1N ,
β0 := βmin ⊗ 1N ,
and
M(x) =: m β0 ,
1 tr [M(x) − m β0 ][M(y) − m β0 ] =: g2 e−|x−y|/ξdis .
r
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
83 / fermio
93
Moreover, we can also always write
M(x) = βmin ⊗ M(x),
B
A
B
A
where the condition
B
det[M(x)] = 0
B
A
defines the boundaries of the domains from
the quantum network model,
each of which can be assigned the index

{−1/2, +1/2},

1 ν := tr sgn[M(x)] ∈ {−1, 0, +1},

2

and so on,
B
A
A
B
B
if N = 1,
if N = 2,
if N > 2,
with [U(x) is unitary]
M =: U † diag (λ1 , . . . , λN ) U,
λ1
λN
†
sgn[M(x)] := U diag
,...,
U.
|λ1 |
|λN |
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
84 / fermio
93
The parameters for the phase diagram: π0 (V ) = Z2
For any of the two AZ symmetry classes with V = R1 , R2 , we can
always choose the parameter space (m, g) ∈ R × [0, ∞[ by selecting
α = αmin ⊗ 1N ,
M(x) = ρmin ⊗ M(x),
with ρmin ⊗ σ2 delivering a representation of the Clifford algebra of rank
rmin ,
M(x) = M † (x),
and
M(x) = 0,
C. Mudry (PSI)
M(x) = −M T (x),
m := Pf [iM(x)],
1
tr [M(x) M(y)] =: g2 e−|x−y|/ξdis .
r
The breakdown of the topological classification Z for gapped phases of noninteracting
85 / fermio
93
B
A
B
A
B
B
A
B
A
A
B
B
Moreover, the condition
det[M(x)] = 0
defines the boundaries of the domains for the quantum network model,
each of which can be assigned the index
Pf [iM(x)]
.
(−1)ν := p
det [iM(x)]
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
86 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
87 / fermio
93
Application to 1D space
AZ symmetry class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
Vd=1,r
C1
C0
R7
R0
R1
R2
R3
R4
R5
R6
(b) AIII, BDI, CII
N odd
ν=−1/2
Insulator
Cut at m = 0
insulating
even-odd
insulating
even-odd
critical
critical
insulating
even-odd
insulating
insulating
(c) D, DIII
N even
ν=1/2
ν=N/2
0
ν=−1
ν=0
ν=1
...
...
...
ν=−N/2
C. Mudry (PSI)
Phase diagram
(a)
(b)
(a)
(b)
(c)
(c)
(a)
(b)
(a)
(a)
π0 (Vd=1,r )
0
Z
0
Z
Z2
,
Z2
0
Z
0
0
...
(a) A, AI, AII, C, CI
rmin
2
2
2
2
2
4
4
4
4
4
ν=−N/2
Insulator
ν=0
Insulator
ν=1
ν=N/2
0
0
The breakdown of the topological classification Z for gapped phases of noninteracting
87 / fermio
93
Application to 2D space
AZ symmetry class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
rmin
2
4
4
4
2
4
4
8
4
8
(a) A, C (N=1)
Vd=2,r
C0
C1
R6
R7
R0
R1
R2
R3
R4
R5
π0 (Vd=2,r )
Z
0
0
0
Z
Z2
Z2
0
Z
0
π1 (Vd=2,r )
0
Z
0
Z
Z2
Z2
0
Z
0
0
Phase diagram
(a)
(e)
(b)
(e)
(c)
(d)
(d)
(f)
(a)
(b)
(b) A, C (N=2)
Cut at m = 0
even-odd
Gade singularity
insulating
Gade singularity
even-odd
metallic
metallic
Gade singularity
even-odd
insulating
(g) AII
Metal
Insulator
ν=−1/2
Insulator
ν=1/2
Insulator
ν=0
Insulator
ν=−1
Insulator
ν=0
Insulator
ν=1
0
0
(c) D (N=1)
(d) D (N=2)
Insulator
ν=1
0
(h) AI, CI
Metal
Metal
Z2 singularity
Insulator
ν=−1/2
Insulator
ν=1/2
Insulator
ν=−1
0
C. Mudry
(e) DIII (N=1)
(PSI)
Insulator
ν=0
Insulator
Insulator
ν=1
0
The breakdown
DIIItopological
(N=2)
of(f)the
0
classification Z for gapped
phases
88 / fermio
93
(i) AIII, BDI,
CII of noninteracting
Application to 2D space
(a) A, C (N=1)
(b) A, C (N=2)
(g) AII
Metal
Insulator
ν=−1/2
Insulator
ν=1/2
Insulator
ν=0
Insulator
ν=−1
Insulator
ν=0
Insulator
ν=1
0
0
(c) D (N=1)
(d) D (N=2)
Insulator
ν=1
0
(h) AI, CI
Metal
Metal
Z2 singularity
Insulator
ν=−1/2
Insulator
ν=1/2
Insulator
ν=0
Insulator
ν=−1
0
0
(e) DIII (N=1)
(f) DIII (N=2)
Metal
Metal
Insulator
ν=0
Insulator
ν=1
C. Mudry (PSI)
0
(i) AIII, BDI, CII
Gade singularity
Insulator
ν=1
Insulator
ν=0
Z2 singularity
0
Insulator
Insulator
ν=1
Insulator
Z2 singularity
0
0
The breakdown of the topological classification Z for gapped phases of noninteracting
89 / fermio
93
Application to 3D space
AZ symmetry class
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
(a) A, AI, BDI, D, C
Metal
Insulator
rmin
4
4
8
8
4
4
4
8
8
8
Vd=3,r
C1
C0
R5
R6
R7
R0
R1
R2
R3
R4
π0 (Vd=3,r )
0
Z
0
0
0
Z
Z2
Z2
0
Z
(b) AIII, DIII, CI (N=1)
Insulator
ν=−1/2
C. Mudry (PSI)
π2 (Vd=3,r )
0
Z
0
Z
Z2
Z2
0
Z
0
0
(c) AIII, DIII, CI (N=2)
Metal
#
1
N+1
1
1
1
N+1
2
2
1
N+1
Insulator
ν=−1
Insulator
ν=0
0
DoS
Nonsing
Sing
Nonsing
Sing
Sing
Sing
Nonsing
Sing
Nonsing
Nonsing
(d) AII, CII
Metal
Metal
Insulator
ν=1/2
0
0
π1 (Vd=3,r )
Z
0
0
0
Z
Z2
Z2
0
Z
0
Insulator
ν=1
Insulator
ν=0
Insulator
ν=1
0
The breakdown of the topological classification Z for gapped phases of noninteracting
90 / fermio
93
1
Main results
2
Introduction
3
Strategy
4
Examples
5
Application to the surfaces of SnTe
6
Summary
7
Appendices
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
91 / fermio
93
Summary
We have considered (noninteracting) random Dirac Hamiltonians
for which we have explored the physics of Anderson localization.
Dimensionality of space and the AZ symmetry classes (tenfold
way) play an essential role.
Localization requires the existence of random Dirac masses.
There are four possibilities in a given AZ symmetry class:
1
2
3
No random Dirac masses are permitted for sufficiently small rank,
these are the delocalizated boundary states of a topological
insulator.
At least two anticommuting Dirac mass matrices are permitted for 5
symmetry classes when the rank is rmin . The insulating phase is
unique.
A unique (up to a sign) Dirac mass matrix is permitted for 3+2
symmetry classes when the rank is rmin .
1
2
For 3 of the symmetry classes, a unique (up to a sign) Dirac mass
matrix is permitted when the rank is rmin N for any N = 1, 2, 3 · · · .
For 2 of the symmetry classes, a unique (up to a sign) Dirac mass
matrix is permitted iff the rank is rmin N for any N = 1, 3, 5, · · · .
C. Mudry (PSI)
The breakdown of the topological classification Z for gapped phases of noninteracting
91 / fermio
93
In any dimension, the density of states can be singular at the band
center in 5 out of the 10 AZ symmetry classes:
In any dimension, the three chiral classes always support point
defects in their Dirac masses labeled by a topological index in Z.
In any dimension, 2 of the remaining 7 symmetry classes support
point defects in their Dirac masses labeled by a topological index
in Z2 .
Case d = 1
Five AZ symmetry classes
Three AZ symmetry classes
(A)
(B)
g
g
g
Two AZ symmetry classes
N=2 and so on
m
C. Mudry (PSI)
0
m
Insulator
ν=+1
0
(C)
g
Insulator
ν=0
Insulator Insulator
ν=−1/2 ν=+1/2
ν=−1
Insulator
Insulator
0
(B)
N=1
m
Insulator
ν=0
Insulator
ν=+1
0
m
The breakdown of the topological classification Z for gapped phases of noninteracting
92 / fermio
93
Case d = 3
Five AZ symmetry classes
Three AZ symmetry classes
(A)
(B)
g
g
Metal
(B)
N=1
g
Two AZ symmetry classes
N=2 and so on
(C)
g
Metal
Metal
Metal
ν=0
ν=−1/2 ν=+1/2
Insulator
0
m
C. Mudry (PSI)
0
m
ν=−1
ν=+1
0
ν=0
m
ν=+1
0
m
The breakdown of the topological classification Z for gapped phases of noninteracting
93 / fermio
93