### Time-reversal anomaly in SPT phases

Shinsei Ryu

University of Chicago

December 9, 2018

• Symmetry in nature can be broken in at least three different
*ways: explicitly, spontaneously, and anomalously*

• Spontaneous symmetry breaking (SSB) is a key ingredient for the Landau-Ginzburg-Wilson paradigm describing phases of matter and phase transitions.

• Symmetry can be also broken by quantum effects – quantum anomalies. They play a key role in describing topological phases – phases of matter which defy the description by the symmetry breaking paradigm.

### Phases of matter;

ordered v.s. disordered and gappped v.s. gaplessSymmetry protected topological phases

(SPT phases)

Topologically ordered phases Long-range entangled states Trivial phases

Short-range entangled states (a.k.a "invertible" states) No topological

order

Topological order

Symmetry enriched topological phases (SET phases) Symmetry

Gapped Gapless

No spontaneous symmetry breaking Spontaneous

symmetry breaking

Gapless Gapped

Continuous symmetry-broken

phases

Discrete symmetry-broken

phases

Symmetry Quantum critical

disordered phases and critical points

Other phases Symmetry breaking coexists

with topological order ...

Ordered phases Quantum disordered phases

Gapped quantum disordered phases Phases of matter

### Ordered v.s. disordered

Today, I am interested in disordered phases (phases without any order parameter).

### Phases of matter;

ordered v.s. disordered and gappped v.s. gaplessSymmetry protected topological phases

(SPT phases)

Topologically ordered phases Long-range entangled states Trivial phases

Short-range entangled states (a.k.a "invertible" states) No topological

order

Topological order

Symmetry enriched topological phases (SET phases) Symmetry

Gapped Gapless

No spontaneous symmetry breaking Spontaneous

symmetry breaking

Gapless Gapped

Continuous symmetry-broken

phases

Discrete symmetry-broken

phases

Symmetry Quantum critical

disordered phases and critical points

Other phases Symmetry breaking coexists

with topological order ...

Ordered phases Quantum disordered phases

Gapped quantum disordered phases Phases of matter

### Gapless v.s. gapped, topological order

• For this talk, I will be interested in gapped phases of matter.

• Furthermore, I will not consider topologically ordered phases.

Energy

Ground state Degenerate ground states Gap

(a) (b) (c)

### Phases of matter;

ordered v.s. disordered and gappped v.s. gaplessSymmetry protected topological phases

(SPT phases)

Topologically ordered phases Long-range entangled states Trivial phases

Short-range entangled states (a.k.a "invertible" states) No topological

order

Topological order

Symmetry enriched topological phases (SET phases) Symmetry

Gapped Gapless

No spontaneous symmetry breaking Spontaneous

symmetry breaking

Gapless Gapped

Continuous symmetry-broken

phases

Discrete symmetry-broken

phases

Symmetry Quantum critical

disordered phases and critical points

Other phases Symmetry breaking coexists

with topological order ...

Ordered phases Quantum disordered phases

Gapped quantum disordered phases Phases of matter

### Phases of matter;

ordered v.s. disordered and gappped v.s. gaplessSymmetry protected topological phases

(SPT phases)

Topologically ordered phases Long-range entangled states Trivial phases

Short-range entangled states (a.k.a "invertible" states) No topological

order

Topological order

Symmetry enriched topological phases (SET phases) Symmetry

Gapped Gapless

No spontaneous symmetry breaking Spontaneous

symmetry breaking

Gapless Gapped

Continuous symmetry-broken

phases

Discrete symmetry-broken

phases

Symmetry Quantum critical

disordered phases and critical points

Other phases Symmetry breaking coexists

with topological order ...

Ordered phases Quantum disordered phases

Gapped quantum disordered phases Phases of matter

### Phases of matter;

ordered v.s. disordered and gappped v.s. gaplessSymmetry protected topological phases

(SPT phases)

Topologically ordered phases Long-range entangled states Trivial phases

Short-range entangled states (a.k.a "invertible" states) No topological

order

Topological order

Symmetry enriched topological phases (SET phases) Symmetry

Gapped Gapless

No spontaneous symmetry breaking Spontaneous

symmetry breaking

Gapless Gapped

Continuous symmetry-broken

phases

Discrete symmetry-broken

phases

Symmetry Quantum critical

disordered phases and critical points

Other phases Symmetry breaking coexists

with topological order ...

Ordered phases Quantum disordered phases

Gapped quantum disordered phases Phases of matter

### Symmetry-protected topological phases (SPT phases)

• *In the absence of symmetry, SPT phases are adiabatically*
connected to a trivial phase

• *Trivial phase = product states |Ψi = |φi|φi · · · |φi*

• Unique ground state on any manifold

• *Nevertheless, if we impose symmetry, SPT phases are*
topologically distinct

### Symmetry-protected topological phases (SPT phases)

• Examples: time-reversal symmetric topological insulators; the Haldane phase in 1d spin chains [Haldane, 2016 Nobel Prize]

• No local order parameter: symmetry-breaking paradigm cannot be applied:

### Topological phases and anomalies

• Questions:

• How do we classify SPT phases?

• How do we characterize/detect/diagnose SPT phase?

Effects of interactions?

• Main “tools/concepts”:

• Response theory (topological quantum field theories)

• Quantum anomalies

• Bulk-boundary correspondence

### Response theory

• *When ∃ global symmetry G (unitary, on-site),*
*Z(M, A) =*

Z

*D[φ] exp[−S(φ, M, A)]*

*A : background G-gauge field,* *φ : “matter field”*

• Even when no global symmetry,
*Z(M ) =*

Z

*D[φ] exp[−S(φ, M )]*

*M : closed spacetime manifold*

• (More data maybe needed for other situations.)

### Response theory for SPT phases

• For topological (SPT) phases,

• *(i) Z(M, A) is expected to have a pure imaginary part:*

*Z(M, A) = exp[iI**top**(M, A)]*

• *(ii) I**top**(M, A) is expected to be topological (metric*
independent).

• *(iii) I**top**(M, A) is not gauge invariant in the presence of*
boundary.

• *I**top* serves as a “non-local order parameter”.

• Generic approach, but very powerful for SPT phases because of unique ground states. [Kapustin et al. (14), Freed (14-16), Witten (15)]

### Example: Quantum Hall effect

### Example: QHE

• *U (1) particle number conservation; can couple the system*
*with an external (probe) gauge field A*^{ex}* _{µ}* .

• Response of the system is encoded in the effective action:

*Z(A** ^{ex}*) =
Z

*D[ψ*^{†}*, ψ]e*^{−S(A}^{ex}^{,ψ}^{†}^{,ψ)}*= e*^{−I}^{eff}^{(A}^{ex}^{)}

• *In the QHE, I** _{eff}* has a topological contribution; the
Chern-Simons term, which is imaginary:

*I*_{eff}*(A) =* *ik*
*4π*

Z

*dτ dxdy ε*^{µνλ}*A*_{µ}*∂*_{ν}*A*_{λ}*,* *k = integer*
Independent of the metric.

### Bulk-boundary correspondence

• In the presence of boundary, the Chern-Simons term is not gauge invariant.

• Necessary to have boundary degrees of freedom which cancel the non-invariance.

• Boundary theory is anomalous.

• They cannot be gapped trivially while preserving symmetry;

Gapless or topologically ordered

• *More generally: Bulk (d + 1)-dim G SPT supports d-dim*
*boundary theory, which has G ’t Hooft anomaly.*

### Example: QHE

• Chiral edge theory:

L = 1

*2πψ*^{†}*i(∂**t**+ ∂*_{x}*)ψ*
Twisted boundary conditions:

*ψ(t, x + L) = e*^{2πia}*ψ(t, x),* *ψ(t + β, x) = e*^{2πib}*ψ(t, x)*

### Example: QHE

• Classical system (Lagrangian + b.c.) is invariant under
*a → a + 1 and b → b + 1 (large gauge transformation)*

• Quantum mechanics:

*Z([a, b]) =*
Z

*D[ψ*^{†}*, ψ]e** ^{−S}* = Tr

_{a}

*e*^{−βH}*e*^{2πi b+}^{1}^{2}

*N*

• Tr_{a}*: Spatial b.c. twisted by the phase e*^{2πia}

• *e*^{2πi b+}^{1}^{2}

*Q*

*: Temporal b.c. is twisted by the phase e** ^{2πib}*:

• Large gauge anomaly:

*Z([a, b]) 6= Z([a, b + 1])* or *Z([a, b]) 6= Z([a + 1, b]).*

### Other symmetries?

• Discrete on-site unitary symmetry[“group cohomology approach”:

Dijkgraaf-Witten Chen-Liu-Gu-Wen (11)]

• Anti unitary on-site unitary symmetry, e.g., Time-reversal, reflection, etc.

• Crystalline symmetries

Today: Orientation-reversing symmetry:

• Time-reversal, spatial reflection,

### Example: Topological insulator

### Example: Topological insulator

### Example:CP symmetry topological insulator

• *System charge U (1) and CP symmetry: P (x, y) → (−x, y).*

• “CPT”-dual of (2+1)d topological insulator

• Edge theory (for CP symmetric edge)

*H =*
Z

*dx*h

*ψ*_{L}^{†}*i∂**x**ψ*_{L}*− ψ*_{R}^{†}*i∂**x**ψ** _{R}*i

• Under CP symmetry

U*CP**ψ*_{L}*(x)U*_{CP}^{−1} *= ψ*_{R}^{†}*(−x),* U*CP**ψ*_{R}*(x)U*_{CP}^{−1} *= ψ*_{L}^{†}*(−x),*
no mass terms are allowed.

• Topological phases with (protected by) time-reversal;

⇒

• *Is there any anomaly associate to time-reversal symmetry?*

• *How can we develop response theory? Or how can we “gauge”*

*time-reversal?*

### Anomaly on unoriented surface

[Hsieh-Sule-Cho-SR-Leigh (14)]

• Twisting by parity symmetry:

• C.f. Twisting by on-site symmetry:

[Hsieh-Sule-Cho-SR-Leigh (14)]

• Klein bottle partition function: twisting by CP and U(1):

*ψ*_{L}*(t + T, x) = ψ*^{†}_{R}*(L − x, t),* *ψ*_{R}*(t + T, x) = ψ*_{L}^{†}*(L − x, t)*
*ψ**L**(t, x + L) = e*^{2πia}*ψ**L**(x, t),* *ψ**R**(t, x + L) = e*^{2πia}*ψ**R**(x, t)*

• *Klein bottle (KB) partition func (CP twisted partition func)*
*Z(KB, a) = Tr*_{a}^{h}U_{CP}*e*^{−βH}^{i}

• *Large gauge anomaly under a → a + 1:*

*Z(KB, a + 1) = (−1)Z(KB, a).*

• C.f. Old work by[Brunner-Hori (03)]

### How about the bulk?

• *The partition function on Klein bottle × S*^{1} with flux:

*Z(KB × S*^{1}*, A) = (−1)*

• This Z2 response is the fundamental response characterizing topological insulators, even in the presence of interactions.

### Many-body Z

2### topological invariant for (2+1)d topological insulators

[Shiozaki-Shapourian-SR (17)]

• Setup:

• Formula: (T1 = fermionic partial transpose)
*Z = Tr*_{R}_{1}*∪R*3

h*ρ*^{+}_{R}

1*∪R*_{3}*C*_{T}^{I}^{1}*[ρ*^{−}_{R}

1*∪R*_{3}]^{T}^{1}*[C*_{T}^{I}^{1}]^{†}^{i}*,*
*ρ*^{±}_{R}

1*∪R*_{3} = Tr_{R}

1*∪R*3

h*e*^{±}
P

**r∈R2***2πiy*

*Ly* **n(r)**

| {z }

*partial U (1) twist*

*|GSihGS|*^{i}

*C** _{T}* ∼ spin flip unitary

### Many-body Z

2### topological invariant for (2+1)d topological insulators

• *Z is the partition function on Klein bottle × S*^{1} with flux.

• *Phase of Z computed numerically on a lattice:*

### 3D example:

^{3}

### He B

• B-phase of ^{3}

• BdG hamiltonian:

*H =*
Z

*d*^{3}k Ψ^{†}*(k)H(k)Ψ(k),* H(k) =

"

*k*^{2}

*2m**− µ* *∆σ · k*

*∆σ · k* −_{2m}^{k}^{2} *+ µ*

#

*Ψ(k) = (ψ*_{↑k}*, ψ*_{↓k}*, ψ*_{↓,−k}^{†} *, −ψ*_{↑,−k}^{†} )^{T}

• SPT phase protected by TRS or spatial inversion
*Iψ*_{σ}^{†}*(r)I*^{−1} *= iψ*_{σ}^{†}(−r)

• When non-interaction, characterized by an integer topological
*invariant ν.*

### Surface Majorna states

• By bulk-boundary correspondence;surface Majorna cones

• Detected by surface acoustic impedance measurement

[Murakawa et al (09)]:

### Z

^{16}

### classification

• The non-interacting classification breaks down to Z16 by interaction. Surface topological order, etc. [Fidkowski et al (13), Metlitski et al (14), Wang-Senthil (14), Morimoto-Furusaki-Mudry (15), ..]

• *Initial calculation of surface anomaly on T*^{3} with parity twist
reveals Z8 [Hsieh-Cho-SR (15)]

### Many-body topological invariant

• (3+1)d DIII topological superconductors are expected to be
*detected by RP*^{4}. [Kapustin et al (14-15), Freed-Hopkins (14-15),
Witten (15), ...]

• *We consider partial inversion I**part* *on a ball D:*

*Z = hΨ|I** _{part}*|Ψi = Tr

_{D}*(I*

_{part}*ρ*

*)*

_{D}• The spacetime is effectively four-dimensional projective plane,
*RP*^{4}.

### Bulk calculations

• Numerics on a lattice:

−4 −2 0 2 4

µ/t

−1.0

−0.5 0.0 0.5

6Z/(π/4)

Top. II Trivial

Trivial Top. I Top. I

• *Z(RP*^{4}*) = exp[2πiν/16] with ν = 0, . . . , 15 is the*

fundamental response of (3+1) topological superconductors protected by orientation-reversing symmetry.

### Boundary calculations

• The topological invariant can be computed from the Majorana surface theory[Shiozaki-Shapourian-SR(16)]

*Z = Tr**D**(I*_{part}*ρ**D*) = Tr_{∂D}*(I*_{part}*e*^{−H}* ^{surf}*)
Tr

_{∂D}*(e*

^{−H}*)*

^{surf}*where H**surf* is the entanglement Hamiltonian ' physical
surface Hamiltonian

• *Result when ν = 1:*

*Z = exp*

"

−*iπ*
8 + 1

12ln(2) −21

16*ζ(3)** R*
*ξ*

^{2}
+ · · ·

#

### Boundary calculations

• With interactions, TSC surface can be gapped topologically ordered. [Senthil-Vishwanath, Fidkowski-Chen-Vishwanath (13),

Wang-Senthil (15), Metlitski-Fidkowski-Chen-Vishwanath (14), ...]

• When surface is topologically ordered: [Wang-Levin, Tachikawa-Yonekura, Barkeshli et al (16)]

*Z =* Tr_{∂D}*(I*_{part}*e*^{−H}* ^{surf}*)
Tr

_{∂D}*(e*

^{−H}*)*

^{surf}=^{X}

*p*

*e*^{−2πih}^{p}*η**p**d**p* = exp

*2πiν*
16

*where sum is over symmetric anyons with topological spin h**p*,
*quantum dimension d*_{p}*, and eigenvalues of T*^{2}.

### Summary

• Topological insulators and topological superconductors protected by orientation reversing symmetry can be

detected/defined by their coupling to unoriented spacetime.

• Constructed explicit many-body topological invariants.

• Essentially the same construction of many-body topological invariants for other cases, e.g., the Kitaev Majorana chain, etc.

• Numerically useful?