Symmetry-protected topological phases: Projective...

SPT@π

Peng Ye

General review

Projectiveconstruction of2d SPT

3d SPT bulk fieldtheory withb ∧ b term

Summary

Symmetry-protected topological phases:Projective construction & Bulk field theory

Peng Ye

SPT enthusiasts @ 3.14159

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SPT@π

Peng Ye

General review

Summary

Collaboration with great colleagues, friends below:

Xiao-Gang Wen (MIT) Zheng-Cheng Gu (PI) Zheng-Xin Liu (Tsinghua) Jia-Wei Mei (PI)

Focused references:

• Projective construction of 2d SPT: arXiv:1212.2121, 1408.1676

• Bulk field theory of 3d SPT: arXiv:1410.2594.

Related but not included here:

• Proj. constr. of 3d SPT: 1303.3572 [see my Princeton talk in Mar.];

• Response theory: 1306.3695;

• 1d SPT order of doped spin chain: 1310.6496;

• 3d SPT: solvable lattice model (Ye, Gu, in preparation)

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SPT@π

Peng Ye

General review

Summary

Preface-[1]: Triviality

• My colleague asked me: You ever told me SPT is trivial. right?

• I replied without doubt: Yes!

• My colleague: But, what do you mean by “trivial SPT” in yourlengthy paper?!!!

• I: .......... Mmmm,... a trivial man is still a man not

a cat.

Defining triviality is quite nontrivial. −→ Our development ofphysics is to continuously redefine triviality by deeper anddeeper probes / insights.

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Peng Ye

General review

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a cat.

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Peng Ye

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a cat.

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Peng Ye

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a cat.

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Peng Ye

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a cat.

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SPT@π

Peng Ye

General review

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a cat.

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SPT@π

Peng Ye

General review

Summary

Preface-[2]: Condensation

• My colleague: You actually belong to cold-atom community

• I: Why?

• My colleague: holon condensation leads to high-Tc; spinoncondensation leads to antiferromagnetism of weakly dopedcuprate; monopole condensation leads to charge localization atweakly doped cuprates; dyon condensation leads to 3d bosonicTI; now, string condensation leads to more general 3d bosonicTI. What are you gonna condense next time?!!!!

• I: .......... ,...,,,.Maybe “membrane”, ......

We are “condensed” matter theorists.

Indeed condensing topological defects generically drives aphase transition to exotic states of matter out from aconventional state.

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Peng Ye

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• I: Why?

• I: .......... ,...,,,.Maybe “membrane”, ......

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SPT@π

Peng Ye

General review

Summary

• I: Why?

• I: .......... ,...,,,.Maybe “membrane”, ......

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SPT@π

Peng Ye

General review

Summary

• I: Why?

• I: .......... ,...,,,.Maybe “membrane”, ......

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SPT@π

Peng Ye

General review

Summary

• I: Why?

• I: .......... ,...,,,.Maybe “membrane”, ......

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SPT@π

Peng Ye

General review

Summary

• I: Why?

• I: .......... ,...,,,.Maybe “membrane”, ......

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Peng Ye

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Contents

1 General review

2 Projective construction of 2d SPT

3 3d SPT bulk field theory with b ∧ b term

4 Summary

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Peng Ye

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Section 1 General review

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Peng Ye

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Hundred years’ hard Condensed Matter (Solid State)

Physics in progress... Great triumph in 1980s.

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SPT@π

Peng Ye

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Summary

A central concept: “order” , a way of organization in many-body system.

1 What is the “order” that organizes the observed phenomena?

2 How to classify those “orders”?

Symmetry-breaking order G −→ H

• Symmetry-breaking order (Landau theory) works very well for a longtime.

• Local order parameter, as a function, is determined by G/H.

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SPT@π

Peng Ye

General review

Summary

A central concept: “order” , a way of organization in many-body system.

1 What is the “order” that organizes the observed phenomena?

2 How to classify those “orders”?

Symmetry-breaking order G −→ H

• Symmetry-breaking order (Landau theory) works very well for a longtime.

• Local order parameter, as a function, is determined by G/H.

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Peng Ye

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Fractional Quantum Hall states and topological order

Experimental conditions: Clean 2DEG (µ ∼ 105cm2/V·s), low electron

density (n ∼ 1011/cm2); low temperature, high magnetic field.Unexpected results: Wigner crystal or CDW? in Prof. Tsui’s seminal paper.Now we know: Topological order— (Wen) non-symmetry-breaking order;long-range entanglement. (Gu, Wen, Chen, et.al.) −→ quantuminformation, computing,...

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Peng Ye

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Symmetry-protected topological phases (SPT)

In absence of topological order and symmetry-breaking order, we haveSPT order.

Some star examples in more realistic electronic materials: (“fermionicSPT”)

1 Topological insulator (3d, time-reversal)

2 Quantum spin Hall insulator (2d TI, time-reversal)

3 Topological crystalline insulator. (space group, see, also, S. Ryu’stalk)

4 ......

99% can be modelled by band theory, or, weak interacting but has awell-defined free-fermion counterpart.

Classification of free-fermion nontrivial SPT is well-known now (Ryu,Schnyder, Furusaki, Ludwig; Kitaev, · · · )

Classification of interacting-fermion nontrivial SPT is under debate (Gu,Wen, Senthil, Kapustin, Ryu, Qi....many people) !!A frontier of research!!.

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4 ......

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4 ......

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SPT in bosons/spin

In boson SPT, strong interaction and strong correlation effects arenecessary!Avoid boson condensation (boson system); avoid spin order (spinparamagnets)

Some examples with concrete lattice models:

1 Haldane phase in spin-1 AKLT (or Heisenberg) is a bosonic SPT, ashort-range entangled state. (Pollmann, et.al. Gu, Wen, et.al.)

2 Topological Ising paramagnet (Levin, Gu 2012). A nontrivial 2dSPT protected by Z2 symmetry.

3 Bosonic topological superconductor (Burnell, Chen, et.al.1302.7072). A nontrivial 3d SPT protected by TR sym.

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SPT@π

Peng Ye

General review

Summary

SPT in bosons/spin

In boson SPT, strong interaction and strong correlation effects arenecessary!Avoid boson condensation (boson system); avoid spin order (spinparamagnets)Some examples with concrete lattice models:

1 Haldane phase in spin-1 AKLT (or Heisenberg) is a bosonic SPT, ashort-range entangled state. (Pollmann, et.al. Gu, Wen, et.al.)

2 Topological Ising paramagnet (Levin, Gu 2012). A nontrivial 2dSPT protected by Z2 symmetry.

3 Bosonic topological superconductor (Burnell, Chen, et.al.1302.7072). A nontrivial 3d SPT protected by TR sym.

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Progress in SPT

Intensive study in North America (quite incomplete list. sorry for missing.)

• Group cohomology classification and fixed-point Lagrangian (XieChen, Gu, Liu, Wen)

• Classification based on Chern-Simons theory (Lu, Vishwananth)

• Classification based on non-linear σ field theory with θ-term (C. Xu’sgroup)

• Classification based on “cobordism” (Kapustin, et.al.)

• SPT—gauge theory duality (Levin, Gu; Wen, et.al.)

• 3d bosonic TI (Vishwanath, Senthil, Metlitski, Ye, Wen, et.al.)

• 2d projective construction (Ye, Wen; Lu, Lee, et.al.)

• Boundary anomaly (S. Ryu’s group; Wang, Wen, et.al.)

• 3d field theory (Xu, Senthil; Ye, Gu; ...)

• · · · See review article by T. Senthil arXiv: 1405.4015 and referencestherein.

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Section 2 Projective construction of 2d SPT states

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Projective construction in non-Abelian QH states

At least, two useful analytic ways to construct non-Abelian QH states:

1 Correlation functions in CFT: edge states (Blok, Wen, Wu, Nayak,Wilczek, Read, Rezayi, Cappelli, et.al.), topologically invariant, twodifferent bdr (cut through plane or through time)

2 Projective (parton) construction: both edge and bulk field theories.(Wen, Blok, Barkeshli, et.al.)

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Peng Ye

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Projective construction in non-Abelian QH states

At least, two useful analytic ways to construct non-Abelian QH states:

1 Correlation functions in CFT: edge states (Blok, Wen, Wu, Nayak,Wilczek, Read, Rezayi, Cappelli, et.al.), topologically invariant, twodifferent bdr (cut through plane or through time)

2 Projective (parton) construction: both edge and bulk field theories.(Wen, Blok, Barkeshli, et.al.)

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Proj. constr. of non-Abelian QH: Wen’s recipe ’99

1 Introduce a few partons ψa (a = 1, · · · , n), each carries Qa charge:

Lparton = iψ†a ∂tψa + 1

2mψ†a (∂i − iQaAi )2ψa

2 Introduce an “electron” operator: Ψe =∑

∏a ψ

a (z),

where, n(m)a = 0, 1, and,

∑a Qan

(m)a = e.

3 ψa →Wabψb leaves Ψe and electron current operators Je unchanged.Thus, W †QW = Q, where, Q = diag(Q1,Q2, · · · ). W ∈ G (gaugegroup).

4 Consider Gc ∈ G, a connected piece of G, which contains identity.Thus, define an emergent gauge field aµ ∈ LGc , i.e. Lie algebra ofGc . Thus, bulk theory becomes:

L = iψ†a [δab∂t − i(a0)ab ]ψb + 1

2mψ†a (∂i − iQAi − iai )2

5 To ensure the bulk is gapped, one may consider partons form IQHstates with filling νa = ma. Integrating partons leads to:

S = 14π

Tr(M(a ∧ da + 23a ∧ a ∧ a)) + Tr(MQ2)

4πA ∧ dA

where, crossing term is eliminated by demanding: Tr(tMQ) = 0, witharbitrary matrix t ∈ LGc . M = diag(m1,m2, · · · )

6 Deriving edge states (Kac-Moody algebra) via coset theory or OPE.

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Proj. constr. of SPT: Ye-Wen’s program ’12

To understand symmetry transformation on matter fields, we preservepartons d.o.f..

• Parton field current J Iµ = 1

2πεµνλ∂νa

Iλ . Automatically resolve

∂µJ Iµ = 0. (aI , at this step, is a non-compact, 1-form U(1) gauge

connection, defined on the real axis.)

• A free gapped parton theory is described by a K -matrix CS theory

• Projection is done by imposing a “constraint” on parton currents (i.e.fluxes) in path-integral measure:

F (J1µ, J

2µ, · · · ) = 0 −→ F (a1

µ, a2µ, · · · ) = 0

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2πεµνλ∂νa

F (J1µ, J

2µ, · · · ) = 0 −→ F (a1

µ, a2µ, · · · ) = 0

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SPT@π

Peng Ye

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2πεµνλ∂νa

F (J1µ, J

2µ, · · · ) = 0 −→ F (a1

µ, a2µ, · · · ) = 0

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• Eventually, the bulk theory is described by a new K -Chern-Simonstheory. (assuming the constraint is a linear equation)

1 If detK = 0, then, apply GL transformation to separate the zeroeigenvalue.

1 Spontaneous Symmetry-breaking order with Goldstone modes ifzero eigenvalues carry symmetry charge. Monopole issuppressed. Maxwell term sets in.

2 If not, zero eigenvalues are in Polyakov confined phase(monopole condensation). Directly Remove it!!

2 If |detK | = 1, then, this state is a gapped state without topo.order. (might be a nontrivial SPT)

3 If |detK | > 1, then, the state is a topologically ordered Abelianstate. might be a nontrivial SET

• Edge reconstruction (Sometimes, chiral edge −→ non-chiral edge!! )

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Proj. constr. of U(1)× U(1) SPT states

Start with a spin-1 system, S = f †Γf , where Γ are three angularmomentum generators. Partons: f = (f+1, f0, f−1).

• Fill three Chern bands on a Kagome lattice (with Chern numberC1,C0,C−1).

• Minimal symmetry is: U(1)×U(1) (conservation of both∑

i Szi and∑

i (Szi )2)

• Projection is done by imposing a constraint on gauge fluxes:J+1µ + J0

µ + J−1µ = Kµ, where, Kµ = (1, 0, 0).

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SPT@π

Peng Ye

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Start with a spin-1 system, S = f †Γf , where Γ are three angularmomentum generators. Partons: f = (f+1, f0, f−1).

Consider C1 = 1,C0 = −1,C−1 = 1: The free parton theory is a“fermionic” Chern-Simons theory with chiral edge (1− 1 + 1 6= 0)

4π( a1µ a0

µ a−1µ )

(1 0 00 −1 00 0 1

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν ( 1 0 −1 )

a−1λ

)εµνλ

After Gutzwiller projection, da1 + da−1 + da0 = 0, we end up with:

4π( a1µ a−1

0 −1−1 0

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν( 1 −1 )

a−1λ

)εµνλ

[1] Groundstate wavefunction: PG |free partons〉 (Monte Carlo testable)[2] Edge is reconstructed to be non-chiral, with even-quantized nontrivialspin Hall conductance:

2π× 2.

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SPT@π

Peng Ye

General review

Summary

Start with a spin-1 system, S = f †Γf , where Γ are three angularmomentum generators. Partons: f = (f+1, f0, f−1).Consider C1 = 1,C0 = −1,C−1 = 1: The free parton theory is a“fermionic” Chern-Simons theory with chiral edge (1− 1 + 1 6= 0)

4π( a1µ a0

µ a−1µ )

(1 0 00 −1 00 0 1

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν ( 1 0 −1 )

a−1λ

)εµνλ

4π( a1µ a−1

0 −1−1 0

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν( 1 −1 )

a−1λ

)εµνλ

2π× 2.

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SPT@π

Peng Ye

General review

Summary

Start with a spin-1 system, S = f †Γf , where Γ are three angularmomentum generators. Partons: f = (f+1, f0, f−1).Consider C1 = 1,C0 = −1,C−1 = 1: The free parton theory is a“fermionic” Chern-Simons theory with chiral edge (1− 1 + 1 6= 0)

4π( a1µ a0

µ a−1µ )

(1 0 00 −1 00 0 1

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν ( 1 0 −1 )

a−1λ

)εµνλ

4π( a1µ a−1

0 −1−1 0

)∂ν

a−1λ

)εµνλ +

2πASz

µ ∂ν( 1 −1 )

a−1λ

)εµνλ

2π× 2.

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Numerics: Even-quantized Hall conductance

0 0.2 0.4 0.6 0.8 10

Φ S z/2π

Szpump

PG |1-11⟩

0 0.2 0.4 0.6 0.8 1−2

−1.5

−0.5

Φ S 2z/2π

S2 zpump

PG |1-11⟩

(b)(a)

One coin, two sides:Non-zero even quantized Hall response ←→ All gapping terms on theboundary necessarily break symmetry ←→ nontrivial U(1) SPT order.

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Gapping conditions

Understand more mathematically:

• In terms of bosonization, perturbation terms have the form of∼ cos(`Tφ)

• Is there a solution ` of equation `TK−1` = 0? (Haldane’s null vectorcondition)

• No. “intrinsic” state (chiral states and some nonchiral states(i.e. ν = 2/3 FQH))

• Yes. is there a solution ` further satisfying `TK−1q = 0?• Yes, trivial SPT. The perturbation term gaps edge without

breaking symmetry.• No, nontrivial SPT. All gapping terms break U(1) symmetry.−→ one can show: qTK−1q is always even-integer (note thatdiagonal entries of K is even for boson systems and |detK | = 1).

• A side note:

# of symmetry-breaking perturbation � # of gapping terms.

For example, in-plane Zeeman term can not gap edge.BSx ∼ cos(2φ1 + φ−1) + cos(2φ−1 + φ1).

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Numerical test

−1.5 −1 −0.5 0−7

l n[ s i n(πx/Lx) ]

x r+x)

1 2 3 4 5 6 7 8 9 100

MC data

power law fitting

−1.5 −1 −0.5 0−7

l n[ s i n(πx/Lx) ]

x r+x)

y = − 2.036*x − 6.935

1 2 3 4 5 6 7 8 9 100

⟨Qx(r)Q

MC datapower law fitting

−1.5 −1 −0.5−8

ln [ s in (πx/L ) ]

ln|⟨Q

x)⟩|

y = − 2.23*x − 7.41

Figure : Correlation function. (1) power-law decay (2) still power-lawdecay after in-plane Zeeman is imposed. Bx = 0.4

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Numerics: Gapped, and, trivial topological order

2 4 6 8 10−16

02 x 10−3

⟨Qx rQ

x r+x⟩

2 4 6 8 10−16

02 x 10−3

⟨Sz rS

z r+x⟩

(b)(a)

Figure : Bulk correlation function. “quadropole” Qx = (Sx)2 − (Sy )2.

0 1 2 3 4 5 6 7 8 9 10 110

Lx (Ly = 10)

EntanglementEntropyS

y = 1.5351*x - 0.029561

MC data

linear fitt ing

8 10 12 14 16−0.05

M C data

ex p fi tt ing

Figure : Topological entanglement entropy is zero.23 / 41

SPT@π

Peng Ye

General review

Summary

Short summary of proj. constr. of SPT

What have been shown:

• Proj. Constr. provides ground state wave functions of SPT order withAbelian continuous symmetry, such as U(1)×U(1). (Ye, Wen1212.2121; Liu, Mei, Ye, Wen 1408.1676)

What have been not shown here:

1 Proj. Constr. can also construct SPT order with non-Abeliansymmetry, such as SU(2), SO(3) (see Ye, Wen arXiv:1212.2121)

2 In 3d, techniques become complicated (∵ monopole/dyon might ormight not be condensed in 3+1d gauge theory.) (see Ye, WenarXiv:1303.3572)

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Section 3 Bulk field theory of 3d SPT states

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Physicists like surface

Totem and Motto of Surface Science

Figure : Physics is like a pizza, all the good stuff is on the surface! (pizzacopied from Prof. Yayu Wang’s lecture note)

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However, bulk definition is more fundamental

• Bulk theory is a “master equation”: e.g. CS theory of QH −→ chiralLuttinger liquid.

• Two concerns on “surface-only research”:

• Many-to-one correspondence between surface and bulk isgenerically possible.e.g. QH at high Landau level (ν=8,12, see Cano,et.al. PRB 2014)

• Anomaly might not be fully canceled.

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However, bulk definition is more fundamental

• Bulk theory is a “master equation”: e.g. CS theory of QH −→ chiralLuttinger liquid.

• Two concerns on “surface-only research”:

• Many-to-one correspondence between surface and bulk isgenerically possible.e.g. QH at high Landau level (ν=8,12, see Cano,et.al. PRB 2014)

• Anomaly might not be fully canceled.

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Review of bosonic topological insulators (BTI)

Focus on BTI only.

Symmetry group G = U(1) o ZT2 (ZT

2 : time-reversal symmetry, T 2 = 1.)

1 Classification from group cohomology with U(1) coefficient:

H4 (G,U(1)) = Z2 × Z2 (Chen, Gu, Liu, Wen 2013)

2 Recently, people found more: Z2 × Z2 × Z2, by surface argument(Vishwanath, Senthil, PRX2013), composed by three “root states”:

1 Z2: Surface Z2 topo order. Both e and m carry half-charge;Witten effect with Θ = 2πmod4π (so-called by eCmC state) (in

TI it is π response, see seminal work Qi-Hughes-Zhang 2008)

2 Z2: Surface Z2 topo order. Both e and m carry Kramer’sdegeneracy. (so-called eTmT state)

3 Z2: Surface Z2 topo order. e, m, and ε are fermions. Beyondgroup cohomology. (so-called efmf state)

3 However, their bulk field theory is, STILL, not well established yet.

4 We aim to solve this problem and do more than that!! (see: Ye,Gu 1410.2594)

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Focus on BTI only.

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Focus on BTI only.

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“Derivation” of a field theory (hydrodynamic

approach)

Following CS derivation in FQH, hydrodynamic approach. (early progressin FQH: Fradkin, Wen,.......See also G. Y. Cho’s slides today)

• Key assumption: (1) Gapped; (2) density & current fluctuation aredominant d.o.f. at low energies and long wavelengths.

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Main results: All BTI obtained by condensing 2π

vortex-lines in superfluid

Superfluid

BTI (presence of b∧b term)

BTI (unusual symmetry transformation)

Trivial Mott Insulator

• Right-corner: one BTI beyond group cohomology.(∼∫K IJbI ∧ daJ +

∫ΛIJbI ∧ bJ). (focused in this talk)

• Left-corner: two BTI within group cohomology. (∫K IJbI ∧ daJ).

Either U(1) or ZT2 is transformed in an unusual way.

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From Superfluid to trivial Mott insulator: Main steps

• Duality: Superfluid L = ρ2

(∂µθ)2 ←→ QED of 2-form gauge theorybµν coupled to vortex-lines Σµν .

48π2ρhµνλhµνλ +

2bµνΣµν ,

where the field strength hµνλ is a rank-3 antisymmetric tensor:

hµνλdef.

==== ∂µbνλ + ∂νbλµ + ∂λbµν ; Σµν def.==== 1

2πεµνλρ∂λ∂ρθ

• Condensing 2π vortex-line Σµν leads to a string condensate coupledto a compact 2-form gauge field bµν , i.e. a Higgs phase (techniques

from string theory are involved, S. J. Rey 1989, M. Franz 2007):

(∂[µΘν] − bµν

)2+ · · ·

• Duality: Higgs term ←→ b ∧ da term with non-compact 1-form U(1)gauge field aµ and compact 2-form U(1) gauge field bµν :

L = i1

4πεµνλρbµν∂λaρ + iaµj

[1] 14π

indicates GSD=1 on a 3-torus T3.

[2] Symmetry transformation is also defined in a usual way. (btransforms axial-like; a transforms polar-like.)

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2bµνΣµν ,

hµνλdef.

)2+ · · ·

L = i1

[1] 14π

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2bµνΣµν ,

hµνλdef.

)2+ · · ·

L = i1

[1] 14π

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From Superfluid to BTI (beyond group cohomology)

• Introduce a new quadratic term in the vortex-line condensate(induced by some interactions):

L ∼1

(∂[µΘs

ν] − bµν)2−i Λ

16πεµνλρ

(∂[µΘs

ν] − bµν) (∂[λΘs

ρ] − bλρ),

• Duality: integrate out smooth phase Θsµ and resolve constraint by

introducing aµ:

Ltop =i1

4πεµνλρbµν∂λaρ+i

16πεµνλρbµνbλρ

• Gauge Invariance:

bµν −→ bµν + ∂[µξν] , aµ −→ aµ − Λξµ .

The gauge transformation of aµ generalizes the usual one by includingtransverse components of ξµ.

• Physical meanings of b ∧ b: unlinking-linking process (collision)

(a) (b) (c) (d)

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L ∼1

(∂[µΘs

16πεµνλρ

(∂[µΘs

ρ] − bλρ),

introducing aµ:

Ltop =i1

(a) (b) (c) (d)

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L ∼1

(∂[µΘs

16πεµνλρ

(∂[µΘs

ρ] − bλρ),

introducing aµ:

Ltop =i1

(a) (b) (c) (d)

32 / 41

SPT@π

Peng Ye

General review

Summary

L ∼1

(∂[µΘs

16πεµνλρ

(∂[µΘs

ρ] − bλρ),

introducing aµ:

Ltop =i1

(a) (b) (c) (d) 32 / 41

SPT@π

Peng Ye

General review

Summary

• Generalize to a multi-component theory:

Ltop = iK IJ

4πεµνλρbI

µν∂λaJρ + i

16πεµνλρbI

µνbJλρ,

It is sufficient to consider symmetric matrix ΛIJ and assume K IJ to bean identity matrix of rank-N, i.e.

K = diag(1, 1, · · · , 1)N×N = I

with I , J = 1, 2, · · · ,N. {aIµ} are non-compact 1-form U(1) gaugefields and {bI

µν} are compact 2-form U(1) gauge fields, respectively.

• Two independent GL(N,Z) transformations:

bIµν = (W−1)IJbJ

µν , aIµ = (M−1)IJaJµ ,

K = W TKM ,Λ = W TΛW ,La = MTLa ,Lb = W TLb .

Here, La and Lb are two qp vectors labelling excitations of particlesand strings.

33 / 41

SPT@π

Peng Ye

General review

Summary

• Generalize to a multi-component theory:

Ltop = iK IJ

4πεµνλρbI

µν∂λaJρ + i

16πεµνλρbI

µνbJλρ,

It is sufficient to consider symmetric matrix ΛIJ and assume K IJ to bean identity matrix of rank-N, i.e.

K = diag(1, 1, · · · , 1)N×N = I

with I , J = 1, 2, · · · ,N. {aIµ} are non-compact 1-form U(1) gaugefields and {bI

µν} are compact 2-form U(1) gauge fields, respectively.

• Two independent GL(N,Z) transformations:

bIµν = (W−1)IJbJ

µν , aIµ = (M−1)IJaJµ ,

K = W TKM ,Λ = W TΛW ,La = MTLa ,Lb = W TLb .

Here, La and Lb are two qp vectors labelling excitations of particlesand strings.

33 / 41

SPT@π

Peng Ye

General review

Summary

In absence of time-reversal,

• Flat-connection condition of b in an orientable manifold leads toΛIJ → Λ + 1. No additional quantization condition. Λ ∈ [0, 1), anynonzero Λ can be continuously tuned to zero. back to trivial Mottinsulator.

• Conclusion: U(1) SPT in 3d is trivial, agreement to

H4(U(1),U(1)) = Z1 . Λ term has no effect.

Imposing time-reversal in a usual way• Definition of time-reversal in a usual way:

T aI0T−1 = Ta

IJ aJ0 , T bI0iT

−1 = TbIJ b

J0i , T aIi T

−1 = −TaIJ a

Ji , T bIijT

−1 = −TbIJ b

Ta = −Tb = diag(1, 1, · · · , 1)N×N

• Under the transformation, Λ term becomes −Λ term.

• Flat-connection condition of b in a non-orientable manifold leads to:ΛII → ΛII + 4 and ΛIJ → ΛIJ + 2 (I 6= J)∴ Λ-quantization induced by time-reversal: (compared to U(1) SPT in 3d)

ΛII = 0,±2 ,ΛIJ = 0,±1 (for I 6= J) .

A time-reversal protected quantization!

34 / 41

SPT@π

Peng Ye

General review

Summary

In absence of time-reversal,

• Flat-connection condition of b in an orientable manifold leads toΛIJ → Λ + 1. No additional quantization condition. Λ ∈ [0, 1), anynonzero Λ can be continuously tuned to zero. back to trivial Mottinsulator.

• Conclusion: U(1) SPT in 3d is trivial, agreement to

H4(U(1),U(1)) = Z1 . Λ term has no effect.

Imposing time-reversal in a usual way• Definition of time-reversal in a usual way:

T aI0T−1 = Ta

IJ aJ0 , T bI0iT

−1 = TbIJ b

J0i , T aIi T

−1 = −TaIJ a

Ji , T bIijT

−1 = −TbIJ b

Ta = −Tb = diag(1, 1, · · · , 1)N×N

• Under the transformation, Λ term becomes −Λ term.

• Flat-connection condition of b in a non-orientable manifold leads to:ΛII → ΛII + 4 and ΛIJ → ΛIJ + 2 (I 6= J)∴ Λ-quantization induced by time-reversal: (compared to U(1) SPT in 3d)

ΛII = 0,±2 ,ΛIJ = 0,±1 (for I 6= J) .

A time-reversal protected quantization!

34 / 41

SPT@π

Peng Ye

General review

Summary

• There are infinite # of Λ matrices satisfying the condition.

• What is nontrivial Λ?

• Flat-connection condition leads to a ”CS” like surface theory withlevel Λ-matrix (“very abnormal” CS!! Gauss-Milgram sum ismeaningless.)

• Two key statements toward ”BTI”:

DefinitionPhysical observables of the surface theory are composed by: groundstate degeneracy, self-statistics and mutual statistics of gappedquasiparticles. chiral central charge c− is not an observable on the surface.

DefinitionBy “obstruction”, we mean that the set of physical observables cannot bereproduced on a 2d plane by any local bosonic lattice model with symmetry.Otherwise, the obstruction is free. Obstruction leads to nontrivial SPT.

35 / 41

SPT@π

Peng Ye

General review

Summary

• There are infinite # of Λ matrices satisfying the condition.

• What is nontrivial Λ?

• Flat-connection condition leads to a ”CS” like surface theory withlevel Λ-matrix (“very abnormal” CS!! Gauss-Milgram sum ismeaningless.)

• Two key statements toward ”BTI”:

DefinitionPhysical observables of the surface theory are composed by: groundstate degeneracy, self-statistics and mutual statistics of gappedquasiparticles. chiral central charge c− is not an observable on the surface.

DefinitionBy “obstruction”, we mean that the set of physical observables cannot bereproduced on a 2d plane by any local bosonic lattice model with symmetry.Otherwise, the obstruction is free. Obstruction leads to nontrivial SPT.

35 / 41

SPT@π

Peng Ye

General review

Summary

Under the time-reversal protected quantization conditionΛII = 0,±2 ,ΛIJ = 0,±1 (for I 6= J):

§ All states with |detΛ| = 1 are trivial (i.e. no obstruction):

• All are generated by two fundamental blocks up to GL transformation:

Λt1 =

(0 11 0

), Λt2 =

2 1 0 0 0 0 0 01 2 1 0 0 0 0 00 1 2 1 0 0 0 10 0 1 2 1 0 0 00 0 0 1 2 1 0 00 0 0 0 1 2 1 00 0 0 0 0 1 2 00 0 1 0 0 0 0 2

where, Λt2 also doesn’t break TR since c− = 8 is not an observable.

• Stacking Recipe for all such states c− = 0,±8,±16, · · · : Apply ∓Λt2

to remove c−.) Surface physical observables are unaffected!

§ All states with |detΛ| > 1 and W TΛW = −Λ, ∃W ∈ GL(N,Z) arealso trivial. (such GL can be regularly defined on a 2d plane.)

36 / 41

SPT@π

Peng Ye

General review

Summary

Under the time-reversal protected quantization conditionΛII = 0,±2 ,ΛIJ = 0,±1 (for I 6= J):

§ All states with |detΛ| = 1 are trivial (i.e. no obstruction):

• All are generated by two fundamental blocks up to GL transformation:

Λt1 =

(0 11 0

), Λt2 =

2 1 0 0 0 0 0 01 2 1 0 0 0 0 00 1 2 1 0 0 0 10 0 1 2 1 0 0 00 0 0 1 2 1 0 00 0 0 0 1 2 1 00 0 0 0 0 1 2 00 0 1 0 0 0 0 2

where, Λt2 also doesn’t break TR since c− = 8 is not an observable.

• Stacking Recipe for all such states c− = 0,±8,±16, · · · : Apply ∓Λt2

to remove c−.) Surface physical observables are unaffected!

§ All states with |detΛ| > 1 and W TΛW = −Λ, ∃W ∈ GL(N,Z) arealso trivial. (such GL can be regularly defined on a 2d plane.)

36 / 41

SPT@π

Peng Ye

General review

Summary

• Only possible obstruction can be realized by a modified GLtransformation (realizable merely on surface):

W T (Λ⊕ Λt1 ⊕ Λt1 · · · )W = (−Λ)⊕±Λt2 ⊕±Λt2 · · · , ∃W ∈ GL(N ′,Z)

which shifts chiral central charge, leaving surface physical observablesunaffected.

• There is only one irreducible solution, namely, the Cartan matrix ofSO(8) group (Wikipedia.org), denoted by Λso8:

Λso8 =

(2 1 1 11 2 0 01 0 2 01 0 0 2

More precisely, the following transformation exists:

W T (Λso8

4∑⊕Λt1)W = (−Λso8)⊕ Λt2,

∃W ∈ GL(12,Z) .

• A nontrivial SPT: efmf : Surface Z2 topo order. e, m, and ε arefermions. Beyond group cohomology.

• Break TR symmetry if put on 2d plane (Stacking Recipe cannotcompletely remove c−.)

37 / 41

SPT@π

Peng Ye

General review

Summary

• Only possible obstruction can be realized by a modified GLtransformation (realizable merely on surface):

W T (Λ⊕ Λt1 ⊕ Λt1 · · · )W = (−Λ)⊕±Λt2 ⊕±Λt2 · · · , ∃W ∈ GL(N ′,Z)

which shifts chiral central charge, leaving surface physical observablesunaffected.

• There is only one irreducible solution, namely, the Cartan matrix ofSO(8) group (Wikipedia.org), denoted by Λso8:

Λso8 =

(2 1 1 11 2 0 01 0 2 01 0 0 2

More precisely, the following transformation exists:

W T (Λso8

4∑⊕Λt1)W = (−Λso8)⊕ Λt2,

∃W ∈ GL(12,Z) .

• A nontrivial SPT: efmf : Surface Z2 topo order. e, m, and ε arefermions. Beyond group cohomology.

• Break TR symmetry if put on 2d plane (Stacking Recipe cannotcompletely remove c−.) 37 / 41

SPT@π

Peng Ye

General review

Summary

Eventually arrive at the right-corner of phase diagram

Superfluid

BTI (presence of b∧b term)

BTI (unusual symmetry transformation)

Trivial Mott Insulator

• Left-corner: two BTI within group cohomology. (b ∧ da).Either U(1) or ZT

2 is transformed in an unusual way.

• Right-corner: one BTI beyond group cohomology.(∼ K IJbI ∧ daJ + ΛIJbI ∧ bJ , Λ = ΛSO(8)).

38 / 41

SPT@π

Peng Ye

General review

Summary

Last story: A new Z2 SPT beyond group cohomology

• H4(Z2,U(1)) = Z1, all Z2 SPT states in 3d are trivial ( groupcohomology classification).

• Our Conclusion: If bf + bb theory also describes a Z2 SPT state, weconclude: Λ = 0 mod 4 is trivial while Λ = 2 mod 4 is a nontrivialSPT :

L = i1

4πaµ∂νbλρε

µνλρ + iΛ

16πbµνbλρε

µνλρ

• Strategy: by adding coupling terms with Bµν , gauging Z2 symmetry:

Lcoupling = i 14πBµν∂λaρε

µνλρ + i 24πBµν∂λAρε

µνλρ

where, Aµ is used for higgsing Bµν down to Z2. Then, integrating outSPT d.o.f., ends up with:

∫d4x

4πBµν∂λAρε

µνλρ + i

∫d4x

16πBµνBλρε

µνλρ

∫d4x

8πBµνFλρε

µνλρ + i

∫d4x

16πBµνBλρε

µνλρ

def.====i

∫B ∧ dA + i

∫B ∧ B .

39 / 41

SPT@π

Peng Ye

General review

Summary

• Flat-connection condition leads to: Λ→ Λ + 22

• Invariance under large gauge transformation leads to: Λ/2 ∈ Z• Thus, Λ = 0, or 2.

Thus, before gauging, a corresponding new Z2 SPT !

This probe theory is complete or consistent? an open question.

40 / 41

SPT@π

Peng Ye

General review

Summary

• Flat-connection condition leads to: Λ→ Λ + 22

• Invariance under large gauge transformation leads to: Λ/2 ∈ Z• Thus, Λ = 0, or 2.

Thus, before gauging, a corresponding new Z2 SPT !

This probe theory is complete or consistent? an open question.

40 / 41

SPT@π

Peng Ye

General review

Summary

Summary and open questions

§ Summary in brief :

1 Projective construction leads to a large classes of 2d SPT groundstates. arXiv:1212.2121; 1408.1676;

2 Bulk field theory of 3d bosonic topological insulator is constructed ina physical way. 1410.2594

3 Bulk definition is far more important than pure boundary argument.

§ Open questions:

1 Realistic spin lattice models that support the ground state from themethod of “projective construction” (e.g. Mei, Wen,arXiv:1407.0869)

2 3d field theory of generic SPT states, other symmetries.

3 A conclusive result of 3d field theory of fermion SPT (free andinteracting), e.g. fermionic topological insulators. (previous effortsever made by Cho, Moore, Ryu, · · · ) spin manifold?

Thank collaborators (X.-G. Wen, Z.-C. Gu, Z.-X. Liu, J.-W. Mei) and mentor S.-S. Lee in PI.

Thank PI for financial supports. Discussion with G. Baskaran about superfluid is

acknowledged.Thanks for your attentions!

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Symmetry-protected topological phases: Projective...

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