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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
1 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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)
2 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
3 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
4 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Contents
1 General review
2 Projective construction of 2d SPT
3 3d SPT bulk field theory with b ∧ b term
4 Summary
5 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Section 1 General review
6 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Hundred years’ hard Condensed Matter (Solid State)
Physics in progress... Great triumph in 1980s.
7 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Order
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.
8 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Order
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.
8 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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,...
9 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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!!.
10 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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!!.
10 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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!!.
10 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
11 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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.
11 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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.
12 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Section 2 Projective construction of 2d SPT states
13 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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.)
14 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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.)
14 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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 =∑
m Cm
∏a ψ
n(m)a
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
abψb
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.
15 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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
16 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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
16 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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
16 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Proj. constr. of SPT: Ye-Wen’s program ’12
To understand symmetry transformation on matter fields, we preservepartons d.o.f..
• 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!! )
17 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Proj. constr. of SPT: Ye-Wen’s program ’12
To understand symmetry transformation on matter fields, we preservepartons d.o.f..
• 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!! )
17 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Proj. constr. of SPT: Ye-Wen’s program ’12
To understand symmetry transformation on matter fields, we preservepartons d.o.f..
• 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!! )
17 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Proj. constr. of SPT: Ye-Wen’s program ’12
To understand symmetry transformation on matter fields, we preservepartons d.o.f..
• 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!! )
17 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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).
18 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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).
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)
L =1
4π( a1µ a0
µ a−1µ )
(1 0 00 −1 00 0 1
)∂ν
(a1λ
a0λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν ( 1 0 −1 )
(a1λ
a0λ
a−1λ
)εµνλ
After Gutzwiller projection, da1 + da−1 + da0 = 0, we end up with:
L =1
4π( a1µ a−1
µ )(
0 −1−1 0
)∂ν
(a1λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν( 1 −1 )
(a1λ
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:
σSz
H =1
2π× 2.
19 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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).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)
L =1
4π( a1µ a0
µ a−1µ )
(1 0 00 −1 00 0 1
)∂ν
(a1λ
a0λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν ( 1 0 −1 )
(a1λ
a0λ
a−1λ
)εµνλ
After Gutzwiller projection, da1 + da−1 + da0 = 0, we end up with:
L =1
4π( a1µ a−1
µ )(
0 −1−1 0
)∂ν
(a1λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν( 1 −1 )
(a1λ
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:
σSz
H =1
2π× 2.
19 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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).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)
L =1
4π( a1µ a0
µ a−1µ )
(1 0 00 −1 00 0 1
)∂ν
(a1λ
a0λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν ( 1 0 −1 )
(a1λ
a0λ
a−1λ
)εµνλ
After Gutzwiller projection, da1 + da−1 + da0 = 0, we end up with:
L =1
4π( a1µ a−1
µ )(
0 −1−1 0
)∂ν
(a1λ
a−1λ
)εµνλ +
1
2πASz
µ ∂ν( 1 −1 )
(a1λ
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:
σSz
H =1
2π× 2.
19 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Numerics: Even-quantized Hall conductance
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Φ S z/2π
Szpump
PG |1-11⟩
0 0.2 0.4 0.6 0.8 1−2
−1.5
−1
−0.5
0
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.
20 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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).
21 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Numerical test
−1.5 −1 −0.5 0−7
−6
−5
−4
−3
l n[ s i n(πx/Lx) ]
ln(Q
x rQ
x r+x)
1 2 3 4 5 6 7 8 9 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Qx rQ
x r+x
x
MC data
power law fitting
−1.5 −1 −0.5 0−7
−6
−5
−4
−3
l n[ s i n(πx/Lx) ]
ln(Q
x rQ
x r+x)
y = − 2.036*x − 6.935
1 2 3 4 5 6 7 8 9 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
x
⟨Qx(r)Q
x(r+
x)⟩
MC datapower law fitting
−1.5 −1 −0.5−8
−7
−6
−5
−4
−3
ln [ s in (πx/L ) ]
ln|⟨Q
x(r)Q
x(r+
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
22 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Numerics: Gapped, and, trivial topological order
2 4 6 8 10−16
−12
−8
−4
02 x 10−3
x
⟨Qx rQ
x r+x⟩
2 4 6 8 10−16
−12
−8
−4
02 x 10−3
x
⟨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
2
4
6
8
10
12
14
16
18
Lx (Ly = 10)
EntanglementEntropyS
(2)
y = 1.5351*x - 0.029561
MC data
linear fitt ing
8 10 12 14 16−0.05
0
0.05
0.1
0.15
Ly
TEE
γ
M C data
ex p fi tt ing
Figure : Topological entanglement entropy is zero.23 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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)
24 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Section 3 Bulk field theory of 3d SPT states
25 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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)
26 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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.
27 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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.
27 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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)
28 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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)
28 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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)
28 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
“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.
29 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Main results: All BTI obtained by condensing 2π
vortex-lines in superfluid
Superfluid
BTI (presence of b∧b term)
ZT2
1
2
BTI (unusual symmetry transformation)
Trivial Mott Insulator
3
• 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.
30 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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 .
L =1
48π2ρhµνλhµνλ +
i
2bµνΣµν ,
where the field strength hµνλ is a rank-3 antisymmetric tensor:
hµνλdef.
==== ∂µbνλ + ∂νbλµ + ∂λbµν ; Σµν def.==== 1
2πεµνλρ∂λ∂ρθ
v
• 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):
L =1
2φ2
0
(∂[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
µ .
[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.)
31 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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 .
L =1
48π2ρhµνλhµνλ +
i
2bµνΣµν ,
where the field strength hµνλ is a rank-3 antisymmetric tensor:
hµνλdef.
==== ∂µbνλ + ∂νbλµ + ∂λbµν ; Σµν def.==== 1
2πεµνλρ∂λ∂ρθ
v
• 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):
L =1
2φ2
0
(∂[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
µ .
[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.)
31 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
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 .
L =1
48π2ρhµνλhµνλ +
i
2bµνΣµν ,
where the field strength hµνλ is a rank-3 antisymmetric tensor:
hµνλdef.
==== ∂µbνλ + ∂νbλµ + ∂λbµν ; Σµν def.==== 1
2πεµνλρ∂λ∂ρθ
v
• 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):
L =1
2φ2
0
(∂[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
µ .
[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.)
31 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Introduce a new quadratic term in the vortex-line condensate(induced by some interactions):
L ∼1
2φ2
0
(∂[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
Λ
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)
z
x y
(a) (b) (c) (d)
32 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Introduce a new quadratic term in the vortex-line condensate(induced by some interactions):
L ∼1
2φ2
0
(∂[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
Λ
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)
z
x y
(a) (b) (c) (d)
32 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Introduce a new quadratic term in the vortex-line condensate(induced by some interactions):
L ∼1
2φ2
0
(∂[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
Λ
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)
z
x y
(a) (b) (c) (d)
32 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Introduce a new quadratic term in the vortex-line condensate(induced by some interactions):
L ∼1
2φ2
0
(∂[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
Λ
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)
z
x y
(a) (b) (c) (d) 32 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Generalize to a multi-component theory:
Ltop = iK IJ
4πεµνλρbI
µν∂λaJρ + i
ΛIJ
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 ,
where, W ,M are two N by N matrices with integer-valued entriesand |detW | = |detM| = 1. A new set of parameters (K ,Λ,La,Lb) canbe introduced via (∴ |detK | is invariant):
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• Generalize to a multi-component theory:
Ltop = iK IJ
4πεµνλρbI
µν∂λaJρ + i
ΛIJ
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 ,
where, W ,M are two N by N matrices with integer-valued entriesand |detW | = |detM| = 1. A new set of parameters (K ,Λ,La,Lb) canbe introduced via (∴ |detK | is invariant):
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
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
Jij ,
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
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
Jij ,
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
From Superfluid to BTI (beyond group cohomology)
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Eventually arrive at the right-corner of phase diagram
Superfluid
BTI (presence of b∧b term)
ZT2
1
2
BTI (unusual symmetry transformation)
Trivial Mott Insulator
3
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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:
S =i
∫d4x
2
4πBµν∂λAρε
µνλρ + i
∫d4x
Λ
16πBµνBλρε
µνλρ
=i
∫d4x
2
8πBµνFλρε
µνλρ + i
∫d4x
Λ
16πBµνBλρε
µνλρ
def.====i
2
2π
∫B ∧ dA + i
Λ
4π
∫B ∧ B .
39 / 41
SPT@π
Peng Ye
General review
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Last story: A new Z2 SPT beyond group cohomology
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
Summary
Last story: A new Z2 SPT beyond group cohomology
• 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
Projectiveconstruction of2d SPT
3d SPT bulk fieldtheory withb ∧ b term
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!
41 / 41
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