Anti-de Sitter Space as Topological Insulator and Holography

Anti-de Sitter Space as Topological Insulator and Holography

S.H. Ho (NCTS)

Work with Prof. Feng-Li Lin (NTNU)

based on arXiv:1205.4185

• Introduction

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Topological orders and symmetry protected topological orders

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SPT phases and topological insulators/superconductors

• Kitaev’s K-thoery classification scheme

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General idea to classify SPT phases

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Complex fermions

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charge conjugation and time reversal symmetries

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Real fermions

• Bulk/edge correspondence and AdS/CFT

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Bulk/edge correspondence

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Holographic fermions

• AdS space as a topological insulator

• ^Discussion

Outline

Introduction

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Phase transition:

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symmetry breaking: order parameters, Nambu-Goldstone/Higgs modes

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non-symmetry breaking: topological orders (gapped system), Fermi liquids...etc.

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e.g. fractional quantum Hall effect: lots of different phases with the same symmetry

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can be characterized by gapless boundary modes

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robust under perturbations and smooth change of parameters

QFT of many-body systems, X.G. Wen

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Symmetry-protected topological (SPT) order:

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gapped quantum phase with certain symmetries

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cannot distinguish these phases if we remove the symmetries (all can be smoothly connected to the same trivial phase if no symmetries)

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e.g. non-interacting (gapped) free fermion systems such as topological insulator/superconductor

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topological insulator can be characterized by the gapless boundary modes protected by time-reversal symmetry

• General idea for classifying topological orders

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only know how to classify gapped system

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gapless modes between different topological phases in the configuration space

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topological phase is an equivalence class of perturbations which do not close the gap

Kitaev’s K-thoery classification scheme

Physically the gapless modes on the phase boundary in Fig. 1 cannot propagate in the bulk because the systems are gapped by definition. To host these modes we must provide some physical support to localize them.

• How to classify? An intuitive example

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Start from d=0 (spatial dimension) gapped system with one orbital the number of different phases is two: occupied and empty (1 or 0)

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Mathematically, consider

the number of positive and negative energy levels are l and m, n=l+m up to an unitary transformation

but is not a one-to-one labeling the H also invariant under

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The classification space is , large n limit

X.-G. Wen, “Symmetry protected topological phases in non-interacting fermion systems,”

Phys. Rev. B 85, 085103 (2012)

not stable under background influences (spectator electrons)

adding n orbitals (large n) topological

consideration

number of disconnected pieces of

connected part

For d=1, we have flat band condition and have to satisfy with and hermitian.

positive and negative eigenvalues always paired because H invariant under

For more general dimension d=p, first pick p fixed hermitian matrix with is the space of hermitian matrix with condition

• Kitaev’s K-theory analysis

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classify the configuration space of mass matrix M of free Dirac

Hamiltonian through the Clifford algebra formed by mass matrix M and symmetry operators (if any).

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Flat-band condition: only care about the number of positive and negative eigenvalue of Hamiltonian for topological consideration

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Clifford algebra

fixed find all possible solution for

• Complex fermions

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if no charge conjugation (C) and time reversal (T)

whole story

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Clifford algebra

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configuration space denoted

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the bulk phase labelled by some integer Z

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the defect , d independent

Bott periodicity

the space of different ways to gap the system (the ways to write down )

Gauss law:

cover the defect by a closed surface

• C and T symmetries

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charge conjugation:

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time reversal:

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C and T are symmetries, define

• Real fermions

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C and T will not join to form an enlarged Clifford algebra in complex case

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first assume are all real

1. a

2. may combine to form a generator of Clifford algebra, 3. the configuration space determined by

4. for the Clifford algebra , the classification space denoted by

only these two are relevant in our consideration

Bott periodicity

• Boundary excitation classified by:

• Modified Kiteav’s K-theory in high energy physics

1. Lorentz invariance

2. assume no non-trivial constituents for fermions

reality condition unitary and

automaticallyly satisfied

exist a basis in which

non-trivial fractionalization of the electrons are commonly postulated when considering the topological orders

•

Consider the Hamiltonian

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In the basis (Majorana representation),

1. trivial

2. only relevant if exists.

3. ？ irrelevant since

real

imaginary redefine

spatial dimension

1. Q: when (real representation exists)？

2. Q: when exists？ real rep.

pseudo-real rep. real rep. for Dirac gamma matrix

pseudo-real rep. for Dirac gamma matrix combined to form a real rep.

real Dirac gamma matrix d=0,1,2,3,7 mod 8 pseudo-real Dirac gamma matrix d=3,4,5,6, 7 mod 8

real Dirac gamma matrix exists when d=1,3,7 mod 8 pseudo-real Dirac gamma matrix exists when d=3,5,7 mod 8

Bulk/edge correspondence and AdS/CFT

• Bulk/edge correspondence

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an idea that relates the topological properties of the bulk to the number of gapless modes on the boundary

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examples: quantum Hall effect

Figures from M. Z. Hasan, C. L. Kane, “Topological Insulators,” Rev. Mod. Phys. 82, 3045 (2010)

the difference between right moving and left moving mode is fixed by the bulk topology information, here is the change of the Chern

number across the boundary.

• Holographic fermions

To have a well-defined variation principle, the boundary term is added

Here we only consider the AdS metric

corresponds to impose Dirichlet boundary condition on (standard and alternative quantization scheme)

• The equation of motion in Hamiltonian form:

1. interpreted as the energy of the boundary fermion theory by the holographic dictionary

2. solution used to construct the Green’s function of boundary fermion operator

Q: CFT is a gapless system by definition, its dual (AdS space) should also be a gapless system. How do we use K-theory to classify a

gapless (bulk) system ？

1. Topological orders can be characterized by gapless boundary modes

2. Cosmological constant as an external field which breaks the translational invariance along radial direction

create a co-dimensional one boundary defect

• AdS space as a topological insulator

5. Bulk/edge correspondence:

edge gapless mode topological properties of the bulk here the bulk/edge correspondence is manifest

3. The Dirichlet condition imposed in AdS/CFT picks up the non-normalizable zero modes ( ). localized mode on AdS boundary

the defect is classified by also the classification of bulk topological phases the defect is classified by , is the spatial dimension of the defect

here

4. These non-normalizable modes corresponds to dual CFT operators, which should be topological sector of CFT.

information about the bulk topological phases hidden in the mass matrix

• Holographic fermions and eigenvalue of mass matrix

zero mode equation

conformal dimension of dual operator the asymptotic behavior of the solution:

1. we consider many fermion flavors

2. for topological consideration, only the sign of the eigenvalue of m relevant

3. , non-normalizable, normalizable 4. , non-normalizable, normalizable

m becomes a matrix

which to choose？

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Standard description of AdS/CFT tells us which pattern is localized on the boundary How？

by fixing the boundary action, “standard” or the “alternative” quantization schemes (Dirichlet boundary condition imposed on the (non-normalizable) mode)

e.g. for m=1, we chose standard quantization scheme

Dirichlet BC on identify as a localized mode

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Once the boundary action is chosen, we know which (zero) modes localized on the boundary because we know the eigenvalues of m.

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These edge modes are topological since m encodes the topological information of the bulk phases.

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To consider the real fermions, we just make sure is also real.

Majorana (real) representation & real

• ^Summary

1. start with Minkowski bulk free fermion (gapped system) 2. turn on cosmological constant: Minkowski AdS

3. solution of Dirac equation in AdS space

4. the non-normalizable part of solutions can be viewed as localized modes on the AdS time-like boundary localized edge mode

5. Bulk/edge correspondence tells us the number of localized gapless modes should be encoded in the pattern of mass matrix (topological info)

note m are eigenvalues of mass matrix with

6. how do we know which should be localized mode？

the quantization scheme in AdS/CFT tells us how many localized boundary mode are e.g. standard quantization is chosen is picked up (for m=1)

7. number of positive/negative eigenvalues of mass matrix = number of localized edge modes manifest bulk/edge correspondence

8. in AdS/CFT description, these edge modes act as sources coupled to the dual CFT

fermionic operators. These fermionic operators should correspond to the topological sector

• Discussions

1. The massive free fermions of many flavors in AdS space as a topological insulator with a co-dimensional one defect.

2. The non-normalizable (zero) modes picked up by the AdS/CFT description localized on the AdS boundary, which corresponds to dual CFT operators and should be topological sector of CFT.

3. Generally, we only know how to classify gapped systems. With the help of AdS/CFT and the bulk/edge correspondence, this scenario can be used to classify SPT of holographic CFTs.

Gapped bulk phase in Minkowski space

edge modes on the AdS boundary

gapless system (holographic CFTs)