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Mimicking the edge of 2d topological superconductors and insulators
in 1d lattice modelsMax Metlitski
MIT
Simons Collaboration on Ultra Quantum Matter: Kickoff Meeting.Harvard, September 12, 2019
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Robert Jones,MIT
R. A. Jones and MM, arXiv:1902.05957MM, arXiv:1908.08958
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Symmetry protected topological phases (SPTs)
• Protected by symmetry G
• Gapped bulk
- unique ground state on a closedspatial manifold
- no anyon exciations
• Surface must be non-trivial
-t’Hooft anomaly (сan’t couple to a background gauge-field).
Trivial SPT
Symmetry breaking
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Q: Can we realize/mimick the boundary in a lattice model without the bulk?
Motivation:
i) Simulating edge of SPT without the bulk overhead
ii) A more lattice based understanding of anomalies
symmetry protected topological phase
(SPT)
``anomalous’’ surface
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Q: Can we realize/mimick the boundary in a lattice model without the bulk?
A: No, subject to
i)
ii)
(On-site symmetry)- Can always be gauged!
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Cohomology phases
i)
ii)
- Class of boson SPT phases
- Classified by (when G is gauged – Dijkgraaf-Witten theories)
- Edge can be mimicked if ii) is dropped (discrete G)
- 1+1D boundary: extract anomaly
X. Chen, Z.-C. Gu, Z.-X, Liu, X.G.-Wen (2012)
Chen, Liu, Wen (2011); D. Else, C. Nayak (2014)
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Example• G = Z2
• H3(G, U(1)) = Z2
• 1+1D boundaryS = 1/2
X. Chen, Z. X. Liu, X.-G. Wen (2012)M. Levin, Z. C. Gu (2012)
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Cohomology phases
i)
ii)
- Extends to “super-cohomology” phases of fermions
(explicit construction for 1+1D boundary and discrete G in MM, arXiv:1908.08958)
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• Beyond (super)cohomology phases:- 2+1D fermion topological superconductors
This talk
R. A. Jones and MM, arXiv:1902.05957
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• (Super)cohomology phases with a continuous symmetry group G
- infinite dimensional site Hilbert space?
This talk
• 2+1D fermion topological insulators (quantum spin Hall)
MM, arXiv:1908.08958
See also J. H. Son and J. Alicea, 1906.11846
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Beyond (super)-cohomology
• Ex: 2+1D, fermions, ,
p+ip
p-ip
Kramers-Wannier duality:
High T low T
• Mimick in 1+1D?
Gu, Levin (14)
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Kramers-Wannier
• Fermionize:
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i)
ii)
- not enough to drop this if
• Starting point: Exactly solvable bulk model of Tarantino and Fidkowski (2016)
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Intuition
-
+ or
+ + - - - + + - - + - + - - - - +
• Restore symmetry – Majorana liquid
1 1 𝜎𝜎 𝜎𝜎 𝜎𝜎 f f 𝜎𝜎 𝜎𝜎 f 𝜎𝜎 1 𝜎𝜎 𝜎𝜎 𝜎𝜎 𝜎𝜎 1
Link labels: ,
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Symmetry action
+ + - - - + + - - + - + - - - - + 1f f 1
1 1 𝜎𝜎 𝜎𝜎 𝜎𝜎 f f 𝜎𝜎 𝜎𝜎 f 𝜎𝜎 1 𝜎𝜎 𝜎𝜎 𝜎𝜎 𝜎𝜎 1
- - + + + - - + + - + - + + + + -
• Strings:
-superposition
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Hamiltonian
• - solid numerical evidence (ED & DMRG) for Ising CFT
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Exact diagonalization
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Exact diagonalization
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DMRG
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Lessons & open questions
• Mimick edge of 2+1D beyond super-cohomology fermion SPT
- constrained Hilbert space
- similar to 3+1D, U(1)xT fermion SPTs
- generalizes to 2+1D fSPTs with
- obstruction?
- other dimensions?
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Quantum spin-Hall
• 2+1D, fermions
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QSH edge
- inspired by Tantivasadakarn and Vishwanath (18)
MM, arXiv:1908.08958
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Properties
• Kramers parity switching
• domain wall: + - +
• Symmetric Hamiltonians- analyze by Jordan-Wigner- Luttinger liquid of QSH edge
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• Can extract anomaly data:
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Open questions
• Mimick edge of 2+1D quantum spin-Hall
- mimick other SPTs with continuous symmetry groups?
- continous vs discrete anomalies?
Thank you!