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Renormalization group constructions of topological quantum liquids
Brian Swingle with John McGreevy, Isaac Kim, and
Mark Van Raamsdonk 1407.8203, 1407.2658, 1405.2933
1. Full formal Hilbert space
2. States realized in nature
3. (Reasonable) ground states
4. States with no entanglement
Local regulated quantum many-body systems
Long term goal: local theory of quantum matter
Aside/Example: reconstruction from local data
Local data:
Proof of 1:
[BGS-Kim, Cramer et al. ‘10, Markov case: Petz, Hayden et al.]
Local regulated quantum many-body systems
Tensor networks
Entanglement = Geometry
Einstein’s equations “practical holography”
Simulation
TODAY
[BGS, Evenbly-Vidal, Nozaki et al., Van Raamsdonk, …]
[Lashkari-McDermott-van Raamsdonk, Faulkner et al., BGS-van Raamsdonk, …]
minimal surface
[Ryu-Takayanagi, Faulkner-Lewkowycz-Maldacena, Sorkin, …]
[finite: Susskind-Uglum, Cooperman-Luty, Satz-Jacobson, Bianchi-Myers, …]
Aside: Einstein’s equations from qubits
Microscopic degrees of freedom (qubits)
Einstein’s equations
Tensor networks
Geometry and curved space QFT
[BGS coming soon]
[BGS-van Raamsdonk]
Local regulated quantum many-body systems
Tensor networks
Entanglement = Geometry
Einstein’s equations
Simulation
TODAY
This talk
A new idea: s source RG fixed point 1. Focuses on the quantum state 2. RG transformation L 2L 3. Resource oriented perspective
Some payoffs: 1. Classification scheme 2. Rigorous area laws for entanglement 3. Tensor network (MERA) representations 4. Notion of short-range entanglement
[BGS-McGreevy]
The setting
Local Hamiltonian with: 1. an energy gap to all excitations, 2. ground state degeneracy , 3. locally indistinguishable ground states, 4. stable to all perturbations, 5. and a low temperature free energy
going like
d spatial dimensions, system size L, subsystem size R
Still hard to understand!
s source RG fixed point
A d-dimensional s source RG fixed point is a system where a ground state on sites can be constructed from s copies of ground states on sites plus some unentangled degrees of freedom by acting with a quasi-local unitary
d=2, s=1
L black sites are intercalated with L blue sites using a quasi-local unitary. The output is the black state on 2L sites.
d=1, s=1 example
Quasi-locality
local sum of quasi-local operators
sum of strictly local operators
supported on disks
rapidly decaying norm
Example 1: “trivial insulators” s=0
gapped
product ground state ground state of interest
Examples: 1. Ground state of diamond 2. Ground state of QCD
quasi-local s=0
Why quasi-local: decoupled spins?
1.Adiabatic interpolation: single site gs gs prob. , total gs gs prob.
2.Quasi-adiabatic interpolation: the correct transformation is produced by
Example 2: chiral insulators, s=1
Examples: 1. Integer quantum Hall states, 2. Massive Dirac fermion, d=2 (same as IQH)
Example 3: gauge theory, s=1
Examples: 1. Discrete gauge theories, 2. Fractional quantum Hall states (also chiral)
[Aguado-Vidal, Gu-Levin-BGS-Wen]
[BGS-McGreevy]
Topological quantum liquid: insensitive to arbitrary smooth deformations of space, aka gapped field theories, but also lattice defn.
Expanding universe construction:
Some useful lemmas
Recursive entropy bounds:
Ground state degeneracy lemma:
result uses [Van Acoleyen-Marien-Verstraete]
Classification scheme (gapped) s
s=0
s=1
s=2 d-1
s>2 d-1
ruled out with thermo argument
ruled out with bound S < log(G)
s<2 d-1
Field theory
Not empty, but unusual
Tensor network representations
d=1: 1. Gapped ground states obey the area law 2. Gapped ground states have MPS
representations d>1: 1. Gapped ground states have exp(poly(log(L)))
bond dimension PEPS 2. Non-chiral topological states 3. Chiral states challenging [Zaletel-Mong, Gu et al., Beri-Cooper,
Dubail-Read]
[Hastings, Arad-Landau-Vazirani]
[White, Hastings]
[Gu-Levin-BGS-Wen, Vidal et al.]
[Hastings, Molnar et al.]
MPS:
What is a MERA?
Consider the ground state of H: We would like a way to represent the ground state (and other states) which 1. makes clear the physics at different length
scales, 2. explicitly takes entanglement into account,
and 3. represents entanglement geometrically.
entanglement network [Vidal, BGS]
Constant bond dimension
Suppose we are willing to make extensive errors but with the intensive error controlled
Excitation energy density size L
Energy density added per step
appeal to stability!
Applications
1. Provable exponential speedup, plausible double exponential(!) speedup of classical simulation of a wide class of problems (e.g. QCD, FQH)
2. Strong approximation results for ground states, e.g. gs is 1/poly(L) close to a state with log(Schmidt rank)
3. Novel algorithms for finding MERAs, no variational calculation needed!
Area law s
s=0
s=1
s=2 d-1
s>2 d-1
ruled out with thermo argument
ruled out with bound S < log(G)
s<2 d-1
[generalizes Hastings argument]
Entanglement and thermodynamics s
s=0
s=1
s=2 d-1
s>2 d-1
ruled out with thermo argument
ruled out with bound S < log(G)
s<2 d-1
Invertible states
How to define short-range entanglement? East-coast: short circuit West-coast: no anyons, etc.
Our definition (also Kitaev): a state is short-range entangled if it has an inverse state:
they don’t agree!
Weak area law
WAL: All phases with a unique gs on every closed manifold obey the area law and have an inverse state
CFTs
Strong conjecture: CFTs are s=1 fixed points Some evidence: 1. Same entanglement structure as non-trivial
topological quantum liquids 2. Correlations easy to include 3. Approximation results in 1d
We are close to some rigorous results …
[Verstraete-Cirac]
Summary
A new idea: s source RG fixed point 1. Focuses on the quantum state 2. RG transformation L 2L 3. resource oriented perspective
Some payoffs: 1. Classification scheme 2. Rigorous area laws for entanglement 3. Tensor network (MERA) representations 4. Notion of short-range entanglement
Questions
Technical: 1. Improve MERA bond dimension to poly(L)? 2. Rigorous constant bond dimension results? 3. Extensions to gapless systems (soon) 4. Algorithms! Einstein from qubits: 1. Role of large N/strong coupling? 2. Conformal data and tensors? Local reconstruction: 1. Better (not ad hoc) notion of error? 2. Actual q info tasks, difficulty of reconstruction? Local theory of quantum matter: 1. Wide open …