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 POINTS

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

Quasi-adiabatic continuation

1. F is rapidly decaying 2. 3. 4.

[Hastings-Wen]

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

MERA REPRESENTATIONS, S=1

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]

Entanglement Renormalization (MERA) [Vidal]

“bulk” degrees of freedom

“boundary” degrees of freedom

Quasi-local unitary circuit d=1

1

2

3

d=2

exp(poly(log(L)) bond dimension

bond dimension (Hilbert space dim.)

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!

BONUS GOODIES

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 …