Many-body quantum physics through the lens of quantum
entanglement
Shinsei Ryu

December 3, 2021

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Many-body quantum systems
N >1023

H=X  p2ii=12m+ v(ri ) +H=JX

V (|ri − rj |)i,jn.n.
X

Si · Sji,j

LQCD = ψ̄ (γµ Dµ − m) ψ −1tr Gµν Gµν
2..
.
• May exhibit a rich varieties of phenomena: spontaneous symmetry

breaking, topological phases of matter, non-equilibrium physics, eigenstate
thermalization, quantum information scrambling, etc.
• Known to be very hard: dim H ∼ eN
• May exhibit (ground) states with non-trivial quantum entanglement

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Quantum entanglement• Product states v.s. entangled states:|Ψi = | ↑i| ↑ivs.

|Ψi =| ↑i| ↓i − | ↓i| ↑i√2• Conventional v.s. topological order:|Ψi = | ↑↑↑↑ · · · i

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“Conventional” phases of matter• Many phases of condensed matter can be characterized by spontaneous

symmetry breaking and local order parameters.

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Topological phases of matter• Topological phases escape from the symmetry breaking paradigm (i.e., noorder parameter).
• Instead, characterized by their quantized responses and/or existence ofexotic excitations• E.g., the Hall conductance in the quantum Hall effect• Other examples: topological insulators, topological superconductors,

Haldane spin chain, etc.

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Topologically-ordered phases in (2+1)D• Phases that support anyons• Anyons are neither bosons nor fermions; have non-trivial exchange(braiding) statistics• Abelian/Non-Abelian topological order• Not characterized by the symmetry-breaking paradigm, but by theproperties of anyons (fusion, braiding, etc.)

Ground states
Excitations
Effective theory

SSB phases
Degeneracy w/ SSB
Nambu-Goldstone bosons
Landau-Ginzburg

Topologically-ordered phases
Topological degeneracyAnyons
Topological field theory• Bulk-boundary correspondence

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Fractional quantum Hall effect (↓)

Spin-orbital-assisted Mott insulator α-RuCl3 (“Kitaev spin liquid”) (↑)
Rydberg atoms on a Kagome lattice (↓) [Semeghini1 et al (21)]7 / 33• Conventional phases can be detected by conventional probes: (local)

correlation functions hO1 O2 · · · i• How can we extract/measure topological data?

Direct observation of abelian braiding statistics [Nakamura et al (20), Bartolomei et al(20)]• Other measurable quantities? Topological invariants?

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Entanglement entropy• The von-Neumann entanglement entropy:

SA := −TrA (ρA log ρA )for the reduced density matrix ρA = TrB |ΨihΨ|• For topologically-ordered states in (2+1)D [Levin-Wen, Kitaev-Preskill (05)]SA = const. × ` − log D“Topological entanglement entropy” γ = log D carries universal data

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Entanglement spectrum• The spectrum of the reduced density matrix ρA• ρA ∼ e−Hedge for topological phases. “Bulk-boundary correspondence”[Li-Haldane (08)]• E.g. Chern insulators (single-particle entanglement spectrum)[SR-Hatsugai (06)]10 / 33

Entanglement as a probe for dynamics• Time evolution of von Neumann or Rényi entropy

SA = −Tr ρA log ρA ,(2)SA = − log Tr ρ2A

in non-equilibrium process. E.g., quantum quench: |Ψ(t)i = e−iHt |Ψinit i• Thermalization, quantum information scrambling, many-body localization,etc.

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“Effective” descriptions of entanglement propagation“Hydrodynamics” of entanglement
Two canonical models for entanglement propagation:
Quasi-particle picture v.s Membrane picture[Figures: Jonay-Huse-Nahum (18)]12 / 33

Can we measure entanglement entropy?
• [Islam et al (15)][Kaufman et al (16)][Lukin et al (18)]• Randomized measurement [Brydges (19)]13 / 33

• Introduction; topology and dynamics in quantum many-body systems

detected by entanglement• Negativity and reflected entropy• Tripartitioning topological liquid• Summary and outlook

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Going beyond entanglement entropy for bipartition• Go beyond bipartition, and study entanglement quantities• Multipartite entanglement?

1√ [| ↑↑↑i + | ↓↓↓i] ,21√ [| ↑↑↓i + | ↑↓↑i + | ↓↑↑i] ,3• Entanglement negativity and reflected entropy

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Entanglement negativity• How to quantify quantum entanglement between A and B when ρA∪B ismixed ? E.g., finite temperature; A, B is a part of bigger system.
• The entanglement entropy is an entanglement measure only for purestates (not monotone under LOCC).
• Entanglement negativity and logarithmic negativity, using partial transpose[Peres (96), Horodecki-Horodecki-Horodecki (96), Vidal-Werner (02), Plenio (05) ...]

N (ρ) :=X
λi <0|λi | =
1||ρTB ||1 − 1 ,2E(ρ) := log(2N (ρ) + 1) = log ||ρTB ||1 .
• Good entanglement measure since LOCC monotone.
• For mixed states, negativity can extract quantum correlations only.

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Partial transpose• Definition: for an operator M , its partial transpose M TB is(A) (B)(A) (B)(A) (B)(A) (B)ej |M TB |ek el i := hei el |M |ek ej ihei(A,B)

where |ei

i is the basis of HA,B .

• For fermionic (anyonic) systems, we need to take into account particle

statistics properly.

[Shapourian-Shiozaki-SR (16); Shapourian-SR (18); Shapourian-Mong-SR (20)]17 / 33

Reflected Entropy[Dutta-Faulkner (19)]• The von-Neumann entropy of a canonical purification:
ρAB =Xpi |ψi i hψi |ABiX√√→ | ρAB iAA∗ BB ∗ ≡pi |ψi iAB |ψi∗ iA∗ B ∗ .
i

RA:B := SvN (A ∪ A∗ )• Satisfies IA:B ≤ RA:B ≤ 2min [S(A), S(B)].
• Admits holographic dual RA:B = 2EW[Dutta-Faulkner (19)]18 / 33

• Introduction; topology and dynamics in quantum many-body systems

detected by entanglement• Negativity and reflected entropy• Tripartitioning topological liquid• Summary and outlook

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Going beyond entanglement entropy for bipartition• What can we learn by studying entanglement negativity and reflected

entropy?
• Will show results for the following two setups:[Wen-Matsuura-SR (16)][Liu-Sohal-Kudler-Flam-SR (21)]20 / 33

Negativity for topological liquid[Lee-Vidal (13), Castelnovo (13), Wen-Matsuura-SR (16), Wen-Chang-SR (16) Lim-Asasi-Teo-Mulligan (21)]• Generic state on a torus: |ψi =Pa

ψa |ha ii• Mutual information and negativity:

IA1 :A2 =X
X
πc l2− 2 ln D + 2|ψa |2 ln da −|ψa |2 ln |ψa |212 aaXπc l2EA1 :A2 =− ln D + ln|ψa |2 ln da16 a

E is dependent on ψa only for non-Abelian topological order (for Abelian
topological order, da = 1 for all a).
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Negativity and reflected entropy in tripartition setup[Liu-Sohal-Kudler-Flam-SR (21)]• Studied chiral p-wave superconductor (the Ising TQFT) with c = 1/2 andthe integer quantum Hal state with c = 1.

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Negativity spectrum
B• For fermionic systems, the eigenvalues of ρTA∪B are complex.
B• Nontrivial distribution of the eigenvalues of ρTA∪B• C.f. (1+1)D fermionic CFTs (6-fold structure) [Shapourian-Ruggiero-SR-Calabrese(19)](1+1)D topological superconductor (8-fold structure)[Inamura-Kobayashi-SR (19)]23 / 33

Reflected entropy• The “Markov gap” hA:B = RA:B − IA:B seems universal and given byhA:B =cln 23

where c is the (total) central charge. Agrees with the recent claim [Zou, Siva,Soejima, Mong, Zaletel (2110.11965)] (hA:B = (c/3) ln 2 = 0.116, 0.231 for c = 1
and c = 1/2).

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Central charge c• Chiral central charge can be measured by the thermal conductance in theedge [Kane-Fisher (96)] :2πkBT×c
6
E.g. half-filled Landau level [Banerjee (18)] , Kitaev spin liquid [Kasahara (18)]κ=• Modular commutator ih[KAB , KBC ]i = πc/3 :Kim-Shi-Kato-Albert (21)25 / 33

Multipartition topological phases using string field theory• String field theory = many-body (second quantized) string theory• Interaction vertices in string field theory[Witten (86), Gross-Jevicki (87), LeClair-Peskin-Preitschopf (89), ...]• Vertex state |V i ' Topological ground state |Ψi near the entangling

boundary by utilized bulk-boundary correspondence• Generalized [Qi-Katsura-Ludwig (12)] for tripartition geometry
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Lattice Chern insulator calculations• Lattice fermion model:
hi−i X X†H=fr† τµ fr+aµ − fr+a
τfµr
µ2r

µ=x,y

1X X+2rµ=x,yhi†fr† τz fr+aµ + fr+a
τ z fr + uµXfr† τz fr ,r27 / 33

Negativity and negativity spectrum
Entropy and Negativity

Negativity spectrum• “Trivial” for |u| → ∞• “Circular distribution”

deep in the Ch=1 phase

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Reflected entropy• hA:B is minimal in the topological phase around u = 1.34
• hA:B ∼ (c/3) ln 2 × 2.3• Note that there are four trijunctions, as opposed to two in the edge theorycalculations. May result in a factor of 2.

• See also:[Zou, Siva, Soejima, Mong, Zaletel (2110.11965)]

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Summary/Outlook• New tripartition setup and new calculations of entanglement quantities• They may capture topological data beyond topological entanglement

entropy, e.g., Abelian v.s. Non-abelian, total central charge.
• Finite-T topological transition can be detected by negativity[Hart-Castelnovo(18);Lu-Hsieh-Grover(19)]• May have an implication on numerics (tensor-networks)• May have an implication in string field theory?
• Other entanglement quantities, such as odd entropy, entanglement ofpurification, etc?
• Experiments: Many-body interference or randomized measurements [Islam etal (15)][Kaufman et al (16)][Lukin et al (18)][Brydges (19)]30 / 33

Negativity/reflected entropy in quantum dynamics• Quantum quench in integrable/chaotic (1+1)D CFT [Kudler-Flam-Kusuki-SR (20)]• Many-body localizing dynamics [MacCormack-Tan-Kudler-Flam-SR (19)]31 / 33

Thank you for your attention!

• Paper: arXiv:2110.11980• Collaborators:

Yuhan Liu, Ramanjit Sohal, Jonah Kudler-Flam• Thanks also:

Hassan Shapourian, Ken Shiozaki, Yuya Kusuki, Laimei Nie, Masahiro
Nozaki, Ryohei Kobayashi, Kansei Inamura, Paola Ruggiero, Pasquale
Calabrese.

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Edge theory approach to entanglement• ρA obtained from a ground state |GSi by tracing out half-space can beobtained from conformal boundary state |Bi:

[Qi-Katsura-Ludwig (12)][T (σ) − T̄ (σ)] |Bi = 0• Near the entangling boundary, |GSi ∼ e−Hedge |Bi so that the reduceddensity matrix isρA ∝ TrB e−Hedge |BihB|e−Hedge

• “Physical picture”: healing the cut; gapped edge by potential

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