Hierarchical Growth of Entangled States Part 2

University:
University of California, Berkeley
Course:
MPS | Mathematics and Physical Sciences
Academic year:

2022
Views:

359

Pages:

11
Author:

Clint W.

Reason to care #4: U MERA A MERA is a representation of the groundstate which: I I I [Vidal] allows efficient computation of observables (few contractions) organizes the information by scale (like Wilson and AdS/CFT taught us to do) geometrizes the entanglement structure [Swingle] (Best representation of 1d critical states, very hard to find in d > 1.) MERA representations of s = 1 fixed points Trotter Quasilocal U → low-depth circuit: |ψL i ' Ucircuit |ψL/2 i|0iL/2 1 finite overlap requires `ˆ ∼ log1+δ (L) 2 =⇒ ˆd bond dimension ∼ e ` ∼ e c log d(1+δ) 3 (L) Crucial point: This construction of Ucircuit d = 2: requires no variational sweeps on large system. Numerical implementation...? [getting started with Snir Gazit] d = 3: Further payoff: Invertible states I A robust notion of ‘short-range-entangled’ Related ideas: [Kitaev, Freed] ‘Invertible states,’ |ψi means ∃|ψ −1 i, U s.t. d |ψi ⊗ |ψ −1 i = U|0i⊗2L has s = 0. I Weak area law: a unique groundstate on any closed manifold (no topological order, but can still be interesting as SPTs) implies the existence of an inverse state and the area law. Graphical proof of weak area law: Disks  hemispheres Create microscopic wormholes (white holes) side view of H + Hrev + Expand wormholes wormholes : Glue at equator Expand wormholes Decoupled disks with edge states Gapped sphere Hamiltonian Pinch off microscopic links (grey links) step 1: ‘edge inverse’ step 2: make adiabatic path kills edge states to |0i⊗ on T d Gapless states and s-sourcery I ‘Entanglement Thermodynamics’ constrains area law violation by gapless states I and gives a relation between s and scaling exponents (s = 2θ ). I Examples of RG circuits for nontrivial critical points. Entanglement bounds for gapless states The area law is violated in groundstates of metals: S ∼ R d−1 log kF R. This violation is a symptom of many low-energy extended modes. =⇒ can be seen in thermodynamics. Result: [Swingle-JM, 1505.07106, PRB] If: thermal entropy of a d−θ scale-invariant state is s(T ) ∼ T z z ≡ dynamical exponent θ ≡ hyperscaling violation exponent (anomalous dimension of Ttt ) Then: the groundstate EE obeys the area law when θ < d − 1 and 0 < z < ∞. (Recall: a Fermi surface has θ = d − 1.) Entanglement thermodynamics Idea: Recast EE as localPthermodynamics problem (T = Tx ) 1 P Find σA ' Z −1 e − x Tx Hx (H ≡ x Hx . local Gibbs state) S(σA ) ≥ S(ρA ) . such that Who is σA ? State of max entropy consistent Hx in A. (hence S(σA ) ≥ S(ρA )) [Cramer et al 2010, Swingle-Kim 2014] =⇒ σA ∝ e − P x∈A Hx /Tx 1/Tx = Lagrange multipliers trHA σA = trHA ρA = Eg ,A +O (|∂A|`|Hx |) |{z} gs energy of HA =⇒ σA is a state with excitations localized at ∂A, Tx → 0 in interior of A. Entanglement thermodynamics Crucial Fact (local thermodynamics): For scaling purposes, Z trHA σA ' Eg ,A + d d x e(Tx ) A Z d −trσA log σA ' d x s(Tx ) A e(Tx ) = Ts(Tx ), bulk thermodynamic densities at temp Tx . Why: True if 1 But: let ∇Tx Tx · ξx (for all x) (ξx ≡ local correlation length). 1 σA (τ ) ≡ Z (τ )−1 e − τ P x H̃x /Tx τ →1 → σA . This state has temperature Tx (τ ) = τ Tx , =⇒ ξx (τ ) ∼ Tx (τ )−1/z ∝ τ 1/z So (unless z = ∞!) the figure of merit for local thermo in state σA (τ ) is 1 ∇Tx (τ ) τ →∞ · ξx (τ ) → 0. T (τ ) | {z } | x{z } ∼τ −1/z ∼τ 0 S(σA (τ )) = τ d−θ z S(σA ) =⇒ scales the same way with region size. Scaling in strip geometry To use local thermo, we need Tx . Our question is local. Choose convenient geometry. Translation invariant in d − 1 dims (PBC). R w a. Scale invariance =⇒  −z  x Tx ∼ ∞ (no)   0 (sometimes: frustration free H) =⇒ SA ≤ −trσA ln σA ∼ R d−1 e(Tx ) ∼ x −z+θ−d , s(Tx ) ∼ x θ−d ∼ R d−1 Rw a dx x −d+θ a−d+θ+1 − w −d+θ+1 w →∞ → ∞ only if d < 1 + θ Hence: scale invariant states with θ < d − 1 obey the area law. Connection to s-sourcery [Swingle-JM, 1505.07106] If our scaling theory is an s-source RG fixed point S(2R) ≤ sS(R) + kR d−1 . Assume saturated (if not, can use smaller s) =⇒ SA = Rw a ' d−1 logX 2 (w /a) s n R k a 2d−1 d−1 n=0 R a d−1−log2 s k 1− + ··· a w Compare subleading terms in EE of strip: s = 2θ (Fermi surface has θ = d − 1, hence s = 2d−1 , marginally violates area law. X) Gapless states with explicit s = 1 RG circuits Expectation: CFTs are s = 1 fixed points. ∞ many examples of d = 2 quantum critical points which are exact s = 1 fixed points: ‘Square-root states’ • Classical stat mech model in d space dimensions • configurations s • Boltzmann weight e −βh(s) P −βh(s) Z ≡ se • coolness β = 1/T −→ −→ −→ −→ e.g. near-neighbor P Ising model: h(s) = hiji si sj −→ correlations hZr Zr 0 i = classical critical point • real-space RG scheme −→ −→ [Kimball 1979] • Quantum system in d space dimensions • states |si (orthonormal) • g.s. wavefunction P |h, βi = Z −1/2 s e −βh(s)/2 |si • coupling Zi |si = si |si. Parent Hamiltonian: P P H = i −Xi + e −βZi hi|ji Zj correlations of diagonal operators hgs|Zr Zr 0 |gsi quantum critical point • quantum RG circuit with s = 1 RG circuits for square root states 2d classical Ising TRG scheme: Z = P abcd... Tabc Tade ··· Two parts of classical RG step [Levin-Nave]: 1: P P e Tabe Tcde = f Sacf Sbdf P P = f e Quantum version: U1 ⊗ |0if = 2: P P abc = abc P f 0 Sakc Scjb Sbia = Tijk ⊗ |0ie . U2 P abc = ⊗ |000i Q Q U = U2 U1 Fixed point of classical TRG =⇒ s = 1 fixed point. [JM, B Swingle, Shenglong Xu, 1602.02805, PRB]