in condensed matter Grgoire Misguich (IPhT, CEA Saclay) - PowerPoint PPT Presentation

Schmidt decomposition (3) – Optimal approximation  What is the best approximation to 𝜔 with a given lower Schmidt rank 𝜓 < 𝑠 ? 𝑠 𝜔 = 𝑁 𝑙𝑚 𝑏 𝑙 𝑐 𝑙 = 𝜇 𝑙 𝑣 𝑙 𝑤 𝑙 𝜇 1 ≥ 𝜇 2 ≥ ⋯ ≥ 𝜇 𝑠 ≥ 0 𝑙𝑚 𝑙=1 SVD: 𝑁 = 𝑉𝑇𝑊 †  Answer: truncate the Schmidt decomposition above, keeping only the 𝜓 largest values (and re-normalize the state): 𝜓 1 𝜇 𝑙 𝑣 𝑙 𝑤 𝑙 𝜔 𝜓 = 𝜓 𝜇 𝑙 2 1 𝑙=1 1 𝑠 𝜔 𝜔 𝜓 = 1 − 𝜇 𝑙 2 𝑙=𝜓+1  « Proof »: Note that the Hilbert space norm is equivalent to Frobenius norm ⋯ 𝐺 for the • 2 = Tr 𝑁𝑁 † = 𝑁 𝑙𝑚 2 matrix 𝑁 : 𝑁 𝐺 = 𝜔 𝜔 𝑙𝑚 • Use the Eckart and Young theorem (1936), which states that the best approximation (in the sense of ⋯ 𝐺 ) of rank 𝜓 ≤ 𝑠 to the matrix 𝑁 is the matrix 𝑁 𝜓 obtained by truncating the SVD decomposition of 𝑁 to its 𝜓 largest singular values. Remark: the proof of this theorem is somewhat “tricky” and not discussed here... Note that this solution to the optimization problem is also optimal for the norm ⋯ 2 -- for which the proof is simpler. 12

Schmidt decomposition (4) – SVD on a computer  Complexity for an 𝑁 ∗ 𝑂 matrix: 𝒫 min 𝑁 ∗ 𝑂 2 , 𝑂 ∗ 𝑁 2 NB: To be compared with a two-step procedures: i) Computing 𝜍 𝐵 : 𝒫 𝑂 ∗ 𝑁 2 and then ii) diagonalizing 𝜍 𝐵 : 𝒫 𝑁 3 .  LAPACK: dgesvd  GSL: gsl_linalg_SV_decomp  Numerically stable  Sometimes the SVD is overkill if the Schmidt values are not needed (just some orthogonal basis on one “side”, as often in DMRG for instance) and a QR factorization is enough (faster). 13

SVD, decay of the singular values & data compression (2) 14

SVD, decay of the singular values & data compression (1) An (color) image viewed as a (three) matrix(ces) 15

Von Neumann entropy (definition)  Von Neumann/entanglement entropy 𝑇 𝐵 = −Tr 𝐶 𝜍 𝐵 log 𝜍 𝐵 = − 𝑞 𝑗 log 𝑞 𝑗 𝑗 With 𝜍 𝐵 = Tr 𝐶 𝜍 𝐵𝐶 , and 𝑞 𝑗 the eigenvalues of 𝜍 𝐵 .  𝑇 𝐵 = 0 ⟺ 𝜍 𝐵 is a projector ⟺ 𝜔 is a product state ⟺ region 𝐵 is in a pure state Remark: For a thermal density matrix 𝜍~𝑓 −𝛾𝐼 , the same formula gives the Boltzmann- Gibbs entropy “ You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage. ” J. Von Neumann, suggesting to Claude Shannon a name for his new uncertainty function, as quoted in Scientific American 225, 3, p180 (1971). 16

Von Neumann entropy – basic properties  𝑇 𝜍 is zero if and only if 𝜍 represents a pure state (projector)  𝑇 𝜍 is maximal and equal to log 𝑂 for a maximally mixed state, 𝑂 being the dimension of the Hilbert space.  𝑇 𝜍 is invariant under changes in the basis of 𝜍 , that is, 𝑇 𝜍 = 𝑇 𝑉𝜍𝑉 † , with 𝑉 unitary.  Concavity 𝑇 𝜇 𝑗 𝜍 𝑗 ≥ 𝜇 𝑗 𝑇 𝜍 𝑗 𝑗 𝑗  Independent systems : 𝑇 𝜍 𝐵 ⊗ 𝜍 𝐶 = 𝑇 𝜍 𝐵 + 𝑇 𝜍 𝐶  If 𝜍 𝐵𝐶 describe a pure state, 𝑇 𝜍 𝐶 = 𝑇 𝜍 𝐵 (proof using the Schmidt decomposition, see next)  A,B,C: 3 parts, without intersection Strong sub-additivity [SS] 𝑇 𝜍 𝐵𝐶𝐷 + 𝑇 𝜍 𝐶 ≤ 𝑇 𝜍 𝐵𝐶 + 𝑇 𝜍 𝐶𝐷 • • Equivalent formulation: 𝑇 𝑌∪𝑍 + 𝑇 𝑌∩𝑍 ≤ 𝑇 𝑌 + 𝑇 𝑍 Proof: difficult ! E. H. Lieb, M. B. Ruskai, “Proof of the Strong Subadditivity of Quantum Mechanichal Entropy ” , J. Math. Phys. 14, 1938 (1973) • Subadditivity 𝑇 𝜍 𝐵𝐷 ≤ 𝑇 𝜍 𝐵 + 𝑇 𝜍 𝐷 (obtained by setting 𝐶 = ∅ in SS above, but direct proof is possible a much much simpler than SS)  Araki-Lieb triangular inequality: 𝑇 𝜍 𝐵𝐷 ≥ 𝑇 𝜍 𝐵 − 𝑇 𝜍 𝐷 Simple proof using i) some auxiliary pure state 𝜍 𝐵𝐶𝐷 in some enlarged space such that Tr B 𝜍 𝐵𝐶𝐷 = 𝜍 𝐵𝐷 , Tr 𝐶𝐷 𝜍 𝐵𝐶𝐷 = 𝜍 𝐵 and Tr 𝐵𝐶 𝜍 𝐵𝐶𝐷 = 𝜍 𝐷 , ii) the sub-additivity above for 𝜍 𝐶𝐷 and iii) 𝑇 𝜍 𝐶𝐷 = 𝑇 𝜍 𝐵 and 𝑇 𝜍 𝐶 = 𝑇 𝜍 𝐵𝐷 (since 𝜍 𝐵𝐶𝐷 is pure). 17

A simple exercise with strong subadditivity  Let 𝑇(𝑦) be the entanglement entropy of a segment of length 𝑦 in a periodic and translation invariant spin chain of total length 𝑀 (in a pure state).  How to prove that 𝑇(𝑦) is a concave function of 𝑦 and that it is maximal for 𝑦 = 𝑀/2 ? (unless it is constant and equal to zero) . 𝑇(𝑦) 𝑦 x  Answer: use the strong sub-additivity with the three following consecutive segments 𝐵 = {0} , 𝐶 = {1,2, … , 𝑦 − 1} and 𝐷 = {𝑦} : 𝑇 𝐵𝐶𝐷 + 𝑇 𝐶 ≤ 𝑇 𝐵𝐶 + 𝑇 𝐶𝐷 𝑇 x + 1 + S x − 1 ≤ 2𝑇 𝑦 → concavity + symmetry 𝑇(𝑦) = 𝑇(𝑀 − 𝑦) 18

Rényi entropies & Replica trick  If you can compute the spectrum of 𝜍 𝐵 … then you get 𝑇 𝐵 by summing over all the eigenvalues: 𝑇 𝐵 = −Tr 𝜍 𝐵 log 𝜍 𝐵 = − 𝑞 𝑗 log 𝑞 𝑗 𝑗  If not… you can compute Tr 𝜍 𝐵𝑜 for integer 𝑜 ≥ 2 (easier than computing the spectrum) • (cross your fingers &) analytically continue the result to n=1 • • use: 1 1 − 𝑜 log Tr 𝜍 𝐵𝑜 −Tr 𝜍 𝐵 log 𝜍 𝐵 = lim 𝑜→1  Rényi entropies are also often interesting, and simpler to compute & measure [in principle] 1 1 − 𝑜 log Tr 𝜍 𝐵𝑜 𝑇 𝐵 (𝑜) = NB: Rényi entropies (integer 𝑜 ≥ 2 ) can be measured in Quantum Monte Carlo: “ Measuring Rényi Entanglement Entropy in Quantum Monte Carlo Simulations ” M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 104, 157201 (2010)  They share some properties with the Von Neumann entropy: Positive 𝑇 𝐵 𝑜 ≥ 0 • Additive for uncorrelated systems 𝑇 𝐵𝐶 𝑜 = 𝑇 𝐵 𝑜 + 𝑇 𝐶 𝑜 if 𝜍 𝐵𝐶 = 𝜍 𝐵 ⨂𝜍 𝐶 • • But no subadditivity 19

Entanglement spectrum - definition 𝜔 = 𝜇 𝑗 𝑏 𝑗 𝑐 𝑗 Schmidt decomposition 𝑗 𝑞 𝑗 = 𝜇 𝑗 2 , 𝜇 𝑗 > 0 , 𝑞 𝑗 = 1 𝑗 Interpret the eigenvalues of 𝜍 𝐵 𝑞 𝑗 = exp −𝐹 𝑗 with Z 𝑜 = 𝑓 −𝑜𝐹 𝑗 as classical Boltzmann weights. 𝑎 1 This defines some “energies” 𝐹 𝑗 𝑗 Tr 𝜍 𝐵𝑜 = 𝑞 𝑗 𝑜 = 𝑎 𝑜 𝑎 𝑜 : Partition function at inverse 𝑎 1 1 𝑗 temperature 𝑈 = 𝛾 = 𝑜 1 𝑓 −𝐹 𝑗 2 𝑏 𝑗 𝑐 𝑗 𝜔 = 𝑎 1 𝑗 Free energy 𝐺 𝑜 = − 1 = − 1 = 1 n 𝐺 1 − 1 n log 𝑎 1 Tr 𝜍 𝐵𝑜 n log Tr 𝜍 𝐵𝑜 n log Z n Rényi entropy ↔ free energy difference 1 1 1 − n log Tr 𝜍 𝐵𝑜 𝑇 𝐵 𝑜 = = 𝑜 − 1 𝑜𝐺 𝑜 − 𝐺 1 20

HOW TO MEASURE ENTANGLEMENT ENTROPIES ? at least in principle… 21

Experimental measurement of entanglement ?  Few q- bits → measure sufficiently many correlations/observables in order to reconstruct the complete density matrix (quantum state tomography)  Examples: nuclear spins I=1/2 or photon polarizations (H/V): “Photon entanglement detection by a single atom” J. Huwer et al 2013 New J. Phys. 15 025033 “Solving Quantum Ground-State Problems with NMR” Zhaokai Li et al., Scientific Reports 1, 88 (2012)  What about the entanglement entropy of a large system ? 22

Measurement of entanglement entropies ? (1)  A few proposals based on coupling 𝑜 copies of the system to measure the entropy 𝑇 𝑜  Example : “Measuring Entanglement Entropy of a Generic Many-Body System with a Quantum Switch” D. A. Abanin & E. Demler, Phys. Rev. Lett. 109, 020504 (2012) We describe here the simplest case, with 𝑜 = 2 . L R Finite chain & left/right 𝜔 = 𝜇 𝑗 𝑚 𝑗 ⨂ 𝑠 𝑗 Schmidt decomposition 𝑗 2 = − log 𝜇 𝑗 4  Goal: “measure” the Rényi entropy 𝑇 𝑜=2 , that is −log Tr 𝑆 𝜍 𝑀 𝑗  4 half-chains, that can be connected in two different ways: L1 L1 R1 R1 or R2 L2 L2 R2 𝐻 = 𝜇 𝑗 𝑚 𝑗 𝑀1 ⨂ 𝑠 𝑗 𝑆1 ⨂ 𝜇 𝑘 𝑚 𝑘 𝑀2 ⨂ 𝑠 𝑘 𝑆2 𝐻′ = 𝜇 𝑗 𝑚 𝑗 𝑀1 ⨂ 𝑠 𝑗 𝑆2 ⨂ 𝜇 𝑘 𝑚 𝑘 𝑀2 ⨂ 𝑠 𝑘 𝑆1 𝑗 𝑘 𝑗 𝑘 “quantum switch” 23

Measurement of entanglement entropies ? (2)  Rényi entropy 𝑇 𝑜 and scalar product L1 L1 R1 R1 or R2 L2 L2 R2 𝐻 = 𝜇 𝑗 𝑚 𝑗 𝑀1 ⨂ 𝑠 𝑗 𝑆1 ⨂ 𝜇 𝑘 𝑚 𝑘 𝑀2 ⨂ 𝑠 𝑘 𝑆2 𝐻′ = 𝜇 𝑙 𝑚 𝑙 𝑀1 ⨂ 𝑠 𝑙 𝑆2 ⨂ 𝜇 𝑚 𝑚 𝑚 𝑀2 ⨂ 𝑠 𝑚 𝑆1 𝑗 𝑘 𝑙 𝑚 𝑘𝑙 = 𝜇 𝑗 4 2 𝐻 𝐻′ = 𝜇 𝑗 𝜇 𝑘 𝜇 𝑙 𝜇 𝑚 𝜀 𝑗𝑙 𝜀 𝑗𝑚 𝜀 𝑘𝑚 𝜀 = Tr 𝑆 𝜍 𝑀 = exp −𝑇 2 𝑗𝑘𝑙𝑚 𝑗  Introduce a weak transverse field on the central spin (quantum switch ) +𝐾𝜏 𝑦 0 𝐾 𝐻 𝐻′ 𝐼 eff = 𝐾 𝐻 𝐻′ 0 Degenerate perturbation theory Δ Δ Eigenvalues 𝜕 = ±𝐾 𝐻 𝐻′ ( 𝐾 ≪ Δ ) 𝐻 𝐻′ Measure the oscillation frequency 𝜕 of 𝜏 𝑨 (𝑢) → access to exp −𝑇 2 𝜏 𝑨 = −1 𝜏 𝑨 = +1 generalization: couple 𝑜 systems and a 2-state switch to measure 𝑇 𝑜 24

Summary of lecture #1 B A • Context: lattice quantum many- body problems (spin, fermions, …) Two large subsystems in a pure state 𝜔 ∈ ℋ 𝐵 ⨂ℋ 𝐶 . • The total density matrix 𝜍 𝐵𝐶 = 𝜔 𝜔 is a projector • Reduced density matrix: 𝜍 𝐵 = Tr 𝐶 𝜍 𝐵𝐶 Tr A 𝜍 𝐵 = Tr A𝐶 𝜍 𝐵𝐶 = 1 • Shmidt decomposition of 𝜔 : • 𝜔 = 𝑁 𝑗𝑘 𝑏 𝑗 ⨂ 𝑐 𝑘 =→ SVD of 𝑁 →= 𝜇 𝑙 𝑣 𝑙 ⨂ 𝑤 𝑙 𝑗𝑘 𝑙 Spectrum of the RDM: 𝜍 𝐵 = 𝜇 𝑙 2 𝑣 𝑙 𝑣 𝑙 • 𝑙 • Von Neumann entropy 𝑇 𝐵 = −Tr 𝐶 𝜍 𝐵 log 𝜍 𝐵 = − 𝜇 𝑙 2 log 𝜇 𝑙 2 𝑗 𝑇 𝐵 quantifies the uncertainty on the state of 𝐵 if we do not observe the region 𝐶 . • 𝑇 𝐵 = 0 ⟺ 𝜍 𝐵 is a projector ⟺ 𝜔 = 𝐵 ⨂ 𝐶 is a product state • Strong sub-additivity of the Von Neumann entropy: 𝑇 𝑌∪𝑍 + 𝑇 𝑌∩𝑍 ≤ 𝑇 𝑌 + 𝑇 𝑍 • 1 1−𝑜 log Tr 𝜍 𝐵𝑜 Useful generalization: Rényi entropies 𝑇 𝐵 (𝑜) = • • 𝑇 𝐵 (𝑜) is often simpler to compute (and possibly to measure experimentally) than 𝑇 𝐵 when 𝑜 is an integer ≥ 2 (replicas, etc.). 𝑜 plays the role of an inverse temperature. Finite correlation length → area law 𝑇 𝐵 ∼ 𝒫 Area of 𝜖𝐵 = 𝒫 𝑀 𝑒−1 . • 25

AREA/BOUNDARY LAW for the entanglement entropy of low-energy states of short-ranged Hamiltonians. Decay of Schmidt eigenvalues. 26

Area law for the entanglement entropy “ Quantum source of entropy for black holes ” L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34, 373 (1986) « Entropy and area », M. Srednicki, Phys. Rev. Lett. 71 , 666 (1993) “ Area Laws in Quantum Systems: Mutual Information and Correlations ” M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008) “Colloquium : Area laws for the entanglement entropy” J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010)  The ground-state (and low-energy excitations) of (many) Hamiltonians with short- L ranged interactions have an entanglement entropy which scale like the area of the boundary of the subsystem A 𝑇 𝐵 ∼ 𝒫 Area of 𝜖𝐵 = 𝒫 𝑀 𝑒−1 Appears to be valid for all gapped systems (in 𝒆 = 𝟐 it’s a theorem), • B and some gapless systems in 𝒆 > 𝟐 (not all). • Known gapless systems which violate the area law: critical systems in 𝑒 = 1 o systems in 𝑒 > 1 with a Fermi surface, where 𝑇 𝐵 ∼ 𝒫 𝑀 𝑒−1 log 𝑀 o • Can be proved in any dimension if we make a strong hypothesis on the decay of all correlations (Wolf et al. 2008 ), see a few slide below. 27

Area law for the entanglement entropy  Simple (hand-waving!) argument Pure state 𝜔 , ground state of some local Hamiltonian in • spatial dimension 𝑒 . Asumme that all connected correlation functions in 𝜔 • decay exponentially in space, with some finite correlation length 𝜊 • Subsystem: some spatial region 𝐵 of typical size L ≫ 𝜊 . ( 𝐶 =complement of 𝐵 ). L Assume that the entanglement between 𝐵 and 𝐶 • is entirely due to local correlations (not a very precise statement…) A  Correlations between degrees of freedom located inside 𝐵 do not contribute to 𝑇 𝐵 . Same for correlations inside the region 𝐶 . 𝝄  The only contributions to the entanglement entropy 𝑇 𝐵 are B those originating from correlations taking place across the boundary between A and B → Area/ Boundary law for the entanglement entropy : 𝑇 𝐵 ~size of 𝜖𝐵 ~𝑀 𝑒−1 28

Area law for the entanglement entropy B A  Variant of the intuitive argument … L Correlation length 𝜊 𝑗 = 1 ⋯ ~2 𝜊𝑀 configurations for the magenta sites ( ∈ A) 𝑘 = 1 ⋯ ~2 𝜊𝑀 configurations for the blue sites ( ∈ B) Assume the wave function can be approximated by : 𝐵 ⨂ 𝜔 𝑘 𝐶 𝜔 ∼ 𝑁 𝑗,𝑘 𝜔 𝑗 𝑗,𝑘 Which means, that, once we project on a particular state (i,j) of the « boundary region » (of width ~𝜊 ), the regions A and B are no longer correlated ( → product state). Schmidt decompostion (=SVD of M) → number of non-zero values ~2 𝜊𝑀 𝑇 𝐵 ≤ 𝜊𝑀 log 2 → boundary law 29

Remark about the area law & entanglement spectrum  𝑇 𝐵 = thermodynamic entropy of the entanglement spectrum 𝐹 𝑗 (by definition) Assume these 𝐹 𝑗 are energies of some “fictitious system” associated to the bi - partition A/B.  Since 𝑇 𝐵 ~𝑀 𝑒−1 can be interpreted as a “volume law” (as usual for thermodynamic entropy) for a system in 𝑒 − 1 spatial dimension, the “fictitious system” probably lives at the boundary between A & B . We will see a few explicit examples later (“bulk - edge correspondence”) Note: this is consistent with the fact that the spectrum of 𝜍 𝐵 (hence the 𝐹 𝑗 ) is unchanged if we exchange A and B.  If the number of Schmidt eigenvalues which contribute to entropy is a finite fraction of the dimension dim (𝐼 𝐵 ) (assume region B is much larger, for simplicity) , we expect a volume-law behavior. If, instead, 𝑇 𝐵 ~𝑀 𝑒−1 , then we expect that most of the eigenvalues of 𝜍 𝐵 are much smaller than 1/dim (𝐼 𝐵 ) . 30

Mutual information  Definition: 𝐵 𝐽 𝐵: 𝐶 = 𝑇 𝜍 A + 𝑇 𝜍 𝐶 − 𝑇 𝜍 AB 𝐶 (with 𝑇 𝜍 = −Tr 𝜍 log 𝜍 ) .  Alternatively: 𝐽 𝐵: 𝐶 = Tr 𝜍 AB log 𝜍 AB − log 𝜍 A ⨂𝜍 𝐶  Thanks to the sub-additivity of the Von Neumann entropy we have 𝐽 𝐵: 𝐶 ≥ 0 . 1 2 (=Pinsker inequality). A stronger result is: 𝐽 𝐵: 𝐶 ≥ 2 𝜍 AB − 𝜍 A ⨂𝜍 𝐶 1 → 𝐽 𝐵: 𝐶 can be viewed as some kind of “distance” between 𝜍 AB and 𝜍 A ⨂𝜍 𝐶 . In particular: 𝐽 𝐵: 𝐶 = 0 ⇔ 𝜍 AB = 𝜍 A ⨂𝜍 𝐶 .  𝐽 𝐵: 𝐶 measures the total amount of correlations between A and B. Indeed, 𝐽 𝐵: 𝐶 can be shown (using the Pinsker inequality above → exercise !) to give a bound on correlators: 2 𝑃 𝐵 𝑃 𝐶 − 𝑃 𝐵 𝑃 𝐶 𝐽 𝐵: 𝐶 ≥ 2 𝑃 𝐶 1 2 2 𝑃 𝐵 1  𝐽 𝐵: 𝐶 = 0 then all correlations between 𝐵 and 𝐶 must vanish. If 𝐽 𝐵: 𝐶 decays exponentially with their distance, it will be also the case for any correlation between 𝐵 and 𝐶 .  Remark 1: if 𝐵𝐶 is in a pure state ( 𝜍 AB = 𝜔 𝜔 ) we have 𝑇 𝜍 AB = 0 . So: 𝐽 𝐵: 𝐶 = 2𝑇 𝜍 A = 2𝑇 𝜍 B is equivalent to the Von Neumann entropy.  Remark 2: the “volume” contributions to the entropy cancel out in 𝐽 𝐵: 𝐶 . → An area law is expected (see next slide) 31

Mutual information & boundary law at T>0 “ Area Laws in Quantum Systems: Mutual Information and Correlations ”, M. M. Wolf, F. 𝐶 Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008) 𝐵 𝐼 𝑗𝑜𝑢 1 𝛾 𝑇(𝜍) is minimized by 𝜍 𝐵𝐶 ~𝑓 −𝛾𝐼  Free energy 𝐺 𝜍 = 𝑉 − 𝑈𝑇 = Tr 𝐼𝜍 − with 𝐼 = 𝐼 𝐵 + 𝐼 𝐶 + 𝐼 𝑗𝑜𝑢 . 𝐼 𝑗𝑜𝑢 contains all the terms which couple 𝐵 and 𝐶 .  In particular: 𝐺 𝜍 ≤ 𝐺 𝜍 A ⨂𝜍 B . So: Tr 𝐼𝜍 − 1 − 1 𝛾 𝑇 𝜍 ≤ Tr 𝐼 𝐵 + 𝐼 𝐶 + 𝐼 𝑗𝑜𝑢 𝜍 𝐵 ⨂𝜍 𝐶 𝛾 𝑇 𝜍 A + 𝑇 𝜍 B 𝐽 𝐵: 𝐶 ≤ Tr 𝐼 𝑗𝑜𝑢 𝜍 𝐵 ⨂𝜍 𝐶 − 𝜍 ~𝒫(Area) 𝛾  This demonstrates an area law behavior for 𝐽 𝐵: 𝐶 at finite temperature .  Remark: no general theorem for the area law at 𝑈 = 0 in d>1, although it is verified by a large class of systems (notable exception: states with Fermi surface). See however the argument on next slide. 32

Area law at 𝑈 = 0 “ Area Laws in Quantum Systems: Mutual Information and Correlations ”, M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008)  Geometry: 𝐵 and 𝐶 are separated by some distance 𝑀 . The “shell” in 𝐶 between is the region 𝐷 . Define 𝐽 𝑀 = 𝐽 𝐵: 𝐶 = mutual information 𝐷  Using strong subadditivity one can show that 𝐽 𝑀 is a decreasing function of 𝑀 𝑀 (exercise!). Define the correlation length 𝝄 as the minimal distance 𝐵 𝑆 𝑱 𝟏 which insures 𝑱 𝑴 ≤ 𝟑 for all 𝑺 .  Remarks: • This correlation length incorporates all types of correlations between 𝐵 and the rest of the system. • It may be infinite in some cases. From now we assume it is finite, and take 𝑀 = 𝜊.  Use sub-additivity and the Araki-Lieb triangular inequality to show (exercise!): 𝐽 𝐵: 𝐶𝐷 ≤ 𝐽 𝐵: 𝐶 + 2𝑇 𝐷 𝐽 0  By construction 𝐽(𝐵: 𝐶𝐷) = 𝐽 0 and 𝐽 𝐵: 𝐶 = 𝐽 𝑀 . By def. of 𝜊 we have 𝐽 𝑀 ≤ 2 , so: 𝐽 0 ≤ 𝐽 𝑀 + 2𝑇 𝐷 ≤ 𝐽 0 2 + 2𝑇 𝐷 . Hence 𝐽 0 ≤ 4𝑇 𝐷 . Since 𝑇 𝐷 is bounded by its volume 𝐷 ∼ 𝜊 𝜖𝐵 . We get 𝐽 0 ≤ 4𝜊 𝜖𝐵 . Now if the entire system 𝐵𝐶𝐷 in a pure state (no necessary so far) we have 𝐽 𝐵: 𝐶𝐷 = 𝐽 0 = 2𝑇 𝐵 and finally: 𝑇 𝐵 ≤ 2𝜊 𝜖𝐵 33

Area/boundary law 1d gapped system M B Hastings J. Stat. Mech. (2007) P08024 34

Universal violations & corrections to the area law A few examples (non-exhaustive list)  Violation (multiplicative log: ∼ L 𝑒−1 log 𝑀 ) Critical systems in 𝑒 = 1 • Holzhey-Larsen-Wilczek 1994, Vidal-Latorre-Rico-Kitaev 2003, Calabrese-Cardy 2004, … • Fermi surface (Wolf 2006, Gioev & Klich 2006)  Corrections (additive log: 𝒫 𝑀 𝑒−1 + 𝒫 log 𝑀 ) • Some critical systems in 𝑒 = 2 , with sharp-corners (Fradkin-Moore 2006) • Countinuous sym. breaking (Nambu-Golstone modes) Metlitsky-Grover 2011, Luitz-Plat-Alet-Laflorencie 2015  Correction (constant terms: 𝒫 𝑀 𝑒−1 + 𝒫 𝑀 0 ) Discrete spontaneous sym. breaking (→ contributtion S 0 = log degeneracy ) • Topological order in 𝑒 = 2 ( Kitaev-Preskill 2006, Levin-Wen 2006 ) • Some (Lorentz-invariant) critical systems in 𝑒 = 2 (Casini-Huerta 2007,+ many others ...) • Some critical systems in 𝑒 = 2 (Hsu et al. 2009, Stéphan-Furukawa-GM-Pasquier 2009) • Concerning critical systems, see the (very) recent work by B. Swingle & J. McGreevy, arXiv:1505.07106 35

VOLUME LAW for high-energy states, relation between entanglement and thermal entropies 36

Comparison with random pure states  “Average entropy of a subsystem”, D. Page, Phys. Rev. Lett. (1993) … = … whole Hilbert space ℋ 𝐵 ⨂ℋ 𝐶 (Haar measure) Assume dim ℋ 𝐵 ≤ dim ℋ 𝐶 : 2dim ℋ 𝐶 ~𝒫 𝑀 𝑒 = Volume law dim ℋ 𝐵 𝑇 𝐵 = − Tr 𝜍 𝐵 log 𝜍 𝐵 = log dim ℋ − 𝐵  For the full probability distribution of 𝑇 𝐵 , see C. Nadal, S. N. Majumdar, and M. Vergassola, Phys Rev Lett 104, 110501 (2010) (random matrix model methods) →The probability distribution is highly peaked around the average value 𝐵 −2 ) ) and the scaling of the typical entropy is a (variance Var(S A )~𝒫(dim ℋ also a volume law. → States satisfying an area law are rare  Choose a subsystem 𝐵 such that 1 ≪ dim ℋ 𝐵 = 𝑛 ≤ dim ℋ 𝐶 = 𝑜 . 𝐵 ⨂ℋ 𝐶 , and you will find 𝑇 𝐵 ~ log 𝑛 − 𝑛 1. Pick a pure state at random in ℋ 2𝑜 . 2. Now consider the infinite-temperature density matrix for the region 𝐵 : 𝑈=∞ = Id m 𝑈=∞ = log 𝑛 . 𝜍 𝐵 m . It gives 𝑇 𝐵 𝑛 3. Conclusion: when the region 𝐶 is large (i.e. 𝑜 ≪ 1 ) you can hardly distinguish the two situations. 37

ETH, entanglement entropy & thermal entropy (1)  Hamiltonian 𝐼 = 𝐹 𝛽 𝛽 𝛽 𝛽  The initial state is a linear combination of eigenstates which lie in a small energy windows 𝐹 − Δ𝐹, 𝐹 + Δ𝐹 ( 𝐹 is extensive (above the ground state) whereas Δ𝐹~𝒫(1) ): 𝜔 = 𝑑 𝛽 𝛽 𝛽  Time evolution: 𝜔(𝑢) = 𝑑 𝛽 𝑓 −𝑗𝐹 𝛽 𝑢 𝛽 𝛽  Expectation value of some observable 𝐵: + 𝐵 𝛽𝛾 𝑑 𝛽 𝑑 𝛾 𝑓 −𝑗(𝐹 𝛾 −𝐹 𝛽 )𝑢 𝜔(𝑢) 𝐵 𝜔(𝑢) = 𝑑 𝛽 2 𝐵 𝛽𝛽 𝛽 𝛽≠𝛾  Long-time average (assume no degeneracy): 𝑓 −𝑗(𝐹 𝛾 −𝐹 𝛽 )𝜐 − 1 𝜐 1 = 𝑑 𝛽 2 𝐵 𝛽𝛽 𝜐 𝜔(𝑢) 𝐵 𝜔(𝑢) 𝑒𝑢 + 𝑗 𝐵 𝛽𝛾 𝑑 𝛽 𝑑 𝛾 𝜐 0 𝛽 𝛽≠𝛾 𝜐 1 = 𝑑 𝛽 2 𝐵 𝛽𝛽 𝐵 = lim 𝜐 𝜔(𝑢) 𝐵 𝜔(𝑢) 𝑒𝑢 𝜐→∞ 0 𝛽  If 𝜔(𝑢) 𝐵 𝜔(𝑢) relaxes to some limit at long times, the long time limit must coincide with 𝐵 . This corresponds to the so-called diagonal ensemble : 𝜍 diag 𝑑 𝛽 = 𝑑 𝛽 2 𝛽 𝛽 𝛽 38

ETH, entanglement entropy & thermal entropy (2)  Assumption: the observable 𝐵 “thermalizes” , which means that 𝜔(𝑢) 𝐵 𝜔(𝑢) converges to the prediction of some thermodynamical ensemble (hence independent of the details of the initial state, except for the energy 𝐹 0 ) , here the micro-canonical one: 1 Ω 𝑑 𝛽 2 𝐵 𝛽𝛽 = Tr 𝜍 diag 𝑑 𝛽 𝐵 = Tr 𝜍 micro 𝐹 𝐵 = 𝐵 𝛽𝛽 𝒪(𝐹, ΔE) 𝛽 Ω 𝛽 𝐹 𝛽 −𝐹 <Δ𝐹 1 𝛽 𝛽 𝜍 micro 𝐹 = 𝒪(𝐹, ΔE) 𝛽 𝐹 𝛽 −𝐹 <Δ𝐹  Remark: 𝜍 diag 𝑑 𝛽 depends on the initial conditions, but 𝜍 micro 𝐹 does not…How can that be ?  A possible/plausible explanation: “ Eigenstate thermalization Hypothesis (ETH)” : the diagonal matrix elements 𝐵 𝛽𝛽 are (in thermodynamic limit) smooth functions of the energy • density, and independent of the choice of the eigenstate (in a sufficiently narrow energy window) . Hence 𝐵 = 𝐵 micro 𝐹 . Off diagonal elements 𝐵 𝛽𝛾 ( 𝛽 ≠ 𝛾 ) vanish (in the thermodynamic limit) •  References on ETH: [1] J. M. Deutsch “ Quantum statistical mechanics in a closed system ”, Phys. Rev. A 43, 2046 (1991) [2] M. Srednicki “ Chaos and quantum thermalization ”, Phys. Rev. E 50, 888 (1994) [3] M. Rigol, V. Dunjko, Vanja & M. Olshanii "Thermalization and its mechanism for generic isolated quantum systems ” , Nature 452 854 (2008) 39

ETH, entanglement entropy & thermal entropy (3) J. R. Garrison & T. Grover arXiv:1503.00729 1 ≪ vol Ω ≪ vol Ω  Strong form of ETH: all the observables in Ω satisfy the ETH Ω Consequences: the RDM computed in some arbitrary eigenstate 𝛽 • Ω at energy 𝐹 𝛽 ~𝐹 becomes “thermal” in the thermodynamic limit: 𝜍 Ω 𝛽 = Tr Ω 𝛽 𝛽 ~Tr Ω 𝜍 𝑛𝑗𝑑𝑠𝑝 𝐹 (with 𝐹 𝛽 ~𝐹 ) • the Von Neumann/entanglement entropy of an excited eigenstate is (asymptotically) the thermodynamic entropy of the subsystem • The volume law for the thermodynamic entropy at 𝑈 > 0 implies a volume law for the Von Neumann entropy of high-energy eigenstates.  Assume further that the thermodynamic free energy density 𝑔(𝛾) can equivalently be obtained exp −𝛾𝐼 Ω Ω from 𝜍 𝑛𝑗𝑑𝑠𝑝 𝐹 = Tr Ω 𝜍 𝑛𝑗𝑑𝑠𝑝 𝐹 or 𝜍 cano ~ (adjust the inverse temperature 𝛾 to match the energy E) . 𝑎(𝛾) which describe thes region Ω , isolated , and at thermal equilibrium. Then : vol Ω 𝑔 𝑜𝛾 = − 1 𝑜𝛾 log Tr exp −𝛾𝐼 Ω 𝑜 ~ − 1 𝑜 𝑜𝛾 log Tr 𝑎 𝛾 𝑜 𝜍 𝑛𝑗𝑑𝑠𝑝 Ω 𝐹 ~ − 1 𝛾 log 𝑎 𝛾 − 1 𝑜𝛾 log Tr 𝜍 Ω 𝛽 𝑜 = − 1 𝛾 log 𝑎 𝛾 − 1 − 𝑜 𝑜 𝛽 𝑜𝛾 𝑇 Rényi 𝑜−1 𝑜 𝛽 The Rényi entanglement entropies of a single eigenstate Finally: 𝑔 𝑜𝛾 − 𝑔 𝛾 = 𝑜𝛾vol Ω 𝑇 Rényi gives the thermodynamic free energy at all temperatures ! 40

FREE PARTICLES Computing & diagonalizing reduced density matrices in free fermion/boson problems 41

Free particles & Peschel’s trick (1) I Peschel and M.-C. Chung, J. Phys. A: Math. Gen. 32 8419 (1999) I. Peschel, J. Phys. A: Math. Gen. 36 L205 (2003)  Free fermions or free bosons on a lattice † 𝑑 † 𝑑 † Most general (lattice) quadratic Hamiltonian: 𝐼 0 = 𝐵 𝑗,𝑘 𝑑 𝑗 + B 𝑗,𝑘 𝑑 𝑗 + 𝐼. 𝑑. 𝑗,𝑘 𝑘 𝑗,𝑘 𝑘 𝐼 0 can be diagonalized using a Bogoliubov transformation (see next slides), and the 𝑈 = 0 (as well as T>0 ) correlations can be calculated easily. † 𝑑  𝑑 𝑗 𝑘 → Wick theorem → any correlator † 𝑑 𝐷 𝑗𝑘 = 𝑑 𝑗 𝑘 𝑗,𝑘∈𝐵 → any observable in region 𝐵 → 𝜍 𝐵 The reduced density 𝜍 𝐵 is completely determined by two-point functions in the region of interest  Correlators in region 𝐵 obeys Wick’s theorem → 𝜍 𝐵 is Gaussian → ∃ quadratic « Hamiltonian » 𝐼 such that 𝜍 𝐵 = 1 𝑎 exp −𝐼 † 𝑑 † 𝑑 † with 𝐼 = ℎ 𝑗,𝑘 𝑑 𝑗 + Δ 𝑗,𝑘 𝑑 𝑗 + H. c. 𝑗,𝑘 𝑘 𝑗,𝑘 𝑘 Remark: 𝐼 depends on the subsystem 𝐵 (should be noted 𝐼 𝐵 …), it is not the physical Hamiltonian 𝐼 0 . → The entanglement entropy 𝑇 𝐵 = −Tr 𝜍 𝐵 log 𝜍 𝐵 is the thermodynamic entropy of 𝐼 at (fictitious) temperature 1. 42

Free particles & Peschel’s trick (2)  Correlation matrix † 𝑑 † 𝑑 † 𝑑 𝑗 𝑑 𝑗 𝑘 𝑘 𝑑 † 𝐵 𝐸 † , 𝐵 𝑗𝑘 = 𝑑 𝑗 † 𝑑 † 𝑑 𝑑 † dim. 2𝑂 × 2𝑂, 𝐷 = 𝐷 = = 1 ± 𝐵 𝑢 , 𝐸 𝑗𝑘 = 𝑑 𝑗 𝑑 𝑘 𝐸 † 𝑘 𝑑 † 𝑑 𝑗 𝑑 𝑑 𝑗 𝑑 𝑘 𝑘  We look for new creation/annihilation operators 𝑏 , 𝑏 † (=Bogoliubov transformation): 𝑏 † = 𝑉 𝑑 † 𝑑 with 𝑉 ∓1 1 𝑉 † = ∓1 0 0 1 = Σ Upper sign for boson, lower sign for fermions 0 0 𝑏 Such that the correlations of the new “particles” are diagonal : † 𝑏 𝑙′ † 𝑏 𝑙′ † 𝑏 𝑙 𝑏 𝑙 𝜇 𝑙 𝜀 𝑙,𝑙′ 0 = Λ = 𝑉𝐷𝑉 † † 0 1 ± 𝜇 𝑙 𝜀 𝑙,𝑙′ 𝑏 𝑙 𝑏 𝑙′ 𝑏 𝑙 𝑏 𝑙′ So, the linear algebra problem we have to solve is 𝑉𝐷𝑉 † = Λ 𝑉Σ𝑉 † = Σ † 𝑑 † 𝑑 † ∓ 𝑑 𝑗 𝑑 𝑗 𝑘 𝑘  One method is to diagonalize the matrix 𝐷Σ = . Indeed, combining the † ∓ 𝑑 𝑘 𝑑 𝑑 𝑗 𝑑 𝑘 𝑘 equations above we get 𝑉𝐷Σ𝑉 −1 = ΛΣ . By construction, the correlations of the new particles † 𝑏 𝑙′ = 𝜀 𝑙,𝑙′ 𝜇 𝑙 and 𝑏 𝑙 𝑏 𝑙′ = 𝑏 𝑙 † 𝑏 𝑙′ † are very simple: 𝑏 𝑙 = 0 . We have found some independent “correlation modes”. 43

Free particles & Peschel’s trick (3) † 𝑑 † 𝑑 † 𝑑 𝑗 𝑑 𝑗 𝑘 𝑘 𝐵 𝐸  𝐷 = = 1 ± 𝐵 𝑢 𝐸 † † 𝑑 𝑘 𝑑 𝑑 𝑗 𝑑 𝑘 𝑘  Exercise: When 𝐵 and 𝐸 are real, show that the following (smaller: 𝑂 × 𝑂 ) diagonalization gives the eigenvalues λ : 2 Eigenvalues of 𝐵 ± 1 𝐵 ± 1 𝜇 𝑙 ± 1 2 + 𝐸 2 − 𝐸 = 2 cf. Lieb, Schultz & Mattis, Ann. Phys., NY 16 407 (1961) 44

Free particles & Peschel’s trick (4) † 𝑏 𝑙′ = 𝜀 𝑙,𝑙′ 𝜇 𝑙 & 𝑏 𝑙 𝑏 𝑙′ = 𝑏 𝑙 † 𝑏 𝑙′ † Recall: 𝑏 𝑙 = 0  Question: what is the (quadratic) “Hamiltonian” 𝐼 such that the correlations above can be 1 𝑎 Tr ⋯ 𝑓 −𝐼 ? obtained through some “thermal” average: ⋯ =  Answer: Since the modes are uncorrelated, 𝐼 must be diagonal in terms of the 𝑏 and 𝑏 † and take † 𝑏 𝑙 † 𝑏 𝑙 = 𝜇 𝑙 : the form 𝐼 = 𝜗 𝑙 𝑏 𝑙 . One then has to adjust the « pseudo energies » 𝜗 𝑙 so that 𝑏 𝑙 𝑙 1 bosons ⟹ 𝜗 𝑙 = log 1 ± 𝜇 𝑙 𝑓 𝜗 𝑙 − 1 † 𝑏 𝑙 = 𝜇 𝑙 = 𝑏 𝑙 1 𝜇 𝑙 fermions 𝑓 𝜗 𝑙 + 1  Which finally gives the RDM: 𝜍 𝐵 ~ exp − log 1 ± 𝜇 𝑙 † 𝑏 𝑙 𝑏 𝑙 𝜇 𝑙 𝑙  And the (entanglement) entropy is given by the sum of the contributions of each mode: 𝜗 − log 1 − 𝑓 −𝜗 + bosons 𝑓 𝜗 − 1 𝜗 𝑇 = 𝑇 𝜗 𝑙 with 𝑇(𝜗) = + log 1 + 𝑓 −𝜗 + 𝑓 𝜗 + 1 fermions 𝑙 = −𝜇 log 𝜇 − 1 − 𝜇 log 1 − 𝜇  Remark: For fermions, the eigenvalues 𝜇 = 0 or 1 do not contribute to the entropy ( 𝜁 = ±∞ ). For bosons, 𝜇 = 0 do not contribute. 45

Entanglement entropy after a local “quench” Example of an application of Peschel’s trick to compute the entanglement in a time-dependent situation Initial (product) state 𝜔 0 = 𝑏 ⨂ 𝑐 L=200 sites (100+100) A B † 𝑑 𝑗+1 + ℎ. 𝑑. ) H A and H B : free fermions ( = 𝑑 𝑗 𝑗 𝑏 = g.s. of H A 𝑐 = g.s. of H B Unitary evolution to 𝑢 > 0 : 𝜔(𝑢) = exp −𝑗𝑢𝐼 𝜔 0 𝐼 = 𝐼 𝐵 + 𝐼 𝐶 + 𝐼 𝑗𝑜𝑢 𝐼 𝑗𝑜𝑢 : hopping between the A & B B 𝜔 𝑢 𝜔 𝑢 remains “Gaussian” 𝜍 𝐵 𝑢 = 𝑈𝑠 𝑇 𝐵 (𝑢) = −Tr 𝐶 𝜍 𝐵 𝑢 log 𝜍 𝐵 𝑢 can be computed with Peschel’s trick, using the (time- dependent) 2-point correlations. Remark: we observe a logarithmic growth of 𝑇 𝐵 (𝑢) (holds as long as t ≲ 𝑀/2 ) 46

Free fermions on a chain – correlation matrix  Ground-state on a periodic chain 𝐼 = − 1 † 𝑑 𝑜+1 + 𝑑 𝑜+1 † † 𝑑 𝑙 2 𝑑 𝑜 𝑑 𝑜 = − cos 𝑙 𝑑 𝑙 𝑜 𝑙 † vacuum 𝜔 = 𝑑 𝑙 𝜗 𝑙 = − cos 𝑙 −𝑙 𝐺 ≤𝑙≤𝑙 𝐺 𝑀−1 † = 1 † 𝑓 −𝑗𝑙𝑛 𝑑 𝑛 𝑙 𝑑 𝑙 𝑀 −𝜌 𝜌 𝑜=0  Two- point correlations → matrix 𝐵 𝑜,𝑛 = 1 † 𝑑 𝑛 𝜔 = † 0 0 𝑑 𝑛 𝑑 𝑙 † 0 𝑓 𝑗𝑙(𝑜−𝑛) 𝜔 𝑑 𝑜 0 𝑑 𝑙 𝑑 𝑜 𝑀 −𝑙 𝐺 ≤𝑙≤𝑙 𝐺 −𝑙 𝐺 ≤𝑙≤𝑙 𝐺 𝑙 𝐺 2𝜌 𝑓 𝑗𝑙(𝑜−𝑛) = sin 𝑙 𝐺 (𝑜 − 𝑛) 𝑒𝑙 𝑀→∞ 𝜔 𝑑 𝑜† 𝑑 𝑛 𝜔 = 𝐵 𝑜,𝑛 = lim 𝜌(𝑜 − 𝑛) −𝑙 𝐺 47

Free fermions on a chain – determinant Jin & Korepin, J. Stat Phys. 116 , 79 (2004)  Recall: 2𝜌 𝑓 𝑗𝑙(𝑜−𝑛) = sin 𝑙 𝐺 (𝑜−𝑛) 𝑙 𝐺 𝑒𝑙 † 𝑑 𝑛 𝜔 = 𝑀→∞ 𝜔 𝑑 𝑜 Correlation matrix: 𝐵 𝑜,𝑛 = lim −𝑙 𝐺 𝜌(𝑜−𝑛) Entanglement entropy: 𝑇 = 𝑇 𝜇 𝑙 = sum over the eigenvalues of 𝐵 𝑙 and 𝑇 𝜇 = −𝜇 log 𝜇 − 1 − 𝜇 log 1 − 𝜇  Equivalent formulation in terms of a contour integral , with poles at each eigenvalue: det 𝜇𝕁 − 𝐵 = 𝜇 − 𝜇 𝑙 𝑙 Im( λ ) log det 𝜇𝕁 − 𝐵 = log 𝜇 − 𝜇 𝑙 0 1 𝑙 𝑒 log det 𝜇𝕁 − 𝐵 1 = Re( λ ) 𝑒𝜇 𝜇 − 𝜇 𝑙 𝑙 = 𝑒𝜇 2𝑗𝜌 𝑇 𝜇 𝑒 log det 𝜇𝕁 − 𝐵 𝑇 = 𝑇 𝜇 𝑙 𝑒𝜇 𝑙 48

Fisher-Hartwig conjecture Asymptotic behavior of the det. of Toeplitz matrices with singular symbol (simplified version …) 2𝜌 𝑒𝜄 2𝜌 𝜚 𝜄 𝑓 −𝑗𝑙𝜄  𝐵 𝑛,𝑜 = 𝐵 𝑛 − 𝑜 parametrized as Fourier coefficients: A 𝑙 = 0 𝜚 is the called the symbol of the Toeplitz matrix 𝐵 . Remark: in our (free fermion) case, the symbol 𝜚 of the correlation matrix 𝐵 has two discontinuities (details in a few slides…)  Parametrize de symbol’s discontinuities with some numbers (complex) 𝛾 𝑠 and (real) 𝜄 𝑠 : 𝑆 exp −𝑗𝛾 𝑠 ∗ arg 𝑓 𝑗 𝜌− 𝜄+𝜄 𝑠 𝜚 𝜄 = 𝜔 𝜄 with arg ∈] − 𝜌, 𝜌] 𝑠=1 𝜔 𝜄 : smooth, and no winding → pointwise singularity at each 𝜄 𝑠 : 𝜚 𝜄 = 𝜄 𝑠 + 𝜗 𝜚 𝜄 = 𝜄 𝑠 − 𝜗 = exp (−𝑗𝛾 𝑠 ∗ (𝜌 + 𝜗)) (−𝑗𝛾 𝑠 ∗ (−𝜌 + 𝜗)) → exp −2𝑗𝛾 𝑠 exp −2𝑗𝛾 𝑠 = log 𝜚 𝜄 𝑠 + 𝜗 𝜚 𝜄 𝑠 − 𝜗  The Fisher- Hartwig’s conjecture describes the symptotic behavior of the det 𝐵 : 𝑆 2𝜌 𝑒𝜄 + −log 𝑀 𝛾 𝑠 2 log det 𝐵 = 𝑀 2𝜌 log 𝜚 𝜄 + ⋯ 0 𝑠=1 49

Fisher-Hartwig conjecture & 1d Fermi sea (1) 𝑙 𝐺 𝑒𝑙 sin 𝑙 𝐺 (𝑜−𝑛) 2𝜌 𝑒𝑙 2𝜌 𝑓 𝑗𝑙(𝑜−𝑛) 2𝜌 𝜚 𝜄 𝑓 𝑗𝑙(𝑜−𝑛)  Recall: 𝐵 𝑜,𝑛 = = = −𝑙 𝐺 0 𝜌(𝑜−𝑛) → Symbol for 𝜇𝕁 − 𝐵 : 𝜚 𝜄 = 𝜇 − 1 if 𝜄 ∈ −𝑙 𝐺 , 𝑙 𝐺 𝜇 otherwise → 2 discontinuities in 𝜄 = −𝑙 𝐺 and 𝜄 = +𝑙 𝐺 : 𝛾 1 = 1 2𝑗𝜌 log 𝜇 − 1 𝜇 𝛾 2 = 1 𝜇 2𝑗𝜌 log = −𝛾 1 𝜇 − 1  Use Fisher-Hartwig: = 1 2𝜌 2𝑙 𝐺 log 𝜇 − 1 + 2(𝜌 − 𝑙 𝐺 ) 𝑀 − 2 𝛾 1 2 log 𝑀 + ⋯ log det 𝜇 − 𝐵 𝑒log det 𝜇 − 𝐵 = 1 𝜇 − 1 + 2(𝜌 − 𝑙 𝐺 ) 2𝑙 𝐺 𝑒𝛾 1 𝑀 − 4𝛾 1 𝑒𝜇 log 𝑀 + ⋯ 𝑒𝜇 2𝜌 𝜇 𝑒𝛾 1 1 𝑒𝜇 = 2𝑗𝜌 𝜇 𝜇 − 1 1 𝑒log det 𝜇 − 𝐵 1 2𝑙 𝐺 2(𝜌 − 𝑙 𝐺 ) 1 𝜇 − 1 1 = = 𝜇 − 1 + 𝑀 − 4 2𝑗𝜌 2 log 𝜇 𝜇 − 1 log 𝑀 + ⋯ 𝜇 − 𝜇 𝑙 𝑒𝜇 2𝜌 𝜇 𝜇 𝑙 Remark: 2 poles in 𝜇 = 0 and 𝜇 = 1 with extensive residues (~ 𝑀 ) Physical meaning: an extensive number of single- particle “states” of the segment are either completely filled ( 𝜇 𝑙 = † 𝑏 𝑙 =1) or completely empty ( 𝜇 𝑙 = 𝑏 𝑙 † 𝑏 𝑙 =0 ) and do not contribute to the entanglement. 𝑏 𝑙 50

Fisher-Hartwig conjecture & 1d Fermi sea (2) 𝑒log det 𝜇 − 𝐵 = 1 𝜇 − 1 + 2(𝜌 − 𝑙 𝐺 ) 2𝑙 𝐺 2𝑗𝜌 2 log 𝜇 − 1 1 1 𝑀 − 4 𝜇 𝜇 − 1 log 𝑀 + ⋯ 𝑒𝜇 2𝜌 𝜇 𝜇 𝑇 = 𝑒𝜇 (1 − 𝜇) 𝑒log det 𝜇 − 𝐵 2𝑗𝜌 −𝜇 log 𝜇 − 1 − 𝜇 log 𝑒𝜇 Im( λ )  The only contribution to this contour integral is the 0 1 discontinuity of log 𝜇 − 1 on the real axis: log 𝜇 − 1 + 𝑗0 − − log 𝜇 − 1 + 𝑗0 + = −2𝑗𝜌 Re( λ ) 1 𝑇 = 0 × 𝑀 − 4 −2𝑗𝜌 2𝑗𝜌 3 log 𝑀 𝑒𝜇 −𝜇 log 𝜇 − 1 − 𝜇 log 1 − 𝜇 𝜇 𝜇 − 1 0 = − 𝜌 2 𝑇 = + 1 3 3 log 𝑀 + ⋯ NB: the poles in 0 and 1 do not contribute since the residue vanishes → no 𝒫(𝑀) (“volume”) term Origin of the log (𝑀) term: discontinuity of the symbol ← discontinuity of the fermion occupation number in Fourier space ← algebraic decay of the correlations 51

Entanglement in a free fermion chain & S~log(L)/3 Total length: L subsystem A x Same data (red dots), compared with the log of the “chord” distance ( green curve) : 𝑒 𝑦 = 2𝑀 sin 𝜌𝑦 𝑀 𝑒 𝑦 52

Summary of lecture #2  Mutual information 𝐽 𝐵: 𝐶 = 𝑇 𝜍 A + 𝑇 𝜍 𝐶 − 𝑇 𝜍 AB . Encodes all the correlations (quantum or classical) between the regions 𝐵 and 𝐶 .  Using 𝐽 𝐵: 𝐶 one can define an “all - correlations” length 𝝄 . If it is finite, the system obeys an area law for the entanglement entropy (argument by Wolf et al , PRL 2008)  Random pure states have a large entanglement entropy (volume law, and close to the maximal possible value log dim ℋ ) 𝐵  Relations between thermodynamics, the eigenstate thermalization hypothesis (ETH) and the A A ( 𝛽 ) entanglement in highly excited pure states: 𝑇 thermo (E) = S VonNeumann  Free particle systems (fermions or bosons): Reduced density matrices are Gaussian and fully determined by the 2- point correlation functions (Wick’s theorem). The Von Neumann entropy is a simple function of the eigenvalues of the correlation matrix.  Application to the calculation of the entropy of a segment in a free Fermion chains (Jin & Korepin 2004): A Map S VonNeumann • to a contour integral of a determinant. Toepliz matrix with a discontinous “symbol” (↔ discontinuity of the fermion occupation number at the Fermi points) • Fisher-Hartwig → asymptotics of the determinant has a log(L) term. 1 𝑇(𝑀) = 3 log (𝑀) . Universal coefficient (only depends on the number of Fermi points, not on • the details of the dispersion relation nor the density). 53

Entanglement in free fermion chain: gap versus gapless Total length: 𝑀 ≫ 𝑦 Dimerized free-fermion chain (2-site unit cell) : † 𝑑 2𝑜+1 + 𝑑 2𝑜+1 † † † H = −𝒖 𝑑 2𝑜 𝑑 2𝑜 − 𝒖′ 𝑑 2𝑜+1 𝑑 2𝑜+2 + 𝑑 2𝑜+2 𝑑 2𝑜+1 subsystem A 𝑜 𝑜 𝒖 𝒖′ 𝑦 2𝑜 2𝑜 + 1 Band structure 𝑇 𝐵 is qualitatively different in the gapped 𝐹 𝑙 ∼ 1 and gapless cases 3 log 𝑦 − 𝜌 + 𝜌 2 2 Δ 𝑻 𝑩 (𝒚) ∼ constant 𝑙 𝑢 − 𝑢 ′ 2 + 4𝑢𝑢 ′ cos 𝑙 2 𝐹 𝑙 = ± gap Δ = 2 𝑢 − 𝑢′ when the chemical potential is at 𝐹 = 0 → band insulator at half -filling if 𝑢 ≠ 𝑢′ 𝒚 54

Log(L) term in presence of a Fermi surface in 𝑒 ≥ 2 “ Violation of the Entropic Area Law for Fermions ” M. M. Wolf, Phys. Rev. Lett. 96, 010404 (2006) “ Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture ” D. Gioev &I. Klich, Phys. Rev. Lett. 96, 100503 (2006) 55

Boundary law violation in presence of a Fermi surface Simple geometric argument : “Entanglement Entropy and the Fermi Surface” B. Swingle, Phys. Rev. Lett. 105, 050502 (2010) Real space 1 st Brillouin zone 𝑜 𝑙 subsystem A  Free-fermion tight-binding model 𝑒𝑚 𝑒𝑙 † 𝑑 † 𝑑 𝑙 𝐼 = 𝑢 𝑗𝑘 𝑑 𝑗 𝑘 + h. c. = 𝜗 𝑙 𝑑 𝑙 Fermi sea 𝑙∈1 st BZ 𝑗,𝑘 𝑜 𝑌  Contribution of the modes 𝑒𝑙 and boundary element 𝑒𝑚 to the entanglement entropy 𝑇 𝐵 ? → Idea: model this contribution by decoupled chains 𝑜 𝑙 running parallel to the 𝑜 𝑙 direction 𝑒𝑚 (insures the same propagation direction for the low-energy modes): Lattice spacing 𝑏 56

Boundary law violation in presence of a Fermi surface 𝑜 𝑙 Real space 1 st Brillouin zone 𝑜 𝑙 subsystem A 𝑒𝑚 𝑒𝑙 Fermi sea Lattice 𝑜 𝑌 spacing 𝑏 𝑒𝑚 • Number of chains crossing the boundary: 𝑒𝑂 = 𝑒𝑚 𝑏 𝑜 𝑙 . 𝑜 𝑌 • Entropy contribution of each chain: decoup. chains = 1 𝑒𝑇 𝐵 6 log 𝑀 × 𝑒𝑂 • Correct by the length of the element along the Fermi surface (relative to that of the decoup. Chain model) 𝑒𝑙 = 1 12 log 𝑀 × 𝑒𝑚 𝑏 𝑜 𝑙 . 𝑜 𝑌 × 𝑏𝑒𝑙 decoup. chains × 𝑒𝑇 𝐵 = 𝑒𝑇 𝐵 2𝜌 2 × 2𝜌 𝑇 𝐵 = 1 12 log 𝑀 𝑒𝑚 𝑒𝑙 2𝜌 𝑜 𝑙 . 𝑜 𝑌 𝑏 Integrating on the real space boundary and Fermi surface: Turns out to be exact ! (and related to Windom’s conjecture) ~ 𝒫 𝑀 log 𝑀 57

Boundary law violation in presence of a Fermi surface “ Entanglement scaling in critical two-dimensional fermionic and bosonic systems ”, T . Barthel, M.-C. Chung, & U. Schollwöck Phys. Rev. A 74, 022329 (2006) The prefactor c ( μ ) in the entanglement entropy scaling law as a function of the chemical potential μ for the ground- state of the two-dimensional fermionic tight-binding model in comparison to the result of Gioev and Klich. Insets show the hopping parameters and the Fermi surfaces for μ =−3,−2,−1,0 . [From Barthel et al. 2006 ] 58

Critical systems in 1d & CFT … the celebrated S~ 𝑑 3 log 𝑀 formula 59

Entanglement & CFT Nucl. Phys B424 (1994) S~ 𝑑 3 log 𝑀 J. Stat. Mech 2004 +Review: Calabrese & Cardy, J. Phys. A 42, 504005 (2009) 60

Numerics & entanglement in critical spin chains “ Entanglement in Quantum Critical Phenomena ” G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, (2003) Critical XX chain 𝑇~ 1 3 log 𝑀 , 𝑑 = 1 c=1 (XX) S A c=1/2 (ICTF) 1 1 Critical Ising chain 𝑇~ 6 log 𝑀 , 𝑑 = 2 Gapped (ICTF) Non-critical Ising chain 𝑇~𝑑𝑡𝑢. L 61

Quantum system in d=1 & partition functions in d=2 Functional/path integral ↔ imaginary time evolution 𝑎 exp −𝛾𝐼 , with 𝑎 = Tr 𝑓 −𝛾𝐼 𝑏  Thermal density matrix 𝜍 = 1 Imaginary time 𝛾 𝑏 𝜍 𝑐 = 1 𝑎 × cylinder “partition function” with boundary conditions 𝑏 and 𝑐 𝑎 : torus partition function 𝑑 𝑐 L 𝛾 → ∞ ~ 𝑑 𝜔  Ground-state wave function given by infinitely long cylinder partition functions : 𝛾→∞ exp −𝛾𝐼 = 𝑓 −𝛾𝐹 0 𝜔 𝜔 lim 𝜔 ~ lim (−𝛾𝐼) 0 𝛾→∞ exp Any state with some overlap 0 with the ground-state 62

Critical system in d=1 & CFT  Critical 1d system: • gapless in the thermodynamic limit • (some) correlation functions decay algebraically with distance  Examples 𝑨 𝑨 𝑇 𝑜+1 𝑦 Critical Ising chain in transverse field 𝐼 = − 𝜏 𝑜 − h 𝑇 𝑜 with ℎ = 1 • 𝑜 𝑜 1 𝑧 𝑇 𝑜+1 𝑧 𝑦 𝑨 𝑦 𝑇 𝑜+1 𝑨 𝑇 𝑜+1 2 XXZ spin chain 𝐼 = 𝑇 𝑜 + 𝑇 𝑜 + Δ 𝑇 𝑜 with Δ ∈] − 1,1] • Spin- 𝑜 𝑜 † 𝑑 ↑,𝑜+1 + 𝑑 ↓,𝑜 † 𝑑 ↓,𝑜+1 + H. c. 1d Hubbard model 𝐼 = −𝑢 𝑑 ↑,𝑜 • 𝑜 Luttinger liquids † 𝑑 ↑,𝑜 + 𝑑 ↓,𝑜 † 𝑑 ↓,𝑜 + H. c. +𝑉 𝑑 ↑,𝑜 𝑜 • Edge of a quantum 2d Hall system • …  Continuum limit and CFT 𝑏 The universal / long-distance properties of the system are obtained 1 by replacing the microscopic density matrix 𝜍 = 𝑎 exp −𝛾𝐼 by the corresponding cylinder CFT partition function 𝛾 𝑏 exp −𝛾𝐼 𝑐 ≈ (with the appropriate boundary conditions, corresponding to “coarse grained” versions of the states 𝑏 and 𝑐 ) 𝑐 63 L

Reduced density matrix & 2d partition functions 𝜍 𝐵 = Tr 𝐶 𝜍 𝑏 𝜍 𝐵 𝑏′ = 𝑏𝑐 𝜍 𝑏 ′ 𝑐 𝑐 𝑐 𝑏 𝛾 A B A B 𝑏 periodic 𝑏 𝜍 𝐵 𝑏′ = 1 𝑏𝑐 𝜍 𝑏 ′ 𝑐 = 1 𝑐 𝑎 𝑎 𝑏′ 𝑏′ 𝑐 L Tracing out the region B 𝜍 𝐵 = 1 A 𝑎 64

Rényi entropy & partition functions  Reduced density matrix to the power n 𝜍 𝐵 = 1 A 𝑎 : cylinder partition function 𝑎 n 1 2 … 1 Tr 𝜍 𝐵𝑜 = nB nA 𝑎 𝑜 2A 2B 1A 1B  Some remarks: • The slits of the n cylinders are “glued” cyclically to obtain the trace → Riemann surface with n sheets. • Consider a path encircling one end of the segment A → moves to the next cylinder. n turns are needed to get back to the origin. 65

Entanglement & CFT (2) Holzhey, Larsen & Wilcek, Nucl. Phys B 424 (1994) 𝛾 → ∞ 𝑥 = − sin 𝜌 𝜂 − 𝑦 𝑀 Im(𝑥) 𝜂 ∈ half−infinite Re(𝑥) sin 𝜌 𝜂 𝑀 cylinder −𝑦 𝜗 1 𝑏𝑐 𝜔 ~ 𝑦 𝜗 1 𝜗 2 𝑦 L half-plane 𝜗 2 𝜗 1 A B 𝑦 𝑀 − 𝑦 𝜗 1 & 𝜗 2 : UV cut-off (~lattice spacing) Assume 𝑀 ≫ 𝑦 for simplicity Remarks: • Without the cut-offs, there would be infinitely many degrees of freedom close to the boundary between A and B, and their contribution to the entanglement entropy would diverge. • The actual conformal mapping which maps the cylinder without the excluded regions to the ½ annulus (right picture) is complicated, but it’s precise form is not needed in what follows. 66

Entanglement & CFT (3) – mapping to a conical singularity Holzhey, Larsen & Wilcek, Nucl. Phys B424 (1994) 𝑐 𝑏 −𝑦 𝜗 1 𝑦 𝜗 1 𝑏′ 𝜗 2 𝑦 𝑏 −𝑦 𝜗 1 𝑦 𝜗 1 Wave function 𝜗 2 𝑦 𝑏𝑐 𝜔 ~ Density matrix 𝑏 𝜍 𝐵 𝑏′ ~ 𝑎 cone angle 2𝑜 𝜌 Conical singularity 𝑜=1−𝛽 = Tr 𝜍 𝐵 𝜗 𝑦 → 0) , with angle 𝑎 disk 𝑜 ( 𝑜 ∼ deficit 2𝛽 𝜌 . = 𝑑 1 𝑜 − 𝑜 log 𝑦 𝜍 𝐵 𝑜 log Tr 𝜍 𝐵 𝜗 6 (n levels) (this can be calculated by mapping the disk to the cone using 𝑨 → w z = z n , and compute the associated Shwarzian derivative – see next slide) 67

Entanglement & CFT (4) – Free energy of a cone 𝑥 disk cone 𝑨 2 𝑆/𝑠 = 𝑦 𝑆 = 𝑦/𝜗 𝜗 𝑠 = 𝜗/𝑦 Mapping from the disk to the cone: 𝑨 → 𝑥 𝑨 = 𝑨 𝑜 𝑥 ′ = 𝑜𝑨 𝑜−1 𝑥 ′′ = 𝑜 𝑜 − 1 𝑨 𝑜−2 Stress energy tensor (holomorphic part) in the disk geometry: 𝑈 𝑒 (𝑨) . Stress In the cone geometry: 𝑈 𝑑 (𝑥) They are related to each other through a standard CFT transformation law, which involves the Scharzian derivative: 2 𝑥 ′′′ 𝑥 ′′ 𝑥′ 2 𝑈 𝑒 𝑨 − 𝑑 1 𝑥′ − 3 𝑈 𝑑 𝑥 = 12 2 𝑥 ′ 𝑈 𝑒 (𝑨) 𝑑 1 Here we find 𝑈 𝑑 𝑥 = 𝑥′ 2 + 24𝑥 2 1 − 𝑜 2 . Integrating the stress energy tensor (times 𝑥 ) along the dashed line gives the variation of log𝑎 𝑑 (𝑜) with respect to the outer radius 𝑆 (keeping the inner radius 𝑠 fixed) : 𝜖 log𝑎 𝑑 (𝑜) = 𝑗 2𝜌 𝑥𝑒𝑥 𝑈 𝑑 𝑥 + H. c. 𝜖 log 𝑆 And from the relation above between 𝑈 𝑒 (𝑨) and 𝑈 𝑑 𝑥 one can show that : 𝜖 log 𝑎 𝑑 (𝑜)/𝑎 𝑒 = 𝑗 24𝑥 2 1 − 1 𝑑 + H. c. = c 1 2𝜌 𝑥𝑒𝑥 𝑜 − 𝑜 𝑜 2 𝜖 log 𝑆 12 c 1 𝑆 So we have log 𝑎 𝑑 (𝑜)/𝑎 𝑒 = 𝑜 − 𝑜 log𝑆 . By conformal invariance, this should in fact be a function or 𝑠 and 12 𝑜 − 𝑜 log 𝑦 c 1 𝑆 c 1 therefore: log 𝑎 𝑑 (𝑜)/𝑎 𝑒 = 𝑜 − 𝑜 log 𝑠 = 𝜗 . This is the result announced on the previous slide. 12 6 68

Entanglement spectrum in a 1d critical system “Entanglement spectrum in one-dimensional systems” P. Calabrese & A. Lefevre, Phys. Rev. A 78, 032329 (2008) The previous calculation gave Tr 𝜍 𝐵𝑜 = 𝑎(𝑜)/𝑎(1) 𝑜 log Tr 𝜍 𝐵𝑜 = log 𝑆 𝑜 = log 𝑎(𝑜) − 𝑜 log 𝑎 1 ~ 𝑑 1 𝑜 − 𝑜 log 𝑦 𝜗 6  From this, what can be said about the eigenvalues of 𝜍 𝐵 ? Density of “states”: 𝑄 𝜇 = 𝜀(𝜇 − 𝜇 𝑗 ) , with 𝜇 𝑗 : eigenvalues of 𝜍 𝐵 • 𝑗 𝑜 The moments 𝑆 𝑜 = 𝜇 𝑗 have a relatively simple dependence on 𝑜 : 𝑗 𝑆 𝑜 = e −𝑐 𝑜− 1 𝑜 with 𝑐 = 𝑑 6 log 𝑦 𝜗 • After a some mathematical manipulations… one obtains the (CFT) density of eigenvalues : − 𝑑 with 𝜇 max = 𝑓 −𝑐 = 𝑦 6 (largest eigenvalue) 𝜗 69

Entanglement spectrum in a 1d critical system P. Calabrese & A. Lefevre, Phys. Rev. A 78, 032329 (2008) 𝑡 𝑁 = 𝜇 𝑗 𝑗=1…𝑁 By construction: 𝑡 𝑁 → ∞ = 1 If one keeps only the M first eigenvalues, the discarded weight is 1 − 𝑡 𝑁 . How do the properties of the of the approximate (truncated) wave-function vary with 𝑁 ? → “finite - entanglement scaling” L. Tagliacozzo et al. PRB 78, 024410 (2008) + many others… 70

CFT & entropy of two disjoint intervals  S. Furukawa, V. Pasquier, and J. Shiraishi. “ Mutual Information and boson radius in a c=1 critical system in one dimension” . Phys. Rev. Lett., 102, 170602, 2009 The mutual information of two disjoint intervals contains more information about the long-distance properties than just the central charge (as for a single interval). In this example of critical spin chains with c=1 (Tomonaga-Luttinger liquid phase in the XXZ spin chain) 𝐽(𝐵: 𝐶) is a function of the so- callled “ compactification radius” (related to the exponent of several spin-spin correlation functions). 71

Matrix-product states to describe weakly-entangled states in 1d, canonical (G. Vidal’s ) form 72

Matrix-product states Review on DMRG (and MPS) Ann. of Phys. 326 , 96-192 (2011) “A practical introduction to tensor networks: Matrix product states and projected entangled pair states” Román Orús, Ann. of Phys. 349, 117 (2014) Original paper: “Density matrix formulation for quantum renormalization groups”, S. R. White, Phys. Rev. Lett. 69, 2863 (1992) >2500 citations in WoS 73

MPS & canonical Vidal’s form (1) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003) Start from the wave function on an open chain of length 𝑀 (here a spin-1/2 example for simplicity): • 𝜔 = 𝑁 𝜏 1 ,…,𝜏 𝑀 𝜏 1 , … , 𝜏 𝑀 𝑀 1 𝜏 1 ,…,𝜏 𝑀 ,=↑,↓ 𝑁 Split the chain in two parts [1 … 𝑜] − [𝑜 + 1 … 𝑀] , • 2 𝑀 coefficients and define the associated Shmidt basis and singular values: 𝑜 Φ 𝑚 [1⋯𝑜] ⊗ Φ 𝑚 [𝑜+1⋯𝑀] 𝜔 = 𝜇 𝑚 𝑚 • Graphically: 𝑜 + 1 … 𝑀 1 𝑜 𝑜 [1⋯𝑜] [𝑜+1⋯𝑀] 𝜔 = Φ 𝑚 Φ 𝑚 𝜇 𝑚 • In practice this decomposition can be obtained by « reshaping » M as rectangular matrix of size 2 𝑜 ∗ 2 𝑀−𝑜 : 𝑁 𝜏 1 ,…,𝜏 𝑀 = 𝑁 𝜏 1 ,…,𝜏 𝑜 ,(𝜏 𝑜+1 ,…,𝜏 𝑀 ) and performing its singular value decomposition (SVD). We assume that 𝜔 can be approximated using some low rank truncation • with at most 𝜓 Schmidt values. 74

MPS & canonical Vidal’s form (2) • Compare the Schmidt decompositions on two successives bonds: 1 𝑜 − 1 𝑜 𝑜 + 1 … 𝑀 𝜔 = [1⋯𝑜] [𝑜+1⋯𝑀] Φ 𝑛 Φ 𝑛 𝑜 𝜇 𝑛 1 𝑜 − 1 𝑜 𝑜 + 1 … 𝑀 [1⋯𝑜−1] [𝑜⋯𝑀] Φ 𝑚 Φ 𝑚 𝑜−1 𝜇 𝑚 𝑜 Φ 𝑚 [𝑜⋯𝑀] in the orthogonal basis ↑ 𝑜 , ↓ 𝑜 ⨂ 𝜇 𝑛 [𝑜+1⋯𝑀] Write the Schmidt state Φ 𝑚 • This defines on (each site 𝑜 ) two matrices Γ 𝑜 ↑ and Γ 𝑜 ↓ of dimension 𝜓 ∗ 𝜓 : • 𝑜 Φ 𝑛 𝑜 𝜏 𝜏 𝑜 𝜇 𝑛 [𝑜⋯𝑀] = [𝑜+1⋯𝑀] Φ 𝑚 Γ 𝑚,𝑛 𝜏=↑,↓ 𝑛=1 …𝜓 75

MPS & canonical Vidal’s form (2) • Graphically : 𝜏 𝑜 + 1 … 𝑀 𝑛 𝑚 𝑚 𝑛 𝑜 𝜏 𝑜 [𝑜+1⋯𝑀] [𝑜⋯𝑀] Γ 𝑚,𝑛 𝜇 𝑛 Φ 𝑛 Φ 𝑚 = =Schmidt vector # 𝑚 for the [1 … 𝑜 − 1][𝑜 … 𝑀] partition =Schmidt vector # 𝑛 for the [1 … 𝑜][𝑜 + 1 … 𝑀] partition Repeat the procedure along the chain to construct all the matrices Γ 𝑜 ↑ and Γ 𝑜 ↓ from • the Schmidt basis. 𝜏 1 𝜏 2 𝜏 3 𝜏 𝑀 𝜇 [𝑀−1] 𝜇 [2] 𝜇 [3] 𝜇 [1] … 𝑨 𝑨 • 𝑙 𝑙 𝑚 𝑚 Γ 3 𝑛 𝑛 Finally: 𝜔 = Γ 𝑀 Γ 1 Γ 2 𝜏 1 ,𝜏 2 ,…,𝜏 𝑀 =↑,↓ 𝑙,𝑚,𝑛,…,𝑨=1…𝜓 1 Γ k,l 2 Γ l,𝑛 3 ⋯ 𝜇 𝑨 𝑀−1 Γ 𝑨 1 𝜏 1 𝜇 𝑙 2 𝜏 2 𝜇 𝑚 3 𝜏 2 𝜇 𝑛 𝑀−1 𝜏 𝑀 𝜏 1 𝜏 2 ⋯ 𝜏 𝑀 = Γ 𝑙 𝜏 1 ,𝜏 2 ,…,𝜏 𝑀 =↑,↓ 𝑙,𝑚,𝑛,…,𝑨=1…𝜓 • Encodes all the left-right Schmidt decomposition Storage : ∼ 𝑀 ∗ 𝜓 + 2𝜓 2 ≪ 2 L if 𝜓 can be kept 𝒫 1 [gapped system] or, at worse, 𝒫 𝑀 𝛽 • [critical] 76

MPS & canonical Vidal’s form (3) 𝜏 1 𝜏 2 𝜏 3 𝜏 𝑀 𝜇 [𝑀−1] 𝜇 [2] 𝜇 [3] 𝜇 [1] … 𝑨 𝑨 𝑙 𝑙 𝑚 𝑚 Γ 3 𝑛 𝑛 Γ 𝑀 Γ 1 Γ 2 =Canonical MPS form  Allows to reconstruct the Schmidt decomposition for any left/right partition 𝜏 𝑜 𝜏 𝑀 n … L 𝑙 Γ 𝑜 𝑙 … [𝑜…𝑀] Γ 𝑀 Φ 𝑙 =  Orthogonality of the Schmidt vectors 𝑙 Γ 𝑜 … Γ 𝑀 [𝑜…𝑀] Φ 𝑙′ [𝑜…𝑀] = Φ 𝑙 = 𝜀 𝑙𝑙′ 𝑙′ Γ 𝑜 … Γ 𝑀 𝜏 𝑜 ,𝜏 𝑜+1 ,…,𝜏 𝑀 =↑,↓ nb: automatically insures 𝜔 𝜔 = 1 77

MPS & canonical Vidal’s form (4) 𝜏 𝑗 𝜏 𝑘 𝑗 ⋅ 𝑇 𝑗+1 𝜏 𝑗 𝜏 𝑗 ⋅ 𝑇 𝑇 = 𝜏′ 𝑗 𝜏′ 𝑘 𝑇 𝑘  Local observables. Example of a 2-spin operator 𝑘 𝜏′ 𝑘 𝜏′ 𝑗 … Γ 𝑀 Γ 1 Γ 2 Γ 3 𝜏′ 2 𝜏′ 3 2 ⋅ 𝑇 3 𝑇 𝑗 ⋅ 𝑇 𝑗+1 𝜔 = 𝜔 𝑇 𝜏 1 𝜏 2 𝜏 3 𝜏 𝑀 … Γ 𝑀 Γ 1 Γ 2 Γ 3 Γ 2 Γ 3 𝜏′ 2 𝜏′ 3 2 ⋅ 𝑇 3 𝑇 = Only local operations required 𝜏 2 𝜏 3 Γ 2 Γ 3 78

MPS Alogorithms  Many algorithms exist to compute et manipulate MPS on long chains : • Variationnal algorithms: successively optimize the tensors to lower the energy and obtain the ground-state of a given Hamiltonian (DMRG) • Alternative approach: perform an imaginary-time evolution to get the ground-state (TEBD) • Perform the (unitary) time evolution starting from an arbitrary state (t-DMRG & TEBD) • Infinite-chain methods (iTEBD) • Extension to finite-temperature (i.e. MPS to describe mixed states)  Example: two-site unitary operation • Consider unitary “gate” 𝑉 𝑜,𝑜+1 acting on sites 𝑜 and 𝑜 + 1 : 𝜏 𝑜 𝜏 𝑜+1 [1⋯𝑜−1] [𝑜+2⋯𝑀] Φ 𝑚 Φ 𝑚 𝑉 𝑜,𝑜+1 𝑜−1 𝑜 𝑜+1 Γ 𝑜−1 Γ 𝑜 Γ 𝑜+1 Γ 𝑜+2 𝑉 𝑜,𝑜+1 𝜔 = 𝜇 𝑚 𝜇 𝑛 𝜇 𝑛 𝑜−1 and Schmidt basis Φ 𝑚 [1⋯𝑜−1] are not modified by 𝑉 𝑜,𝑜+1 The schmidt values 𝜇 𝑚 • 𝑜+1 and the basis Φ 𝑚 [𝑜+2⋯𝑀] Same for 𝜇 𝑛 •  Only Γ 𝑜 , Γ 𝑜+1 𝜇 𝑛 𝑜 need to be updated → fast local updates 𝒫 𝜓 3  Using small time steps, the observation above can be used to compute the real time evolution (here for nearest-neighbor spin-spin interactions): 79

Summary of lecture #3 𝐹 𝑙  Dimerized (free fermion) chain: “metal versus band insulator” − 𝜌 + 𝜌 2 2 𝒖 𝒖′ Δ 𝑙  Fermi surface contribution to the entanglement in 𝑒 > 1 [Gioev & Klich , 2006]  Entanglement in critical 1d systems & conformal field theory [Holzhey-Larsen Wilckek 1994] 𝑇~ 𝑑 3 log 𝑀  Entanglement spectrum from CFT – decay of the Schmidt values [Calabrese Lefevre 2008]  Two intervals: more information than the central charge [Furukawa-Pasquier-Shiraishi 2009]  Matrix product states: a powerful way to encode (weakly entangled) states 𝜏 1 𝜏 2 𝜏 3 𝜏 𝑀 𝜇 [𝑀−1] 𝜇 [3] 𝜇 [2] 𝜇 [1] … 𝑨 𝑨 𝑙 𝑙 𝑚 𝑚 Γ 3 𝑛 𝑛 Γ 𝑀 Γ 1 Γ 2 80

Tensor-product states in d>1 How to generalize MPS to d>1 ?  “Snake” MPS • Quite powerful in practice (produced new results on several frustrated spin systems) • But… problem with the area law → requires and exponential growth the the matrix dimension with the transverse dimension • Reference: “ Studying Two-Dimensional Systems with the Density Matrix Renormalization Group ” E. M. Stoudenmire and S. R. White Ann. Rev. of Cond. Mat. Phys. 3, 111 (2012)  Use tensor networks (=more than two virtual indices) • Finite-rank tensor can reproduce area laws • High computation cost in practice (to perform contractions, tensor optimizations, …) , but very prosmissing. “ A practical introduction to tensor networks: Matrix product states and projected entangled pair states ” Román Orús, Ann. of Phys. 349, 117 (2014) 81

Multi-scale Entanglement Renormalization Ansatz “Entanglement Renormalization”, G. Vidal Phys. Rev. Lett. 99, 220405 (2007); “Class of Quantum Many-Body States That Can Be Efficiently Simulated”, G. Vidal Phys. Rev. Lett. 101, 110501 (2008).  Can reproduce the 𝑇~ 𝑑 3 log 𝑚 behavior with constant tensor dimensions → adapted for 1d critical systems (and generalizations exists in 𝑒 > 1 )  The different “layers” of the network correspond to different length scales (RG idea) 82

Corrections to the area law in 2d systems 2 examples showing (universal) subleading corrections: a magnet with gapless Goldstone modes (spin waves) and a quantum dimer model in a “topological” ( Z 2 ) phase 83

Area law in 2d – spin ½ Heisenberg model “Anomalies in the entanglement properties of the square-lattice Heisenberg model” A. B. Kallin, M. B. Hastings, R. G. Melko & R. R. P. Singh, Phys. Rev. B 84, 165134 (2011) +see also: D. J. Luitz, X. Plat, F. Alet, & N. Laflorencie, Phys. Rev. B 91, 155145 (2015) Magnetic long-range order ↔ spontaneous SU(2) symmetry braking → gapless spin waves (Goldstone modes) → additive log(L) correction to the entanglement entropy. area law coeff. 𝑂 𝐻 Metlitsky-Grover [2011] →the coefficient of the log (𝑚) term is 𝑑 = 2 (in 𝑒 = 2 ) where 𝑂 𝐻 is the number of Nambu-Goldstone modes (consistent with the numerics above). 84

Area law in 2d – quantum dimer model S. Furukawa & GM, Phys Rev B 2007 Quantum dimer model wave-function J.-M. Stéphan, GM & V. Pasquier J. Stat. Mech. 2012 𝜔 = 𝑑 = Equal amplitude superposition Local hard-core constraint all the hard−core dimer coverings 𝑑∈ of the triangular lattice 𝑀 𝑦 → ∞ A=½-infinite A=disk 𝑀 cylinder 𝑆 t: fugacity for dimers on horizontal bonds Universal subleading Term -log(2) radius 𝑆 circunference 𝑀 85

Bulk-edge correspondance in 2d Relation between the entanglement `Hamiltonian ’ and physical edge modes. Topological entanglement entropy 86

Topological phases of matter  Ground state properties  No spontaneously broken symmetry, no local order parameter (“quantum liquids”)  The ground-state degeneracy depends on the topology  Degenerate ground-state are locally undistinguishable E E E Example of a Z 2 Liquid :  Elementary excitations • Excitations are gapped (at least in the bulk) • Quantum number fractionalization (elementary excitations must be created in pairs, and can then be separated far away) Examples: fractional electric charges (FQHE), or spin-1/2 excitations in magnetic insulators • Exotic statistics in 2d (can be different from fermions & bosons, and can even be non-Abelian)  Examples • Theoretical realizations: many models with fermions, bosons, spins, strings, dimers … • Experimental realizations: • fractional quantum Hall effect • some exotic superconductors ? • some magnetic insulators (spin liquids) ?  Closely related phases : • Integer quantum Hall effect • Topological insulators • … 87

Fractional quantum Hall effect  «Two-Dimensional Magnetotransport in the Extreme Quantum Limit » D. C. Tsui, H. L. Störmer, and A. C. Gossard, Phys. Rev. Lett. 48 , 1559 (1982) [Web od Science: ~2100 citations]  “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations”, R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, R. B. Laughlin Phys. Rev. Lett. 50, 1395 &J. H. English, Phys. Rev. Lett. 59, 1776 (1987) (1983) [Web od Science: ~3000 citations] exp − 1 𝜉 = 1 𝑛 2 𝜔 Laughlin 𝑨 𝑗 = 𝑨 𝑘 − 𝑨 𝑘 4 𝑨 𝑘 𝑛 𝑛: odd integer 𝑘<𝑙 𝑘 88

Gapless edge modes All the excitation in the bulk are gapped Gapless excitations exists along the edge. This gaplessness is “protected” by some topological properties of the wave function in the bulk (and/or some symmetries) Examples: Quantum hall phases, Chiral spins liquids, Topological insulators, … 89

Bulk-edge correspondence in 2d (0)  H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008)  Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig, Phys. Rev. Lett. 108, 196402 (2012) 90

Bulk-edge correspondence in 2d (1) Qi-Katsura-Ludwig argument in presence of gapless edge excitations • Assumption 1: the system is gapped in the bulk, but with gapless edge states Examples: integer Hall effect, fractional Hall effect, topological insulators, chiral spin liquids, … Treat the coupling 𝜇 between A and B perturbatively. Assumption 2: there is • some adiabatic continuity from 𝜇 = 1 (homogenous system) and 0 < 𝜇 ≪ 1 (two weakly coupled cylinders), and the result for the spectrum of the reduced density matrix will not qualitatively change. Due to the presence of a gap in the bulk, the (perturbative) ground state 𝐻 • can be described in the space of the low energy edge modes of A & B (this should at least be true if Δ is sent to ∞) 𝐹 𝐵 𝐹 𝑐 Two (weakly) coupled Bulk excitations Bulk excitations edges In region A In region B Δ Gapless edge states Gapless edge states “Left” modes 𝑜, 𝑀 “Right” modes 𝑜, 𝑆 91

Bulk-edge correspondence in 2d (2) Qi-Katsura-Ludwig argument: mapping to a quantum quench problem 𝑏 (topo. Sector) 𝐼 𝑀 Calabrese-Cardy (2006) result about the a global quench in a critical 1d system: 𝜇 = 0 - to obtain the long time & long-distance 𝐼 𝑆 correlations, the initial state can be replaced by 𝑓 −𝜐(𝐼 𝑀 +𝐼 𝑆 ) 𝐻 ∗ 𝑢 = 0 𝑢 > 0 - 𝜐 is a finite non-universal constant 𝜔(𝑢) = 𝑓 −𝑗𝐼𝑢 𝐻 𝜔(0) = 𝐻 (“extrapolation length”). - 𝐻 ∗ : scale/conformally-invariant Ground-state of 𝜍 𝑀 (𝑢) = 𝑓 −𝑗𝐼 𝑀 𝑢 𝜍 𝑀 (0)𝑓 +𝑗𝐼 𝑀 𝑢 𝐼 𝑀 + 𝜇𝐼 𝑗𝑜𝑢 + 𝐼 𝑆 boundary state (fixed point of an RG flow 𝜍 𝑀 = 𝑈𝑠 𝑆 𝐻 𝐻 starting from 𝐻 ). Entanglement entropy & spectrum ( 𝑀 and 𝑆 are entangled) of 𝜍 𝑀 (𝑢) : independent of 𝑢 . - Rational CFT: 𝐻 ∗ is a linear combination of all excited states, 𝐻 ∗,𝑏 ~ 𝑙, 𝑘, 𝑏 𝑀 𝑙, 𝑘, 𝑏 𝑆 maximally entangled combination between the L and R edges. 𝑙 𝑘 𝑓 −2𝜐𝑤𝑙 𝑙, 𝑘, 𝑏 𝑀 −𝑙, 𝑘, 𝑏 𝑆 𝑓 −𝜐(𝐼 𝑀 +𝐼 𝑆 ) 𝐻 ∗,𝑏 ~ momentum 𝑙>0 𝑘 Entanglement Hamiltonian 𝜍 𝑀,𝑏 ~ 𝑓 −4𝜐𝑤𝑙 𝑙, 𝑘, 𝑏 𝑀 𝑙, 𝑘, 𝑏 𝑀 ~exp (−4𝜐𝐼 𝑀 ) (universal part of) : ~𝐼 𝑀 𝑙>0 𝑘 92

(real space) Entanglement spectrum & quantum Hall effect 𝜉 = 1 (Integer Q. Hall effect) 1 Laughlin 𝜉 = 3 (8 particles) Low “energy” part of the 1 Laughlin 𝜉 = 3 entanglement spectrum is gapless (in the thermodynamic limit) has the 1 Coulomb 𝜉 = same structure (linear 3 dispersion & degeneracies) as that of a massless chiral free boson, which also describe the physical edge modes 1 (12 particles, 𝜉 = 3 Laughlin’s state) J. Dubail, N. Read, and E. H. Rezayi, A. Sterdyniak, A. Chandran, N. Regnault, Phys. Rev. B 85, 115321 (2012) B. A. Bernevig, and P. Bonderson, Phys. Rev. B 85 , 125308 (2012) 93

Bulk-edge correspondence in 2d – Free fermion edge (1) Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] 𝜁 = 𝑤𝑙 † 𝑑 𝑙 † 𝑑 𝑙 𝐼 𝑆 = 𝑤 𝑙𝑑 𝑙 𝑙 𝑙 Particle-hole excitations 𝜁 = −𝑤𝑙 𝜖𝜗 𝜖𝑙 = ±𝑤 propagate at velocity † 𝑒 𝑙 † 𝑒 𝑙 𝐼 𝑀 = −𝑤 𝑙𝑒 𝑙 𝑙 𝑙 † 𝑒 𝑙 + 𝑒 𝑙 † 𝑑 𝑙 𝐼 int = Δ 𝑑 𝑙 = tunneling from one edge to another, with momentum conservation 𝑙 𝑑 𝑙 𝑤𝑙 Δ † † 𝐼 = 𝐼 𝑆 + 𝐼 𝑀 + 𝐼 int = 𝑑 𝑙 𝑒 𝑙 𝑒 𝑙 Δ −𝑤𝑙 𝑙 Diagonalization using new fermionic creation/annihilation operators: 𝑤𝑙 2 + Δ 2 ( → gapped spectrum) † 𝑏 𝑙 − 𝑐 𝑙 † 𝑐 𝑙 𝐼 = 𝐹 𝑙 𝑏 𝑙 , 𝐹 𝑙 = 𝑙 𝑏 𝑙 𝑑 𝑙 𝛽 𝑙 𝛾 𝑙 𝐹 𝑙 +𝑤𝑙 𝐹 𝑙 −𝑤𝑙 𝑐 𝑙 = 𝑒 𝑙 with 𝛽 𝑙 = 2𝐹 𝑙 and 𝛾 𝑙 = 2𝐹 𝑙 −𝛾 𝑙 𝛽 𝑙 94

Bulk-edge correspondence in 2d – Free fermion edge (2) Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] 𝑏 𝑙 𝑑 𝑙 𝛽 𝑙 𝛾 𝑙 𝐹 𝑙 +𝑤𝑙 𝐹 𝑙 −𝑤𝑙 𝑐 𝑙 = 𝑒 𝑙 with 𝛽 𝑙 = 2𝐹 𝑙 and 𝛾 𝑙 = 2𝐹 𝑙 −𝛾 𝑙 𝛽 𝑙 Since the Hamiltonian is quadratic in the Fermion operators, it is “Gaussian” and can be written as some exponential of a quadratic form acting on some vacuum. In the present case we want to write the ground-state 𝐻 of the two coupled edges in the form: 𝐻 = exp −𝐶 𝐻 𝑀 ⨂ 𝐻 𝑆 where 𝐻 𝑀/𝑆 is the ground-state of 𝐼 𝑀/𝑆 . 𝐶 should contain terms that “dress” the two edges by particle hole-excitations, keeping the total momentum as well as the total number of Fermions. 𝐶 should therefore have the following form: (using the 𝑙 ⟷ −k & L ⟷ 𝑆 symmetry) † 𝑑 −𝑙 † 𝑒 𝑙 + 𝑒 −𝑙 𝜁 𝐶 = 𝜇 𝑙 𝑑 𝑙 𝑙>0 𝑙 † ). How to determine 𝜇 𝑙 ? Insure that 𝐻 is annihilated by 𝑏 𝑙 (and 𝑐 𝑙 Commute 𝑏 𝑙 and 𝑓 −𝐶 : 𝜁 𝐶, 𝑏 𝑙 = 𝐶, 𝛽 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 = −𝜇 𝑙 𝛽 𝑙 𝑒 𝑙 if 𝑙 > 0 if 𝑙 < 0 𝜇 −𝑙 𝛾 𝑙 𝑑 𝑙 𝑙 𝐶, 𝐶, 𝑏 𝑙 = 0 → Baker – Campbell – Hausdorff formula gives: 𝑏 𝑙 exp −𝐶 = exp −𝐶 𝑏 𝑙 + 𝐶, 𝑏 𝑙 95

Bulk-edge correspondence in 2d – Free fermion edge (3) Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] 𝜁 𝐻 𝑆 𝐶, 𝑏 𝑙 = 𝐶, 𝛽 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 = −𝜇 𝑙 𝛽 𝑙 𝑒 𝑙 if 𝑙 > 0 if 𝑙 < 0 𝜇 −𝑙 𝛾 𝑙 𝑑 𝑙 𝑙 So: 𝑏 𝑙 exp −𝐶 = exp −𝐶 𝛿 𝑙 𝜁𝑙 With 𝛿 𝑙 = 𝑏 𝑙 + 𝐶, 𝑏 𝑙 = 𝛽 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 − 𝜇 𝑙 𝛽 𝑙 𝑒 𝑙 if 𝑙 > 0 𝐻 𝑀 if 𝑙 < 0 𝛽 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 + 𝜇 −𝑙 𝛾 𝑙 𝑑 𝑙 𝑙 We determine 𝜇 𝑙 by requiring that 𝛿 𝑙 annihilates 𝐻 𝑀 ⨂ 𝐻 𝑆 : 𝛿 𝑙 = 𝛽 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 𝛾 𝑙 − 𝜇 𝑙 𝛽 𝑙 if 𝑙 > 0 if 𝑙 < 0 𝑑 𝑙 𝛽 𝑙 + 𝜇 −𝑙 𝛾 𝑙 𝑑 𝑙 + 𝛾 𝑙 𝑒 𝑙 (note that the 2 conditions above are equivalent since 𝛽 𝑙 = 𝛾 −𝑙 ) 𝐹 𝑙 2 − 𝑤𝑙 2 𝜇 𝑙 = 𝛾 𝑙 𝐹 𝑙 − 𝑤𝑙 Δ = 𝐹 𝑙 + 𝑤𝑙 = = 𝐹 𝑙 + 𝑤𝑙 𝛽 𝑙 𝐹 𝑙 + 𝑤𝑙 Final expression for the ground-state: Δ † 𝑑 −𝑙 † 𝑒 𝑙 + 𝑒 −𝑙 𝐻 = exp − 𝐻 𝑀 ⨂ 𝐻 𝑆 𝐹 𝑙 + 𝑤𝑙 𝑑 𝑙 𝑙>0 Reduced density matrix for the L-edge ? 96

Bulk-edge correspondence in 2d – Free fermion edge (4) Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] 𝜁 1 −𝑙,𝑆 0 𝑙,𝑆 𝐻 𝑆 𝑙 Δ † 𝑑 −𝑙 † 𝑒 𝑙 + 𝑒 −𝑙 𝐻 = exp − 𝐹 𝑙 + 𝑤𝑙 𝑑 𝑙 𝐻 𝑀 ⨂ 𝐻 𝑆 𝑙>0 𝜁𝑙 Remark: modes with different 𝑙 are independent, and all the terms 0 −𝑙,𝑀 1 𝑙,𝑀 𝐻 𝑀 𝑙 in the exponential above commute with each other. † 𝑑 −𝑙 0 −𝑙,𝑀 1 −𝑙,𝑆 † 𝑒 𝑙 1 𝑙,𝑀 0 𝑙,𝑆 𝐻 = 1 − 𝜇 𝑙 𝑑 𝑙 1 − 𝜇 𝑙 𝑒 −𝑙 𝑙>0 𝑙>0 𝐻 = 1 𝑙,𝑀 0 𝑙,𝑆 − 𝜇 𝑙 0 𝑙,𝑀 1 𝑙,𝑆 0 −𝑙,𝑀 1 −𝑙,𝑆 − 𝜇 𝑙 1 𝑙,𝑀 0 𝑙,𝑆 𝑙>0 𝑙>0 Normalize -> Schmidt decomposition : 1 𝜇 𝑙 1 𝐻 = 1 + 𝜇 𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 − 1 + 𝜇 𝑙 2 0 𝑙,𝑀 1 𝑙,𝑆 1 + 𝜇 𝑙 2 0 −𝑙,𝑀 1 −𝑙,𝑆 𝑙>0 𝑙>0 𝜇 𝑙 − 1 + 𝜇 𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 97

Bulk-edge correspondence in 2d – Free fermion edge (3) Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] Schmidt decomposition : 1 𝜇 𝑙 1 𝐻 = 1 + 𝜇 𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 − 1 + 𝜇 𝑙 2 0 𝑙,𝑀 1 𝑙,𝑆 1 + 𝜇 𝑙 2 0 −𝑙,𝑀 1 −𝑙,𝑆 𝑙>0 𝑙>0 𝜇 𝑙 1 + 𝜇 𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 − Reduced density matrix for the L -edge: 1 1 1 + 𝜇 𝑙 2 0 0 𝑙,𝑆 + 𝜇 𝑙 2 1 1 𝑙,𝑆 1 + 𝜇 𝑙 2 1 1 −𝑙,𝑆 + 𝜇 𝑙 2 0 0 −𝑙,𝑆 𝜍 𝑆 = 𝑙>0 𝑙>0 1 1 𝜇 𝑙 2 𝑑 𝑙 𝜇 𝑙 2 𝑑 −𝑙 𝑑 −𝑙 † 𝑑 𝑙 † 𝜍 𝑆 = 1 + 𝜇 𝑙 2 exp log 1 + 𝜇 𝑙 2 exp log 𝑙>0 𝑙>0 Δ 1 𝑤𝑙 𝜇 𝑙 2 ≈ −2 𝑤𝑙 Expand for 𝑙 → 0 : 𝜇 𝑙 = 𝐹 𝑙 +𝑤𝑙 ≈ ≈ 1 − Δ → log Δ 1+ 𝑤𝑙 Δ 𝜍 𝑆 ~ exp −2 𝑤𝑙 = exp −2 𝑤𝑙 † 𝑑 −𝑙 + 1 † 𝑑 𝑙 + 𝑑 −𝑙 𝑑 −𝑙 † † 𝑑 𝑙 − 𝑑 −𝑙 𝑑 𝑙 𝑑 𝑙 Δ Δ 𝑙>0 𝑙>0 = exp −2 𝑤𝑙 exp − 2 † 𝑑 𝑙 + 𝑑𝑡𝑢. Δ 𝑑 𝑙 ~ Δ 𝐼 𝑆 𝑙 Δ → The 𝑆 -edge is seen at some effective temperature T eff = 2 . Entanglement Hamiltonian ~ 𝐼 𝑆 98

Bulk-edge correspondence in a spin ladder  Spin ladder model A 𝜇 𝑲  “Entanglement Spectra of Quantum Heisenberg Ladders” , D. Poilblanc, Phys. Rev. Lett. 105 , 077202 (2010).  Numerical observation: the ES of 𝐵 (upper chain) is very similar 1 to the (energy) spectrum of the Heisenberg spin- 2 chain . NB: Des Cloizeaux Pearson [1962] dispersion relation for a single chain: 𝜌 𝜗 𝑙 = 2 sin 𝑙 . 99

Bulk-edge correspondence in a spin ladder  “ On the relation between entanglement and subsystem Hamiltonians ” , I. Peschel and M.-C. Chung, EPL 96 , 50006 (2011). Idea: compute the entanglement spectrum (of the region 𝐵 ) perturbatively in 𝜇 . 𝜇 A 𝑲 𝑇 𝑗𝑏 ⋅ 𝑇 𝑗𝑐 𝐼 = 𝑲 𝑇 + 𝜇 𝐼 𝐵 + 𝐼 𝐶 𝑗=1… 𝑀 𝑗𝑏 ⋅ 𝑇 𝑗+1 𝑏 𝑗𝑐 ⋅ 𝑇 𝑗+1 𝑐 𝐼 𝐵 = 𝑇 𝐼 𝐶 = 𝑇 𝑗 𝑗 1 4𝜇 𝐾 𝐼 𝐵 + 𝒫 𝜇 2  Result: 𝜍 𝐵 ≃ 𝑎 exp − 𝜍 𝐵 =Thermal density matrix (for small 𝜇 ) • 𝐾 Effective temperature 𝑈 eff = • 4𝜇 Entanglement Hamiltonian (defined by −log (𝜍 𝐵 )) is proportional • to the real Hamiltonian 𝐼 𝐵 of the upper chain. 100