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

Quantum entanglement in condensed matter

Grégoire Misguich (IPhT, CEA Saclay)

ipht.cea.fr/Pisp/gregoire.misguich gregoire.misguich@cea.fr

IPhT Lectures

& Ecole doctorale “Physique en île de France” (ED-PIF)

May 22th & 29th, June 5th and 12th, 2015

Version of June 11, 2015

2

INTRODUCTION

motivations, plan basic definitions: reduced density matrix, entanglement entropy, Schmidt decomposition, Rényi entropies, …

3

Why studying quantum entanglement ?

(in the many-body systems encountered in condensed matter)  A powerful theoretical probe for quantum many-body systems. In particular, the scaling of the entanglement entropy can reveals most of the long-distance properties of the system. Celebrated example: 𝑇Von Neumann ∼

𝑑 3 log

(𝑚) for critical quantum chains, relating the central 𝑑 charge of the underlying CFT to the ground-state entanglement.  Understanding entanglement is useful to construct new algorithms to compute & store efficiently quantum (many-body) states in a computer, to simulate their evolution, or construct some variational anstaz (tensor networks etc.).  Quantum entanglement is at the core of quantum information processing (quantum cryptography and quantum computation, adiabatic quantum computation) But…  It is very hard to measure experimentally

𝑚

4

Answering condensed-matter questions by studying entanglement

 “Entanglement-based” approach  “Conventional” approach

Scaling of 𝑇𝐵 = −Tr𝐶 𝜍𝐵 log 𝜍𝐵 with the size and shape of A Spectrum of 𝜍𝐵 Characterize the state of matter

(long distance & low energy properties)

• Long-range order

and spontaneous symmetry breaking ?

• Criticality / algebraic correlations ?
• Excitation spectrum, gap / gapless ?
• Unconventional (i.e. topological) order ?
• Fractional excitations ?
• Gapless edge modes ?

Ground-state 𝜔 Choose some observable 𝑃 Correlations 𝜔 𝑃 𝑦 𝑃 (𝑧) 𝜔 Hamiltonian 𝐼 Low energy spectrum

& quantum numbers

Choice of a (large) subsystem A RDM: 𝜍𝐵 = Tr𝐶 𝜔 𝜔

5

Focus of these lectures …

 Entanglement between two spatial regions of the system : In most cases the total system A+B will be in a pure state (wave-function 𝜔 ). We will not discuss multipartite entanglement, or other partition schemes (particles, momentum space, …)  Situations where A & B contain a large number of degrees of freedom  State 𝜔 : ground state of some condensed-matter lattice model (ex: spins, fermions, …)  We will not discuss :

 disordered systems and many-body localization  the theory of quantum measurements (w.-f. collapse, etc)  physical mechanisms and characterization of decoherence (system interacting

with an environment). More generally, will we say very little about experiments 

 quantum computers and the role of entanglement in quantum algorithms

A B 𝜔

6

Outline

I. Introduction - why studying quantum entanglement ? II. Basic definitions Reduced density matrix (RDM), entanglement (Von Neumann) entropy Schmidt decomposition, singular-value decomposition, replica trick, Rényi entropies, … III. A proposal to measure (Rényi) entanglement entropies

• IV. Area law:

I. General idea II. Mutual information & area law at T>0 III. T=0 & mutual information correlation length (Wolf et al. 2008)

• IV. Hastings’ theorem in 1d (2007)

V. High energy states: volume law for the entanglement entropy, thermal entropy and the eigenstate thermalization hypothesis (ETH)

• VI. Free fermions and bosons

1. Gaussian reduced density matrix & correlation functions (Peschel 2003) 2. Exactly solvable example: entropy of a segment in a free fermion chain (1d Fermi sea) 3. Violation of the boundary law (multiplicative log) in presence of a Fermi surface in d>1 (Gioev &I. Klich 2006)

• VII. CFT & Entanglement in critical 1d systems (Holzhey et al. ‘94)
• VIII. Matrix product states

I. Canonical MPS form & entanglement II. Real time evolution (Vidal’s TEBD)

• IX. Some examples of (universal) corrections to the area law in 2d

X. Bulk-edge correspondence 1. Bulk-edge correspondence in 2d (Li-Haldane 2008 & Qi-Katsura-Ludwig 2012) 2. Example of a spin ladder (Poilblanc 2010, Pechel & Chung 2011) 3. Topological entanglement entropy (Kitaev-Preskill & Levin-Wen 2006) Lecture #1 Lecture #2 Lecture #3 Lecture #4

7

Some reviews

 “Entanglement in many-body systems”

• L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008)

 “ Quantum entanglement”

• R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki,
• Rev. Mod. Phys. 81, 865 (2009)

 “ Entanglement entropy in extended quantum systems” , editors: P. Calabrese, J. Cardy & B. Doyon, J. Phys A, Special issue (2009)  “Quantum Entanglement in Condensed Matter Physics”, editors: S. Rachel, M. Haque, A. Bernevig, A. Laeuchli and E. Fradkin

• J. Stat. Mech.: Th. and Experiment, Special Issue (2014-2015)

8

Product states & entanglement

 Separate the degrees of freedom into two blocks, and two Hilbert spaces ℋ = ℋ

𝐵⨂ℋ𝐶

 A and B are not entangled ⇔ 𝜔 is a product state 𝜔 = 𝑏 ⨂ 𝑐  A and B are Entangled ⇔ 𝜔 cannot be written as a product state  Standard 2-spin examples

• Products:

𝜔 = ↑1↑2 𝜔 = ↑↑ + ↑↓ + ↓↑ + ↓↓ = ↑ + ↓ 1⨂ ↑ + ↓ 2

• Not products:

𝜔 = ↑↑ + ↓↓ 𝜔 = ↑↓ − ↓↑

• What about 𝜔 = 𝛽 ↑↑ + 𝛾 ↑↓ + 𝛿 ↓↑ + 𝜀 ↓↓ ?

 What if 𝜔 is the ground state of the spin-1/2 Heisenberg model on a chain

• f length L ?

L A B 𝐼 = 𝑇 𝑜 ⋅ 𝑇 𝑜+1

𝑜

𝐼 𝜔 = 𝐹0 𝜔 → Some way to quantify the amount of (bi partite) entanglement is needed

9

Reduced density matrix (definition)

 Definition: 𝜍𝐵 = Tr𝐶 𝜍𝐵𝐶 “tracing out the degrees of freedom of B”  2-spin example: General pure state 𝜔 = 𝛽 ↑↑ + 𝛾 ↑↓ + 𝛿 ↓↑ + 𝜀 ↓↓ Reduced density matrix for 𝐵 =spin 1: 𝜍spin 1 = TrB=spin 2 𝜔 𝜔 Matrix elements: 𝜏1 ρ1 𝜏1′ = 𝜏1𝜏2 𝜔 𝜔 𝜏1′𝜏2

𝜏2=↑,↓

𝜏1 ρ1 𝜏1′ = 𝜏1 ↑ 𝜔 𝜔 𝜏1′ ↑ + 𝜏1 ↓ 𝜔 𝜔 𝜏1′ ↓ 𝜍1 = 𝛽 2 + 𝛾 2 𝛿 𝛽 + 𝜀 𝛾 𝛽 𝛿 + 𝛾 𝜀 𝛿 2 + 𝜀 2  General formula: 𝜔 = 𝑁𝑗𝑘 𝑏𝑗 ⨂ 𝑐

𝑘 𝑗𝑘

→ 𝑏𝑗 𝜍𝐵 𝑏𝑗′ = 𝑁𝑗𝑘𝑁𝑗′𝑘

𝑘

= 𝑁𝑁†

𝑗𝑗′

Remark: 𝑁𝑁† → positive eigenvalues. Since Tr𝐵 𝜍𝐵 = 1, all the eigenvalues must be ∈ [0,1].

 Basic properties

 Hermitian  Expectation values of observables acting on subsystem 𝐵 can be computed using 𝜍1:

𝜔 𝑃𝐵 𝜔 = Tr𝐵,𝐶 𝜔 𝜔 𝑃𝐵 = TrA ρ𝐵𝑃𝐵

 TrA 𝜍𝐵 = TrA𝐶 𝜍𝐵𝐶 = 1. Example: Trspin 1 ρ1 = 𝛽 2 + 𝛾 2 + 𝛿 2 + 𝜀 2 = 𝜔 𝜔 = 1  The two subsystems are not entangled ⇔ 𝜔 = 𝑏 ⨂ 𝑐 ⇔ 𝜍1is a projector

(𝜍1 = 𝑏 𝑏 ) ⇔ eigenvalues 𝑞𝑗 of ρ1 are 1 (non-degenerate) and 0.

10

Schmidt decompostion (1)

Pure state 𝜔 ∈ ℋ

𝐵⨂ℋ𝐶, 𝑏𝑗 basis of ℋ 𝐵, 𝑐 𝑘 basis of 𝐼𝐶

𝜔 = 𝑁𝑗𝑘 𝑏𝑗 ⨂ 𝑐

𝑘 𝑗𝑘

𝑁 rectangular matrix: m = dim ℋ

𝐵

× 𝑜 = dim ℋ𝐶 Singular value decomposition (SVD) of 𝑁: ∃ unitary matrices 𝑉 (size 𝑛) and 𝑊 (size 𝑜), and a “diagonal” one 𝜇 (=singular

values of 𝑁, real an non-negative)

such that 𝑁 = 𝑉𝜇𝑊† We can rewrite 𝜔 as: 𝜔 = 𝑉𝑗𝑙𝜇𝑙𝑊

𝑙𝑘 † 𝑏𝑗 ⨂ 𝑐 𝑘 𝑗𝑘𝑙

Then we define 𝑣𝑙 = 𝑉𝑗𝑙 𝑏𝑗

𝑙

, 𝑤𝑙 = 𝑊

𝑘𝑙 𝑐𝑗 𝑙

. Orthogonality property: 𝑣𝑙 𝑣𝑙′ = 𝜀𝑙,𝑙′= 𝑤𝑙 𝑤𝑙′ An finally we get the Schmidt decomposition of 𝜔 : 𝜔 = 𝜇𝑙 𝑣𝑙 ⨂ 𝑤𝑙

𝑙

A B 𝜔 𝜇𝑙 𝑛 𝑜 𝑛 = 𝑜 𝑜 𝑜 × × 𝑉 𝑛 𝑊† 𝑁 𝑏𝑗 𝑐

𝑘

𝜔 𝑣𝑙 𝑤𝑙

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Schmidt decomposition (2)

Recall : 𝜔 = 𝜇𝑙 𝑣𝑙 ⨂ 𝑤𝑙

𝑙

 Reduced density matrices & their spectrum 𝜍𝐵 = Tr𝐶 𝜔 𝜔 = 𝜇𝑙 2 𝑣𝑙 𝑣𝑙

𝑙

→ the eigenvalues of 𝜍𝐵 are the 𝜇𝑙 2 𝜍𝐶 = Tr𝐵 𝜔 𝜔 = 𝜇𝑙 2 𝑤𝑙 𝑤𝑙

𝑙

→ the eigenvalues of 𝜍𝐶 are the 𝜇𝑙 2 → Both RDM have the same non-zero eigenvalues → symmetry of the Von Neumann entropy: 𝑇𝐵 = 𝑇𝐶  “Purification”: if 𝜍𝐵 describes some mixed state of some system 𝐵, one can construct a pure state 𝜔 in some enlarged system 𝐵𝐵’ such that 𝜍𝐵 = 𝑈𝑠A′ 𝜔 𝜔 . Solution: 𝐵’=“copy of 𝐵”. Diagonalize 𝜍𝐵: 𝜍𝐵 = 𝑞𝑙 𝑙 𝑙

𝑙

and define the entangled wave-function: 𝜔 = 𝑞𝑙

𝑙

𝑙 𝐵⨂ 𝑙 𝐵′

 A separate unitary “evolution” of regions 𝐵 and 𝐶 does not change the Schmidt eigenvalues: 𝑓−𝑗𝑢 𝐼𝐵+𝐼𝐶 𝜔 = 𝑉𝐵𝑉𝐶 𝜔 = 𝜇𝑙 𝑉𝐵 𝑣𝑙 ⨂ 𝑉𝐶 𝑤𝑙

𝑙

=still a valid Schmidt decomposition → 𝜇𝑙 unchanged, and constant entanglement entropy

12

Schmidt decomposition (3) – Optimal approximation

 What is the best approximation to 𝜔 with a given lower Schmidt rank𝜓 < 𝑠 ? 𝜔 = 𝑁𝑙𝑚 𝑏𝑙 𝑐𝑙

𝑙𝑚

= 𝜇𝑙 𝑣𝑙 𝑤𝑙

𝑠 𝑙=1

𝜇1 ≥ 𝜇2 ≥ ⋯ ≥ 𝜇𝑠 ≥ 0 SVD: 𝑁 = 𝑉𝑇𝑊†  Answer: truncate the Schmidt decomposition above, keeping only the 𝜓 largest values (and re-normalize the state):

𝜔𝜓 = 1 𝜇𝑙 2

𝜓 1

1 𝜇𝑙 𝑣𝑙 𝑤𝑙

𝜓 𝑙=1

𝜔 𝜔𝜓 = 1 − 𝜇𝑙 2

𝑠 𝑙=𝜓+1

 « Proof »:

• Note that the Hilbert space norm is equivalent to Frobenius norm ⋯

𝐺 for the

matrix 𝑁: 𝑁 𝐺

2 = Tr 𝑁𝑁† =

𝑁𝑙𝑚 2

𝑙𝑚

= 𝜔 𝜔

• 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

• ptimization problem is also optimal for the norm ⋯

2 -- for which the proof is simpler.

13

Schmidt decomposition (4) – SVD on a computer

 Complexity for an 𝑁 ∗ 𝑂 matrix: 𝒫 min 𝑁 ∗ 𝑂2, 𝑂 ∗ 𝑁2

NB: To be compared with a two-step procedures: i) Computing 𝜍𝐵: 𝒫 𝑂 ∗ 𝑁2 and then ii) diagonalizing 𝜍𝐵: 𝒫 𝑁3 .

 LAPACK: dgesvd  GSL: gsl_linalg_SV_decomp  Numerically stable  Sometimes the SVD is overkill if the Schmidt values are not needed (just some

• rthogonal basis on one “side”, as often in DMRG for instance) and a QR

factorization is enough (faster).

14

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

15

SVD, decay of the singular values & data compression (1)

An (color) image viewed as a (three) matrix(ces)

16

Von Neumann entropy (definition)

 Von Neumann/entanglement entropy 𝑇𝐵 = −Tr𝐶 𝜍𝐵 log 𝜍𝐵 = − 𝑞𝑗 log 𝑞𝑗

𝑗

With 𝜍𝐵 = Tr𝐶 𝜍𝐵𝐶 , and 𝑞𝑗 the eigenvalues of 𝜍𝐵.  𝑇𝐵 = 0 ⟺ 𝜍𝐵 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).

17

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

• f 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 𝜇𝑗𝜍𝑗

𝑗

≥ 𝜇𝑗𝑇 𝜍𝑗

𝑗

 Independent systems : 𝑇 𝜍𝐵 ⊗ 𝜍𝐶 = 𝑇 𝜍𝐵 + 𝑇 𝜍𝐶  If 𝜍𝐵𝐶 describe a pure state, 𝑇 𝜍𝐶 = 𝑇 𝜍𝐵 (proof using the Schmidt decomposition, see next)  A,B,C: 3 parts, without intersection

• Strong sub-additivity [SS] 𝑇 𝜍𝐵𝐶𝐷 + 𝑇 𝜍𝐶 ≤ 𝑇 𝜍𝐵𝐶 + 𝑇 𝜍𝐶𝐷
• Equivalent formulation: 𝑇𝑌∪𝑍 + 𝑇𝑌∩𝑍 ≤ 𝑇𝑌 + 𝑇𝑍

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 𝜍𝐵𝐷 ≤ 𝑇 𝜍𝐵 + 𝑇 𝜍𝐷 (obtained by setting 𝐶 = ∅ in SS above, but direct proof is

possible a much much simpler than SS)

 Araki-Lieb triangular inequality: 𝑇 𝜍𝐵𝐷 ≥ 𝑇 𝜍𝐵 − 𝑇 𝜍𝐷

Simple proof using i) some auxiliary pure state 𝜍𝐵𝐶𝐷 in some enlarged space such that TrB 𝜍𝐵𝐶𝐷 = 𝜍𝐵𝐷, Tr𝐶𝐷 𝜍𝐵𝐶𝐷 = 𝜍𝐵 and Tr𝐵𝐶 𝜍𝐵𝐶𝐷 = 𝜍𝐷, ii) the sub-additivity above for 𝜍𝐶𝐷 and iii) 𝑇 𝜍𝐶𝐷 = 𝑇 𝜍𝐵 and 𝑇 𝜍𝐶 = 𝑇 𝜍𝐵𝐷 (since 𝜍𝐵𝐶𝐷 is pure).

18

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). x 𝑇(𝑦) 𝑦  Answer: use the strong sub-additivity with the three following consecutive segments 𝐵 = {0}, 𝐶 = {1,2, … , 𝑦 − 1} and 𝐷 = {𝑦}: 𝑇𝐵𝐶𝐷 + 𝑇𝐶 ≤ 𝑇𝐵𝐶 + 𝑇𝐶𝐷 𝑇 x + 1 + S x − 1 ≤ 2𝑇 𝑦 → concavity + symmetry 𝑇(𝑦) = 𝑇(𝑀 − 𝑦)

19

Rényi entropies & Replica trick

 If you can compute the spectrum of 𝜍𝐵… then you get 𝑇𝐵 by summing over all the eigenvalues: 𝑇𝐵 = −Tr 𝜍𝐵 log 𝜍𝐵 = − 𝑞𝑗 log 𝑞𝑗

𝑗

 If not… you can

• compute Tr 𝜍𝐵𝑜 for integer 𝑜 ≥ 2 (easier than computing the spectrum)
• (cross your fingers &) analytically continue the result to n=1
• use:

−Tr 𝜍𝐵 log 𝜍𝐵 = lim

𝑜→1

1 1 − 𝑜 log Tr 𝜍𝐵𝑜  Rényi entropies are also often interesting, and simpler to compute & measure [in principle] 𝑇𝐵(𝑜) = 1 1 − 𝑜 log Tr 𝜍𝐵𝑜

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 𝑇𝐵 𝑜 ≥ 0
• Additive for uncorrelated systems 𝑇𝐵𝐶 𝑜 = 𝑇𝐵 𝑜 + 𝑇𝐶 𝑜 if 𝜍𝐵𝐶 = 𝜍𝐵⨂𝜍𝐶
• But no subadditivity

20

Entanglement spectrum - definition

𝜔 = 𝜇𝑗 𝑏𝑗 𝑐𝑗

𝑗

𝜇𝑗 > 0 , 𝑞𝑗 = 𝜇𝑗 2 , 𝑞𝑗

𝑗

= 1 𝑞𝑗 = exp −𝐹𝑗 𝑎 1 with Z 𝑜 = 𝑓−𝑜𝐹𝑗

𝑗

Schmidt decomposition Interpret the eigenvalues of 𝜍𝐵 as classical Boltzmann weights. This defines some “energies” 𝐹𝑗 Tr 𝜍𝐵𝑜 = 𝑞𝑗 𝑜 = 𝑎 𝑜 𝑎 1

𝑗

𝜔 = 1 𝑎 1 𝑓−𝐹𝑗

2 𝑏𝑗 𝑐𝑗 𝑗

Free energy 𝐺 𝑜 = − 1 n log Z n = − 1 n log 𝑎 1 Tr 𝜍𝐵𝑜 = 1 n 𝐺 1 − 1 n log Tr 𝜍𝐵𝑜 Rényi entropy ↔ free energy difference 𝑇𝐵 𝑜 = 1 1 − n log Tr 𝜍𝐵𝑜 = 1 𝑜 − 1 𝑜𝐺 𝑜 − 𝐺 1 𝑎 𝑜 : Partition function at inverse temperature

1 𝑈 = 𝛾 = 𝑜

21

HOW TO MEASURE ENTANGLEMENT ENTROPIES ?

at least in principle…

22

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

 What about the entanglement entropy of a large system ?

“Solving Quantum Ground-State Problems with NMR” Zhaokai Li et al., Scientific Reports 1, 88 (2012)

23

Measurement of entanglement entropies ? (1)

 A few proposals based on coupling 𝑜 copies of the system to measure the entropy 𝑇𝑜  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. 𝜔 = 𝜇𝑗 𝑚𝑗 ⨂ 𝑠𝑗

𝑗

L R Finite chain & left/right Schmidt decomposition  Goal: “measure” the Rényi entropy 𝑇𝑜=2, that is −log Tr𝑆 𝜍𝑀

2 = − log 𝜇𝑗 4 𝑗

 4 half-chains, that can be connected in two different ways: L1 L2 R1 R2

𝐻 = 𝜇𝑗 𝑚𝑗 𝑀1⨂ 𝑠

𝑗 𝑆1 𝑗

⨂ 𝜇𝑘 𝑚𝑘 𝑀2 ⨂ 𝑠

𝑘 𝑆2 𝑘

L1 R1

• r

𝐻′ = 𝜇𝑗 𝑚𝑗 𝑀1⨂ 𝑠

𝑗 𝑆2 𝑗

⨂ 𝜇𝑘 𝑚𝑘 𝑀2 ⨂ 𝑠

𝑘 𝑆1 𝑘

L2 R2 “quantum switch”

24

Measurement of entanglement entropies ? (2)

L1 L2 R1 R2

𝐻 = 𝜇𝑗 𝑚𝑗 𝑀1⨂ 𝑠

𝑗 𝑆1 𝑗

⨂ 𝜇𝑘 𝑚𝑘 𝑀2 ⨂ 𝑠

𝑘 𝑆2 𝑘

L1 R1

• r

𝐻′ = 𝜇𝑙 𝑚𝑙 𝑀1⨂ 𝑠

𝑙 𝑆2 𝑙

⨂ 𝜇𝑚 𝑚𝑚 𝑀2⨂ 𝑠

𝑚 𝑆1 𝑚

L2 R2 𝐻 𝐻′ = 𝜇𝑗𝜇𝑘𝜇𝑙𝜇𝑚𝜀𝑗𝑙𝜀𝑗𝑚𝜀

𝑘𝑚𝜀 𝑘𝑙 = 𝜇𝑗 4 𝑗

= Tr𝑆 𝜍𝑀

2 𝑗𝑘𝑙𝑚

= exp −𝑇2  Rényi entropy 𝑇𝑜 and scalar product  Introduce a weak transverse field on the central spin (quantum switch) 𝐻 𝐻′ +𝐾𝜏𝑦 𝜏𝑨 = +1 𝜏𝑨 = −1

Degenerate perturbation theory (𝐾 ≪ Δ)

𝐼eff = 𝐾 𝐻 𝐻′ 𝐾 𝐻 𝐻′ Eigenvalues 𝜕 = ±𝐾 𝐻 𝐻′ Measure the oscillation frequency 𝜕 of 𝜏𝑨(𝑢) → access to exp −𝑇2 generalization: couple 𝑜 systems and a 2-state switch to measure 𝑇𝑜 Δ Δ

25

Summary of lecture #1

A B

• Context: lattice quantum many-body problems (spin, fermions, …)
• Two large subsystems in a pure state 𝜔 ∈ ℋ

𝐵⨂ℋ𝐶.

• The total density matrix 𝜍𝐵𝐶 = 𝜔 𝜔 is a projector
• Reduced density matrix: 𝜍𝐵 = Tr𝐶 𝜍𝐵𝐶 TrA 𝜍𝐵 = TrA𝐶 𝜍𝐵𝐶 = 1
• Shmidt decomposition of 𝜔 :

𝜔 = 𝑁𝑗𝑘 𝑏𝑗 ⨂ 𝑐

𝑘 𝑗𝑘

=→ SVD of 𝑁 →= 𝜇𝑙 𝑣𝑙 ⨂ 𝑤𝑙

𝑙

• Spectrum of the RDM: 𝜍𝐵 = 𝜇𝑙 2 𝑣𝑙 𝑣𝑙

𝑙

• Von Neumann entropy

𝑇𝐵 = −Tr𝐶 𝜍𝐵 log 𝜍𝐵 = − 𝜇𝑙 2 log 𝜇𝑙 2

𝑗

• 𝑇𝐵 quantifies the uncertainty on the state of 𝐵 if we do not observe the region 𝐶.
• 𝑇𝐵 = 0 ⟺ 𝜍𝐵is a projector ⟺ 𝜔 = 𝐵 ⨂ 𝐶 is a product state
• Strong sub-additivity of the Von Neumann entropy: 𝑇𝑌∪𝑍 + 𝑇𝑌∩𝑍 ≤ 𝑇𝑌 + 𝑇𝑍
• Useful generalization: Rényi entropies 𝑇𝐵(𝑜) =

1 1−𝑜 log Tr 𝜍𝐵𝑜

• 𝑇𝐵(𝑜) is often simpler to compute (and possibly to measure experimentally) than 𝑇𝐵

when 𝑜 is an integer ≥ 2 (replicas, etc.). 𝑜 plays the role of an inverse temperature.

• Finite correlation length → area law 𝑇𝐵 ∼ 𝒫 Area of 𝜖𝐵 = 𝒫 𝑀𝑒−1 .

26

AREA/BOUNDARY LAW

for the entanglement entropy of low-energy states

• f short-ranged Hamiltonians. Decay of Schmidt

eigenvalues.

27

Area law for the entanglement entropy

 The ground-state (and low-energy excitations) of (many) Hamiltonians with short- ranged interactions have an entanglement entropy which scale like the area of the boundary of the subsystem

𝑇𝐵 ∼ 𝒫 Area of 𝜖𝐵 = 𝒫 𝑀𝑒−1

• Appears to be valid for all gapped systems (in 𝒆 = 𝟐 it’s a theorem),

and some gapless systems in 𝒆 > 𝟐(not all).

• Known gapless systems which violate the area law:
• critical systems in 𝑒 = 1
• systems in 𝑒 > 1 with a Fermi surface, where 𝑇𝐵 ∼ 𝒫 𝑀𝑒−1 log 𝑀
• 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.

“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)

A

L

B

28

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).

• Assume that the entanglement between 𝐵 and 𝐶

is entirely due to local correlations (not a very precise statement…)

• Correlations between degrees of freedom located inside 𝐵

do not contribute to 𝑇𝐵. Same for correlations inside the region 𝐶.

• The only contributions to the entanglement entropy 𝑇𝐵 are

those originating from correlations taking place across the boundary between A and B → Area/Boundary law for the entanglement entropy : 𝑇𝐵~size of 𝜖𝐵 ~𝑀𝑒−1

A B

L 𝝄

29

Area law for the entanglement entropy

 Variant of the intuitive argument … L Correlation length 𝜊

A

B 𝑗 = 1 ⋯ ~2𝜊𝑀configurations for the magenta sites ( ∈ A) 𝑘 = 1 ⋯ ~2𝜊𝑀configurations for the blue sites ( ∈ B) Assume the wave function can be approximated by : 𝜔 ∼ 𝑁𝑗,𝑘

𝑗,𝑘

𝜔𝑗

𝐵 ⨂ 𝜔𝑘 𝐶

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~2𝜊𝑀 𝑇𝐵 ≤ 𝜊𝑀 log 2 → boundary law

30

Remark about the area law & entanglement spectrum

 𝑇𝐵 = thermodynamic entropy of the entanglement spectrum 𝐹𝑗 (by definition) Assume these 𝐹𝑗 are energies of some “fictitious system” associated to the bi- partition A/B.  Since 𝑇𝐵~𝑀𝑒−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 𝜍𝐵 (hence the 𝐹𝑗 ) 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 (𝐼𝐵) (assume region B is much larger, for simplicity), we expect a volume-law behavior. If, instead, 𝑇𝐵~𝑀𝑒−1, then we expect that most of the eigenvalues of 𝜍𝐵 are much smaller than 1/dim (𝐼𝐵).

31

Mutual information

𝐵

 Definition: 𝐽 𝐵: 𝐶 = 𝑇 𝜍A + 𝑇 𝜍𝐶 − 𝑇 𝜍AB

(with 𝑇 𝜍 = −Tr 𝜍 log 𝜍 ).

 Alternatively: 𝐽 𝐵: 𝐶 = Tr 𝜍AB log 𝜍AB − log 𝜍A⨂𝜍𝐶  Thanks to the sub-additivity of the Von Neumann entropy we have 𝐽 𝐵: 𝐶 ≥ 0. A stronger result is: 𝐽 𝐵: 𝐶 ≥

1 2 𝜍AB − 𝜍A⨂𝜍𝐶 1 2 (=Pinsker inequality).

→ 𝐽 𝐵: 𝐶 can be viewed as some kind of “distance” between 𝜍AB and 𝜍A⨂𝜍𝐶. In particular: 𝐽 𝐵: 𝐶 = 0 ⇔ 𝜍AB = 𝜍A⨂𝜍𝐶.  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:

𝐽 𝐵: 𝐶 ≥ 𝑃𝐵𝑃𝐶 − 𝑃𝐵 𝑃𝐶

2

2 𝑃𝐵 1

2 𝑃𝐶 1 2

 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 (𝜍AB = 𝜔 𝜔 ) we have 𝑇 𝜍AB = 0. So: 𝐽 𝐵: 𝐶 = 2𝑇 𝜍A = 2𝑇 𝜍B 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)

𝐶

32

Mutual information & boundary law at T>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)

𝐵 𝐶  Free energy 𝐺 𝜍 = 𝑉 − 𝑈𝑇 = Tr 𝐼𝜍 −

1 𝛾 𝑇(𝜍) is minimized by 𝜍𝐵𝐶~𝑓−𝛾𝐼

with 𝐼 = 𝐼𝐵 + 𝐼𝐶 + 𝐼𝑗𝑜𝑢. 𝐼𝑗𝑜𝑢 contains all the terms which couple 𝐵 and 𝐶.  In particular: 𝐺 𝜍 ≤ 𝐺 𝜍A⨂𝜍B . So: Tr 𝐼𝜍 − 1 𝛾 𝑇 𝜍 ≤ Tr 𝐼𝐵 + 𝐼𝐶 + 𝐼𝑗𝑜𝑢 𝜍𝐵⨂𝜍𝐶 − 1 𝛾 𝑇 𝜍A + 𝑇 𝜍B 𝐽 𝐵: 𝐶 𝛾 ≤ Tr 𝐼𝑗𝑜𝑢 𝜍𝐵⨂𝜍𝐶 − 𝜍 ~𝒫(Area)

• 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. 𝐼𝑗𝑜𝑢

33

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 between is the region 𝐷. Define 𝐽𝑀 = 𝐽 𝐵: 𝐶 = mutual information  Using strong subadditivity one can show that 𝐽𝑀 is a decreasing function of 𝑀 (exercise!). Define the correlation length 𝝄 as the minimal distance which insures 𝑱𝑴 ≤

𝑱𝟏 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.

𝐶 𝐵

𝑆 𝐷

𝑀

 Use sub-additivity and the Araki-Lieb triangular inequality to show (exercise!): 𝐽 𝐵: 𝐶𝐷 ≤ 𝐽 𝐵: 𝐶 + 2𝑇𝐷  By construction 𝐽(𝐵: 𝐶𝐷) = 𝐽0 and 𝐽 𝐵: 𝐶 = 𝐽𝑀. By def. of 𝜊 we have 𝐽𝑀 ≤

𝐽0 2, so: 𝐽0 ≤ 𝐽𝑀 + 2𝑇𝐷 ≤ 𝐽0 2 + 2𝑇𝐷. Hence 𝐽0 ≤ 4𝑇𝐷. Since 𝑇𝐷 is bounded by its volume 𝐷 ∼ 𝜊 𝜖𝐵 . We get 𝐽0 ≤ 4𝜊 𝜖𝐵 .

Now if the entire system 𝐵𝐶𝐷 in a pure state (no necessary so far) we have 𝐽 𝐵: 𝐶𝐷 = 𝐽0 = 2𝑇𝐵 and finally: 𝑇𝐵 ≤ 2𝜊 𝜖𝐵

34

Area/boundary law 1d gapped system

M B Hastings J. Stat. Mech. (2007) P08024

35

Universal violations & corrections to the area law

 Violation (multiplicative log: ∼ L𝑒−1log 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 𝑀𝑒−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 𝑀𝑒−1 + 𝒫 𝑀0 )

• Discrete spontaneous sym. breaking (→contributtion S0 = 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

A few examples (non-exhaustive list)

36

VOLUME LAW

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

37

Comparison with random pure states

 “Average entropy of a subsystem”, D. Page, Phys. Rev. Lett. (1993) … = … whole Hilbert space ℋ𝐵⨂ℋ𝐶 (Haar measure) Assume dim ℋ

𝐵 ≤ dim ℋ𝐶 :

𝑇𝐵 = − Tr 𝜍𝐵 log 𝜍𝐵 = log dim ℋ

𝐵

−

dim ℋ𝐵 2dim ℋ𝐶 ~𝒫 𝑀𝑒 = Volume law

 For the full probability distribution of 𝑇𝐵, 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 (variance Var(SA)~𝒫(dim ℋ

𝐵 −2)) and the scaling of the typical entropy is a

also a volume law. → States satisfying an area law are rare  Choose a subsystem 𝐵 such that 1 ≪ dim ℋ

𝐵 = 𝑛 ≤ dim ℋ𝐶 = 𝑜.

• 1. Pick a pure state at random in ℋ

𝐵⨂ℋ𝐶, and you will find 𝑇𝐵~ log 𝑛 − 𝑛 2𝑜.

• 2. Now consider the infinite-temperature density matrix for the region 𝐵:

𝜍𝐵

𝑈=∞ = Idm m . It gives 𝑇𝐵 𝑈=∞ = log 𝑛.

• 3. Conclusion: when the region 𝐶 is large (i.e.

𝑛 𝑜 ≪ 1) you can hardly

distinguish the two situations.

38

ETH, entanglement entropy & thermal entropy (1)

 Hamiltonian 𝐼 = 𝐹𝛽

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 = 𝑑𝛽 𝛽

𝛽

 Time evolution: 𝜔(𝑢) = 𝑑𝛽𝑓−𝑗𝐹𝛽𝑢 𝛽

𝛽

 Expectation value of some observable 𝐵: 𝜔(𝑢) 𝐵 𝜔(𝑢) = 𝑑𝛽 2𝐵𝛽𝛽

𝛽

+ 𝐵𝛽𝛾𝑑𝛽𝑑𝛾𝑓−𝑗(𝐹𝛾−𝐹𝛽)𝑢

𝛽≠𝛾

 Long-time average (assume no degeneracy): 1 𝜐 𝜔(𝑢) 𝐵 𝜔(𝑢) 𝑒𝑢

𝜐

= 𝑑𝛽 2𝐵𝛽𝛽

𝛽

+ 𝑗 𝐵𝛽𝛾𝑑𝛽𝑑𝛾 𝑓−𝑗(𝐹𝛾−𝐹𝛽)𝜐 − 1 𝜐

𝛽≠𝛾

𝐵 = lim

𝜐→∞

1 𝜐 𝜔(𝑢) 𝐵 𝜔(𝑢) 𝑒𝑢

𝜐

= 𝑑𝛽 2𝐵𝛽𝛽

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: 𝜍diag 𝑑𝛽 = 𝑑𝛽 2

𝛽

𝛽 𝛽

39

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 𝐹0), here the micro-canonical one:

𝑑𝛽 2𝐵𝛽𝛽

𝛽

= Tr 𝜍diag 𝑑𝛽 𝐵 = Tr 𝜍micro 𝐹 𝐵 = 1 𝒪(𝐹, ΔE) 𝐵𝛽𝛽

𝛽 𝐹𝛽−𝐹 <Δ𝐹

𝜍micro 𝐹 = 1 𝒪(𝐹, ΔE) 𝛽 𝛽

𝛽 𝐹𝛽−𝐹 <Δ𝐹

 Remark:𝜍diag 𝑑𝛽 depends on the initial conditions, but 𝜍micro 𝐹 does not…How can that be ?  A possible/plausible explanation: “Eigenstate thermalization Hypothesis (ETH)” :

• the diagonal matrix elements 𝐵𝛽𝛽 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 𝐵𝛽𝛾 (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)

Ω Ω

40

ETH, entanglement entropy & thermal entropy (3)

 Strong form of ETH: all the observables in Ω satisfy the ETH Consequences:

• the RDM computed in some arbitrary eigenstate 𝛽

at energy 𝐹𝛽~𝐹 becomes “thermal” in the thermodynamic limit: 𝜍Ω 𝛽 = TrΩ

𝛽 𝛽 ~TrΩ 𝜍𝑛𝑗𝑑𝑠𝑝 𝐹

(with 𝐹𝛽~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 from 𝜍𝑛𝑗𝑑𝑠𝑝

Ω

𝐹 = TrΩ

𝜍𝑛𝑗𝑑𝑠𝑝 𝐹

• r 𝜍cano~

exp −𝛾𝐼Ω 𝑎(𝛾)

(adjust the inverse temperature 𝛾 to match the energy E). which describe thes region Ω, isolated , and at thermal equilibrium. Then : vol Ω 𝑔 𝑜𝛾 = − 1 𝑜𝛾 log Tr exp −𝛾𝐼Ω 𝑜 ~ − 1 𝑜𝛾 log Tr 𝑎 𝛾 𝑜 𝜍𝑛𝑗𝑑𝑠𝑝

Ω

𝐹

𝑜

~ − 1 𝛾 log 𝑎 𝛾 − 1 𝑜𝛾 log Tr 𝜍Ω 𝛽 𝑜 = − 1 𝛾 log 𝑎 𝛾 − 1 − 𝑜 𝑜𝛾 𝑇Rényi

𝑜

𝛽 Finally: 𝑔 𝑜𝛾 − 𝑔 𝛾 =

𝑜−1 𝑜𝛾vol Ω 𝑇Rényi 𝑜

𝛽 The Rényi entanglement entropies of a single eigenstate gives the thermodynamic free energy at all temperatures !

Ω Ω

1 ≪ vol Ω ≪ vol Ω

• J. R. Garrison & T. Grover arXiv:1503.00729

41

FREE PARTICLES

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

42

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

Most general (lattice) quadratic Hamiltonian: 𝐼0 = 𝐵𝑗,𝑘𝑑𝑗

†𝑑 𝑘 𝑗,𝑘

+ B𝑗,𝑘𝑑𝑗

†𝑑 𝑘 † 𝑗,𝑘

+ 𝐼. 𝑑. 𝐼0 can be diagonalized using a Bogoliubov transformation (see next slides), and the 𝑈 = 0 (as well as T>0) correlations can be calculated easily.

 𝑑𝑗

†𝑑 𝑘 → Wick theorem → any correlator

𝐷𝑗𝑘 = 𝑑𝑗

†𝑑 𝑘 𝑗,𝑘∈𝐵 → any observable in region 𝐵 → 𝜍𝐵

The reduced density 𝜍𝐵 is completely determined by two-point functions in the region

• f interest

 Correlators in region 𝐵 obeys Wick’s theorem → 𝜍𝐵 is Gaussian → ∃ quadratic « Hamiltonian » 𝐼 such that 𝜍𝐵 = 1

𝑎 exp −𝐼

with 𝐼 = ℎ𝑗,𝑘𝑑𝑗

†𝑑 𝑘 𝑗,𝑘

+ Δ𝑗,𝑘𝑑𝑗

†𝑑 𝑘 † 𝑗,𝑘

+ H. c.

Remark: 𝐼 depends on the subsystem 𝐵 (should be noted 𝐼

𝐵…), it is not the physical Hamiltonian 𝐼0.

→ The entanglement entropy 𝑇𝐵 = −Tr 𝜍𝐵 log 𝜍𝐵 is the thermodynamic entropy of 𝐼 at (fictitious) temperature 1.

43

Free particles & Peschel’s trick (2)

 Correlation matrix 𝐷 = 𝑑𝑗

†𝑑 𝑘

𝑑𝑗

†𝑑 𝑘 †

𝑑𝑗𝑑

𝑘

𝑑𝑗𝑑

𝑘 †

= 𝑑† 𝑑 𝑑 𝑑† dim. 2𝑂 × 2𝑂, 𝐷 = 𝐵 𝐸 𝐸† 1 ± 𝐵𝑢 , 𝐸𝑗𝑘 = 𝑑𝑗

†𝑑 𝑘 † , 𝐵𝑗𝑘 = 𝑑𝑗 †𝑑 𝑘

 We look for new creation/annihilation operators 𝑏,𝑏† (=Bogoliubov transformation): 𝑏† 𝑏 = 𝑉 𝑑† 𝑑 with 𝑉 ∓1 1 𝑉† = ∓1 1 = Σ Upper sign for boson, lower sign for fermions Such that the correlations of the new “particles” are diagonal : 𝑏𝑙

†𝑏𝑙′

𝑏𝑙

†𝑏𝑙′ †

𝑏𝑙𝑏𝑙′ 𝑏𝑙𝑏𝑙′

†

𝜇𝑙𝜀𝑙,𝑙′ 1 ± 𝜇𝑙 𝜀𝑙,𝑙′ = Λ = 𝑉𝐷𝑉† So, the linear algebra problem we have to solve is 𝑉𝐷𝑉† = Λ 𝑉Σ𝑉† = Σ  One method is to diagonalize the matrix 𝐷Σ = ∓ 𝑑𝑗

†𝑑 𝑘

𝑑𝑗

†𝑑 𝑘 †

∓ 𝑑

𝑘𝑑 𝑘

𝑑𝑗𝑑

𝑘 †

. Indeed, combining the equations above we get 𝑉𝐷Σ𝑉−1 = ΛΣ. By construction, the correlations of the new particles are very simple: 𝑏𝑙

†𝑏𝑙′ = 𝜀𝑙,𝑙′𝜇𝑙 and 𝑏𝑙𝑏𝑙′ = 𝑏𝑙 †𝑏𝑙′ †

= 0. We have found some independent “correlation modes”.

44

Free particles & Peschel’s trick (3)

 Exercise: When 𝐵 and 𝐸 are real, show that the following (smaller: 𝑂 × 𝑂) diagonalization gives the eigenvalues λ : Eigenvalues of 𝐵 ± 1 2 + 𝐸 𝐵 ± 1 2 − 𝐸 = 𝜇𝑙 ± 1 2

2

• cf. Lieb, Schultz & Mattis, Ann. Phys., NY 16 407 (1961)

 𝐷 = 𝑑𝑗

†𝑑 𝑘

𝑑𝑗

†𝑑 𝑘 †

𝑑

𝑘𝑑 𝑘

𝑑𝑗𝑑

𝑘 †

= 𝐵 𝐸 𝐸† 1 ± 𝐵𝑢

45

Free particles & Peschel’s trick (4)

Recall: 𝑏𝑙

†𝑏𝑙′ = 𝜀𝑙,𝑙′𝜇𝑙 & 𝑏𝑙𝑏𝑙′ = 𝑏𝑙 †𝑏𝑙′ †

= 0  Question: what is the (quadratic) “Hamiltonian” 𝐼 such that the correlations above can be

• btained through some “thermal” average: ⋯

=

1 𝑎 Tr ⋯ 𝑓−𝐼 ?

 Answer: Since the modes are uncorrelated, 𝐼 must be diagonal in terms of the 𝑏 and 𝑏† and take the form 𝐼 = 𝜗𝑙𝑏𝑙

†𝑏𝑙 𝑙

. One then has to adjust the « pseudo energies » 𝜗𝑙 so that 𝑏𝑙

†𝑏𝑙 = 𝜇𝑙:

𝑏𝑙

†𝑏𝑙 = 𝜇𝑙 =

1 𝑓𝜗𝑙 − 1 bosons 1 𝑓𝜗𝑙 + 1 fermions ⟹ 𝜗𝑙= log 1 ± 𝜇𝑙 𝜇𝑙  Which finally gives the RDM: 𝜍𝐵~ exp − log 1 ± 𝜇𝑙 𝜇𝑙

𝑙

𝑏𝑙

†𝑏𝑙

 And the (entanglement) entropy is given by the sum of the contributions of each mode: 𝑇 = 𝑇 𝜗𝑙

𝑙

with 𝑇(𝜗) = − log 1 − 𝑓−𝜗 + 𝜗 𝑓𝜗 − 1 bosons + log 1 + 𝑓−𝜗 + 𝜗 𝑓𝜗 + 1 = −𝜇 log 𝜇 − 1 − 𝜇 log 1 − 𝜇 fermions  Remark: For fermions, the eigenvalues 𝜇 = 0 or 1 do not contribute to the entropy (𝜁 = ±∞). For bosons, 𝜇 = 0 do not contribute.

46

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 𝜔0 = 𝑏 ⨂ 𝑐 A B

HA and HB: free fermions (= 𝑑𝑗

†𝑑𝑗+1 + ℎ. 𝑑. ) 𝑗

𝑏 = g.s. of HA 𝑐 = g.s. of HB Unitary evolution to 𝑢 > 0 : 𝜔(𝑢) = exp −𝑗𝑢𝐼 𝜔0 𝐼 = 𝐼

𝐵 + 𝐼𝐶 + 𝐼𝑗𝑜𝑢

𝐼𝑗𝑜𝑢: hopping between the A & B 𝜍𝐵 𝑢 = 𝑈𝑠

B 𝜔 𝑢 𝜔 𝑢 remains “Gaussian”

𝑇𝐵(𝑢) = −Tr𝐶 𝜍𝐵 𝑢 log 𝜍𝐵 𝑢 can be computed with Peschel’s trick, using the (time- dependent) 2-point correlations.

L=200 sites (100+100)

Remark: we observe a logarithmic growth of 𝑇𝐵(𝑢) (holds as long as t ≲ 𝑀/2)

47

Free fermions on a chain – correlation matrix

 Ground-state on a periodic chain 𝐼 = − 1 2 𝑑𝑜

†𝑑𝑜+1 + 𝑑𝑜+1 †

𝑑𝑜

𝑜

= − cos 𝑙 𝑑𝑙

†𝑑𝑙 𝑙

𝜔 = 𝑑𝑙

† vacuum −𝑙𝐺≤𝑙≤𝑙𝐺

𝑑𝑙

† = 1

𝑀 𝑓−𝑗𝑙𝑛𝑑𝑛

† 𝑀−1 𝑜=0

 Two-point correlations → matrix 𝐵𝑜,𝑛 𝜔 𝑑𝑜

†𝑑𝑛 𝜔 =

0 𝑑𝑙𝑑𝑜

† 0 0 𝑑𝑛𝑑𝑙 † 0 −𝑙𝐺≤𝑙≤𝑙𝐺

= 1 𝑀 𝑓𝑗𝑙(𝑜−𝑛)

−𝑙𝐺≤𝑙≤𝑙𝐺

𝐵𝑜,𝑛 = lim

𝑀→∞ 𝜔 𝑑𝑜†𝑑𝑛 𝜔 =

𝑒𝑙 2𝜌 𝑓𝑗𝑙(𝑜−𝑛) = sin 𝑙𝐺(𝑜 − 𝑛) 𝜌(𝑜 − 𝑛)

𝑙𝐺 −𝑙𝐺

𝑙

𝜌 −𝜌 𝜗𝑙 = − cos 𝑙

48

Free fermions on a chain – determinant

Jin & Korepin, J. Stat Phys. 116, 79 (2004)

 Recall: Correlation matrix: 𝐵𝑜,𝑛 = lim

𝑀→∞ 𝜔 𝑑𝑜 †𝑑𝑛 𝜔 = 𝑒𝑙 2𝜌 𝑓𝑗𝑙(𝑜−𝑛) = sin 𝑙𝐺(𝑜−𝑛) 𝜌(𝑜−𝑛) 𝑙𝐺 −𝑙𝐺

Entanglement entropy: 𝑇 = 𝑇 𝜇𝑙

𝑙

= sum over the eigenvalues of 𝐵 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 − 𝜇𝑙

𝑙

log det 𝜇𝕁 − 𝐵 = log 𝜇 − 𝜇𝑙

𝑙

𝑒 log det 𝜇𝕁 − 𝐵 𝑒𝜇 = 1 𝜇 − 𝜇𝑙

𝑙

𝑇 = 𝑇 𝜇𝑙

𝑙

= 𝑒𝜇 2𝑗𝜌 𝑇 𝜇 𝑒 log det 𝜇𝕁 − 𝐵 𝑒𝜇 Im(λ) Re(λ) 1

49

Fisher-Hartwig conjecture

Asymptotic behavior of the det. of Toeplitz matrices with singular symbol (simplified version…)  𝐵𝑛,𝑜 = 𝐵 𝑛 − 𝑜 parametrized as Fourier coefficients: A 𝑙 =

𝑒𝜄 2𝜌 𝜚 𝜄 𝑓−𝑗𝑙𝜄 2𝜌

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) 𝛾𝑠and (real) 𝜄𝑠: 𝜚 𝜄 = 𝜔 𝜄 exp −𝑗𝛾𝑠 ∗ arg 𝑓𝑗 𝜌− 𝜄+𝜄𝑠

𝑆 𝑠=1

with arg ∈] − 𝜌, 𝜌] 𝜔 𝜄 : smooth, and no winding → pointwise singularity at each 𝜄𝑠: 𝜚 𝜄 = 𝜄𝑠 + 𝜗 𝜚 𝜄 = 𝜄𝑠 − 𝜗 = exp (−𝑗𝛾𝑠 ∗ (𝜌 + 𝜗)) exp (−𝑗𝛾𝑠 ∗ (−𝜌 + 𝜗)) → exp −2𝑗𝛾𝑠 −2𝑗𝛾𝑠 = log 𝜚 𝜄𝑠 + 𝜗 𝜚 𝜄𝑠 − 𝜗  The Fisher-Hartwig’s conjecture describes the symptotic behavior of the det 𝐵 : log det 𝐵 = 𝑀 𝑒𝜄 2𝜌 log 𝜚 𝜄

2𝜌

+ −log 𝑀 𝛾𝑠 2

𝑆 𝑠=1

+ ⋯

50

Fisher-Hartwig conjecture & 1d Fermi sea (1)

 Recall: 𝐵𝑜,𝑛 =

𝑒𝑙 2𝜌 𝑓𝑗𝑙(𝑜−𝑛) 𝑙𝐺 −𝑙𝐺

=

sin 𝑙𝐺(𝑜−𝑛) 𝜌(𝑜−𝑛)

=

𝑒𝑙 2𝜌 𝜚 𝜄 𝑓𝑗𝑙(𝑜−𝑛) 2𝜌

→ Symbol for 𝜇𝕁 − 𝐵: 𝜚 𝜄 = 𝜇 − 1 if 𝜄 ∈ −𝑙𝐺, 𝑙𝐺 𝜇 otherwise → 2 discontinuities in𝜄 = −𝑙𝐺 and𝜄 = +𝑙𝐺: 𝛾1 = 1 2𝑗𝜌 log 𝜇 − 1 𝜇 𝛾2 = 1 2𝑗𝜌 log 𝜇 𝜇 − 1 = −𝛾1  Use Fisher-Hartwig: log det 𝜇 − 𝐵 = 1 2𝜌 2𝑙𝐺 log 𝜇 − 1 + 2(𝜌 − 𝑙𝐺) 𝑀 − 2 𝛾1 2 log 𝑀 + ⋯ 𝑒log det 𝜇 − 𝐵 𝑒𝜇 = 1 2𝜌 2𝑙𝐺 𝜇 − 1 + 2(𝜌 − 𝑙𝐺) 𝜇 𝑀 − 4𝛾1 𝑒𝛾1 𝑒𝜇 log 𝑀 + ⋯ 𝑒𝛾1 𝑒𝜇 = 1 2𝑗𝜌 𝜇 𝜇 − 1 1 𝜇 − 𝜇𝑙

𝑙

=

𝑒log det 𝜇 − 𝐵 𝑒𝜇 = 1 2𝜌 2𝑙𝐺 𝜇 − 1 + 2(𝜌 − 𝑙𝐺) 𝜇 𝑀 − 4 1 2𝑗𝜌 2 log 𝜇 − 1 𝜇 1 𝜇 𝜇 − 1 log 𝑀 + ⋯ 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 (𝜇𝑙= 𝑏𝑙

†𝑏𝑙 =1) or completely empty (𝜇𝑙= 𝑏𝑙 †𝑏𝑙 =0) and do not contribute to the entanglement.

51

Fisher-Hartwig conjecture & 1d Fermi sea (2)

𝑒log det 𝜇 − 𝐵 𝑒𝜇 = 1 2𝜌 2𝑙𝐺 𝜇 − 1 + 2(𝜌 − 𝑙𝐺) 𝜇 𝑀 − 4 1 2𝑗𝜌 2 log 𝜇 − 1 𝜇 1 𝜇 𝜇 − 1 log 𝑀 + ⋯ 𝑇 = 𝑒𝜇 2𝑗𝜌 −𝜇 log 𝜇 − 1 − 𝜇 log (1 − 𝜇) 𝑒log det 𝜇 − 𝐵 𝑒𝜇  The only contribution to this contour integral is the discontinuity of log 𝜇 − 1 on the real axis: log 𝜇 − 1 + 𝑗0− − log 𝜇 − 1 + 𝑗0+ = −2𝑗𝜌 𝑇 = 0 × 𝑀 − 4 −2𝑗𝜌 2𝑗𝜌 3 log 𝑀 𝑒𝜇 −𝜇 log 𝜇 − 1 − 𝜇 log 1 − 𝜇 𝜇 𝜇 − 1

1

𝑇 = + 1 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

• ccupation number in Fourier space ← algebraic decay of the correlations

= − 𝜌2 3

Im(λ) Re(λ) 1

52

Entanglement in a free fermion chain & S~log(L)/3

subsystem A x

Total length: L

Same data (red dots), compared with the log of the “chord” distance (green curve) : 𝑒 𝑦 = 2𝑀 sin 𝜌𝑦 𝑀

𝑒 𝑦

53

Summary of lecture #2

 Mutual information 𝐽 𝐵: 𝐶 = 𝑇 𝜍A + 𝑇 𝜍𝐶 − 𝑇 𝜍AB . 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 entanglement in highly excited pure states: 𝑇thermo

A

(E) = SVonNeumann

A

( 𝛽 )  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):

• Map SVonNeumann

A

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.
• 𝑇(𝑀) =

1 3 log

(𝑀). Universal coefficient (only depends on the number of Fermi points, not on the details of the dispersion relation nor the density).

54

Entanglement in free fermion chain: gap versus gapless

subsystem A

𝑦

Total length: 𝑀 ≫ 𝑦

Dimerized free-fermion chain (2-site unit cell): H = −𝒖 𝑑2𝑜

† 𝑑2𝑜+1 + 𝑑2𝑜+1 †

𝑑2𝑜

𝑜

− 𝒖′ 𝑑2𝑜+1

†

𝑑2𝑜+2 + 𝑑2𝑜+2

†

𝑑2𝑜+1

𝑜

𝒖 𝒖′

2𝑜 2𝑜 + 1 𝐹 𝑙 = ± 𝑢 − 𝑢′ 2 + 4𝑢𝑢′ cos 𝑙 2 gap Δ = 2 𝑢 − 𝑢′ when the chemical potential is at 𝐹 = 0 → band insulator at half-filling if 𝑢 ≠ 𝑢′

+ 𝜌 2 𝑙 𝐹 𝑙 − 𝜌 2

Δ

Band structure ∼ constant

∼ 1 3 log 𝑦

𝑻𝑩(𝒚) 𝒚

𝑇𝐵 is qualitatively different in the gapped and gapless cases

55

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)

56

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)

 Free-fermion tight-binding model 𝐼 = 𝑢𝑗𝑘 𝑑𝑗

†𝑑 𝑘 + h. c. 𝑗,𝑘

= 𝜗𝑙𝑑𝑙

†𝑑𝑙 𝑙∈1st BZ

1st Brillouin zone

𝑒𝑙

Fermi sea

𝑜𝑙

subsystem A

𝑒𝑚

Real space

 Contribution of the modes 𝑒𝑙 and boundary element 𝑒𝑚 to the entanglement entropy 𝑇𝐵 ? → Idea: model this contribution by decoupled chains running parallel to the 𝑜𝑙direction (insures the same propagation direction for the low-energy modes): 𝑜𝑌 𝑜𝑙 𝑒𝑚

Lattice spacing 𝑏

57

Boundary law violation in presence of a Fermi surface

subsystem A

𝑒𝑚

Real space 1st Brillouin zone

𝑒𝑙

Fermi sea

𝑜𝑙

• Number of chains crossing the boundary:

𝑒𝑂 = 𝑒𝑚 𝑏 𝑜𝑙. 𝑜𝑌

• Entropy contribution of each chain:

𝑒𝑇𝐵

• decoup. chains = 1

6 log 𝑀 × 𝑒𝑂

• Correct by the length of the element along the Fermi surface

(relative to that of the decoup. Chain model)

𝑒𝑇𝐵 = 𝑒𝑇𝐵

• decoup. chains ×

𝑒𝑙 2 × 2𝜌 𝑏 = 1 12 log 𝑀 × 𝑒𝑚 𝑏 𝑜𝑙. 𝑜𝑌 × 𝑏𝑒𝑙 2𝜌 Integrating on the real space boundary and Fermi surface: Turns out to be exact ! (and related to Windom’s conjecture)

𝑜𝑙 𝑒𝑚

Lattice spacing 𝑏

𝑜𝑌 𝑇𝐵 = 1 12 log 𝑀 𝑒𝑚 𝑒𝑙 2𝜌 𝑜𝑙. 𝑜𝑌 ~ 𝒫 𝑀 log 𝑀

58

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]

59

Critical systems in 1d & CFT

… the celebrated S~ 𝑑

3 log 𝑀 formula

60

Entanglement & CFT

• Nucl. Phys B424 (1994)
• J. Stat. Mech 2004

S~ 𝑑 3 log 𝑀

+Review: Calabrese & Cardy, J. Phys. A 42, 504005 (2009)

61

Numerics & entanglement in critical spin chains

Non-critical Ising chain 𝑇~𝑑𝑡𝑢. Critical Ising chain 𝑇~

1 6 log 𝑀 , 𝑑 = 1 2

Critical XX chain 𝑇~ 1

3 log 𝑀 , 𝑑 = 1

c=1/2 (ICTF) c=1 (XX) Gapped (ICTF)

SA L

“Entanglement in Quantum Critical Phenomena”

• G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, (2003)

62

Quantum system in d=1 & partition functions in d=2

L Imaginary time 𝛾

𝑐 𝑏  Thermal density matrix 𝜍 = 1

𝑎 exp −𝛾𝐼 , with 𝑎 = Tr 𝑓−𝛾𝐼

𝑏 𝜍 𝑐 = 1

𝑎 × cylinder “partition function”

with boundary conditions 𝑏 and 𝑐 𝑎: torus partition function 𝛾 → ∞

𝑑

 Ground-state wave function given by infinitely long cylinder partition functions :

lim

𝛾→∞ exp −𝛾𝐼 = 𝑓−𝛾𝐹0 𝜔 𝜔

𝜔 ~ lim

𝛾→∞ exp

(−𝛾𝐼) 0

Any state with some overlap with the ground-state

~ 𝑑 𝜔

Functional/path integral ↔ imaginary time evolution

63

Critical system in d=1 & CFT

 Critical 1d system:

• gapless in the thermodynamic limit
• (some) correlation functions decay algebraically with distance

 Examples

• Critical Ising chain in transverse field 𝐼 = − 𝜏𝑜

𝑨𝑇𝑜+1 𝑨 𝑜

− h 𝑇𝑜

𝑦 𝑜

with ℎ = 1

• Spin-

1 2 XXZ spin chain 𝐼 =

𝑇𝑜

𝑦𝑇𝑜+1 𝑦

+ 𝑇𝑜

𝑧𝑇𝑜+1 𝑧 𝑜

+ Δ 𝑇𝑜

𝑨𝑇𝑜+1 𝑨 𝑜

with Δ ∈] − 1,1]

• 1d Hubbard model 𝐼 = −𝑢

𝑑↑,𝑜

† 𝑑↑,𝑜+1 + 𝑑↓,𝑜 † 𝑑↓,𝑜+1 + H. c. 𝑜

+𝑉 𝑑↑,𝑜

† 𝑑↑,𝑜 + 𝑑↓,𝑜 † 𝑑↓,𝑜 + H. c. 𝑜

• Edge of a quantum 2d Hall system
• …

 Continuum limit and CFT

Luttinger liquids

L 𝛾

𝑐 𝑏

The universal / long-distance properties of the system are obtained by replacing the microscopic density matrix 𝜍 =

1 𝑎 exp −𝛾𝐼

by the corresponding cylinder CFT partition function (with the appropriate boundary conditions, corresponding to “coarse grained” versions of the states 𝑏 and 𝑐) 𝑏 exp −𝛾𝐼 𝑐 ≈

64

Reduced density matrix & 2d partition functions

𝜍𝐵 = Tr𝐶 𝜍 𝑏 𝜍𝐵 𝑏′ = 𝑏𝑐 𝜍 𝑏′𝑐

𝑐

A B L 𝑏′ 𝑏 𝑐 𝑐 𝑏𝑐 𝜍 𝑏′𝑐 = 1 𝑎 A B

periodic

𝑏 𝜍𝐵 𝑏′ = 1 𝑎 𝑏 𝑏′ 𝑐 𝛾 Tracing out the region B A 𝜍𝐵 = 1 𝑎

65

Rényi entropy & partition functions

 Reduced density matrix to the power n

Tr 𝜍𝐵𝑜 =

1 𝑎𝑜

…

1 2 n  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.

1A 2B 2A 1B nB nA

A 𝜍𝐵 = 1 𝑎

𝑎: cylinder partition function

66

Entanglement & CFT (2)

Holzhey, Larsen & Wilcek, Nucl. Phys B 424 (1994)

𝑥 = − sin 𝜌 𝜂 − 𝑦 𝑀 sin 𝜌 𝜂 𝑀 𝜗2 𝑦 𝑦 𝜗1 −𝑦 𝜗1 𝜗1 & 𝜗2 : UV cut-off (~lattice spacing) Im(𝑥) Re(𝑥)

half-plane Assume 𝑀 ≫ 𝑦 for simplicity 𝜂 ∈ half−infinite cylinder

L 𝛾 → ∞ 𝑦 𝑀 − 𝑦 A B

𝜗2 𝜗1

𝑏𝑐 𝜔 ~

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.

67

Entanglement & CFT (3) – mapping to a conical singularity

Holzhey, Larsen & Wilcek, Nucl. Phys B424 (1994)

𝜗2 𝑦 𝑦 𝜗1 −𝑦 𝜗1

Wave function

𝑏𝑐 𝜔 ~ 𝑏 𝑐

Density matrix

𝑏 𝜍𝐵 𝑏′ ~ 𝜗2 𝑦 𝑦 𝜗1 −𝑦 𝜗1 𝑏′ 𝑏 𝜍𝐵

𝑜 ∼

(n levels) Tr 𝜍𝐵

𝑜=1−𝛽 =

𝑎cone angle 2𝑜 𝜌 𝑎disk 𝑜 log Tr 𝜍𝐵

𝑜

= 𝑑 6 1 𝑜 − 𝑜 log 𝑦 𝜗

(this can be calculated by mapping the disk to the cone using 𝑨 → w z = zn, and compute the associated Shwarzian derivative – see next slide)

Conical singularity (

𝜗 𝑦 → 0), with angle

deficit 2𝛽 𝜌.

68

Entanglement & CFT (4) – Free energy of a cone

Mapping from the disk to the cone: 𝑨 → 𝑥 𝑨 = 𝑨𝑜 𝑥′ = 𝑜𝑨𝑜−1 𝑥′′ = 𝑜 𝑜 − 1 𝑨𝑜−2 Stress energy tensor (holomorphic part) in the disk geometry: 𝑈𝑒(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: 𝑈

𝑑 𝑥 =

1 𝑥′ 2 𝑈𝑒 𝑨 − 𝑑 12 𝑥′′′ 𝑥′ − 3 2 𝑥′′ 𝑥′

2

Here we find 𝑈

𝑑 𝑥 = 𝑈𝑒(𝑨) 𝑥′ 2 + 𝑑 24𝑥2 1 − 1 𝑜2 . Integrating the stress energy tensor (times 𝑥) along the dashed line gives

the variation of log𝑎𝑑 (𝑜) with respect to the outer radius 𝑆 (keeping the inner radius 𝑠 fixed) : 𝜖 log𝑎𝑑(𝑜) 𝜖 log 𝑆 = 𝑗 2𝜌 𝑥𝑒𝑥 𝑈

𝑑 𝑥

+ H. c. And from the relation above between 𝑈𝑒(𝑨) and 𝑈

𝑑 𝑥 one can show that :

𝜖 log 𝑎𝑑(𝑜)/𝑎𝑒 𝜖 log 𝑆 = 𝑗 2𝜌 𝑥𝑒𝑥 𝑑 24𝑥2 1 − 1 𝑜2 + H. c. = c 12 1 𝑜 − 𝑜 So we have log 𝑎𝑑(𝑜)/𝑎𝑒 =

c 12 1 𝑜 − 𝑜 log𝑆. By conformal invariance, this should in fact be a function or 𝑆 𝑠 and

therefore: log 𝑎𝑑(𝑜)/𝑎𝑒 =

c 12 1 𝑜 − 𝑜 log 𝑆 𝑠 = c 6 1 𝑜 − 𝑜 log 𝑦

𝜗. This is the result announced on the previous slide.

𝑆 = 𝑦/𝜗 𝑠 = 𝜗/𝑦 𝑨 𝑥 disk cone 𝑆/𝑠 = 𝑦 𝜗

2

69

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 𝜍𝐵𝑜 = 𝑎(𝑜)/𝑎(1)𝑜 log Tr 𝜍𝐵𝑜 = log 𝑆𝑜 = log 𝑎(𝑜) − 𝑜 log 𝑎 1 ~ 𝑑 6 1 𝑜 − 𝑜 log 𝑦 𝜗  From this, what can be said about the eigenvalues of 𝜍𝐵 ?

• Density of “states”: 𝑄 𝜇 = 𝜀(𝜇 − 𝜇𝑗)

𝑗

, with 𝜇𝑗: eigenvalues of 𝜍𝐵 The moments 𝑆𝑜 = 𝜇𝑗

𝑜 𝑗

have a relatively simple dependence on 𝑜: 𝑆𝑜 = e−𝑐 𝑜−1

𝑜 with 𝑐 = 𝑑

6 log 𝑦 𝜗

• After a some mathematical manipulations… one obtains the (CFT) density of

eigenvalues : with 𝜇max = 𝑓−𝑐 =

𝑦 𝜗 −𝑑

6 (largest eigenvalue)

70

Entanglement spectrum in a 1d critical system

• P. Calabrese & A. Lefevre, Phys. Rev. A 78, 032329 (2008)

𝑡 𝑁 = 𝜇𝑗

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…

71

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).

72

Matrix-product states

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

73

Matrix-product states

• Ann. of Phys. 326, 96-192 (2011)

Review on DMRG (and MPS) “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

74

MPS & canonical Vidal’s form (1)

1 𝑀 𝑁

• 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 = 𝑁𝜏1,…,𝜏𝑀 𝜏1, … , 𝜏𝑀

𝜏1,…,𝜏𝑀,=↑,↓

• Split the chain in two parts [1 … 𝑜] − [𝑜 + 1 … 𝑀],

and define the associated Shmidt basis and singular values: 𝜔 = 𝜇𝑚

𝑜 Φ𝑚 [1⋯𝑜] ⊗ Φ𝑚 [𝑜+1⋯𝑀] 𝑚

• Graphically:
• In practice this decomposition can be obtained by « reshaping » M as rectangular matrix
• f size 2𝑜 ∗ 2𝑀−𝑜 : 𝑁𝜏1,…,𝜏𝑀 = 𝑁 𝜏1,…,𝜏𝑜 ,(𝜏𝑜+1,…,𝜏𝑀) 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. 𝑜 + 1 … 𝑀 1 𝑜 Φ𝑚

[1⋯𝑜]

Φ𝑚

[𝑜+1⋯𝑀]

𝜇𝑚

𝑜

𝜔 = 2𝑀 coefficients

75

MPS & canonical Vidal’s form (2)

• Compare the Schmidt decompositions on two successives bonds:

𝑜 + 1 … 𝑀 𝑜 Φ𝑛

[1⋯𝑜]

Φ𝑛

[𝑜+1⋯𝑀]

𝜇𝑛

𝑜

𝜔 = 1 𝑜 − 1 𝑜 − 1 Φ𝑚

[1⋯𝑜−1]

Φ𝑚

[𝑜⋯𝑀]

𝜇𝑚

𝑜−1

1 𝑜 𝑜 + 1 … 𝑀

• Write the Schmidt state Φ𝑚

[𝑜⋯𝑀] in the orthogonal basis ↑ 𝑜, ↓ 𝑜 ⨂ 𝜇𝑛 𝑜 Φ𝑚 [𝑜+1⋯𝑀]

• This defines on (each site 𝑜) two matrices Γ 𝑜 ↑ and Γ 𝑜 ↓ of dimension 𝜓 ∗ 𝜓:

Φ𝑚

[𝑜⋯𝑀] =

Γ

𝑚,𝑛 𝑜 𝜏 𝜏 𝑜 𝜇𝑛 𝑜 Φ𝑛 [𝑜+1⋯𝑀] 𝜏=↑,↓ 𝑛=1 …𝜓

76

MPS & canonical Vidal’s form (2)

• Graphically :

Φ𝑚

[𝑜⋯𝑀]

𝑚 = 𝑜 + 1 … 𝑀 Φ𝑛

[𝑜+1⋯𝑀]

𝜇𝑛

𝑜

𝑚 𝑛 𝑛 Γ𝑚,𝑛

𝑜 𝜏

𝜏

• Repeat the procedure along the chain to construct all the matrices Γ 𝑜 ↑ and Γ 𝑜 ↓ from

the Schmidt basis.

• Finally:

𝜇[1]

𝑙 𝑙 Γ 1 𝜏1 Γ 2 𝜏2

𝜇[2]

𝑚 𝑚 Γ 3 𝜏3

𝜇[3]

𝑛 𝑛

𝜇[𝑀−1]

Γ 𝑀 𝜏𝑀

… 𝑨 𝑨 𝜔 =

𝜏1,𝜏2,…,𝜏𝑀=↑,↓ 𝑙,𝑚,𝑛,…,𝑨=1…𝜓

= Γ𝑙

1 𝜏1𝜇𝑙 1 Γk,l 2 𝜏2𝜇𝑚 2 Γl,𝑛 3 𝜏2𝜇𝑛 3 ⋯ 𝜇𝑨 𝑀−1 Γ𝑨 𝑀−1 𝜏𝑀 𝜏1 𝜏2 ⋯ 𝜏𝑀 𝜏1,𝜏2,…,𝜏𝑀=↑,↓ 𝑙,𝑚,𝑛,…,𝑨=1…𝜓

• Encodes all the left-right Schmidt decomposition
• Storage: ∼ 𝑀 ∗ 𝜓 + 2𝜓2 ≪ 2L if 𝜓 can be kept 𝒫 1 [gapped system] or, at worse, 𝒫 𝑀𝛽

[critical]

=Schmidt vector #𝑚 for the [1 … 𝑜 − 1][𝑜 … 𝑀] partition =Schmidt vector #𝑛 for the [1 … 𝑜][𝑜 + 1 … 𝑀] partition

77

MPS & canonical Vidal’s form (3)

=Canonical MPS form  Allows to reconstruct the Schmidt decomposition for any left/right partition

𝑙 Γ 𝑜 𝜏𝑜 Γ 𝑀 𝜏𝑀

…

𝑙

n … L

Φ𝑙

[𝑜…𝑀]

=  Orthogonality of the Schmidt vectors

𝑙′ Γ 𝑜 Γ 𝑀

…

𝑙 Γ 𝑜 Γ 𝑀

…

𝜏𝑜,𝜏𝑜+1,…,𝜏𝑀=↑,↓

= 𝜀𝑙𝑙′

nb: automatically insures 𝜔 𝜔 = 1 𝜇[1]

𝑙 𝑙 Γ 1 𝜏1 Γ 2 𝜏2

𝜇[2]

𝑚 𝑚 Γ 3 𝜏3

𝜇[3]

𝑛 𝑛

𝜇[𝑀−1]

Γ 𝑀 𝜏𝑀

… 𝑨 𝑨 Φ𝑙

[𝑜…𝑀] Φ𝑙′ [𝑜…𝑀] =

78

MPS & canonical Vidal’s form (4)

 Local observables. Example of a 2-spin operator = 𝜏′𝑗𝜏′𝑘 𝑇 𝑗 ⋅ 𝑇 𝑗+1 𝜏𝑗𝜏

𝑘

= 𝜔 𝑇 𝑗 ⋅ 𝑇 𝑗+1 𝜔

𝜏1 𝜏2 𝜏3 𝜏𝑀 Γ 1 Γ 2 Γ 3 Γ 𝑀

…

Γ 1 Γ 2 Γ 3 Γ 𝑀

… 𝑇 2 ⋅ 𝑇 3

𝜏′2 𝜏′3

𝜏𝑗 𝜏

𝑘

𝜏′𝑗 𝜏′𝑘 𝑇 𝑗 ⋅ 𝑇

𝑘

Γ 2 Γ 3 𝜏2 𝜏3

=

Γ 2 Γ 3

𝑇 2 ⋅ 𝑇 3

𝜏′2 𝜏′3

Only local operations required

79

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” 𝑉𝑜,𝑜+1 acting on sites 𝑜 and 𝑜 + 1 :

𝜇𝑚

𝑜−1

𝑉𝑜,𝑜+1 𝜔 = Γ 𝑜 𝜇𝑛

𝑜

Γ 𝑜+1 𝜇𝑛

𝑜+1

𝜏𝑜+1

𝑉𝑜,𝑜+1

𝜏𝑜

• The schmidt values 𝜇𝑚

𝑜−1 and Schmidt basis Φ𝑚 [1⋯𝑜−1] are not modified by 𝑉𝑜,𝑜+1

• Same for 𝜇𝑛

𝑜+1 and the basis Φ𝑚 [𝑜+2⋯𝑀]

• Only Γ 𝑜 , Γ 𝑜+1 𝜇𝑛

𝑜 need to be updated → fast local updates 𝒫 𝜓3

 Using small time steps, the observation above can be used to compute the real time evolution (here for nearest-neighbor spin-spin interactions):

Γ 𝑜−1 Γ 𝑜+2 Φ𝑚

[1⋯𝑜−1]

Φ𝑚

[𝑜+2⋯𝑀]

80

Summary of lecture #3

 Dimerized (free fermion) chain: “metal versus band insulator”  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

𝜇[1]

𝑙 𝑙 Γ 1 𝜏1 Γ 2 𝜏2

𝜇[2]

𝑚 𝑚 Γ 3 𝜏3

𝜇[3]

𝑛 𝑛

𝜇[𝑀−1]

Γ 𝑀 𝜏𝑀

… 𝑨 𝑨

+ 𝜌 2 𝑙 𝐹 𝑙 − 𝜌 2

Δ

𝒖 𝒖′

81

Tensor-product states in d>1

“A practical introduction to tensor networks: Matrix product states and projected entangled pair states” Román Orús, Ann. of Phys. 349, 117 (2014)

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.

82

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)

83

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” (Z2) phase

84

Area law in 2d – spin ½ Heisenberg model

Magnetic long-range

• rder ↔ spontaneous

SU(2) symmetry braking → gapless spin waves (Goldstone modes) → additive log(L) correction to the entanglement entropy.

“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)

Metlitsky-Grover [2011] →the coefficient of the log (𝑚) term is 𝑑 =

𝑂𝐻 2 (in 𝑒 = 2) where 𝑂𝐻 is the number

• f Nambu-Goldstone modes (consistent with the numerics above).

area law coeff.

85

Area law in 2d – quantum dimer model

𝜔 = 𝑑

𝑑∈ all the hard−core dimer coverings

• f the triangular lattice

Quantum dimer model wave-function

= Equal amplitude superposition Local hard-core constraint

𝑀 A=½-infinite cylinder

𝑀𝑦 → ∞

• S. Furukawa & GM, Phys Rev B 2007

J.-M. Stéphan, GM & V. Pasquier J. Stat. Mech. 2012

𝑆 A=disk radius 𝑆 circunference 𝑀

t: fugacity for dimers on horizontal bonds

Universal subleading Term -log(2)

86

Bulk-edge correspondance in 2d

Relation between the entanglement `Hamiltonian’ and physical edge modes. Topological entanglement entropy

87

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  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
• …

E E E

Example of a Z2 Liquid :

88

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. B. Laughlin Phys. Rev. Lett. 50, 1395

(1983) [Web od Science: ~3000 citations]

• R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard,

&J. H. English, Phys. Rev. Lett. 59, 1776 (1987)

𝜔Laughlin 𝑨𝑗 = 𝑨𝑘 − 𝑨𝑘

𝑛 𝑘<𝑙

exp − 1 4 𝑨𝑘

2 𝑘

𝜉 = 1 𝑛 𝑛: odd integer

89

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, …

90

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)

91

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 ∞) 𝐹𝐵 Δ

Gapless edge states “Left” modes 𝑜, 𝑀 Bulk excitations In region A

𝐹𝑐

Gapless edge states “Right” modes 𝑜, 𝑆 Bulk excitations In region B

Two (weakly) coupled edges

92

Bulk-edge correspondence in 2d (2)

Qi-Katsura-Ludwig argument: mapping to a quantum quench problem

𝑢 = 0 𝜔(0) = 𝐻 Ground-state of 𝐼𝑀 + 𝜇𝐼𝑗𝑜𝑢 + 𝐼𝑆 𝜍𝑀 = 𝑈𝑠

𝑆 𝐻 𝐻

(𝑀 and 𝑆 are entangled) 𝑢 > 0 𝜔(𝑢) = 𝑓−𝑗𝐼𝑢 𝐻 𝜍𝑀(𝑢) = 𝑓−𝑗𝐼𝑀𝑢𝜍𝑀(0)𝑓+𝑗𝐼𝑀𝑢 Entanglement entropy & spectrum

• f 𝜍𝑀(𝑢): independent of 𝑢.

𝜇 = 0 𝐼𝑀 𝐼𝑆

Calabrese-Cardy (2006) result about the a global quench in a critical 1d system:

• to obtain the long time & long-distance

correlations, the initial state can be replaced by 𝑓−𝜐(𝐼𝑀+𝐼𝑆) 𝐻∗

• 𝜐 is a finite non-universal constant

(“extrapolation length”).

• 𝐻∗ : scale/conformally-invariant

boundary state (fixed point of an RG flow starting from 𝐻 ).

• Rational CFT: 𝐻∗ is a linear

combination of all excited states, maximally entangled combination between the L and R edges.

𝑏 (topo. Sector)

𝐻∗,𝑏 ~ 𝑙, 𝑘, 𝑏 𝑀 𝑙, 𝑘, 𝑏 𝑆

𝑘 𝑙

𝑓−𝜐(𝐼𝑀+𝐼𝑆) 𝐻∗,𝑏 ~ 𝑓−2𝜐𝑤𝑙 𝑙, 𝑘, 𝑏 𝑀 −𝑙, 𝑘, 𝑏 𝑆

𝑘 momentum 𝑙>0

𝜍𝑀,𝑏~ 𝑓−4𝜐𝑤𝑙 𝑙, 𝑘, 𝑏 𝑀 𝑙, 𝑘, 𝑏 𝑀

𝑘 𝑙>0

~exp (−4𝜐𝐼𝑀)

~𝐼𝑀

Entanglement Hamiltonian

(universal part of):

93

(real space) Entanglement spectrum & quantum Hall effect

(12 particles, 𝜉 =

1 3 Laughlin’s state)

• J. Dubail, N. Read, and E. H. Rezayi,
• Phys. Rev. B 85, 115321 (2012)

Low “energy” part of the entanglement spectrum is gapless (in the

thermodynamic limit) has the

same structure (linear

dispersion & degeneracies)

as that of a massless chiral free boson, which also describe the physical edge modes 𝜉 = 1 (Integer Q. Hall effect)

• A. Sterdyniak, A. Chandran, N. Regnault,
• B. A. Bernevig, and P. Bonderson,
• Phys. Rev. B 85, 125308 (2012)

Laughlin 𝜉 =

1 3

(8 particles) Coulomb 𝜉 =

1 3

Laughlin 𝜉 =

1 3

94

Bulk-edge correspondence in 2d – Free fermion edge (1)

Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012] 𝐼𝑆 = 𝑤 𝑙𝑑𝑙

†𝑑𝑙 𝑙

𝐼𝑀 = −𝑤 𝑙𝑒𝑙

†𝑒𝑙 𝑙

𝐼int = Δ 𝑑𝑙

†𝑒𝑙 + 𝑒𝑙 †𝑑𝑙 𝑙

= tunneling from one edge to another, with momentum conservation 𝐼 = 𝐼𝑆 + 𝐼𝑀 + 𝐼int = 𝑑𝑙

†

𝑒𝑙

†

𝑤𝑙 Δ Δ −𝑤𝑙 𝑑𝑙 𝑒𝑙

𝑙

Diagonalization using new fermionic creation/annihilation operators: 𝐼 = 𝐹𝑙 𝑏𝑙

†𝑏𝑙 − 𝑐𝑙 †𝑐𝑙 𝑙

, 𝐹𝑙 = 𝑤𝑙 2 + Δ2 (→ gapped spectrum) 𝑏𝑙 𝑐𝑙 = 𝛽𝑙 𝛾𝑙 −𝛾𝑙 𝛽𝑙 𝑑𝑙 𝑒𝑙 with 𝛽𝑙 =

𝐹𝑙+𝑤𝑙 2𝐹𝑙 and 𝛾𝑙 = 𝐹𝑙−𝑤𝑙 2𝐹𝑙

𝜁 = −𝑤𝑙 𝑙 𝑙 𝜁 = 𝑤𝑙

Particle-hole excitations propagate at velocity

𝜖𝜗 𝜖𝑙 = ±𝑤

𝑑𝑙

†

𝑒𝑙

†

95

Bulk-edge correspondence in 2d – Free fermion edge (2)

Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012]

𝑏𝑙 𝑐𝑙 = 𝛽𝑙 𝛾𝑙 −𝛾𝑙 𝛽𝑙 𝑑𝑙 𝑒𝑙 with 𝛽𝑙 =

𝐹𝑙+𝑤𝑙 2𝐹𝑙 and 𝛾𝑙 = 𝐹𝑙−𝑤𝑙 2𝐹𝑙

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 𝐻𝑀 ⨂ 𝐻𝑆 where 𝐻𝑀/𝑆 is the ground-state of 𝐼𝑀/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)

𝐶 = 𝜇𝑙 𝑑𝑙

†𝑒𝑙 + 𝑒−𝑙 † 𝑑−𝑙 𝑙>0

How to determine 𝜇𝑙 ? Insure that 𝐻 is annihilated by 𝑏𝑙 (and 𝑐𝑙

†).

Commute 𝑏𝑙 and 𝑓−𝐶: 𝐶, 𝑏𝑙 = 𝐶, 𝛽𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 = −𝜇𝑙𝛽𝑙𝑒𝑙 if 𝑙 > 0 𝜇−𝑙𝛾𝑙𝑑𝑙 if 𝑙 < 0 𝐶, 𝐶, 𝑏𝑙 = 0 →Baker–Campbell–Hausdorff formula gives: 𝑏𝑙 exp −𝐶 = exp −𝐶 𝑏𝑙 + 𝐶, 𝑏𝑙 𝜁 𝑙 𝑙 𝜁

96

Bulk-edge correspondence in 2d – Free fermion edge (3)

Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012]

𝐶, 𝑏𝑙 = 𝐶, 𝛽𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 = −𝜇𝑙𝛽𝑙𝑒𝑙 if 𝑙 > 0 𝜇−𝑙𝛾𝑙𝑑𝑙 if 𝑙 < 0 So: 𝑏𝑙 exp −𝐶 = exp −𝐶 𝛿𝑙 With 𝛿𝑙 = 𝑏𝑙 + 𝐶, 𝑏𝑙 = 𝛽𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 − 𝜇𝑙𝛽𝑙𝑒𝑙 if 𝑙 > 0 𝛽𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 + 𝜇−𝑙𝛾𝑙𝑑𝑙 if 𝑙 < 0 We determine 𝜇𝑙by requiring that 𝛿𝑙 annihilates 𝐻𝑀 ⨂ 𝐻𝑆 : 𝛿𝑙 = 𝛽𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 𝛾𝑙 − 𝜇𝑙𝛽𝑙 if 𝑙 > 0 𝑑𝑙 𝛽𝑙 + 𝜇−𝑙𝛾𝑙𝑑𝑙 + 𝛾𝑙𝑒𝑙 if 𝑙 < 0

(note that the 2 conditions above are equivalent since 𝛽𝑙 = 𝛾−𝑙)

𝜇𝑙 = 𝛾𝑙 𝛽𝑙 = 𝐹𝑙 − 𝑤𝑙 𝐹𝑙 + 𝑤𝑙 = 𝐹𝑙 2 − 𝑤𝑙 2 𝐹𝑙 + 𝑤𝑙 = Δ 𝐹𝑙 + 𝑤𝑙 Final expression for the ground-state: 𝐻 = exp − Δ 𝐹𝑙 + 𝑤𝑙 𝑑𝑙

†𝑒𝑙 + 𝑒−𝑙 † 𝑑−𝑙 𝑙>0

𝐻𝑀 ⨂ 𝐻𝑆 Reduced density matrix for the L-edge ? 𝜁𝑙 𝑙 𝑙 𝜁 𝐻𝑀 𝐻𝑆

97

Bulk-edge correspondence in 2d – Free fermion edge (4)

Toy model: free chiral fermions at the edge [Qi, Katsura & Ludwig et al. PRL 2012]

𝐻 = exp − Δ 𝐹𝑙 + 𝑤𝑙 𝑑𝑙

†𝑒𝑙 + 𝑒−𝑙 † 𝑑−𝑙 𝑙>0

𝐻𝑀 ⨂ 𝐻𝑆 Remark: modes with different 𝑙 are independent, and all the terms in the exponential above commute with each other. 𝐻 = 1 − 𝜇𝑙𝑑𝑙

†𝑒𝑙 1 𝑙,𝑀 0 𝑙,𝑆 𝑙>0

1 − 𝜇𝑙𝑒−𝑙

† 𝑑−𝑙 0 −𝑙,𝑀 1 −𝑙,𝑆 𝑙>0

𝐻 = 1 𝑙,𝑀 0 𝑙,𝑆 − 𝜇𝑙 0 𝑙,𝑀 1 𝑙,𝑆

𝑙>0

0 −𝑙,𝑀 1 −𝑙,𝑆 − 𝜇𝑙 1 𝑙,𝑀 0 𝑙,𝑆

𝑙>0

Normalize -> Schmidt decomposition : 𝐻 = 1 1 + 𝜇𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 − 𝜇𝑙 1 + 𝜇𝑙 2 0 𝑙,𝑀 1 𝑙,𝑆

𝑙>0

1 1 + 𝜇𝑙 2 0 −𝑙,𝑀 1 −𝑙,𝑆

𝑙>0

− 𝜇𝑙 1 + 𝜇𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 𝜁𝑙 𝑙 𝑙 𝜁 𝐻𝑀 𝐻𝑆 1 −𝑙,𝑆 1 𝑙,𝑀 0 −𝑙,𝑀 0 𝑙,𝑆

98

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 : 𝐻 = 1 1 + 𝜇𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 − 𝜇𝑙 1 + 𝜇𝑙 2 0 𝑙,𝑀 1 𝑙,𝑆

𝑙>0

1 1 + 𝜇𝑙 2 0 −𝑙,𝑀 1 −𝑙,𝑆

𝑙>0

− 𝜇𝑙 1 + 𝜇𝑙 2 1 𝑙,𝑀 0 𝑙,𝑆 Reduced density matrix for the L-edge: 𝜍𝑆 = 1 1 + 𝜇𝑙 2 0 0 𝑙,𝑆 + 𝜇𝑙 2 1 1 𝑙,𝑆

𝑙>0

1 1 + 𝜇𝑙 2 1 1 −𝑙,𝑆 + 𝜇𝑙 2 0 0 −𝑙,𝑆

𝑙>0

𝜍𝑆 = 1 1 + 𝜇𝑙 2 exp log 𝜇𝑙 2 𝑑𝑙

†𝑑𝑙 𝑙>0

1 1 + 𝜇𝑙 2 exp log 𝜇𝑙 2 𝑑−𝑙𝑑−𝑙

† 𝑙>0

Expand for 𝑙 → 0: 𝜇𝑙 =

Δ 𝐹𝑙+𝑤𝑙 ≈ 1 1+𝑤𝑙

Δ

≈ 1 −

𝑤𝑙 Δ → log

𝜇𝑙 2 ≈ −2

𝑤𝑙 Δ

𝜍𝑆~ exp −2 𝑤𝑙 Δ 𝑑𝑙

†𝑑𝑙 + 𝑑−𝑙𝑑−𝑙 † 𝑙>0

= exp −2 𝑤𝑙 Δ 𝑑𝑙

†𝑑𝑙 − 𝑑−𝑙 † 𝑑−𝑙 + 1 𝑙>0

= exp −2 𝑤𝑙 Δ 𝑑𝑙

†𝑑𝑙 + 𝑑𝑡𝑢.

~

𝑙

exp − 2 Δ 𝐼𝑆 → The 𝑆 -edge is seen at some effective temperature Teff =

Δ

• 2. Entanglement Hamiltonian ~𝐼𝑆

99

Bulk-edge correspondence in a spin ladder

 “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

to the (energy) spectrum of the Heisenberg spin-

1 2 chain.

NB: Des Cloizeaux Pearson [1962] dispersion relation for a single chain:

𝜗𝑙 =

𝜌 2 sin 𝑙 .

A 𝑲 𝜇

 Spin ladder model

100

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.

A 𝑲 𝜇 𝐼 = 𝑲 𝑇 𝑇𝑗𝑏 ⋅ 𝑇 𝑗𝑐

𝑗=1… 𝑀

+ 𝜇 𝐼𝐵 + 𝐼𝐶 𝐼𝐵 = 𝑇 𝑗𝑏 ⋅ 𝑇 𝑗+1 𝑏

𝑗

𝐼𝐶 = 𝑇 𝑗𝑐 ⋅ 𝑇 𝑗+1 𝑐

𝑗

 Result: 𝜍𝐵 ≃

1 𝑎 exp − 4𝜇 𝐾 𝐼𝐵 + 𝒫 𝜇2

• 𝜍𝐵=Thermal density matrix (for small 𝜇)
• Effective temperature 𝑈eff =

𝐾 4𝜇

• Entanglement Hamiltonian (defined by −log

(𝜍𝐵)) is proportional to the real Hamiltonian 𝐼𝐵 of the upper chain.

101

Topological entanglement entropy in 2d (1)

“Topological Entanglement Entropy” Alexei Kitaev and John Preskill

• Phys. Rev. Lett. 96, 110404 (2006)

“Detecting Topological Order in a Ground State Wave Function” Michael Levin and Xiao-Gang Wen

• Phys. Rev. Lett. 96, 110405 (2006)

“Quasiparticle statistics and braiding from ground-state entanglement”

• Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath
• Phys. Rev. B 85, 235151 (2012)

102

Topological entanglement entropy in 2d (2)

 Assume that the (low- “energy” part of the) entanglement spectrum is described by the 1d gapless Hamiltonian describing gapless edge modes [Qi-Katsura-Ludwig argument]: 𝜍𝐵~ exp −𝐼 and H = 𝐼𝑓𝑒𝑕𝑓 ≃ 𝐼𝐷𝐺𝑈 is gapless and describe a 1d quantum system of length 𝑀.  As usual, to compute the entanglement entropy we write Tr 𝜍𝐵

𝑜 = 𝑎𝑏 𝑀,𝑜 𝑎𝑏 𝑀,1 𝑜 with

𝑎𝑏 𝑀, 𝑜 = Tr 𝑓−𝑜𝐼𝑏(𝑀) . 𝑎𝑏 𝑀, 𝑜 is a CFT partition function on a torus of size 𝑜 × 𝑀. Here 𝑜 plays the role of an imaginary time (with respect to 𝐼𝑏(𝑀)).  We are interested in 𝑀 → ∞ and 𝑜 fixed, so it is convenient to invert the space and (imaginary) time directions using a (CFT) modular transformation: 𝑎𝑏 𝑀, 𝑜 = 𝒯𝑏

𝑐𝑎𝑐(𝑜, 𝑀) 𝑐

with 𝒯: modular matrix of the CFT  Since the “time” dimension 𝑀 now goes to ∞, we can approximate the partition function 𝑎𝑐(𝑜, 𝑀) by the contribution of the ground state of 𝐼𝑐 𝑜 : 𝑎𝑐 𝑜, 𝑀 ≃ exp −𝑀𝐹𝑐(𝑜) .  The lowest energy is expected in the “trivial” sector 𝑐=1. Standard finite-size correction to the energy in CFT gives: 𝐹1 𝑜 = 𝑓0 × 𝑜 +

𝜌𝑑 6𝑜 + ℴ 1 𝑜 , where 𝑓0 is the

(non-universal) ground-state energy.

 We get: 𝑎𝑏 𝑀, 𝑜 ≃ 𝒯𝑏

1 exp −𝑀 × 𝑓0 × 𝑜 − 𝑀 𝜌𝑑 6𝑜 + 𝑀ℴ 1 𝑜

and log Tr 𝜍𝐵

𝑜 = 1 − 𝑜 log 𝒯𝑏 1 + 𝒫(𝑀). We thus have a constant term log𝒯𝑏 1 in the Rényi

entropies (+plus the area law 𝒫(𝑀)). If 𝑏=1 (ground state sector), we find 𝒯1

1 = 1/𝒠

where 𝒠 is the so-called total quantum dimension of the CFT. It is related to the fractional excitation content of the phase.  Finally, the universal piece in the entanglement entropy reads 𝑇𝐵 = 𝒫 𝑀 − log 𝒠

A

𝑀 B 𝜔

Fractional excitation of type “a”

𝑜

𝑀 𝑏

PBC

𝑎𝑏 𝑀, 𝑜

𝑀 𝑜

𝑎𝑐(𝑜, 𝑀) 𝑐

𝐼𝑏(𝑀) 𝐼𝑐(𝑜)

PBC

103

Area law in 2d – fractional quantum Hall states

“Entanglement scaling of fractional quantum Hall states through geometric deformations”

• A. M. Lauchli, E. Bergholtz, M. Haque,

New Jour. of Phys, 12, 075004 (2010) 𝒠 = 𝑛 is the total quantum dimension for a Laughlin state at filling fraction 𝜉 =

1 𝑛 (𝑛 odd integer). In the

torus geometry, the topological entanglement entropy is twice that of a disk, hence−2log (𝒠)= −2 log 3 = − log 3 in the example above.

Area law Subleading constant

105

Last remarks…

 Thanks to Vincent Pasquier for numerous discussions on these topics  All the slides are online:

• Home page: http://ipht.cea.fr/Pisp/gregoire.misguich
• r
• http://ipht.cea.fr

Cours / Programme des cours de l'année académique / « Quantum entanglement… »

 Comments & questions are very welcome  Thank you !