Entanglement Entropy Scaling Laws and Eigenstate Thermalization in Many-Particle Systems

Kun YangNational High Magnetic Field Lab

and Florida State University

In collaboration with Hsin-Hua Lai, Wenxin Ding, Nick Bonesteel (NHMFL+FSU)Alex Seidel (Washington U.)

• Entanglement Entropy (EE) has emerged as a key property useful to characterize phases and critical points, especially in strongly correlated systems that support “exotic” phases. Ground state EE mostly obeys area law in D > 1, even in critical systems; free Fermi gas is the only known example that violates area law until recently.

• We use high-D bosonization to show that interacting fermions in the Fermi liquid phase, EE violates area law in a manner identical to free Fermi gas; super-criticality (W. Ding, A. Seidel and KY, PRX 12).

• We find a similar area-law violation in Bose liquids with Bose surfaces (H.-H. Lai, KY and N. Bonesteel, PRL 13).

• EE of highly excited states (with finite excitation energy density) are expected to follow volume law (like thermal entropy), but few results available. We find it is dual to ground state area law, and of the same origin in free fermion systems. We further show that thermalization emerges under appropriate conditions (H.-H. Lai and KY, PRB 15).

Motivation and Summary of Results

Perhaps statistical mechanics and thermal entropy emerge from entanglement in a single eigenstate! Free fermion system an example of this Eigenstate Thermalization Hypothesis.(H.-H. Lai and KY, PRB 15)

Definition of Eigenstate Thermalization Hypothesis (ETH) for this talk: Reduced density matrix of a subsystem approaches thermal density matrix, when the whole system is in an eigenstate (cf. Nandkishore and Huse, arxiv 14)

What is Entanglement?

Area Law ≈Holography!

Known Examples of Area Law ViolationQuantum Critical Points (QCPs) in1-Dimensional Systems

• Asymptotic behavior of entanglement entropy for conformal QCPs:

• Random Singlet and related random QCPs (G. Refael and J. Moore PRL 04; N. Bonesteel and KY PRL 07; …)

• Specific Example: 1D fermions (c=1 CFT) fermionic wavefunction approach for free fermions (Jin & Korepin, 2004, …) bosonic approach: bosonization and calculate the entanglement entropy of the

bosonic theory (Calabrese & Cardy, 2004, both CFT and massive boson field calculation are discussed.)

Interaction can be included in this approach, and it does NOT change the scaling behavior of the entanglement entropy!

Free Fermions with Fermi Surface in D-Dimensions• Scaling behavior of the entanglement entropy (Wolf, 2006; Gioev & Klich, 2006)

SA=(c/3) log L + const.

Decoupled parallel 1D chains

Intuitive Understanding of G&K’s Result via a Toy Model

•   (1/ 3)log LL

1DL

1 1

1

1

ˆ| |1 1log log | |

6 12(2 )

log | |12(2 )

S xx kD D

D

x

A

kD

n dSE L L dS dS

a

LL dS dS

B. Swingle PRL 10; W. Ding, Ph.D. prospectus 08 and arxiv 09.

Allows for inclusion of Fermi liquid interactions, found to have no effect on logarithmic enhancement of entanglement entropy. (W. Ding, A. Seidel and KY, PRX 12).

Obviously existence of extended Fermi surface(s) crucial for area law violation. What about bosonic systems?

Bosons are different; in general they either localize or Bose condense, resulting in Godstone modes that are gapless isolated points. No area law violation in these cases. Emergent Fermi surface possible in spin liquids, leading to area law violation (Zhang, Grover and Vishwanath 11).

There have been theoretical proposals (Lesik Motrunich, Matthew Fisher and many co-workers) of Bose metal phases with Bose surfaces along which gapless bosonic excitations live. Do these phases violate area law? Yes! (H.-H. Lai, KY and N. Bonesteel, PRL 13)

Simplest Example of Bose Metal: Exciton Bose Liquid (EBL)

Bose surfaces (where gapless bosonic excitations live) along kx=0, ky=0

•Low energy theory: Free bosons with an energy dispersion which vanishes linearly on a locus of points in k-space.

• Bosonic harmonic oscillator with factorized dispersion

fx(kx), fy(ky): Periodic functions which vanish linearly as kx, ky→0, such as sin(kx), sin(ky).

short-ranged oscillator coupling!

A. Paramekanti , L. Balents, and M. Fisher, PRB 02

Entanglement Entropy of a Belt Subsystem

Anti-periodic boundary condition

Partial Fourier transform along y-axis

N decoupled 1D chain with dispersion ω1D(kx)=|fx(kx)||fy(ky)|, ky =

(2ny + 1)π/N and ny =0, 1, ..., N−1.

Violation of entanglement area-law in two dimensions due to the existence of “Bose" surface!

Entanglement entropy:

Belt subsystem

What about the entanglement entropy of subsystems with arbitrary shapes? Can we put some bounds? YES!

Bounds for the Entanglement Entropy

Strong subadditivity inequality (Lieb)

Systems A and B:

Inequality:

• Strong subadditivity inequality

• Upper bound:•N

• Lower bound:

Concrete Example

An equally-partitioned square systems

Generalization to equally-partitioned subsystem in d dimension:

Only differ a factor of 2! Logarithmic enhancement!

What about the subsystems with arbitrary shapes?

Argument for arbitrary shape

For a non-rectangular system with at least one boundary parallel to a Cartesian axis

Lattice symmetries

(translation and mirror) + Strong subadditivity inequality

(1) (2) (3)

Take EBL for example:

For subsystems with general smooth boundary: No rigorous argument

• SL R∪ is likely to show logarithmic enhancement. • Results generalisable to similar models with

arbitrary Bose surfaces.Arbitrarily sliced (H.-H. Lai, KY and N. Bonesteel, PRL 13)

Summary Thus Far

• Challenge: Highly excited states (with finite excitation energy density).

• Expectation: entanglement entropy follow volume law, just like a random state in Hilbert space.

• Problem: Almost no explicit result other than limited numerics.

• Entanglement entropy area law widely obeyed by ground states of local Hamiltonians.

• Violations of area law rare above 1D; requires Fermi or Bose surfaces. Reasonably well understood.

• Caution: Finding logarithmic violation of area law in a holographic calculation does not necessarily mean you have found a condensed matter system with Fermi surface(s)!

Position-Momentum Duality for Free Fermion States

Position-Momentum

Duality

Position SpacePartition

Momentum Space Occupation

OriginalSystem

DualSystem

Identical entanglement spectrum!

Huang and Arovas 12; Lee, Ye and Qi 14.Visible in Gieov-Klich formula!

For a Free fermion Hamiltonian:

Reduced density matrix:

In free fermion systems: A given matrix M

Uniquely determine the entanglement spectrum!

Mjl: Two-point fermion Green’s function defined in subsystem A.

Heuristic Derivation of Duality-I

: Projection operator onto A

: Projection operator onto the occupied states in momentum space

the same eigenvalue!

Original system : Dual system :

The two free fermion systems in which the roles of position and momentum exchange have the same entanglement! Huang and Arovas 12, Lee, Ye and Qi 14

Heuristic Derivation of Duality-II

Duality between GS and ES

Number of pockets:

Ground State of Original system

The entanglement entropy scales with the system volume

Highly Excited State in Dual system

From the position-momentum duality, the entanglement entropy of the a highly excited state exhibits volume instead of area law.

The entanglement entropic area law for free fermion ground states and the volume law for highly excited states are of the same origin!

Eigenstate thermalization for a typical excited state

• Typical highly excited state

• Atypical

• Thermodynamic limit: • Coarse-graining process

• Divided into cells

• Coarse-graining (Blurring)

Coarse-Graining Process• At thermodynamic limit, we can divide the Brillouin zone into a

huge number of cells with linear size

• All the momentum points within the same cell approximately share the same phase factor

: Total occupation number in the cell m

: The total number of momentum points in a cell

Searching for the Most Probable Macrostate

• Two constraints for an excited state:

• Introduce two Lagrange multipliers 𝜶 and 𝜷 and examine the fluctuation of the distribution set {Nm}

The same as the Fermi-Dirac distribution from the grand canonical thermal ensemble if we identify = 𝜶− /T and = 1/T!𝝁 𝜷

: single particle energy corresponding to a cell

Schematic Illustration of Eigenstate Thermalization

For “all” elements defined in the subsystem A

(Weak) Eigenstate thermalization in free Fermion systems!Consequence: Entanglement Entropy = Thermal Entropy!

In agreement with numerical work by Storms and Singh, PRB 14

• Entanglement Entropy (EE) has emerged as a key property useful to characterize phases and critical points, especially in strongly correlated systems that support “exotic” phases. (Ground state) EE mostly obeys area law in high D, even in critical systems; free Fermi gas is the only known example that violates area law until recently.

• We use high-D bosonization to show that interacting fermions in the Fermi liquid phase, EE violates area law in a manner identical to free Fermi gas; super-criticality (W. Ding, A. Seidel and KY, PRX 12).

• We find a similar area-law violation in Bose liquids with Bose surfaces (H.-H. Lai, KY and N. Bonesteel, PRL 13).

• EE of highly excited states (with finite excitation energy density) are expected to follow volume law (like thermal entropy), but few results available. We find it is dual to ground state area law, and of the same origin in free fermion systems. We further show that thermalization emerges under appropriate conditions (H.-H. Lai and KY, PRB 15).

Motivation and Summary of Results

If your life is a mess and has high entropy, don’t blame the society, which may well be in a pure and steady (i.e., eigen!) state.

It is (probably) because you are entangled with someone who is having an equally messy life!

Charles Bennet: “A classical house is at least as dirty as its dirtiest room, but a quantum house can be dirty in every room, yet still perfectly clean overall.”