The Entanglement Hamiltonian and others inone-dimensional critical and gapped systems
Shinsei Ryu
Univ. of Illinois, Urbana-Champaign
Outline:- Part I:
Entanglement spectrum in 1+1d a conformal field theories (CFTs)
and the sine-square deformation (SSD)[with Xueda Wen and Andreas Ludwig]- Part II (optional):
Entanglement spectrum in 1+1 d gapped (SPT) phases
and boundary conformal field theories (BCFTs)[with Gil Cho, Ken Shiozaki, Andreas Ludwig]

Hamiltonians in CFT- Let's start from a Hamiltonian of (1+1)d CFTs;
On a lattice (chain), it would look like:- Deformed evolution operator:
envelope function- E.g. Entanglement Hamiltonian:- E.g. Sine-square deformation (SSD):[Gendiar-Krcmar-Nishino (2009) ... ]- Other applications: inhomogenous systems,
quantum energy inequalities, etc.

What is the sine-square deformation (SSD)?
- What it does is to introduce an "optimal" or "infinitely smooth" cutoff.

- This may be of interest as a numerical technique.
Reducing finite-size error, etc.
- Eearly numerical observations:- Correlation functions- Entanglement scaling[Hikihara-Nishino (11)]Key properties of SSD ?
- The ground state of SSD = ground state of periodic chain

Numerics: Hikihara-Nishino (11);Exactly solvable models: Katsura (11), Maruyama-Katsura-Hikihara (11),Okunishi-Katsura (15)Proof within CFT: Katsura (12)- 1/L^2 finite fize scaling of energy levels
was observed numerically.

[Gendiar-Krcmar-Nishino (09), Hotta-Nishimoto-Shibata (13),LSM type analysis by Katsura ... ]References:- Grand canonical numerical analysis -- efficient extraction of physicalquantities in the presence of an applied field.
Shibata and Hotta (11)
Hotta, Nishimoto and Shibata (13)- SSD and string theory:
Tada, arXiv:1404.6346[hep‐th]- Dipolar quantization:
Ishibashi and Tada, arXiv:1504.00138[hep‐th]Ishibashi andTada, arXiv: 1602.01190[hep‐th]- Mobius quantization:
Okunishi arXiv:1603.09543

Strategy- We will disscuss types of "deformations"generated by various conformal maps.
- Put differently, we are interested in deformations which we can "undo"
by conformal maps.
- Will discuss spectral properties (finite size scaling) of

Warm-up-- CFT on the plane- Radial evolution(Dilatation)<--> CFT on a cylinder (quantum lattice model on a circle)<-->

Hamiltonian (1/L scaling)- Angular evolution<--> Hamiltonian with boundary("Rindler" or "Modular"or "Entanglement" Hamiltonian)Finite size scaling of CFT [Cardy]- CFT on a cylinder of circumference L:- Conformal map: cylinder --> plane-- For a given tower of states,
all levels are equally spaced(with degenearcy, which depends
on details of the theory)c =1 compactified free boson
at R= sqrt(2)-- Level spacing scales as 1/L

-- Entanglement Hamiltonian on finite interval [-R, R]<--> Hamiltonian with boundaries- Transforming from strip to plane:- Entanglement spec: 1/Log(R) scaling.
[See, e.g: Casini-Huerta-Myers (11), Cardy @ 2015 KITP conference]What is the evolution orthogonal to the evolution by Entanglement H ?
- Let's take a circle as a Cauchy surf

We have chosen:
Circumference:- Evolution operator- "Regularized" version of the SSD:

"Regularized" SSD- By construction, this operator has the spectrum of CFT on a circle
with level spacing of order one.
- Define:- The envelope function:- "Regularized" version of the SSD:
R, the distance between vortices, is the regularziation parameter.
- Scalilng:(i) fix u0, change R --> 1/L scaling(ii) fix R, change u_0 --> 1/L^2 scaling

The dipolar limit- Can take the dipolar limit

rSSD --> SSD:

- In the dipolar limit, the w-plane (u-v plane) is an infinit plane--> Infinite system length limit, continuum spectrum [Ishibashi-Tada (15,16)]- The prefactor L^2 is indicative of the 1/L^2 scaling seen in numerics.Numerics (rSSD)(numerical)(analytical)

Regularized SSD spectrum

Physical spectrum (PBC)

Numerics (SSD)

SSD

Physical spectrum (PBC)

SSD spectrum does not much physical spectrum,
1/L^2 scalingOther examples-infinite stripe- Engineering conformal map and evolution operator- "square root deformation"- Known in the context of "perfect state transfer"(Thanks: Hosho Katsura)Physical spectrum (OBC)"Square root" deformation

Summary (Part I)- Setup a general discusion of "deformed" Hamiltonians in CFTs
- Proposed a "regularized" version of SSD (rSSD).
- Original SSD can be viewed as a "singular" limit of rSSD
- Spectrum of rSSD is easy to understand.
Shed light on 1/L^2 scaling of SSD.
- Issues:Exciations?
Relation to the classification of conformal vacua [Candelas-Dowker (1979)]

(Part II) Going away from CFTs
- Add a relevant deformation --> go into a massive phase- Consider the entanglement Hamiltonian for the half space;
The entanglement spectrum?

- Repeat the conformal map analysis- Massive perturbation creates an
exponentially growing potential

- Massive perturbation creates an artificial boundary
in the entanglement Hamiltonian- To a good approximation, the entanglement Hamiltonian is the Hamiltonian ofa CFT with boundaries; Boundary CFT.

- Comment: this argument shows the low-energy part of the entanglement spec.
of a massive theory is given by BCFT.
There are integrable massive models, whose corner transfer matrices aregiven exactly by Virasoro characters (BCFT).

Question and result-ES for gapped phases is given by nearby boundary conformal field thoery- Q: Which boundary condition ? <----> Which gapped phase?
- Let's focus on the case when the massive phase is a SPT phase
I.e.: (i) unique ground state(ii) topologically distinct in the presence of some symmetry- Result:
For a given symmetry G, and a given boundary state |B>,
found a method to compute the topoogical invariantof the corresponding SPT phase.
- Related to symmetry-protected degeneracy of ES- Relation to physics of fractional branes

Symmetry-protected degeneracy- E.g. 1d lattice fermion model ("SSH" model)

Symmetry:
Phase diagram:
SPTTrivial- Symmetry-protected contribution to EE [SR-Hatsugai (06)]Symmetry-protected degenerac- (1+1)d SPT phase:
E.g. the Haldane phase, the Kitaev chain- Symmetry-protected degenearcy in ES: [Pollmann-Berg-Turner-Oshikawa (10)]- Symmetry-protected degeneracy --> Vanishing of part. function:- Exchange time and space:- Act with a symmetry on |B>- Symmetry-enforced vanishing of partition function

Anomalous boundary states- Ideal lead obeys B.C. set by SPT

lead (CFT)- Symmetry G acts on fundamental fields- B.C. is invariant under G:- But boundary state may not be:- Z8 classification of TRS Kitaev chain, Haldane phase

SPTAnalysis and result: Fidkowski-Kitaev problem- Ideal lead

lead (CFT)- Symmetry group:- Boundary states

SPT- Symmetry action on fermion number parity:

Anomalous relative sign goes away for 2N copies --> Z2- Time reversal: