Panjin Kim*, Hosho Katsura, Nandini Trivedi, Jung Hoon Han Entanglement and corner Hamiltonian spectra of integrable open spin chains Panjin Kim*, Hosho Katsura, Nandini Trivedi, Jung Hoon Han Department of Physics Sungkyunkwan University, Suwon, Korea (*) pansics@skku.edu April 21, 2016 arXiv preprint 1512.08597

Outline Quantum Entanglement Motivation Numerical results on the SU(N) open spin chain Summary

Quantum Entanglement Can you see the difference? What about these?

Quantum Entanglement Classification of different phases Late 20th, physicists began to realize Landau symmetry breaking theory is not sufficient to describe phases in nature. order parameter (symmetry breaking) + quantum entanglement (topological order)

A B Quantum Entanglement Entanglement entropy & Area law Consider a pure state, Divide the system into two, Although is a pure state, there can exist entanglement between subsystems A and B. The entanglement entropy reads, A B ※ Schmidt decomposition

Motivation (brief history of 1d entanglement entropy) Logarithmic dependence of the entropy on the length of the subsystem in (1+1)-dimensional conformal field theory (CFT)- by Holzhey et al. C. Holzhey, F. Larsen, and F. Wilczek, Nuclear Physics B 424, 443 (1994). Logarithmic dependence of the entropy on the subsystem size in some quantum spin lattice model, with the prefactor proportional to the central charge of the corresponding CFT- by Vidal et al. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003) Generalization of CFT results with open boundaries and finite temperatures- by Calabrese and Cardy P. Calabrese and J. Cardy, Journal of Statistical Mechanics: Theory and Experiment 2004, P06002 (2004).

Motivation Entanglement entropy (EE) in 1D lattice system Gapless: logarithmic dependence on subsystem size Gapped: saturated (area law) G. Vidal et al., PRL, 2003 oscillation Looks like the end of the story, but… Laflorencie et al.- oscillation in entanglement entropy with open boundary condition (OBC) which was not expected. The origin of the oscillation has been an open question. N. Laflorencie, E. S. Sørensen, M.-S. Chang, and I. Affleck, Phys. Rev. Lett. 96, 100603 (2006).

SU(2) critical spin chain Numerical results- entanglement entropy Red solid line: CFT fit with correction for the oscillation Black dots: Entanglement entropy obtained by MPS+DMRG Amplitude of oscillation slowly decays as n approaches the center of the chain.

SU(2) critical spin chain Numerical results- entanglement spectrum Alternating structure in ES does not diminish deep inside the chain. Seeming discrepancy occurs; the largest singular value, which is the weight of the most relevant Schmidt state of the ground state, alternates while its (weighted) trace, the entanglement entropy, does not oscillates deep inside the chain.

SU(2) critical spin chain Numerical results- weighted entanglement spectrum (WES) Sum of WES at given n results in the EE. Comparison of WES at n=1,2 with n=25,26. Different ES structure can give the similar EE. EE is dominated by only a handful of entanglement levels.

SU(2) critical spin chain ES up to n=4 Ground state is a singlet One doublet Composite spin; has to be doublet One singlet, one triplet Composite spins; have to be one singlet and one triplet singlet+singlet=singlet No degeneracy triplet+triplet=singlet+…

Summary Period-N oscillations in the entanglement structure of the SU(N) critical spin chain is re-examined from the perspective of the site dependence of the entanglement spectra. Entanglement spectra continues to retain the periodic structure in the bulk where the entanglement entropy itself has diminishing oscillation amplitudes. Only a handful of entanglement spectra contribute meaningfully to the entropy, and quite different ES patterns realized at different sites can still give rise to the same sum, which is the entanglement entropy.

SU(N) critical spin chain- SU(N) symmetry Example: SU(2) symmetry : generators of SU(2) Generalization to SU(N) -dimensional vector generators of SU(N)

Spin-1 bilinear-biquadratic spin chain Entanglement spectrum and corner spectrum For integrable model, CS-ES correspondence is known to exist away from critical points. Integrable points: ULS (gapless), TB (gapless), PBQ (gapped) A few other points (red dots on the phase diagram) still show CS-ES correspondence.

SU(2) critical spin chain Numerical results- ES and young tableaux SU(2) Young tableaux ES up to n=4 1:1 correspondence between dimensionalities of Young tableaux and the degeneracies in ES

SU(3) critical spin chain EE, ES, and WES

SU(4) critical spin chain EE, ES, and WES