Boundary theories of two-dimensional tensor network states near the AKLT point

S. Yang, L. Lehman, D. Poilblanc, K. Van Acoleyen, F. Verstraete, J.I. Cirac and N. Schuch

The edges of strongly correlated quantum many-body systems are
known to possess interesting properties, which are related to the properties
of the bulk material.
For example, the low energy behaviour of two-dimensional quantum Hall
systems can be described in terms of chiral modes which are
localized at the edge of the system.
Here this so called bulk-boundary correspondence is studied in the setting
of projected entangled-pair states (PEPS) [1],
a class of two-dimensional tensor network states which describe
gapped quantum systems.


Using the tensor network formalism, the low energy Hilbert space of the
bulk Hamiltonian can be identified with the entanglement degrees
of freedom at the boundary of the system [2].
This allows to define a Hermitian operator acting on the boundary and
which can be interpreted as the boundary Hamiltonian
in the lowest order.
The boundary Hamiltonian contains information about the low energy
properties of the bulk Hamiltonian, and the behaviour of the
boundary Hamiltonian can be investigated when the parameters
of the bulk Hamiltonian are varied.


Here these boundary theories are studied numerically using the PEPS
representation of the AKLT state [3], the valence bond
ground state of interacting integer spins in two dimensions.
The strengths of the interaction terms appearing in the boundary
Hamiltonian are investigated when the AKLT Hamiltonian is perturbed by
various interaction terms in the bulk.
The boundary Hamiltonian is shown to be quasi-local with
the range of interactions decreasing exponentially,
and for large systems it consists of nearest-neighbour and
next-nearest-neighbour interactions:
\[
H_B \approx \eta_1\sum_i \mathbf{S}_i \cdot \mathbf{S}_{i+1}
+ \eta_2\sum_i \mathbf{S}_i \cdot \mathbf{S}_{i+2}
\]
A local magnetic field in the bulk induces a local field also at the boundary.
Using the PEPS structure, it can be shown that all local symmetries in
the bulk are also inherited by the boundary Hamiltonian.


An interesting question is whether the low energy physics, described by
the boundary Hamiltonian $H_B$, bears some relation to the entanglement
spectrum of the bulk ground state.
This connection, known as the Li-Haldane conjecture [4], suggests
an intimate relation between the boundary Hamiltonian and the entanglement
Hamiltonian derived from the entanglement spectrum of tensor
network states [5].


References

[1]
F. Verstraete and J.I. Cirac, arXiv:cond-mat/0407066

[2]
S. Yang, L. Lehman, D. Poilblanc, K. Van Acoleyen, F. Verstraete, J.I. Cirac and N. Schuch,
Phys. Rev. Lett. {\bf 112}, 036402 (2014);
arXiv:1309.4596

[3]
I. Affleck, T. Kennedy, E. Lieb and H. Tasaki,
Phys. Rev. Lett. {\bf 59}, 799 (1987)

[4]
H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008);
arXiv: 0805.0332

[5]
J.I. Cirac, D. Poilblanc, N. Schuch and F. Verstraete,
Phys. Rev. B 83, 245134 (2011);
arXiv: 1103.3427