Abstract
We apply the charge pumping argument to fermionic tensor network representations of d-dimensional topological insulators (TIs) to obtain tensor network states for (d+1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair states (PEPS)for a Chern insulator starting from a matrix product state (MPS) in d=1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in d=2 for the obstructed atomic insulator (OAI) of the quadrupole model. The (d+1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the d-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the d-dimensional models, we identify gapped next-nearest neighbour Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively.
Highlights
Higher-order topological insulators (TIs) [1,2,3,4] have recently been introduced as a new class of symmetry-protected topological systems generalizing the framework of TIs with surface states [5]
From the d-dimensional models, we identify gapped next-nearest-neighbor Hamiltonians interpolating between the trivial and obstructed atomic insulator (OAI) phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an matrix product state (MPS) and a projected entangled pair state (PEPS) with a constant bond dimension equal to 2, respectively
In Appendix H, we show that the PEPS with parameters α ≡ αx = αy and β ≡ βx = βy is the unique ground state of an extended version of the quadrupole model HPEPS with C4-symmetric hoppings, with a staggered chemical potential that breaks C4 symmetry and with an additional next-to-nearest-neighbor hopping tτ(2)[aτ†,xaτ,x+x + aτ†,xaτ,x+y + H.c.]
Summary
Higher-order TIs [1,2,3,4] have recently been introduced as a new class of symmetry-protected topological systems generalizing the framework of TIs with surface states [5]. The bond dimension of the real-space local tensor in this direction generically grows with the system size due to the nonlocality of the FT, whereas it is identical to the finite bond dimension of the d-dimensional TNS in the other d directions We apply this construction both to a matrix product state (MPS) [46] for the SSH model in order to obtain a PEPS for a Chern insulator, and to a Gaussian PEPS with finite bond dimension for the topological quadrupole model in order to obtain a three-dimensional GfTNS for the second-order chiral hinge state TI of Refs. Our approach provides us with a gapped TNS with one-dimensional chiral boundary states and a constant finite bond dimension in all but one of the spatial directions This representation is potentially useful for tensor network algorithms.
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