Time-reversal symmetry breaking and gapped surface states due to spontaneous emergence of new order in dd-wave nanoislands

We solve the Bogoliubov-de Gennes equations self-consistently for the $d$-wave order parameter in nanoscale $d$-wave systems with [110] surfaces and show that spontaneous time-reversal symmetry (TRS) breaking occurs at low temperatures due to a spontaneously induced complex order parameter of extended $s$-wave symmetry. The Andreev surface bound states, which are protected by a one-dimensional (1D) topological invariant in the presence of TRS, are gapped by the emergence of this new order parameter. The extended $s$-wave order parameter is localized within a narrow region near the surfaces, which is consistent with the fact that topological protection of the gapless Andreev surface states is characterized by the 1D topological invariant. In this TRS-breaking phase, not only is the complex order parameter induced, but also the $d$-wave order parameter itself becomes complex. Furthermore, the disappearance of topological protection brings about novel vortex phenomena near the surfaces. We show that vortex-antivortex pairs are formed in the extended $s$-wave order parameter along the surfaces if the side length of a nanoisland or the width of an infinitely long nanoribbon is relatively large.Comment: 6 pages, 4 figures + 6 pages (supplemental material), to be published in Phys. Rev. B Rapid communicatio

Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment, we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.Comment: 11 pages, 4 figure

Tensor factorizations of local second-order M{\o}ller Plesset theory

Efficient electronic structure methods can be built around efficient tensor representations of the wavefunction. Here we describe a general view of tensor factorization for the compact representation of electronic wavefunctions. We use these ideas to construct low-complexity representations of the doubles amplitudes in local second order M{\o}ller-Plesset perturbation theory. We introduce two approximations - the direct orbital specific virtual approximation and the full orbital specific virtual approximation. In these approximations, each occupied orbital is associated with a small set of correlating virtual orbitals. Conceptually, the representation lies between the projected atomic orbital representation in Pulay-Saeb{\o} local correlation theories and pair natural orbital correlation theories. We have tested the orbital specific virtual approximations on a variety of systems and properties including total energies, reaction energies, and potential energy curves. Compared to the Pulay-Saeb{\o} ansatz, we find that these approximations exhibit favourable accuracy and computational times, while yielding smooth potential energy curves