Author(s): Gil Young Cho, Rodrigo Soto-Garrido, and Eduardo Fradkin

We show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensiona…

[Phys. Rev. Lett. 113, 256405] Published Fri Dec 19, 2014

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Author(s): Román Orús, Tzu-Chieh Wei, Oliver Buerschaper, and Artur García-Saez

Topological order in two-dimensional (2D) quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here, we show how topological phase transitions in 2D systems can be …

[Phys. Rev. Lett. 113, 257202] Published Fri Dec 19, 2014

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Author(s):

We study a chain of ferromagnetic nano-particles or ferromagnetic
molecule/atoms on a substrate of fully gapped superconductors. We find that
under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$
can spontaneously break. In other words, such a chain can realize a 1+1D
fermionic topologically ordered state and the corresponding two-fold
topological degeneracy on an open chain. Such a topological degeneracy becomes
the so called Majorana zero mode in the non-interacting limit. More
specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D
topological order can appear if (1) the electron hopping $t_{ij}$ between
nano-particles is larger than the energy splitting $\delta E_{eo}$ between the
ground states of even and odd electrons on a nano-particle, (2) the Josephson
coupling $J_i$ between the superconducting substrate and the nano-particle is
larger than or similar to $\delta E_{eo}$, and (3) the electron hopping
amplitude $t_{ij}$ is complex, or more precisely, the phase of gauge invariant
combination $J_i t_{ij}^2 J_j^*$ is not zero.

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Author(s):

Braiding of Majorana zero modes provides a promising platform for quantum
information processing, which is topologically protected against errors.
Strictly speaking, however, the scheme relies on infinite braiding times as it
utilizes the adiabatic limit. Here we show how to minimize nonadiabatic errors
for finite braiding times by finding an optimal protocol for the Majorana
movement. Interestingly, these protocols are characterized by sharp transitions
between Majorana motion at maximal and minimal velocities. We find that these
so-called bang-bang protocols can minimize the nonadiabatic transitions of the
system by orders of magnitude in comparison with naive protocols.

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Author(s):

The challenge of identifying symmetry-protected topological states (SPTs) is
due to their lack of symmetry-breaking order parameters and intrinsic
topological orders. For this reason, it is impossible to formulate SPTs under
Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and
topology-dependent ground state degeneracy. However, the partition functions
from path integrals with various symmetry twists are universal SPT invariants,
fully characterizing SPTs. In this work, we use gauge fields to represent those
symmetry twists in closed spacetimes of any dimensionality and arbitrary
topology. This allows us to express the SPT invariants in terms of continuum
field theory. We show that SPT invariants of pure gauge actions describe the
SPTs predicted by group cohomology, while the mixed gauge-gravity actions
describe the beyond-group-cohomology SPTs, recently observed by Kapustin. We
find new examples of mixed gauge-gravity actions for U(1) SPTs in 4+1D via the
gravitational Chern-Simons term. Field theory representations of SPT invariants
not only serve as tools for classifying SPTs, but also guide us in designing
physical probes for them. In addition, our field theory representations are
independently powerful for studying group cohomology within the mathematical
context.

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Author(s):

Spin chains with tunable interaction range have become available in systems
of trapped ions or atoms coupled to waveguides. The spin chain can be mapped
onto hard-core bosons moving on a higher-dimensional graph, where the
dimensionality of the graph is given by the range of the interactions. We
propose to use shaking techniques to mimic magnetic fields. This can lead to a
fractal energy spectrum, and the appearance of topological phases, which we
reveal by calculating edge states and Chern numbers.

link to article (opens in new tab)

 

Author(s):

Inspired by the recent experimental observation of topological
superconductivity in ferromagnetic chains, we consider a dilute 2D lattice of
magnetic atoms deposited on top of a superconducting surface with a Rashba
spin-orbit coupling. We show that the studied system supports a generalization
of $p_x+ip_y$ superconductivity and that its topological phase diagram contains
Chern numbers higher than $\xi/a$ $(\gg1)$, where $\xi$ is the superconducting
coherence length and $a$ is the distance between the magnetic atoms. The
signatures of nontrivial topology can be observed by STM spectroscopy in
finite-size islands.

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Author(s):

We numerically show that the zero-energy Majorana surface states are
suppressed around a vortex in the three-dimensional topological superconductors
such as CuxBi2Se3 and Sn1-xInxTe. On the other hand, the zero-energy Majorana
bound states along the vortex line are robust against cut by the surface. This
difference of the robustness between the surface- and vortex- bound states is
originating from the fact that the vortex breaks the time-reversal symmetry
which protects the surface states, whereas the surface for the vortex is just
regarded as the wall. The suppression of the surface-bound states around a
vortex can be observed as the unconventional energy dependence of imaging of
the scanning tunneling microscopy/spectroscopy.

link to article (opens in new tab)

 

Author(s):

Projected entangled pair states (PEPS) provide a natural description of the
ground states of gapped, local Hamiltonians in which global characteristics of
a quantum state are encoded in properties of local tensors. We show that
on-site symmetries, as occurring in systems exhibiting symmetry-protected
topological (SPT) quantum order, can be captured by a virtual symmetry of the
tensors expressed as a set of matrix product operators labelled by the
different group elements. A classification of SPT phases can hence be obtained
by studying the topological obstructions to continuously deforming one set of
matrix product operators into another. This leads to the classification of
bosonic SPT states in terms of group cohomology, as originally derived by Chen
et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed
point models, and hence opens up the possibility of studying phase transitions
between different SPT phases. We furthermore show how the global symmetries of
SPT PEPS can be promoted into a set of local gauge constraints by introducing
bosonic degrees of freedom on the links of the PEPS lattice, thereby providing
a natural and general mapping between PEPS in SPT phases and topologically
ordered phases.

link to article (opens in new tab)

 

Author(s):

High-temperature superconductors exhibit a wide variety of novel excitations.
If contacted with a topological insulator, the lifting of spin rotation
symmetry in the surface states can lead to the emergence of unconventional
superconductivity and novel particles. In pursuit of this possibility, we
fabricated high critical-temperature (Tc ~ 85 K) superconductor/topological
insulator (Bi2Sr2CaCu2O8+delta/Bi2Te2Se) junctions. Below 75 K, a zero-bias
conductance peak (ZBCP) emerges in the differential conductance spectra of this
junction. The magnitude of the ZBCP is suppressed at the same rate for magnetic
fields applied parallel or perpendicular to the junction. Furthermore, it can
still be observed and does not split up to at least 8.5 T. The temperature and
magnetic field dependence of the excitation we observe appears to fall outside
the known paradigms for a ZBCP.

link to article (opens in new tab)

 

Author(s):

Authors: Joel Klassen, Xiao-Gang Wen

We study a chain of ferromagnetic nano-particles or ferromagnetic
molecule/atoms on a substrate of fully gapped superconductors. We find that
under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$
can spontaneously break. In other words, such a chain can realize a 1+1D
fermionic topologically ordered state and the corresponding two-fold
topological degeneracy on an open chain. Such a topological degeneracy becomes
the so called Majorana zero mode in the non-interacting limit. More
specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D
topological order can appear if (1) the electron hopping $t_{ij}$ between
nano-particles is larger than the energy splitting $\delta E_{eo}$ between the
ground states of even and odd electrons on a nano-particle, (2) the Josephson
coupling $J_i$ between the superconducting substrate and the nano-particle is
larger than or similar to $\delta E_{eo}$, and (3) the electron hopping
amplitude $t_{ij}$ is complex, or more precisely, the phase of gauge invariant
combination $J_i t_{ij}^2 J_j^*$ is not zero.

link to article (opens in new tab)

 

Author(s):

Authors: Torsten Karzig, Armin Rahmani, Felix von Oppen, Gil Refael

Braiding of Majorana zero modes provides a promising platform for quantum
information processing, which is topologically protected against errors.
Strictly speaking, however, the scheme relies on infinite braiding times as it
utilizes the adiabatic limit. Here we show how to minimize nonadiabatic errors
for finite braiding times by finding an optimal protocol for the Majorana
movement. Interestingly, these protocols are characterized by sharp transitions
between Majorana motion at maximal and minimal velocities. We find that these
so-called bang-bang protocols can minimize the nonadiabatic transitions of the
system by orders of magnitude in comparison with naive protocols.

link to article (opens in new tab)

 

Author(s):

Authors: Juven Wang, Zheng-Cheng Gu, Xiao-Gang Wen

The challenge of identifying symmetry-protected topological states (SPTs) is
due to their lack of symmetry-breaking order parameters and intrinsic
topological orders. For this reason, it is impossible to formulate SPTs under
Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and
topology-dependent ground state degeneracy. However, the partition functions
from path integrals with various symmetry twists are universal SPT invariants,
fully characterizing SPTs. In this work, we use gauge fields to represent those
symmetry twists in closed spacetimes of any dimensionality and arbitrary
topology. This allows us to express the SPT invariants in terms of continuum
field theory. We show that SPT invariants of pure gauge actions describe the
SPTs predicted by group cohomology, while the mixed gauge-gravity actions
describe the beyond-group-cohomology SPTs, recently observed by Kapustin. We
find new examples of mixed gauge-gravity actions for U(1) SPTs in 4+1D via the
gravitational Chern-Simons term. Field theory representations of SPT invariants
not only serve as tools for classifying SPTs, but also guide us in designing
physical probes for them. In addition, our field theory representations are
independently powerful for studying group cohomology within the mathematical
context.

link to article (opens in new tab)

 

Author(s):

Authors: Tobias Grass, Christine Muschik, Alessio Celi, Ravindra Chhajlany, Maciej Lewenstein

Spin chains with tunable interaction range have become available in systems
of trapped ions or atoms coupled to waveguides. The spin chain can be mapped
onto hard-core bosons moving on a higher-dimensional graph, where the
dimensionality of the graph is given by the range of the interactions. We
propose to use shaking techniques to mimic magnetic fields. This can lead to a
fractal energy spectrum, and the appearance of topological phases, which we
reveal by calculating edge states and Chern numbers.

link to article (opens in new tab)

 

Author(s):

Authors: Joel Röntynen, Teemu Ojanen

Inspired by the recent experimental observation of topological
superconductivity in ferromagnetic chains, we consider a dilute 2D lattice of
magnetic atoms deposited on top of a superconducting surface with a Rashba
spin-orbit coupling. We show that the studied system supports a generalization
of $p_x+ip_y$ superconductivity and that its topological phase diagram contains
Chern numbers higher than $\xi/a$ $(\gg1)$, where $\xi$ is the superconducting
coherence length and $a$ is the distance between the magnetic atoms. The
signatures of nontrivial topology can be observed by STM spectroscopy in
finite-size islands.

link to article (opens in new tab)

 

Author(s):

Authors: Yuki Nagai, Hiroki Nakamura, Masahiko Machida

We numerically show that the zero-energy Majorana surface states are
suppressed around a vortex in the three-dimensional topological superconductors
such as CuxBi2Se3 and Sn1-xInxTe. On the other hand, the zero-energy Majorana
bound states along the vortex line are robust against cut by the surface. This
difference of the robustness between the surface- and vortex- bound states is
originating from the fact that the vortex breaks the time-reversal symmetry
which protects the surface states, whereas the surface for the vortex is just
regarded as the wall. The suppression of the surface-bound states around a
vortex can be observed as the unconventional energy dependence of imaging of
the scanning tunneling microscopy/spectroscopy.

link to article (opens in new tab)

 

Author(s):

Authors: Dominic J. Williamson, Nick Bultinck, Michael Mariën, Mehmet B. Sahinoglu, Jutho Haegeman, Frank Verstraete

Projected entangled pair states (PEPS) provide a natural description of the
ground states of gapped, local Hamiltonians in which global characteristics of
a quantum state are encoded in properties of local tensors. We show that
on-site symmetries, as occurring in systems exhibiting symmetry-protected
topological (SPT) quantum order, can be captured by a virtual symmetry of the
tensors expressed as a set of matrix product operators labelled by the
different group elements. A classification of SPT phases can hence be obtained
by studying the topological obstructions to continuously deforming one set of
matrix product operators into another. This leads to the classification of
bosonic SPT states in terms of group cohomology, as originally derived by Chen
et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed
point models, and hence opens up the possibility of studying phase transitions
between different SPT phases. We furthermore show how the global symmetries of
SPT PEPS can be promoted into a set of local gauge constraints by introducing
bosonic degrees of freedom on the links of the PEPS lattice, thereby providing
a natural and general mapping between PEPS in SPT phases and topologically
ordered phases.

link to article (opens in new tab)

 

Author(s):

Authors: Parisa Zareapour, Alex Hayat, Shu Yang F. Zhao, Michael Kreshchuk, Yong Kiat Lee, Anjan A. Reijnders, Achint Jain, Zhijun Xu, T. S. Liu, G.D. Gu, Shuang Jia, Robert J. Cava, Kenneth S. Burch

High-temperature superconductors exhibit a wide variety of novel excitations.
If contacted with a topological insulator, the lifting of spin rotation
symmetry in the surface states can lead to the emergence of unconventional
superconductivity and novel particles. In pursuit of this possibility, we
fabricated high critical-temperature (Tc ~ 85 K) superconductor/topological
insulator (Bi2Sr2CaCu2O8+delta/Bi2Te2Se) junctions. Below 75 K, a zero-bias
conductance peak (ZBCP) emerges in the differential conductance spectra of this
junction. The magnitude of the ZBCP is suppressed at the same rate for magnetic
fields applied parallel or perpendicular to the junction. Furthermore, it can
still be observed and does not split up to at least 8.5 T. The temperature and
magnetic field dependence of the excitation we observe appears to fall outside
the known paradigms for a ZBCP.

link to article (opens in new tab)

 

Author(s):Yu I. Latyshev

There are two types of intrinsic surface states in solids. The first type is formed on the surface of topological insulators. Recently, transport of massless Dirac fermions in the band of “topological” states has been demonstrated. States of the second type were predicted by Tamm and Shockley long ago. They do not have a topological background and are therefore strongly dependent on the properties of the surface. We study the problem of the conductivity of Tamm-Shockley edge states through direct transport experiments. Aharonov-Bohm magneto-oscillations of resistance are found on graphene samples that contain a single nanohole. The effect is explained by the conductivity of the massless Dirac fermions in the edge states cycling around the nanohole. The results demonstrate the deep connection between topological and non-topological edge states in 2D systems of massless Dirac fermions.

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Author(s): Hoi-Yin Hui, Alejandro M. Lobos, Jay D. Sau, and S. Das Sarma

We consider recent experiments on wide superconductor–quantum spin Hall insulator (QSHI)–superconductor Josephson junctions, which have shown preliminary evidence of proximity-induced superconductivity at the edge modes of the QSHI system based on an approximate analysis of the observed Fraunhofer s…

[Phys. Rev. B 90, 224517] Published Thu Dec 18, 2014

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