Tensor network states and topological quantum phases

We apply the charge-pumping argument to fermionic tensor network representations of $d$ -dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for $(d+1)$ -dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (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-neighbor 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.We apply the charge-pumping argument to fermionic tensor network representations of $d$-dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for $(d+1)$-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (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-neighbor 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.

arXiv:1912.08219Physical review. B./Physical review. BMar 19, 20206 citations

Fermionic projected entangled-pair states and topological phases

Nick Bultinck, Dominic J. Williamson, Jutho Haegeman +1 more

We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of symmetry-protected and intrinsic topological phases. The tensor networks states presented here are well-suited for numerical applications and hence open up new possibilities for studying interacting fermionic topological phases.We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of symmetry-protected and intrinsic topological phases. The tensor networks states presented here are well-suited for numerical applications and hence open up new possibilities for studying interacting fermionic topological phases.

arXiv:1707.00470Journal of Physics A Mathematical and TheoreticalNov 10, 201753 citations

Topological Phenomena in Tensor Network States of Quantum Spin Systems

Mohsin Iqbal, Norbert Schuch, Stefan Weßel

The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the $D(\\mathbb{Z}_4)$ quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of $\\mathbb{Z}_2$ -invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of $\\mathbb{Z}_2$ spin liquids. The behavior of the system at the phase boundaries of $\\mathbb{Z}_2$ spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the $\\mathbb{Z}_4$ -invariant tensors with $SU(2)$ symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped $\\mathbb{Z}_2$ spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped $\\mathbb{Z}_2$ spin liquid phase.The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the $D(\\mathbb{Z}_4)$ quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of $\\mathbb{Z}_2$-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of $\\mathbb{Z}_2$ spin liquids. The behavior of the system at the phase boundaries of $\\mathbb{Z}_2$ spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the $\\mathbb{Z}_4$-invariant tensors with $SU(2)$ symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped $\\mathbb{Z}_2$ spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped $\\mathbb{Z}_2$ spin liquid phase.

Jan 1, 20180 citations

Dynamical classification of topological quantum phases

Lin Zhang, Long Zhang, Sen Niu +1 more

arXiv:1802.10061中国科学通报：英文版Sep 29, 2018150 citations

We apply the charge-pumping argument to fermionic tensor network representations of

d

-dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for

(d+1)

-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (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-neighbor 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.

D(\\mathbb{Z}_4)

quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of

\\mathbb{Z}_2

-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of

\\mathbb{Z}_2

spin liquids. The behavior of the system at the phase boundaries of

\\mathbb{Z}_2

spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the

\\mathbb{Z}_4

-invariant tensors with

SU(2)

symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped

\\mathbb{Z}_2

spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped

\\mathbb{Z}_2

spin liquid phase.