Inhomogeneous Two-Dimensional Fermion Systems in External Magnetic Field or Chern-Simons Flux

A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin- $1/2$ magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the $z$ -direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the $z$ -direction, which we also confirm by spin-wave calculations.A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-$1/2$ magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the $z$-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the $z$-direction, which we also confirm by spin-wave calculations.

arXiv:2210.07718Physical review. B./Physical review. BAug 7, 20233 citations

Fermion Systems in Different Dimensions

Leon N. Cooper

Springer eBooksJan 1, 19770 citations

Topological spin ordering via Chern-Simons superconductivity

Tigran Sedrakyan, Victor M. Galitski, Alex Kamenev

We use the Chern-Simons (CS) fermion representation of $s=1/2$ spin operators to construct topological, long-range magnetically ordered states of interacting two-dimensional (2D) quantum spin models. We show that the fermion-fermion interactions mediated by the dynamic CS flux attachment may give rise to Cooper pairing of the fermions. Specifically, in an $XY$ model on the honeycomb lattice, this construction leads to a ``CS superconductor'', which belongs to a topologically nontrivial in 2D symmetry class DIII, with particle-hole and time-reversal symmetries. It is shown that in the original spin language, this state corresponds to a symmetry protected topological state, which coexists with a magnetic long-range order. We discuss physical manifestations of the topological character of the corresponding state.We use the Chern-Simons (CS) fermion representation of $s=1/2$ spin operators to construct topological, long-range magnetically ordered states of interacting two-dimensional (2D) quantum spin models. We show that the fermion-fermion interactions mediated by the dynamic CS flux attachment may give rise to Cooper pairing of the fermions. Specifically, in an $XY$ model on the honeycomb lattice, this construction leads to a ``CS superconductor'', which belongs to a topologically nontrivial in 2D symmetry class DIII, with particle-hole and time-reversal symmetries. It is shown that in the original spin language, this state corresponds to a symmetry protected topological state, which coexists with a magnetic long-range order. We discuss physical manifestations of the topological character of the corresponding state.

arXiv:1606.08473Physical review. B./Physical review. BMar 15, 201723 citations

Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field

Alexander G. Abanov, Andrey Gromov

We compute electromagnetic, gravitational, and mixed linear response functions of two-dimensional free fermions in an external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave vectors and frequencies of the probing electromagnetic and metric fields. In addition to the well-studied $U(1)$ Chern-Simons and Wen-Zee terms we find a gravitational Chern-Simons term that controls the correction to the Hall viscosity due to the background curvature. We relate the coefficient in front of the term with the chiral central charge.We compute electromagnetic, gravitational, and mixed linear response functions of two-dimensional free fermions in an external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave vectors and frequencies of the probing electromagnetic and metric fields. In addition to the well-studied $U(1)$ Chern-Simons and Wen-Zee terms we find a gravitational Chern-Simons term that controls the correction to the Hall viscosity due to the background curvature. We relate the coefficient in front of the term with the chiral central charge.

arXiv:1401.3703Physical Review BJul 29, 2014164 citations

1/2

magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the

z

-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the

z

-direction, which we also confirm by spin-wave calculations.

We use the Chern-Simons (CS) fermion representation of

s=1/2

spin operators to construct topological, long-range magnetically ordered states of interacting two-dimensional (2D) quantum spin models. We show that the fermion-fermion interactions mediated by the dynamic CS flux attachment may give rise to Cooper pairing of the fermions. Specifically, in an

XY

model on the honeycomb lattice, this construction leads to a ``CS superconductor'', which belongs to a topologically nontrivial in 2D symmetry class DIII, with particle-hole and time-reversal symmetries. It is shown that in the original spin language, this state corresponds to a symmetry protected topological state, which coexists with a magnetic long-range order. We discuss physical manifestations of the topological character of the corresponding state.

U(1)

Chern-Simons and Wen-Zee terms we find a gravitational Chern-Simons term that controls the correction to the Hall viscosity due to the background curvature. We relate the coefficient in front of the term with the chiral central charge.