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arXiv:2210.07718 · Physical review. B./Physical review. B

Jordan-Wigner fermionization of quantum spin systems on arbitrary two-dimensional lattices: A mutual Chern-Simons approach

Jagannath Das, Aman Kumar, Avijit Maity, Vikram Tripathi

Published Aug 7, 2023• 3 citations
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Abstract

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.

DOI
10.1103/physrevb.108.085110

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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-1/21/21/2 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 zzz-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 zzz-direction, which we also confirm by spin-wave calculations.
We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'eel state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular, we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin-1/21/21/2 XY models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous U(1)\mathrm{U}(1)U(1) symmetry generating a linearly dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'eel antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of the system with frustration, we show that the competing interactions emerge in the form of long-range interaction vertices mediated by the CS gauge field, which are responsible for restoring the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications for unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons, and discuss the results.
Two-dimensional quantum antiferromagnets host rich physics, including long-range ordering, high-Tc{T}_{c}Tc​ superconductivity, quantum spin-liquid behavior, topological ordering, a variety of other exotic phases, and quantum criticalities. Frustrating perturbations in antiferromagnets may give rise to strong quantum fluctuations, challenging the theoretical understanding of the many-body ground state. Here we develop a method to describe the quantum antiferromagnets using fermionic degrees of freedom. The method is based on a formally exact mapping between spin-exchange models and theories describing fermionic matter with the emergent U(1)\text{U}(1)U(1) Chern-Simons gauge field. For the planar N\'eel state, this mapping self-consistently generates the Chern-Simons superconductor mean-field ground state of introduced spinless fermions. We systematically compare the Chern-Simons superconductor state with the planar N\'eel state at the level of collective modes as well as order parameters. We reveal qualitative and quantitative correspondences between these two states. We demonstrate that such a construction using the fractionalized excitations and Chern-Simons gauge field can not only describe the N\'eel order, but can also be applied to study quantum spin liquids. Furthermore, we show that the confinement-deconfinement transitions from the N\'eel order to quantum spin liquids are signaled and characterized by the instabilities of Chern-Simons superconductors, driven by strong frustration. The results suggest observing and classifying the descendants of antiferromagnets, including other ordered states and unconventional superconductors, as well as emergent quantum spin liquids.
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
We use the Chern-Simons (CS) fermion representation of s=1/2s=1/2s=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 XYXYXY 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.
Frustrated spin systems on kagome lattices have long been considered to be a promising candidate for realizing exotic spin-liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as volborthite and herbertsmithite that have kagomelike structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at m=13m=\frac{1}{3}m=31​ is considered to be the most prominent. Here, we study the problem of the antiferromagnetic spin-12\frac{1}{2}21​ quantum XXZ Heisenberg model on a kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Such a lattice Chern-Simons term had previously only been written for the square lattice and was used to successfully study the unfrustrated Heisenberg antiferromagnet on the square lattice. At a mean-field level, these ideas have also been applied to frustrated systems by ignoring the details of the Chern-Simons term. However, fluctuations are generally strong in these models and are expected to affect the mean-field physics. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated kagome lattice, we can now formalize the Jordan-Wigner transformation on the kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of 13\frac{1}{3}31​ filling (m=13m=\frac{1}{3}m=31​ plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of XY\text{XY}XY anisotropy, the ground state at the 13\frac{1}{3}31​ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 12\frac{1}{2}21​ and that at the 59\frac{5}{9}95​ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction 23\frac{2}{3}32​.
We generalize the Kitaev's spin-1/2 model on the honeycomb by introducing a two-dimensional Z3{Z}_{3}Z3​ clock model on the triangular lattice with three-body interaction. We discuss various properties of this model and show that the low energy theory of the Z3{Z}_{3}Z3​ generalized Kitaev model (GKM) is described by a single Z3{Z}_{3}Z3​ parafermion per lattice site coupled to a Z3{Z}_{3}Z3​ gauge field. We also introduce a slave-fermion approach for this GKM, treat the resulting fermionic Hamiltonian at the mean-field level, solve the mean-field parameters self-consistently, and obtain the low energy effective Chern-Simons (CS) gauge theory. The resulting CS gauge theory is identical to that of a (221)(221)(221) fractional quantum Hall state. We then go beyond the mean-field approximation and demonstrate that fluctuations generate a uniform interlayer pairing for the dual (221)(221)(221) bilayer state. We argue that this perturbed system can undergo a phase transition to the Fibonacci phase by tuning the interlayer pairing strength.