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Interacting potential between spinons in the compact QED3 description of the Heisenberg model

Raoul Dillenschneider, J. Richert

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

We implement a Chern-Simons (CS) contribution into the compact QED3 description of the antiferromagnetic Heisenberg model in two dimensions at zero temperature. The CS term allows for the conservation of the SU(2) symmetry of the quantum spin system and fixes the flux through a plaquette to be a multiple of pi as was shown by Marston. We work out the string tension of the confining potential which acts between the spinons and show that the CS term induces a screening effect on the magnetic field only. The confining potential between spinons is not affected by the CS flux. The strict site-occupation by a single spin-1/2 is enforced by the introduction of an imaginary chemical potential constraint.

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We implement a Chern-Simons (CS) contribution into the compact QED3 description of the antiferromagnetic Heisenberg model in two dimensions at zero temperature. The CS term allows for the conservation of the SU(2) symmetry of the quantum spin system and fixes the flux through a plaquette to be a multiple of pi as was shown by Marston. We work out the string tension of the confining potential which acts between the spinons and show that the CS term induces a screening effect on the magnetic field only. The confining potential between spinons is not affected by the CS flux. The strict site-occupation by a single spin-1/2 is enforced by the introduction of an imaginary chemical potential constraint.
We study the effect of site occupation in two-dimensional quantum spin systems at finite temperature in a $\ensuremath{\pi}$-flux state description at the mean-field level. We impose each lattice site to be occupied by a single SU(2)SU(2)SU(2) spin. This is realized by means of a specific prescription. We consider the low-energy Hamiltonian which is mapped into a QED3{\mathrm{QED}}_{3}QED3​ Lagrangian of spinons. We compare the dynamically generated mass to the one obtained by means of an average site occupation constraint.
A mutual Chern-Simons Lagrangian is derived as a minimal field theory description of the phase-string model for doped antiferromagnets. Such an effective Lagrangian is shown to retain the full symmetries of parity, time reversal, and global SU(2) spin rotation, in contrast to conventional Chern-Simons theories where first two symmetries are usually broken. Two ordered phases---i.e., antiferromagnetic and superconducting states---are found at low temperatures as characterized by ``dual'' Meissner effects and dual-flux-quantization conditions due to the mutual Chern-Simons gauge structure. A ``dual'' confinement in charge and spin degrees of freedom occurs such that no true spin-charge separation is present in these ordered phases, but the spin-charge separation/deconfinement serves as a driving force in the unconventional phase transitions of these ordered states to disordered states.
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.
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.
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 reexamine two-dimensional frustrated quantum magnetism with the aim of exploring new critical points and critical phases. We study easy-plane triangular antiferromagnets using a dual vortex approach, fermionizing the vortices with a Chern-Simons field. Herein we develop this technique for integer-spin systems which generically exhibit a simple paramagnetic phase as well as magnetically ordered phases with coplanar and collinear spin order. Within the fermionized-vortex approach, we derive a low-energy effective theory containing Dirac fermions with two flavors minimally coupled to a U(1) and a Chern-Simons gauge field. At criticality we argue that the Chern-Simons gauge field can be subsumed into the U(1) gauge field, and up to irrelevant interactions one arrives at quantum electrodynamics in 2+12+12+1 dimensions (QED3). Moreover, we conjecture that critical QED3 with full SU(2) flavor symmetry describes the O(4) multicritical point of the spin model where the paramagnet and two magnetically ordered phases merge. The remarkable implication is that QED3 with flavor SU(2) symmetry is dual to ordinary critical ${\ensuremath{\Phi}}^{4}$ field theory with O(4) symmetry. This leads to a number of unexpected, verifiable predictions for QED3. A connection of our fermionized-vortex approach with the dipole interpretation of the $\ensuremath{\nu}=1∕2$ fractional quantum Hall state is also demonstrated. The approach introduced in this paper will be applied to spin-1∕21∕21∕2 systems in a forthcoming publication.