We show that despite the absence of a Hopf term and zero Berry phase terms, the N`eel ordered phase of 2+1 D quantum antiferromagnets have spin 1/2 excitations, i.e. spinons. The spinons are skyrmion excitations of a topological nature. Since skyrmion gap is proportional to the spin stiffness, quantum criticality corresponds to skyrmion gap collapse. We speculate that skyrmions are relevant at criticality and are, perhaps, related to recent suggestions of critical fractionalization.
The familiar spin-
21 quantum Heisenberg antiferromanget in a two-dimensional (2D) square lattice is shown, within the nonlinear sigma model approximations, to be another state of matter that has excitations with fractional quantum numbers above a finite energy gap. The 1-skyrmion with an energy $\ensuremath{\approx}2\ensuremath{\pi}J$ is shown to be made of two ``deconfined spinons'' or ``O(3) vortices.'' The many skyrmion operator and the wave functions that we have found are strikingly similar to quantum Hall quasiparticle operators and wave functions. We also predict the presence of finite energy ``spin-
S spinon'' for a general spin S Heisenberg antiferromagnets in 2D. Some consequences are briefly discussed.
We investigate the quantum phase transition from antiferromagnetism (
AF) to algebraic spin liquid (
ASL). {\it We propose that spin 1/2 fermionic spinons in the
ASL fractionalize into spin 1/2 bosonic spinons and spinless fermions at the quantum critical point (
QCP) between the
AF and the
ASL}. Condensation of the bosonic spinons leads to the
AF, where the condensed bosonic spinons are confined with the spinless fermions to form the fermionic spinons. These fermionic spinons are also confined to make antiferromagnons as elementary excitations in the
AF. {\it Approaching the
QCP from the
AF, spin 1 critical antiferromagnetic fluctuations are expected to break up into spin 1/2 critical bosonic spinons. Then, these bosonic spinons hybridize with spin 1/2 fermionic spinons, making spinless fermions}. As a result the fermionic spinons decay into the bosonic spinons and the spinless fermions. But, the spinless fermions are confined and thus, only the bosonic spinons emerge at the
QCP. This coincides with the recent studies of {\it deconfined quantum criticality}\cite{Laughlin_deconfinement,Senthil_deconfinement,Kim1,Ichinose_de confinement}. When the bosonic spinons are gapped, the
ASL is realized. The bosonic spinons are confined with the spinless fermions to form the fermionic spinons. These fermionic spinons are deconfined to describe the
ASL.
We study the level structure of excitations at the ``deconfined'' critical point separating antiferromagnetic and valence-bond-solid phases in two-dimensional quantum spin systems using the $J\text{\ensuremath{-}}Q$ model as an example. Energy gaps in different spin
(S) and momentum
(k) sectors are extracted from imaginary-time correlation functions obtained in quantum Monte Carlo simulations. We find strong quantitative evidence for deconfined linearly dispersing spinons with gapless points at $\mathbf{k}=(0,0),(\ensuremath{\pi},0),(0,\ensuremath{\pi})$, and $(\ensuremath{\pi},\ensuremath{\pi})$, as inferred from two-spinon excitations (
S=0 and
S=1 states) around these points. We also observe a duality between singlet and triplet excitations at the critical point and inside the ordered phases, in support of an enhanced symmetry, possibly SO(5).
Deconfined quantum criticality of two-dimensional
SU(2) quantum antiferromagnets featuring a transition from an antiferromagnetically ordered ground state to a so-called valence-bond solid state, is governed by a non-compact CP
1 model with a Maxwell term in 2+1 spacetime dimensions. We introduce a new perspective on deconfined quantum criticality within a field-theoretic framework based on an expansion in powers of
ε=4−d for fixed number
N of complex matter fields. We show that in the allegedly weak first-order transition regime, a so-called conformal phase transition leads to a genuine deconfined quantum critical point. In such a transition, the gap vanishes when the critical point is approached from above and diverges when it is approached from below. We also find that the spin stiffness has a universal jump at the critical point.
We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the
S=1∕2 square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the
Z2 spin liquid: the critical theory has deconfined excitations interacting with an emergent
U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
It has been proposed that there are degrees of freedom intrinsic to quantum critical points that can contribute to quantum critical physics. We point out that this conclusion is quite general below the upper critical dimension. We show that in (2+1)D antiferromagnets Skyrmion excitations are stable at criticality and identify them as the critical excitations. We find exact solutions composed of Skyrmion and anti-Skyrmion superpositions, which we call topolons. We include the topolons in the partition function and renormalize by integrating out small size topolons and short wavelength spin waves. We obtain a correlation length exponent nu=0.690 666 and anomalous dimension eta=0.0166.