Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm

TL;DR

<3-5 sentence high-level summary>This work challenges the Landau-Ginzburg-Wilson paradigm by introducing deconfined quantum critical points (DQCPs) in two-dimensional spin-1/2 magnets and related bosonic systems, where criticality is governed by fractionalized spinons coupled to a noncompact U(1) gauge field rather than fluctuations of the local order parameters. The authors develop concrete lattice realizations (the SJ CP^N-1 models) and their easy-plane and isotropic limits, showing monopole (instanton) tunneling is irrelevant at the critical point, yielding emergent topological conservation and two divergent length scales. They connect Néel–valence-bond-solid (VBS) transitions to dual vortex descriptions, demonstrate self-duality in the easy-plane limit, and extend the framework to VBS–spin-liquid transitions and bosonic superfluid–insulator transitions, highlighting broad implications for correlated electron systems and potential experimental probes. The results imply large anomalous dimensions for order parameters and provide a unified language for deconfined criticality, challenging conventional LGW-based analyses of quantum phase transitions in 2D.

Abstract

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 Neel 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. The critical theory is not expressed in terms of the order parameters characterizing either state (as would be the case in Landau-Ginzburg-Wilson theory) 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 Z_2 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.

Figures (6)

• Figure 1: Ground states of the square lattice $S=1/2$ antiferromagnet studied in this paper. The coupling $g$ controls the strength of quantum spin fluctuations about a magnetically ordered state, and appears in Eq. (\ref{['eq:nlsm']}) (the classical limit is $g=0$). There is broken spin rotation invariance in the Néel state for $g<g_c$, described by the order parameter $\vec{N}_r$ in Eq. (\ref{['eq:staggered']}). The VBS ground state appears for $g>g_c$, and is characterized by the order parameter $\psi_{\rm VBS}$ in Eq. (\ref{['eq:VBSorder']}) -- the distinct lines represent distinct values of $\langle \vec{S}_r \cdot \vec{S}_{r'} \rangle$ on each link. The VBS state on the left has "columnar" bond order, while that on the right has "plaquette" order. The theory $\mathcal{L}_z$ in Eq. (\ref{['sz']}) applies only at the QCP $g=g_c$ at its critical point obtained at $s=s_c$.
• Figure 2: A skyrmion configuration of the field $\hat{n}_r$. In (a) we show the vector $(n^x,n^y)$ at different points in the XY plane; note that $\hat{n} \propto (-1)^{x+y} \vec{S}_r$, and so the underlying spins have a rapid sublattice oscillation which is not shown. In (b) we show the vector $(n^x, n^z)$ along a section of (a) on the $x$ axis. Along any other section of (a), a picture similar to (b) pertains, as the former is invariant under rotations about the $z$ axis. The skyrmion above has $\hat{n} (r=0) = (0,0,1)$ and $\hat{n}(|r| \rightarrow \infty) = (0,0,-1)$.
• Figure 3: A monopole event, taken to occur at the origin of spacetime. An equal-time slice of spacetime at the tunnelling time is represented following the conventions of Fig \ref{['skyr']}. So (a) contains the vector $(n^x, n^y)$; the spin configuration is radially symmetric, and consequently a similar picture is obtained along any other plane passing through the origin. Similarly, (b) is the representation of $(n^x, n^z)$ along the $x$ axis, and a similar picture is obtained along any line in spacetime passing through the origin. The monopole above has $\hat{n}_r = r/|r|$.
• Figure 4: Specification of the fixed field $\vartheta = -\zeta/4$. The filled circles are the sites of the direct lattice, and $\vartheta$ resides on the sites of the dual lattice.
• Figure 5: The 'meron' vortices in the easy plane case. There are two such vortices, $\psi_{1,2}$, and $\psi_1$ is represented in (a) and (b), while $\psi_2$ is represented by (a) and (c), following the conventions of Fig \ref{['skyr']}. The $\psi_1$ meron above has $\hat{n} (r=0) = (0,0,1)$ and $\hat{n} ( |r| \rightarrow \infty ) = (x,y,0)/|r|$; the $\psi_2$ meron has $\hat{n} (r=0) = (0,0,-1)$ and the same limit as $|r| \rightarrow \infty$. Each meron above is 'half' the skyrmion in Fig \ref{['skyr']}: this is evident from a comparison of (b) and (c) above with Fig \ref{['skyr']}b. Similarly, one can observe that a composite of $\psi_1$ and $\psi_2^\ast$ makes one skyrmion.