Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm

T. Senthil, Leon Balents, Subir Sachdev, Ashvin Vishwanath, and Matthew P. A. Fisher

Phys. Rev. B 70, 144407 – Published 15 October 2004

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 Néel 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 $Z_{2}$ 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.

Received 26 December 2003

DOI:https://doi.org/10.1103/PhysRevB.70.144407

Authors & Affiliations

T. Senthil¹, Leon Balents², Subir Sachdev³, Ashvin Vishwanath¹, and Matthew P. A. Fisher⁴

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics, University of California, Santa Barbara, California 93106-4030, USA
³Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
⁴Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA

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Vol. 70, Iss. 14 — 1 October 2004

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Images

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. (2.1) (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. (1.2). The VBS ground state appears for $g > g_{c}$ , and is characterized by the order parameter $ψ_{VBS}$ in Eq. (1.4)—the distinct lines represent distinct values of $⟨ {\vec{S}}_{r} \cdot {\vec{S}}_{r^{'}} ⟩$ on each link. The VBS state on the left has “columnar” bond order, while that on the right has “plaquette” order. The theory $L_{z}$ in Eq. (1.7) applies only at the QCP $g = g_{c}$ at its critical point obtained at $s = s_{c}$ .Reuse & Permissions
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 $X Y$ 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 $\hat{n} (∣ r ∣ \to \infty) = (0, 0, - 1)$ .Reuse & Permissions
Figure 3
A monopole event, taken to occur at the origin of space-time. An equal-time slice of space-time at the tunneling time is represented following the conventions of Fig. 2. 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 space-time passing through the origin. The monopole above has ${\hat{n}}_{r} = r ∕ ∣ r ∣$ .Reuse & Permissions
Figure 4
Specification of the fixed field $ϑ = - ζ ∕ 4$ . The filled circles are the sites of the direct lattice, and $ϑ$ resides on the sites of the dual lattice.Reuse & Permissions
Figure 5
The “meron” vortices in the easy-plane case. There are two such vortices, $ψ_{1, 2}$ , and $ψ_{1}$ is represented in (a) and (b), while $ψ_{2}$ is represented by (a) and (c), following the conventions of Fig. 2. The $ψ_{1}$ meron above has $\hat{n} (r = 0) = (0, 0, 1)$ and $\hat{n} (∣ r ∣ \to \infty) = (x, y, 0) ∕ ∣ r ∣$ ; the $ψ_{2}$ meron has $\hat{n} (r = 0) = (0, 0, - 1)$ and the same limit as $∣ r ∣ \to \infty$ . Each meron above is “half” the skyrmion in Fig. 2: this is evident from a comparison of (b) and (c) above with Fig. 2b. Similarly, one can observe that a composite of $ψ_{1}$ and $ψ_{2}^{*}$ makes one skyrmion.Reuse & Permissions
Figure 6
Specification of the nonzero values of the fixed fields (a) $B_{0}$ and (b) $β$ . The circles are the sites of the direct lattice, $j$ , while the crosses are the sites of the dual lattice, $\bar{j}$ ; the latter are also offset by half a lattice spacing in the direction out of the paper (the $μ = τ$ direction). The $B_{0 μ}$ are all zero for $μ = τ, x$ , while the only nonzero values of $B_{0 y}$ are shown in (a). Only the $μ = τ$ components of $β_{μ}$ are nonzero, and these are shown in (b).Reuse & Permissions

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