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arXiv:1710.04668 · Physical review. B./Physical review. B

Deconfined quantum critical point on the triangular lattice

Chao‐Ming Jian, Alex Thomson, Alex Rasmussen, Zhen Bi, Cenke Xu

Published May 10, 2018• 43 citations
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Abstract

In this work we propose a theory for the deconfined quantum critical point (DQCP) for spin-1/2 systems on a triangular lattice, which is a direct unfine-tuned quantum phase transition between the standard ``$\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$'' noncollinear antiferromagnetic order (or the so-called ${120}^{\ensuremath{\circ}}$ state) and the ``$\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$'' valence solid bond (VBS) order, both of which are very standard ordered phases often observed in numerical simulations. This transition is beyond the standard Landau-Ginzburg paradigm and is also fundamentally different from the original DQCP theory on the square lattice due to the very different structures of both the magnetic and VBS order on frustrated lattices. We first propose a topological term in the effective-field theory that captures the ``intertwinement'' between the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ antiferromagnetic order and the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ VBS order. Then using a controlled renormalization-group calculation, we demonstrate that an unfine-tuned direct continuous DQCP exists between the two ordered phases mentioned above. This DQCP is described by the ${N}_{f}=4$ quantum electrodynamics (QED) with an emergent PSU(4)=SU(4)/${Z}_{4}$ symmetry only at the critical point. The aforementioned topological term is also naturally derived from the ${N}_{f}=4$ QED. We also point out that physics around this DQCP is analogous to the boundary of a $3d$ bosonic symmetry- protected topological state with only on-site symmetries.

DOI
10.1103/physrevb.97.195115

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In this work we propose a theory for the deconfined quantum critical point (DQCP) for spin-1/2 systems on a triangular lattice, which is a direct unfine-tuned quantum phase transition between the standard $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$'' noncollinear antiferromagnetic order (or the so-called ${120}^{\ensuremath{\circ}}$ state) and the 12\ifmmode×\else\texttimes\fi12\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}12​\ifmmode×\else\texttimes\fi12​'' valence solid bond (VBS) order, both of which are very standard ordered phases often observed in numerical simulations. This transition is beyond the standard Landau-Ginzburg paradigm and is also fundamentally different from the original DQCP theory on the square lattice due to the very different structures of both the magnetic and VBS order on frustrated lattices. We first propose a topological term in the effective-field theory that captures the ``intertwinement'' between the 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ antiferromagnetic order and the 12\ifmmode×\else\texttimes\fi12\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}12​\ifmmode×\else\texttimes\fi12​ VBS order. Then using a controlled renormalization-group calculation, we demonstrate that an unfine-tuned direct continuous DQCP exists between the two ordered phases mentioned above. This DQCP is described by the Nf=4{N}_{f}=4Nf​=4 quantum electrodynamics (QED) with an emergent PSU(4)=SU(4)/Z4{Z}_{4}Z4​ symmetry only at the critical point. The aforementioned topological term is also naturally derived from the Nf=4{N}_{f}=4Nf​=4 QED. We also point out that physics around this DQCP is analogous to the boundary of a 3d3d3d bosonic symmetry- protected topological state with only on-site symmetries.
Deconfined quantum critical points (DQCPs) are putative phase transitions beyond the Landau paradigm with emergent fractionalized degrees of freedom. The original example of a DQCP is the spin-12\frac{1}{2}21​ quantum antiferromagnet on the square lattice which features a second-order transition between valence bond solid (VBS) and N\'eel order. The VBS order breaks a lattice symmetry, and the corresponding VBS order parameter may couple to lattice distortion modes (phonons) at appropriate momenta. We investigate a field-theoretic description of the DQCP in the presence of such a spin-lattice coupling. We show that treating phonons as classical lattice distortions leads to a relevant monopole-phonon interaction inducing an instability towards a distorted lattice by an analogous mechanism to the spin-Peierls instability in one dimension. Consequently, there is a breakdown of the DQCP which generally becomes a strong first-order transition. Taking into account the full quantum nature of the phonons, we argue that the continuous DQCP persists above a critical phonon frequency. Lastly, we comment on the connection to general gapless, deconfined gauge theories.
Abstract Deconfined quantum critical points (DQCPs) have been proposed as a class of continuous quantum phase transitions occurring between two ordered phases with distinct symmetry-breaking patterns, beyond the conventional framework of Landau-Ginzburg-Wilson (LGW) theory. At the DQCP, the system exhibits emergent gauge fields, fractionalized excitations, and enhanced symmetries. Here we review recent theoretical and experimental progress on exploring DQCPs in condensed matter systems. We first introduce theoretical advancements in the study of DQCPs over the past twenty years, particularly in magnetic models on square lattices, honeycomb lattices, kagome lattices, and one-dimensional spin chains. We then discuss recent progress on experimental realization of DQCP in quantum magnetic systems. Experimentally, the Shastry-Sutherland model, realized in SrCu2(BO3)2, offers a particularly promising platform for realizing DQCPs. The magnetic frustration inherent to this model drives phase transitions between two distinct symmetry-breaking states: a valence bond solid (VBS) phase and a Néel antiferromagnetic phase. Remarkably, SrCu2(BO3)2 has provided the first experimental evidence of a proximate DQCP through a field-induced Bose-Einstein condensation, transitioning from the VBS state to the Néel state. Nevertheless, the direct experimental realization of a DQCP remains a significant challenge. Despite this, it offers a promising platform for exploring emergent phenomena through quantum phase transition in low-dimensional quantum systems.
We propose field theories for a deconfined quantum critical point in SU(3) antiferromagnets on the triangular lattice. In particular we consider the continuous transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU(3) singlet phase. Starting from the magnetically ordered state we derive a critical theory in terms of fractional bosonic degrees of freedom, in close analogy to the well-developed description of the SU(2) N\'eel---valence bond solid (VBS) transition on the square lattice. Our critical theory consists of three coupled CP2C{P}^{2}CP2 models and we study its fixed point structure using a functional renormalization group approach in a suitable large NNN limit. We find a stable critical fixed point and estimate its critical exponents, thereby providing an example of deconfined criticality beyond the universality class of the CPNC{P}^{N}CPN model. In addition we present a complementary route towards the critical field theory by studying topological defects of the trimerized SU(3) singlet phase.
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.
Recent proposal of the duality between the N=2N=2N=2 noncompact QED3_33​ and the easy-plane noncompact CP1^11 (NCCP1^11) model suggests that the deconfined quantum critical point (dQCP) between the easy-plane antiferromagnet and the VBS order on the square lattice may have an emergent O(4) symmetry, due to the self-duality of the N=2N=2N=2 noncompact QED3_33​. Recent numerical progresses suggest that this easy-plane dQCP does exist and it has an emergent O(4) symmetry. But for the O(4) symmetry to really emerge at the dQCP, certain O(4) symmetry breaking perturbations need to be irrelevant at the putative O(4) fixed point. It is more convenient to study these symmetry breaking perturbations in the N=2N=2N=2 noncompact QED3_33​. We demonstrate that a natural large-NNN generalization and a controlled 1/N1/N1/N expansion supports the stability of the O(4) fixed point against the symmetry breaking perturbations. We also develop the theory for two tricritical points close to the easy-plane dQCP. One tricritical point is between the dQCP and a {\it self-dual} Z2Z_2Z2​ topological order; the other is the tricritical point that connects the continuous dQCP and a first order Néel-VBS transition, motivated by recent numerical results.
Deconfined quantum critical points (DQCPs) are proposed as unconventional second-order phase transitions beyond the Landau-Ginzburg-Wilson paradigm. The nature and experimental realizations of DQCPs are crucial issues of importance. We illustrate the potential for DQCPs between the valence bond solid state and the antiferromagnetic phase to arise in optical lattices containing frustrated dipolar bosons subject to hard-core constraints. The emergence of DQCPs is comprehended through the fusion of two Berezinskii-Kosterlitz-Thouless (BKT) transitions. The DQCPs and the BKTs are confirmed by the scaling of ground-state fidelity susceptibilities in finite systems and the analysis of order parameters obtained from infinite systems. The numerical analysis reveals varying critical exponents of the correlation length in DQCPs and the logarithmic scaling in BKTs, respectively. In this paper, we offer a promising platform for realizing DQCPs and provide valuable insights into their nature within the framework of topological phase transitions.