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arXiv:0803.3822 · Physical Review B

Quantum phase transitions beyond the Landau paradigm in a Sp(4) spin system

Yang Qi, Cenke Xu

Published Jul 7, 2008• 13 citations
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Abstract

We propose quantum phase transitions beyond the Landau's paradigm of Sp(4) spin Heisenberg models on the triangular and square lattices motivated by the exact $\text{Sp}(4)\ensuremath{\simeq}\text{SO}(5)$ symmetry of spin-3/2 fermionic cold atomic system with only $s$-wave scattering. On the triangular lattice, we study a phase transition between the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin ordered phase and a ${Z}_{2}$ spin liquid phase; this phase transition is described by an O(8) sigma model in terms of fractionalized spinon fields, with significant anomalous scaling dimensions of spin order parameters. On the square lattice, we propose a deconfined critical point between the Neel order and the valence bond solid (VBS) order, which is described by the CP(3) model, and the monopole effect of the compact U(1) gauge field is expected to be suppressed at the critical point.

DOI
10.1103/physrevb.78.014410

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We propose quantum phase transitions beyond the Landau's paradigm of Sp(4) spin Heisenberg models on the triangular and square lattices motivated by the exact Sp(4)\ensuremath≃SO(5)\text{Sp}(4)\ensuremath{\simeq}\text{SO}(5)Sp(4)\ensuremath≃SO(5) symmetry of spin-3/2 fermionic cold atomic system with only sss-wave scattering. On the triangular lattice, we study a phase transition between the 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ spin ordered phase and a Z2{Z}_{2}Z2​ spin liquid phase; this phase transition is described by an O(8) sigma model in terms of fractionalized spinon fields, with significant anomalous scaling dimensions of spin order parameters. On the square lattice, we propose a deconfined critical point between the Neel order and the valence bond solid (VBS) order, which is described by the CP(3) model, and the monopole effect of the compact U(1) gauge field is expected to be suppressed at the critical point.
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 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.
Quantum fluctuations of the Néel state of the square lattice antiferromagnet are usually described by a CP 1 theory of bosonic spinons coupled to a U(1) gauge field, and with a global SU(2) spin rotation symmetry.Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge-2 Higgs fields, deconfined phases with Z 2 topological order possibly intertwined with discrete broken global symmetries.We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in π flux in each square lattice plaquette.Fluctuations about this π-flux state are described by (2 þ 1)-dimensional quantum chromodynamics (QCD 3 ) with a SU(2) gauge group and N f ¼ 2 flavors of massless Dirac fermions.It has recently been argued by Wang et al. [Deconfined Quantum Critical Points: Symmetries and Dualities, Phys. Rev. X 7, 031051 (2017).]that this QCD 3 theory describes the Néel-VBS quantum phase transition.We introduce adjoint Higgs fields in QCD 3 and obtain fermionic dual descriptions of the phases with Z 2 topological order obtained earlier using the bosonic CP 1 theory.We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state.The global phase diagram of these phases contains multicritical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.
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We study possible quantum ground states of the Sp(N)Sp(N)Sp(N) generalized Heisenberg model on a cubic lattice with nearest-neighbor and next-nearest-neighbor exchange interactions. The phase diagram is obtained in the large-NNN limit and fluctuation effects are considered via appropriate gauge theories. In particular, we find three U(1)U(1)U(1) spin liquid phases with different short-range magnetic correlations. These phases are characterized by deconfined gapped spinons, gapped monopoles, and gapless ``photons.'' As NNN becomes smaller, a confinement transition from these phases to valence-bond solids (VBS) may occur. This transition is studied by using duality and analyzing the resulting theory of monopoles coupled to a noncompact dual gauge field; the condensation of the monopoles leads to VBS phases. We determine the resulting VBS phases emerging from two of the three spin liquid states. On the other hand, the spin liquid state near ${J}_{1}\ensuremath{\approx}{J}_{2}$ appears to be more stable against monopole condensation and could be a promising candidate for a spin liquid state in real systems.
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∕2S=1∕2S=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{Z}_{2}Z2​ spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1)\mathrm{U}(1)U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
We study the classical and quantum phase transitions of Sp(4) spin systems on three-dimensional stacked square and triangular lattices. We present general Ginzburg-Landau field theories for various types of Sp(4) spin orders with different ground-state manifolds such as CP(3), S7/Z2{S}^{7}/{Z}_{2}S7/Z2​, Grassmann manifold G2,5{G}_{2,5}G2,5​, G2,6{G}_{2,6}G2,6​, and so on, based on which the nature of the classical phase transitions are studied, and a global phase diagram is presented. The classical phase transitions close to quantum phase transitions toward spin-liquid states are also discussed based on renormalization group flow. Our results can be directly applied to the simplest Sp(4) and SU(4) Heisenberg models which can be realized using spin-3/2 atoms and alkaline-earth atoms trapped in optical lattice.