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arXiv:cond-mat/0404427 · Physical Review B

Néel-dimer transition in an antiferromagnetic Heisenberg model and deconfinement of spinons at the critical point

Daisuke Yoshioka, Gaku Arakawa, Ikuo Ichinose, Tetsuo Matsui

Published Nov 1, 2004• 22 citations
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Abstract

Quantum phase transition from the N\'eel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter $\ensuremath{\alpha}$ for the exchange coupling that connects the N\'eel $(\ensuremath{\alpha}=0)$ and the dimer $(\ensuremath{\alpha}=1)$ states. We employ the ${\mathit{CP}}^{1}$ representation of the $s=\frac{1}{2}$ spin operator and integrate out the half of the ${\mathit{CP}}^{1}$ variables at odd sites to obtain a ${\mathit{CP}}^{1}$ nonlinear $\ensuremath{\sigma}$ model. The effective coupling constant is a function of $\ensuremath{\alpha}$, and at $\ensuremath{\alpha}=0$, the ${\mathit{CP}}^{1}$ model is in the ordered phase that corresponds to the N\'eel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of ${\ensuremath{\alpha}}_{C}$ as $\ensuremath{\alpha}$ increases. In the N\'eel state, the dynamical composite U(1) gauge field in the ${\mathit{CP}}^{1}$ model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with $s=1$ excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and $s=\frac{1}{2}$ spinons appear as low-energy excitations.

DOI
10.1103/physrevb.70.174407

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Quantum phase transition from the N'eel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter \ensuremathα\ensuremath{\alpha}\ensuremathα for the exchange coupling that connects the N'eel (\ensuremathα=0)(\ensuremath{\alpha}=0)(\ensuremathα=0) and the dimer (\ensuremathα=1)(\ensuremath{\alpha}=1)(\ensuremathα=1) states. We employ the CP1{\mathit{CP}}^{1}CP1 representation of the s=12s=\frac{1}{2}s=21​ spin operator and integrate out the half of the CP1{\mathit{CP}}^{1}CP1 variables at odd sites to obtain a CP1{\mathit{CP}}^{1}CP1 nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model. The effective coupling constant is a function of \ensuremathα\ensuremath{\alpha}\ensuremathα, and at \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0, the CP1{\mathit{CP}}^{1}CP1 model is in the ordered phase that corresponds to the N'eel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of \ensuremathαC{\ensuremath{\alpha}}_{C}\ensuremathαC​ as \ensuremathα\ensuremath{\alpha}\ensuremathα increases. In the N'eel state, the dynamical composite U(1) gauge field in the CP1{\mathit{CP}}^{1}CP1 model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with s=1s=1s=1 excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and s=12s=\frac{1}{2}s=21​ spinons appear as low-energy excitations.
We study the S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on a square lattice with nearest-neighbor and plaquette four-spin exchanges [introduced by A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).] This model undergoes a quantum phase transition from a spontaneously dimerized phase to N\'eel order at a critical coupling. We show that as the critical point is approached from the dimerized side, the system exhibits strong fluctuations in the dimer background, reflected in the presence of a low-energy singlet mode, with a simultaneous rise in the triplet quasiparticle density. We find that both singlet and triplet modes of high density condense at the transition, signaling restoration of lattice symmetry. In our approach, which goes beyond mean-field theory in terms of the triplet excitations, the transition appears sharp; however since our method breaks down near the critical point, we argue that we cannot make a definite conclusion regarding the order of the transition.
The theory for the vanishing of N\'eel order in the spin S=1/2S=1/2S=1/2 square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable Z2{\mathbb{Z}}_{2}Z2​ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing J2/J1{J}_{2}/{J}_{1}J2​/J1​: N\'eel, gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid, and valence bond solid.
Abstract We analyze a Higgs transition from a U(1) Dirac spin liquid to a gapless ℤ2 spin liquid. This ℤ2 spin liquid is of relevance to the spin S = 1/2 square lattice antiferromagnet, where recent numerical studies have given evidence for such a phase existing in the regime of high frustration between nearest neighbor and next-nearest neighbor antiferromagnetic interactions (the J1-J2 model), appearing in a parameter regime between the vanishing of Néel order and the onset of valence bond solid ordering. The proximate Dirac spin liquid is unstable to monopole proliferation on the square lattice, ultimately leading to Néel or valence bond solid ordering. As such, we conjecture that this Higgs transition describes the critical theory separating the gapless ℤ2 spin liquid of the J1-J2 model from one of the two proximate ordered phases. The transition into the other ordered phase can be described in a unified manner via a transition into an unstable SU(2) spin liquid, which we have analyzed in prior work. By studying the deconfined critical theory separating the U(1) Dirac spin liquid from the gapless ℤ2 spin liquid in a 1/Nf expansion, with Nf proportional to the number of fermions, we find a stable fixed point with an anisotropic spinon dispersion and a dynamical critical exponent z ≠ 1. We analyze the consequences of this anisotropic dispersion by calculating the angular profiles of the equal-time Néel and valence bond solid correlation functions, and we find them to be distinct. We also note the influence of the anisotropy on the scaling dimension of monopoles.
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.
Abstract By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor ( J 1 ) and second-neighbor ( J 2 ) couplings, for J 2 < 0.5 J 1 . The ground state of the system shows long-range antiferromagnetic order for J 2 / J 1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J 2 / J 1 ≲ 0.36, and columnar dimer order for J 2 / J 1 ≳ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J 2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J 2 / J 1 is increased and becomes gapless at the transition point, J 2 / J 1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J 2 / J 1 , valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.
Based on a special variant of the plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of a spin- 12\frac{1}{2}21​ Heisenberg antiferromagnet on a square lattice with nearest-heighbor and frustrating next-nearest-neighbor exchange couplings J 1 and J 2. It is well known that a nonmagnetic phase arises in this model for 0.4 ≲ J 2/J 1 ≲ 0.6, sandwiched by two Néel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at J 2 ≈ 0.64 J 1 from the non-magnetic phase to the Néel one. A large gap (≳ 0.4J 1) is found in the singlet spectrum for J 2 < 0.64J 1, which excludes a gapless spin-liquid state for 0.4 ≲ J 2/J 1 ≲ 0.6 and the deconfined quantum criticality scenario for the QPT to another Néel phase. We observe a first-order QPT at J 2 ≈ 0.55J 1, presumably between two nonmagnetic phases.