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arXiv:1202.5723 · Physical Review B

Gutzwiller projected wave functions in the fermionic theory of S=1 spin chains

Zheng-Xin Liu, Yi Zhou, Hong-Hao Tu, Xiao-Gang Wen, Tai Kai Ng

Published May 29, 2012• 26 citations
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Abstract

We study in this paper a series of Gutzwiller projected wave functions for $S=1$ spin chains obtained from a fermionic mean-field theory for general $S>1/2$ spin systems [Liu, Zhou, and Ng, Phys. Rev. B 81, 224417 (2010)] applied to the bilinear-biquadratic ($J$-$K$) model. The free-fermion mean-field states before the projection are 1D paring states. By comparing the energies and correlation functions of the projected pairing states with those obtained from known results, we show that the optimized Gutzwiller projected wave functions are very good trial ground-state wave functions for the antiferromagnetic bilinear-biquadratic model in the regime $K<J$ ($\ensuremath{-}3\ensuremath{\pi}/4<\ensuremath{\theta}<\ensuremath{\pi}/4$). We find that different topological phases of the free-fermion paring states correspond to different spin phases: the weak pairing (topologically nontrivial) state gives rise to the Haldane phase, whereas the strong pairing (topologically trivial) state gives rise to the dimer phase. In particular, the mapping between the Haldane phase and Gutwziller wave function is exact at the Affleck-Kennedy-Lieb-Tasaki (AKLT) point $K/J=1/3$ [$\ensuremath{\theta}={\mathrm{tan}}^{\ensuremath{-}1}(1/3)$]. The transition point between the two phases determined by the optimized Gutzwiller projected wave function is in good agreement with the known result. The effect of ${Z}_{2}$ gauge fluctuations above the mean-field theory is analyzed.

DOI
10.1103/physrevb.85.195144

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We study in this paper a series of Gutzwiller projected wave functions for S=1S=1S=1 spin chains obtained from a fermionic mean-field theory for general S>1/2S>1/2S>1/2 spin systems [Liu, Zhou, and Ng, Phys. Rev. B 81, 224417 (2010)] applied to the bilinear-biquadratic (JJJ-KKK) model. The free-fermion mean-field states before the projection are 1D paring states. By comparing the energies and correlation functions of the projected pairing states with those obtained from known results, we show that the optimized Gutzwiller projected wave functions are very good trial ground-state wave functions for the antiferromagnetic bilinear-biquadratic model in the regime K<JK<JK<J (\ensuremath−3\ensuremathπ/4<\ensuremathθ<\ensuremathπ/4\ensuremath{-}3\ensuremath{\pi}/4<\ensuremath{\theta}<\ensuremath{\pi}/4\ensuremath−3\ensuremathπ/4<\ensuremathθ<\ensuremathπ/4). We find that different topological phases of the free-fermion paring states correspond to different spin phases: the weak pairing (topologically nontrivial) state gives rise to the Haldane phase, whereas the strong pairing (topologically trivial) state gives rise to the dimer phase. In particular, the mapping between the Haldane phase and Gutwziller wave function is exact at the Affleck-Kennedy-Lieb-Tasaki (AKLT) point K/J=1/3K/J=1/3K/J=1/3 [\ensuremathθ=tan\ensuremath−1(1/3)\ensuremath{\theta}={\mathrm{tan}}^{\ensuremath{-}1}(1/3)\ensuremathθ=tan\ensuremath−1(1/3)]. The transition point between the two phases determined by the optimized Gutzwiller projected wave function is in good agreement with the known result. The effect of Z2{Z}_{2}Z2​ gauge fluctuations above the mean-field theory is analyzed.
In a previous paper (Liu et al 2012 Phys. Rev. B 85 195144 ), a variational Monte Carlo method (based on Gutzwiller projected states) was generalized to S = 1 systems. This method provided very good trial ground states for the gapped phases of an S = 1 bilinear-biquadratic (BLBQ) Heisenberg chain. In this paper, we extend the approach to study the low-lying elementary excitations in S = 1 chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in the literature. In our approach, the difference of the excitation spectrum between the Haldane phase and the dimer phase (such as the even/odd size effect) can be understood from their different topologies of the corresponding mean field theory. We especially study the Takhtajan–Babujian critical point. Despite the fact that the 'elementary excitations' are spin-1 magnons, which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ( ) and central charge ( c = 1.45) calculated from our theory agree well with the Bethe ansatz solution and conformal field theory predictions.
SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.
We systematically study the effects of higher-order interactions on the S=32,1S=\frac{3}{2},1S=23​,1 orthogonal dimer chains using exact diagonalization and density matrix renormalization group. Due to frustration and higher spin, there are rich quantum phases, including three Haldane phases, two gapless phases, and several magnetically ordered phases. To characterize these phases and their phase transitions, we study various physical quantities such as energy gap, energy-level crossing, fidelity susceptibility, spin correlation, entanglement spectrum, and central charge. According to our calculations, the biquadratic term can enhance the Haldane phase regions. In particular, we numerically identify that a Haldane phase in S=32S=\frac{3}{2}S=23​ case is adiabatically connected to the exact AKLT (Affleck, Kennedy, Lieb, and Tasaki) point when adding a bicubic term. Our study on the orthogonal dimer model, which is a 1D version of the Shastry-Sutherland model, provides insights into understanding the possible S=32,1S=\frac{3}{2},1S=23​,1 Haldane phases in quasi-1D and 2D frustrated magnetic materials.
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We consider a finite-size scaling function across a topological phase transition in one-dimensional models. For models of noninteracting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition [T. Gulden, M. Janas, Y. Wang, and A. Kamenev, Phys. Rev. Lett. 116, 026402 (2016)]. Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/\ensuremath{\xi}$, where LLL is the chain length and $\ensuremath{\xi}$ is the correlation length, coincides with that of three species of noninteracting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge c=3/2c=3/2c=3/2. We have shown that it still holds away from the conformal point, including the finite-size corrections. We have also observed peculiar differences between even- and odd-size chains, which may be fully accounted for by residual interactions of the edge states.
We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field hhh. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\ensuremath{\omega})$ as a function of hhh. At h=0h=0h=0, the envelope of the lowest-energy excitations goes soft at two points, ${q}_{1}=2\ensuremath{\pi}/3$ and ${q}_{2}=4\ensuremath{\pi}/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point $q=\ensuremath{\pi}$ at ${h}_{c1}\ensuremath{\approx}0.94$. Beyond hc1{h}_{c1}hc1​ the system enters another gapless B phase until the transition at hc2=4{h}_{c2}=4hc2​=4 to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.