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arXiv:1901.02107 · Physical review. B./Physical review. B

Finite-size scaling at a topological transition: Bilinear-biquadratic spin-1 chain

Yuting Wang, Hao Zhang, Alex Kamenev

Published Jun 19, 2020• 7 citations
arXivHTMLPDF

Abstract

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 $L$ 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 $c=3/2$. 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.

DOI
10.1103/physrevb.101.235145

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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/\ensuremathξL/\ensuremath{\xi}L/\ensuremathξ, where LLL is the chain length and \ensuremathξ\ensuremath{\xi}\ensuremathξ 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.
The critical point of a topological phase transition is described by a conformal field theory, where finite-size corrections to energy are uniquely related to its central charge. We investigate the finite-size scaling away from criticality and find a scaling function, which discriminates between phases with different topological indices. This function appears to be universal for all five Altland-Zirnbauer symmetry classes with nontrivial topology in one spatial dimension. We obtain an analytic form of the scaling function and compare it with numerical results.
Scaling aspects of Gaussian topological phase transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy DDD. This model presents a critical line separating the gapped Haldane and large-DDD phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted, in contrast to the standard scaling holding in the Ising-like transition between the gapless antiferromagnetic N\'eel and gapped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.
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Quantum one-dimensional criticality with conformal invariance is known to be characterized by conformal field theory with the inherent central charge (ccc), and universal finite-size amplitudes. Here, the authors demonstrate that around a quantum criticality with ccc=\textonehalf{} and two Majorana boundary modes, the symmetry-breaking effect modifies the finite-size amplitudes from a universal constant to be a universal function of an emergent scale. This finding provides a new perspective to study the interplay of topology and quantum criticality.
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 ($\ensuremath{-}3\ensuremath{\pi}/4<\ensuremath{\theta}<\ensuremath{\pi}/4$). 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=1/3K/J=1/3K/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 Z2{Z}_{2}Z2​ gauge fluctuations above the mean-field theory is analyzed.
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial state with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. Recently, the framework of matrix product states for topological phases with Majorana fermions/parafermions has been established. Here we first generalize these fixed-point matrix product states with the zero correlation length to the more generic ground-state wave functions with a finite correlation length for the general one-dimensional interacting Majorana fermion/parafermion systems. Then we employ the previous method to decode quantum criticality to the interacting Majorana fermion/parafermion matrix product states. The obtained quantum critical spectra are described by the conformal field theories with central charge $c\ensuremath{\le}1$, characterizing the quantum critical theories separating the fermionic/parafermionic topological phase from the trivial phases with the same symmetry.