Comment on ``Frustration and multicriticality in the antiferromagnetic spin-1 chain''

Comment on “Frustration and multicriticality in the antiferromagnetic spin-1 chain”

Natalia Chepiga, Ian Affleck, and Frédéric Mila

Phys. Rev. B 94, 136401 – Published 28 October 2016

Abstract

The phase diagram of the spin-1 chain with bilinear-biquadratic and next-nearest-neighbor interactions, recently investigated by Pixley, Shashi, and Nevidomskyy [Phys. Rev. B 90, 214426 (2014)], has been revisited in the light of results we have recently obtained on a similar model. Combining extensive density-matrix renormalization-group simulations with conformal-field theory arguments, we confirm the presence of the three phases identified by Pixley et al., a Haldane phase, a next-nearest-neighbor (NNN) Haldane phase, and a dimerized phase, but we come to significantly different conclusions regarding the nature of the phase transitions to the dimerized phase: (i) We provide numerical evidence of a continuous Ising transition between the NNN-Haldane phase and the dimerized phase. (ii) We show that the tricritical end point, where the continuous transition between the Haldane phase and the dimerized phase turns into a first-order transition, is distinct from the triple point where the three phases meet. (iii) Finally, we demonstrate that the tricritical end point is in the same Wess-Zumino-Witten $SU {(2)}_{2}$ universality class as the continuous transition line that ends at this point.

Received 28 June 2016

DOI:https://doi.org/10.1103/PhysRevB.94.136401

Physics Subject Headings (PhySH)

Conformal field theory Density matrix renormalization group Spin chains

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Natalia Chepiga¹, Ian Affleck², and Frédéric Mila¹

¹Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
²Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

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Original Article

Frustration and multicriticality in the antiferromagnetic spin-1 chain

J. H. Pixley, Aditya Shashi, and Andriy H. Nevidomskyy

Phys. Rev. B 90, 214426 (2014)

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Vol. 94, Iss. 13 — 1 October 2016

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Images

Figure 1
Phase diagram of the spin-1 chain with next-nearest-neighbor coupling $J_{2}$ and biquadratic interaction $J_{b}$ . The transition from the dimerized phase to the Haldane phase starts at the Takhtajan-Babudjian (TB) point, is continuous along the solid line, with central charge $c = 3 / 2$ , and first order along the dashed line. The transition from the NNN-Haldane phase to the dimerized phase is a continuous transition in the Ising universality class with central charge $c = 1 / 2$ . The transition between the Haldane phase and the NNN-Haldane phase is always first order.
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Figure 2
(a) Log-log plot of the midchain dimerization in open chain as a function of the number of sites $N$ for $J_{2} = 1$ and different values of $J_{b}$ . The linear curve corresponds to the Ising critical point, and its slope to the critical exponent $d$ . This leads to $J_{b} = - 0.223$ and $d = 0.129$ , in good agreement with the prediction $1 / 8$ for Ising. (b) Site dependence of $D (j, N)$ at the critical point fitted to $1 / {(N r)}^{d}$ , where $r = sin (π j / N)$ . Inset: same plot in log-log scale. The data points and the fitting lines overlap for the left and right halves of the chain due to the symmetry of the sinus with respect to $π / 2$ . The fit leads to an exponent $d = 0.129$ , again close to the Ising prediction $1 / 8$ .
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Figure 3
Ground-state and excitation energy at $J_{2} = 1$ and $J_{b} = - 0.223$ , a point that belongs to the Ising critical line. (a),(b) Linear scaling of the ground-state energy per site in an open chain with $1 / N^{2}$ after subtracting the $ɛ_{0}$ and $ɛ_{1}$ terms for even and odd number of sites. (c),(d) Energy gaps in the singlet and triplet sectors for OBCs as a function of $1 / N$ for even and odd number of sites. The slope of the singlet gap gives access to the value of the velocity. Insets: Conformal towers. Grey lines show Ising conformal towers for $I$ ( $N$ even) and for $ε$ ( $N$ odd). Blue symbols correspond to DMRG data.
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Figure 4
(a) Central charge along the critical line as determined from fitting the entanglement entropy of periodic chains with the Calabrese-Cardy formula [10]. (b) Apparent critical exponent along the $SU {(2)}_{2}$ critical line as a function of $J_{2}$ from the slope of the log-log plot $D (N / 2, N)$ as a function of $N$ for the values $J_{b}$ for which it is linear. The grey line is the theoretical value of the exponent, $3 / 8$ . (c) Log-log plot of the midchain dimerization as a function of the number of sites $N$ at the critical point $J_{2} = 0.37$ and $J_{b} = - 0.331$ . The slope corresponds to the critical exponent $d = 0.364$ , in good agreement with $3 / 8$ for WZW $SU {(2)}_{2}$ . (d) Site dependence of $D (j, N)$ at the critical point fitted to $1 / {(N r)}^{d}$ , where $r = sin (π j / N)$ . Inset: same plot in log-log scale. The data points and the fitting lines overlap for the left and right halves of the chain. The fit leads to an exponent $d = 0.351$ , again close to the WZW $SU {(2)}_{2}$ prediction $3 / 8$ .
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Figure 5
Ground-state and excitation energy at $J_{2} = 0.37$ and $J_{b} = - 0.331$ , a point that belongs to the critical line between the Haldane and the Dimerized phases. (a),(b) Linear scaling of the ground-state energy per site in an open chain with $1 / N^{2}$ after subtracting the $ɛ_{0}$ and $ɛ_{1}$ terms for even and odd number of sites. (c),(d) Energy gap between the ground state and the lowest energy states in different sectors of $S_{z}^{tot} = 0, 1, ..., 5$ as a function of $1 / N$ for even and odd number of sites. Insets: Conformal towers. Blue symbols correspond to DMRG data. Red lines are the expected conformal towers [2], with a velocity defined by the finite-size scaling of the ground-state energy for $N$ even.
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Figure 6
Velocity along the critical line between the Haldane phase and the dimerized phase extracted from the gap between various energy levels and the ground state. For clarity, results for $N = 50, 51$ and $N = 30, 31$ are shifted vertically by 1 and 2 respectively.
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