Gapless to gapless phase transitions in quantum spin chains

We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field $h$. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\omega )$ as a function of $h$. At $h=0$, the envelope of the lowest-energy excitations goes soft at two points, $q_1=2\pi /3$ and $q_2=4\pi /3$, 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 $q=\pi$ at $h_{c1}\approx 0.94$. Beyond $h_{c1}$ the system enters another gapless B phase until the transition at $h_{c2}=4$ 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.

Figure 1

Left panel: phase diagram of the spin-1 bilinear-biquadratic (BLBQ) model parametrized by the angle $\theta$: $H={\sum }_{\langle ij\rangle }cos\theta {\mathbf{S}}_i·{\mathbf{S}}_j+sin\theta {({\mathbf{S}}_i·{\mathbf{S}}_j)}^2$. $\mathrm{\Delta }$ denotes the gap which can be zero or finite in different phases. We focus on two representative points: the Affleck-Kennedy-Lieb-Tasaki (AKLT) model at $\theta \approx 0.1024\pi$ ($\beta =\frac{1}{3}$), and the critical ULS model at $\theta =\pi /4$ ($\beta =1$). VBS refers to the valence bond solid ground state of the AKLT model at $h=0$. Right panel: schematic phase diagram of BLBQ parametrized by $\beta$ and $h$ reproduced from Ref. [9]. Phase boundaries are marked by black solid lines. black dashed line marks a crossover to an effective spin-$\frac{1}{2}$ XXZ model in a field within the B phase; at $\beta =1$ the mapping is to an effective spin-$\frac{1}{2}$ Heisenberg model. We obtain the evolution of the static and dynamical correlation functions in the gapless phase A, B phase, and Haldane phase along the two red dashed lines.

Figure 2

Results of AKLT model under a field (a) magnetization per site $s_z$ as a function of $h\ge 0$, showing two second-order phase transitions. Data obtained from different system sizes converge rapidly and coincide with each other. (b) von Neumann entanglement entropy $S_{vN}$ as a function of $h$ at central bond computed for the same set of system sizes. (c) Static structure factor under different external field. (d) Real-space correlation functions at $h=0.0,0.5$ for the VBS state, and $h=0.9,1.0$ for the gapless phase B. Curves of the same phase coincide. (e) Correlation function of longitudinal and transverse components at different fields of phase B. (f) Exponent of real-space correlation function fitted by $S\left(R\right)\sim R^{-\eta }$ in the phase B.

Figure 3

Dynamic results of AKLT model under a field. $S^{+-}(q,\omega )$ at field $h=0$ ($sz=\frac{1}{200}$), $h=0.5$ ($sz=\frac{1}{200}$), $h=1.0$ ($sz\approx 0.3$), and at field $h=1.5$ ($sz\approx 0.5$). (a)–(c) Are within phase A while (d) is within phase B. Dynamical structure factors are obtained by 200-site DMRG under OBC.

Figure 4

Results of ULS model under a field (a) magnetization per site $s_z$ as a function of $h$. Inset is the zoom-in segment of the magnetization in phase A which shows a zigzag pattern regardless of system size. (b) von Neumann entanglement entropy $S_{vN}$ as a function of $h$ for a cut at the central bond for the same lattice and again computed with the DMRG. (c) Static structure factor at $h=0,0.5$ for the first phase, and $h=1.0$ for the intermediate phase, respectively (d) Real-space correlation function for the same set of fields. (e) Correlation function of longitudinal and transverse components at different fields of phase B. (f) Exponent of real-space correlation function fitted by $S\left(R\right)\sim R^{-\eta }$ for phases A and B separated by the vertical dashed line.

Figure 5

$S^{+-}(q,\omega )$ dynamics of ULS model at field $h=0$ ($sz=\frac{1}{200}$), $h=0.5$ ($s_z\approx 0.16$), $h=0.9$ ($s_z\approx 0.50$), and $h=1.5$ ($s_z\approx 0.60$). (a)–(c) Are within phase A while (d) is within phase B. Results of dynamics are obtained by 200-site DMRG under under OBC.

Figure 6

Entanglement entropies as a function of bond position $n$ and fits to the data (red solid line) for (a) the spin-$\frac{1}{2}$ Heisenberg model with $c=1$, which is equivalent to ULS under a field larger than $h_{c1}$, (b) AKLT model, (c) spin-1 Heisenberg model. (d) Shows central charge $c$ as a function extracted from EE of AKLT and spin-1 Heisenberg model within the intermediate phase. Results obtained from 200-site DMRG under OBC.

Figure 7

SMA results (solid lines) compared with $S(q,\omega )$ obtained by DMRG (intensity plot) in (a) AKLT VBS state; (b)–(e) AKLT phase B at $h=0.8$, $h=1.8$, $h=2.8$, $h=3.8>h_{c1}$ calculated for different $L$. The gap of the VBS state is remarkably close to the exact result of ${\mathrm{\Delta }}_{\text{VBS}}=0.350$. Within the phase B of AKLT, at field slightly larger than $h_{c1}$, e.g., $h=0.8$ in (a), the (upper bound of) gap given by ${\omega }_{\text{SMA}}\left(k\right)$ is close to zero and decreases as $L$ increases, indicative of a gapless mode in the thermodynamic limit. However, as $h$ keeps increasing in (b) and (c), the gapless mode is no longer captured by SMA. (f) ULS: SMA and DMRG for the gapless intermediate phase at $h=1.5>h_{c1}\approx 0.94$. The upper bound is not tight enough to capture the gapless mode at $\pi$.