Model of spin liquids with and without time-reversal symmetry

We study a model in (2 + 1)-dimensional space-time that is realized by an array of chains, each of which realizes relativistic Majorana fields in (1 + 1)-dimensional space-time, coupled via current-current interactions. The model is shown to have a lattice realization as an array of coupled quantum spin-1/2 ladders. We study this model both in the presence and in the absence of time-reversal symmetry within a mean-field approximation. We find regimes in coupling space where Abelian and non-Abelian spin-liquid phases are stable. In the case when the Hamiltonian is time-reversal symmetric, we find regimes where gapped Abelian and non-Abelian chiral phases appear as a result of spontaneous breaking of time-reversal symmetry. These gapped phases are separated by a discontinuous phase transition. More interestingly, we find a regime for which a nonchiral gapless non-Abelian spin liquid is stable. The excitations in this regime are described by relativistic Majorana fields in (2 + 1)-dimensional space-time, much as those appearing in the Kitaev honeycomb model but here emerging in a model of coupled spin ladders that do not break the $SU\left(2\right)$ spin-rotation symmetry.

Figure 1

(a) Mean-field phase diagram as a function of the three couplings $\lambda \ge 0,\tilde{\lambda }\ge 0$, and $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x$ for the theory defined in Eq. (2.4) under the assumptions (2.8) $[{\mathrm{\Lambda }}_x$ is a momentum cutoff that is introduced in Eq. (3.24b)]. The yellow surface represents those points in coupling space at which a continuous mean-field transition separates two distinct gapped phases of matter: an Abelian topological order (ATO) phase and a non-Abelian topological order (NATO) phase. The brown surface represents those points at which a discontinuous mean-field phase transition occurs. The acronym “LR” (respectively, “RL”) refers to the chiralities of the gapless edge states, i.e., gapless edge states on the first wire $\mathrm{m}=1$ are left handed (respectively, right handed), whereas those on the last wire $\mathrm{m}=n$ are right handed (respectively, left handed) when open boundary conditions are chosen along the stacking direction of the wires. The quadrant $\lambda =\tilde{\lambda }$ for which Hamiltonian density (2.4) is TRS is colored in gray. (b) For $|m_{\mathrm{t}}^{}/{\mathrm{\Lambda }}_x|\ge 0$ and $\lambda =\tilde{\lambda }\ge 0$, the region bounded by the vertical axis and the green colored continuous line supports a TRS mean-field solution as the minimum of the mean-field potential with the vanishing singlet gap ${\mathrm{\Delta }}_{\mathrm{s}}=0$ and the nonvanishing triplet gap ${\mathrm{\Delta }}_{\mathrm{t}}=2\left|m_{\mathrm{t}}^{}\right|\ne 0$ defined by Eq. ((3.31)). Outside of this region, TRS is spontaneously broken at the mean-field level with nonvanishing singlet ${\mathrm{\Delta }}_{\mathrm{s}}\ne 0$ and triplet ${\mathrm{\Delta }}_{\mathrm{t}}\ne 0$ gaps defined by Eq. ((3.31)). The brown dashed line is a line of discontinuous phase transitions that separates the mean-field snapshot of an ATO from a NATO phase.

Figure 2

Blue curve: We fix $(\lambda ,\tilde{\lambda })=(7,1)$ in Fig. 1 so as to strongly break time-reversal symmetry. Red curve: We fix $(\lambda ,\tilde{\lambda })=4\sqrt{2}(sin\theta ,cos\theta )$ with $\theta =23\pi /90$ in Fig. 1 so as to weakly break time-reversal symmetry. (a) and (b) Continuous and discontinuous dependences on $m_{\mathrm{t}}^{}$ of the stable solution ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}$ to the saddle-point equations ((3.29)). (c) and (d) Continuous and discontinuous dependences on $m_{\mathrm{t}}^{}$ of the singlet gap ${\mathrm{\Delta }}_{\mathrm{s}}$ and the triplet gap ${\mathrm{\Delta }}_{\mathrm{t}}$ given by Eq. ((3.31)). The triplet gap ${\mathrm{\Delta }}_{\mathrm{t}}$ represented by the blue curve vanishes at $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x\approx 0.76$ that signals a continuous quantum phase transition.

Figure 3

(a) and (b) Cut with fixed $\lambda =\tilde{\lambda }=4$ from Fig. 1. The stable mean-field solutions ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}$ are presented in panels (a) and (b) as functions of $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x$, respectively. The $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x$ dependence of the singlet (${\mathrm{\Delta }}_{\mathrm{s}}$) and the triplet (${\mathrm{\Delta }}_{\mathrm{t}}$) gaps are plotted in panels (c) and (d) by making use of Eqs. (3.31a) and (3.31b), respectively.

Figure 4

The stable mean-field solutions ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}$ as a function of $\theta :=\arctan (\lambda /\tilde{\lambda })$ and fixing ${\lambda }^2+{\tilde{\lambda }}^2=32$. When $\theta =\pi /4$, there is a discontinuous (respectively, continuous) phase transition for panels (a) and (c) [respectively, (b)]. (a) Case $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x=0.1$. Here, $|{\mathrm{\Phi }}_{+}|\ne |{\mathrm{\Phi }}_{-}|$ whereas, $||{\mathrm{\Phi }}_{+}|-|{\mathrm{\Phi }}_{-}||\ll 1$. (b) Case $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x=1$. (c) Case $|m_{\mathrm{t}}^{}|/{\mathrm{\Lambda }}_x=3$.

Figure 5

(a) The phase diagram for the ladder model (4.1) as a function of $J_U<0$ and $|J_{\perp }|\le 0.3$. The phase boundary between the columnar-dimer (CD) and the rung-singlet (RS) phases is a continuous phase transition in the Ising universality class. The phase boundary between the Haldane phase (H) and the columnar-dimer phase is a continuous phase transition in the $\widehat{su}{\left(2\right)}_2$ Wess-Zumino-Novikov-Witten (WZNW) universality class. (b) Classical representation for the CD order. (c) Classical representation for the RS order.

Figure 6

(a) Plot for the leg-dimer order parameter (4.3) at the center of the ladder $D_{N/2}$ as a function of $J_{\perp }$ for different system sizes while fixing $J_U=-1$. The extrapolation to the thermodynamic limit is obtained with a second-order polynomial in $1/N$, whereas the dashed curve is a fit to the Ising scaling law $D_{\infty }\left(J_{\perp }\right)\propto {(J_{\perp ,\mathrm{c}}-J_{\perp })}^{1/8}$ in the vicinity of the critical point $J_{\perp ,\mathrm{c}}\approx 0.041$. (b) Fixing $J_U=-1$, this log-log plot shows the scaling of $D_{N/2}$ with $1/N$ for different values of $J_{\perp }$ in the vicinity of the critical point. The Ising scaling law $D_{N/2}\left(J_{\perp ,\mathrm{c}}\right)\propto N^{-1/8}$ fits pretty well the scaling of $D_{N/2}$ at the transition point $J_{\perp ,\mathrm{c}}\approx 0.041$. (c) and (d) Fitting the entanglement entropy from Eq. (4.4) as a function of $x$ with $N=128$ in panel (c) and of $N$ with $x=N/2$ in panel (d) yields $c=0.4692$ and $c=0.4824$, respectively.