Commensurate-incommensurate Mott transition without magnetic field: emergence of nematic Luttinger liquid in XXZ chain

TL;DR

The study analyzes a spin-1/2 XXZ chain with competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchanges at zero magnetization, revealing two Pokrovsky-Talapov type commensurate-incommensurate quantum phase transitions. These transitions separate a gapped period-4 phase from an incommensurate nematic Luttinger liquid and then from a central-charge-$2$ mixed Luttinger liquid, both driven by frustration without doping or magnetic field. The nematic phase features bound two-magnon excitations with one-magnon excitations gapped, while the $c=2$ phase constitutes a coexistence of conventional and quadrupolar Luttinger liquids; the transitions are characterized by PT critical exponents, changes in the Luttinger parameter $K$, and entanglement-based central-charge measurements. Overall, the work demonstrates that frustration alone can induce continuous commensurate-incommensurate Mott transitions and stabilize incommensurate quasi-long-range order in zero-field, broadening the landscape of possible quantum critical phenomena in low-dimensional magnets.

Abstract

We investigate the zero-magnetization phase diagram of a spin-1/2 chain with competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange couplings in the strongly interacting regime. Using density matrix renormalization group (DMRG) simulations, we discover two successive commensurate-incommensurate transitions of the non-conformal Pokrovsky-Talapov universality class, occurring (even) at zero magnetic field. The first transition marks the condensation of bound pairs of magnons into a critical phase with central charge $c=2$, emerging from a gapped period-4 phase. At the second transition, an incommensurate quadrupolar (or nematic) Luttinger liquid forms out of a gapped phase separation state, via the pairwise condensation of domain walls. We argue that both transitions involve the same underlying incommensurate nematic Luttinger liquid, and that the $c=2$ phase can be understood as a coexistence of a conventional (single-magnon type) and quadrupolar (two-magnon type) Luttinger liquids. Our results demonstrate that frustration alone is sufficient to drive continuous commensurate-incommensurate transitions of Mott type and stabilise incommensurate quasi-long-range order without doping.

Figures (8)

• Figure 1: Phase diagram of the system defined in Eq. \ref{['H_spin']} and \ref{['H_fermion']} as a function of the nearest-neighbor ferromagnetic exchange $\Delta_1$ and the next-nearest-neighbor antiferromagnetic exchange $\Delta_2$. Three gapped phases - phase separation, period-4 and bond ordered period-2 phases are shown in blue. The latter two are separated by the Ising transition. The first order transition between phase separation and conventional Luttinger liquid (LL, beige) is characterized by Luttinger coefficient $K \to \infty$ and velocity $u \to 0$. The transition between LL and bond order is of Berezinskii-Kosterlitz-Thouless (BKT) type with $K_c = 1/2$. In the strongly interacting regime $-\Delta_1,\Delta_2\gg 1$ two Pokrovsky-Talapov transitions confine a critical sector that hosts a nematic LL with central charge $c=1$ (pink) and a two-flavor LL with $c=2$ (green), the latter being connected to the conventional LL via a BKT condensation of a nematic component.
• Figure 2: Numerical evidences of Pokrovsky-Talapov transition from $c=2$ Luttinger liquid phase to period-4 phase. (a) Scaling of the correlation length $\xi$ extracted from $\langle S^{+}_iS^{-}_j\rangle$ correlations in the period-4 gapped phase according to Eq. \ref{['OZ']}. The results are in excellent agreement with theory prediction $\nu=1/2$ (dashed line). (b) Friedel oscillations in the $c=2$ gapless phase induced by boundary conditions $S^z_1 = -1/2$, $S^z_N = 1/2$. The fit to Eq. \ref{['FO']} yields values for the wave vector $q\simeq0.918\pi$ and the Luttinger parameter $K\simeq0.359$. (c) Scaling of the deviation of the wave vector from commensurate value $\Delta q=|\pi-q |$ (diamonds) and of the nematic density $m_{\mathrm{nema}}$ (circles) in the $c=2$ gapless phase. The discreteness of the values (spaced by $\sim N^{-1}$) is discussed in the main text. The fit shows good agreement with the critical exponent $\bar{\beta}=1/2$. (d) Luttinger parameter $K$ approaching the critical value $K_c=1/4$ at the transition point $\Delta_1=-5.0,\Delta_2\simeq2.635$.
• Figure 3: Spin flip correlations. (a) Single-spin flip correlations in the nematic Luttinger liquid phase. The exponential decay (see Eq. \ref{['OZ']}) reveals a gap in the excitation spectrum for single-spin flip events, as breaking a bound state of two spins costs a finite energy. (b) Two-spin flip (or nematic) correlations at the same parameters point. These correlations exhibit algebraic decay, reflecting the gapless nature of the two-bound magnon Luttinger liquid. (c) Single-spin and two-spin flip correlations in the $c=2$ Luttinger liquid phase; both are of the algebraic type.
• Figure 4: Scaling of the reduced entanglement entropy $\tilde{S}_N(j)$ as a function of the logarithm of the conformal distance $d_N(j)=\left(2N/\pi\right) \log{\left(\sin{\left(\pi x/N\right)}\right)}$ according to Eq. \ref{['CC']}. As the system size increases, the central charge approaches the value (a) $c=1$ in the nematic LL and (b) $c = 2$ in the mixed LL phase. The very last points for each data set in (a) correspond to $\tilde{S}_N(j)$ values near the middle of the chain where phase separation occurs (see End Matter) and are therefore excluded from the fit.
• Figure 5: Numerical evidences of the Pokrovsky-Talapov transition between the gapped phase separation and nematic Luttinger liquid (nematic LL). (a) $\langle S^z_j\rangle$ profile demonstrating incommensurate phase separation in the nematic LL phase. (b) Friedel oscillations in the nematic LL induced by open boundary conditions. The fit with Eq. \ref{['FO']} in the main text yields values for the wave vector $q\simeq2.32\pi$ and the Luttinger parameter $K\simeq0.243$. (c) Scaling of the deviation of the wave vector (obtained from Friedel oscillations) from commensuration $\Delta q=|2\pi-q |$ and of the density component $m_{\mathrm{nema}}$ in the nematic LL. The fit shows excellent agreement with the critical exponent $\bar{\beta}=1/2$. (d) Luttinger parameter $K$ approaching the critical value $K_c=1$ at the Pokrovsky-Talapov transition point $\Delta_1=-4.0,\Delta_2\simeq1.914$.