Magnetic phase diagram of the spin-12antiferromagnetic zigzag ladder

Abstract

We study the one-dimensional spin-$\frac{1}{2}$ Heisenberg model with antiferromagnetic nearest-neighbor ${J}_{1}$ and next-nearest-neighbor ${J}_{2}$ exchange couplings in magnetic field $h$. With varying dimensionless parameters ${J}_{2}/{J}_{1}$ and $h/{J}_{1}$, the ground state of the model exhibits several phases including three gapped phases (dimer, 1/3-magnetization plateau, and fully polarized phases) and four types of gapless Tomonaga-Luttinger liquid (TLL) phases which we dub TLL1, TLL2, spin-density-wave $({\text{SDW}}_{2})$, and vector chiral phases. From extensive numerical calculations using the density-matrix renormalization-group method, we investigate various (multiple-)spin-correlation functions in detail and determine dominant and subleading correlations in each phase. For the one-component TLLs, i.e., the TLL1, ${\text{SDW}}_{2}$, and vector chiral phases, we fit the numerically obtained correlation functions to those calculated from effective low-energy theories of TLLs and find good agreement between them. The low-energy theory for each critical TLL phase is thus identified, together with TLL parameters which control the exponents of power-law decaying correlation functions. For the TLL2 phase, we develop an effective low-energy theory of two-component TLL consisting of two free bosons (central charge $c=1+1$), which explains numerical results of entanglement entropy and Friedel oscillations of local magnetization. Implications of our results to possible magnetic phase transitions in real quasi-one-dimensional compounds are also discussed.