Magnetic phase diagram of the spin-$\frac{1}{2}$ antiferromagnetic zigzag ladder

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 $\left({\text{SDW}}_2\right)$, 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.

Figure 1

(Color online) Magnetic phase diagram of the spin-$\frac{1}{2}$ antiferromagnetic zigzag ladder (a) in the $J_2/J_1$ versus $M$ plane and (b) in the $J_2/J_1$ versus $h/J_1$ plane. In (a), symbols represent parameter points for which their ground-state phases are identified: open (○) and solid (●) circles represent the TLL1 phase with dominant transverse and longitudinal spin correlations, respectively. Diamonds (◇), triangles (△), and squares (◻), respectively, represent the ${\text{SDW}}_2$, vector chiral, and TLL2 phases. The solid line shows the 1/3-plateau phase. The dotted curves are the guide for the eyes. In (b), symbols P, VC, D, and F indicate the 1/3 plateau, vector chiral, dimer, and fully polarized phases, respectively. The phase boundaries shown by solid lines are obtained in Ref. 21 from the numerical results of magnetization curves, except for the boundaries between the vector chiral and TLL2 phases which are obtained from the analysis of correlation functions in the present paper. The dotted line in the TLL1 phase represents the crossover line between the transverse and longitudinal spin dominant regimes.

Figure 2

(Color online) Typical spatial dependence of correlation functions in critical phases; (a) TLL1 phase (transverse spin-correlation dominant), (b) TLL1 phase (longitudinal spin-correlation dominant), (c) ${\text{SDW}}_2$ phase, (d) vector chiral phase, and (e) TLL2 phase. Absolute values of the averaged correlation functions are plotted.

Figure 3

(Color online) Correlation functions in the antiferromagnetic zigzag ladder with $L=160$ spins in the TLL1 phase; (a) local spin polarization $⟨s_l^z⟩$, (b) longitudinal spin fluctuation $⟨s_l^z{s}_{l^{\prime }}^z⟩-⟨s_l^z⟩⟨s_{l^{\prime }}^z⟩$, and (c) transverse spin-correlation function $⟨s_l^x{s}_{l^{\prime }}^x⟩$. The upper and the lower panels show the results for $\left(J_2/J_1,M\right)=\left(0.1,0.375\right)$ and (0.5,0.125), respectively. The open symbols represent the DMRG data and the solid lines and circles are fits to Eqs. (12, 17, 18). In (b) and (c), the data for $l=L/2-\left[r/2\right]$ and $l^{\prime }=L/2+\left[\left(r+1\right)/2\right]$ are shown as a function of $r=\left|l-l^{\prime }\right|$.

Figure 4

(Color online) $M$ dependence of the exponent $\eta$ for the TLL1 phase estimated from the fitting of $⟨s_l^x{s}_{l^{\prime }}^x⟩$ for the antiferromagnetic zigzag ladder with $L=160$ spins. The error bars represent the difference of the estimates obtained from the fitting of the data of different ranges. The vertical dashed line corresponds to $M=1/6$ where the 1/3 plateau can appear for large $J_2/J_1$. The dotted line at $\eta =1$ represents the boundary between the regimes of dominant staggered transverse spin correlation $\left(\eta >1\right)$ and dominant incommensurate longitudinal spin correlation $\left(\eta <1\right)$; see Eqs. (10, 11). The exponent $\eta$ relates to the parameter $K$ as $\eta =2K$ in the TLL theory for the TLL1 phase.

Figure 5

(Color online) Absolute values of the averaged transverse spin-correlation function, $\left|{⟨s_0^x{s}_r^x⟩}_{\text{av}}\right|$, in the antiferromagnetic zigzag ladder with $L=160$ spins for $M=0.125$ and several $J_2/J_1$.

Figure 6

(Color online) Correlation functions in the antiferromagnetic zigzag ladder with $L=160$ spins in the ${\text{SDW}}_2$ phase; (a) local spin polarization $⟨s_l^z⟩$ and (b) longitudinal spin fluctuation $⟨s_l^z{s}_{l^{\prime }}^z⟩-⟨s_l^z⟩⟨s_{l^{\prime }}^z⟩$. The upper and the lower panels show the results for $\left(J_2/J_1,M\right)=\left(1.5,0.1\right)$ and (1.0,0.1), respectively. The open symbols represent the DMRG data and the solid symbols are the results of fitting to Eqs. (30, 31). In (b), the data for $l=L/2-\left[r/2\right]$ and $l^{\prime }=L/2+\left[\left(r+1\right)/2\right]$ are shown as a function of $r=\left|l-l^{\prime }\right|$.

Figure 7

(Color online) $M$ dependence of the exponent $\eta$ for the ${\text{SDW}}_2$ phase estimated from the fitting of $⟨s_l^z{s}_{l^{\prime }}^z⟩-⟨s_l^z⟩⟨s_{l^{\prime }}^z⟩$ for the antiferromagnetic zigzag ladder with $L=160$ spins. The error bars represent the difference of the estimates obtained from the fitting of the data of different ranges. The vertical dashed line corresponds to $M=1/6$ where the 1/3 plateau can appear for small $J_2/J_1$. The exponent $\eta$ relates to the parameter $K_{+}$ as $\eta =K_{+}$ in the TLL theory for the ${\text{SDW}}_2$ phase.

Figure 8

(Color online) (a) Local magnetization $⟨s_l^z⟩$ clearly shows the up-up-down spin configuration. (b) Semilog plot of the absolute values of the averaged correlation functions in the antiferromagnetic zigzag ladder with $L=120$ spins for $\left(J_2/J_1,M\right)=\left(0.7,20/120\right)$.

Figure 9

(Color online) Averaged vector chiral correlation functions in the antiferromagnetic zigzag ladder with $L=160$ spins for $\left(J_2/J_1,M\right)=\left(1.2,0.35\right)$. Open circles, squares, and triangles, respectively, represent the DMRG data of ${⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(1\right)}⟩}_{\text{av}}$, $-{⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$, and ${⟨{\kappa }_0^{\left(2\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$, where ${\kappa }_r^{\left(n\right)}={\left({\mathit{s}}_r\times {\mathit{s}}_{r+n}\right)}^z$.

Figure 10

(Color online) Amplitude of the vector chiral correlations at a distance $r=L/2$, $\left|{⟨{\kappa }_0^{\left(1\right)}{\kappa }_{L/2}^{\left(1\right)}⟩}_{\text{av}}\right|$, in the antiferromagnetic zigzag ladder with $L=160$ spins.

Figure 11

(Color online) Averaged transverse spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ in the antiferromagnetic zigzag ladder with $L=160$ spins for $\left(J_2/J_1,M\right)=\left(1.2,0.35\right)$. Open circles represent the DMRG data. The fits to Eq. (41) are shown by solid circles.

Figure 12

(Color online) $J_2/J_1$ dependence of the decay exponent $1/\left(4K_{+}\right)$ of the transverse spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ in the vector chiral phase for the antiferromagnetic zigzag ladder with $L=160$ sites.

Figure 13

(Color online) $J_2/J_1$ dependence of the wave number $\tilde{Q}$ of the transverse spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ in the vector chiral phase for the antiferromagnetic zigzag ladder with $L=160$ sites. The dashed curve represents the classical pitch angle $\tilde{Q}=\text{arccos}\left(-J_1/4J_2\right)$.

Figure 14

Four Fermi points located at $k=\pm k_s$ and $\pm k_l$. The density of fermions is related to the magnetization, $k_l-k_s=\pi \left(\frac{1}{2}-M\right)$. The phase transition to the TLL1 phase occurs when $k_s=0$, i.e., when the inner two Fermi points merge at $k=0$.

Figure 15

(Color online) (a) Entanglement entropy of the TLL2 phase $\left(J_2/J_1=0.6,M=0.4\right)$. The horizontal axis is $x=\text{ln}\left[\left(L/\pi \right)\text{sin}\left(\pi l/L\right)\right]$. Solid and dotted lines represent the slope $1/3\left(c=2\right)$ and $1/6\left(c=1\right)$, respectively. (b) Entanglement entropy of the TLL1 phase $\left(J_2/J_1=0.1,M=0.375\right)$ and the vector chiral phase $\left(J_2/J_1=1.2,M=0.35\right)$. Dotted lines indicate the slope $1/6\left(c=1\right)$.

Figure 16

(Color online) (a) Squared modulus of the Fourier transform of the local spin polarization, ${\left|s^z\left(k\right)\right|}^2$, for the antiferromagnetic zigzag ladder with $L=160$ spins and $J_2/J_1=0.6$. (b) $M$ dependence of peak positions of ${\left|s^z\left(k\right)\right|}^2$. Solid line represents the highest peak while the dotted lines correspond to the subdominant peaks in TLL2 phase. Gray horizontal lines show phase boundaries.

Figure 17

(Color online) (a) Squared modulus of the Fourier transform of the local spin polarization, ${\left|s^z\left(k\right)\right|}^2$, for the antiferromagnetic zigzag ladder with $L=160$ spins and $J_2/J_1=0.9$. (b) $M$ dependence of peak positions of ${\left|s^z\left(k\right)\right|}^2$. Solid line represents the highest peak while the dotted lines correspond to the subdominant peaks in TLL2 phase. Gray horizontal lines show phase boundaries.