Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic spin-$\frac{1}{2}$ Heisenberg chain in a magnetic field

We present the phase diagram of the frustrated ferromagnetic $S=\frac{1}{2}$ Heisenberg $J_1-J_2$ chain in a magnetic field, obtained by large scale exact diagonalizations and density matrix renormalization group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin-multipolar Luttinger liquid phases up to hexadecupolar kind are found. We provide numerical evidence for a locking mechanism, which can drive spiral states toward spin-multipolar phases, such as quadrupolar or octupolar phases. Our results also shed light on previously discovered spin-multipolar phases in two-dimensional $S=\frac{1}{2}$ quantum magnets in a magnetic field.

Figure 1

(Color online) Phase diagram of the frustrated ferromagnetic chain (1) in the $J_1/J_2$ vs $m/m_{\text{sat}}$ plane. The gray low-$m$ region exhibits vector chiral long range order. The colored regions denote spin-multipolar Luttinger liquids of bound states of $p=2,3,4$ spin flips. Close to saturation the dominant correlations are multipolar, while below the dashed crossover lines, the dominant correlations are of $\text{SDW}\left(p\right)$ type. The tiny cyan colored region corresponds to an incommensurate $p=2$ phase. The white region denotes a metamagnetic jump. Finally the scribbled region close to the transition $J_1/J_2\to -4$ has not been studied here, but consists most likely of a low field vector chiral phase, followed by a metamagnetic region extending up to saturation magnetization. The inset shows the same diagram in the $J_1/J_2$ vs $h/h_{\text{sat}}$ plane.

Figure 2

(Color online) Squared vector chirality order parameter ${\kappa }^2$ [Eq. (2)] in the low magnetization phase for different values of $J_2/\left|J_1\right|$ as a function of $m/m_{\text{sat}}$: (a) $J_2/\left|J_1\right|=0.27$, (b) $J_2/\left|J_1\right|=0.32$, and (c) $J_2/\left|J_1\right|=0.38$. The order parameter vanishes in the multipolar Luttinger liquids.

Figure 3

(Color online) Wave vector $q$ maximizing the longitudinal (inset: transverse) spin structure factor as a function of the magnetization $m/m_{\text{sat}}$ for selected $J_2/\left|J_1\right|$ parameters, each one belonging to a different $p$ sector at high $m$. The straight lines are guides to the eyes and show $\pi \left(1-m/m_{\text{sat}}\right)/p$. The different symbols denotes ED results on system sizes ranging from 18 to 28 sites.

Figure 4

(Color online) Upper panel: ED results for the domain of stability of bound states of $p$ spin flips above the saturated ferromagnetic state. Lower panel: Location $q_{\text{max}}$ of the maximum of the zero field spin structure factor $\mathcal{S}\left(q\right)$ as a function of $J_2$. The colored lines indicate the construction of the domain boundaries, based on the locking rule Eq. (5). The thin dashed line displays $q_{\text{max}}^{\text{class}}$ for the classical model.