Field-driven quantum phase transitions in $S=\frac{1}{2}$ spin chains

We study the magnetization process of a one-dimensional extended Heisenberg model, the $J\text{−}Q$ model, as a function of an external magnetic field $h$. In this model, $J$ represents the traditional antiferromagnetic Heisenberg exchange and $Q$ is the strength of a competing four-spin interaction. Without external field, this system hosts a twofold-degenerate dimerized (valence-bond solid) state above a critical value $q_c\approx 0.85$ where $q\equiv Q/J$. The dimer order is destroyed and replaced by a partially polarized translationally invariant state at a critical field value. We find magnetization jumps (metamagnetism) between the partially polarized and fully polarized state for $q>q_{\min }$, where we have calculated $q_{\min }=\frac{2}{9}$ exactly. For $q>q_{\min }$, two magnons (flipped spins on a fully polarized background) attract and form a bound state. Quantum Monte Carlo studies confirm that the bound state corresponds to the first step of an instability leading to a finite magnetization jump for $q>q_{\min }$. Our results show that neither geometric frustration nor spin anisotropy are necessary conditions for metamagnetism. Working in the two-magnon subspace, we also find evidence pointing to the existence of metamagnetism in the unfrustrated $J_1\text{−}{J}_2$ chain $\left(J_1>0,J_2<0\right)$, but only if $J_2$ is spin anisotropic. In addition to the studies at zero temperature, we also investigate quantum-critical scaling near the transition into the fully polarized state for $q\le q_{\min }$ at $T>0$. While the expected “zero-scale-factor” universality is clearly seen for $q=0$ and $q\ll q_{\min }$, for $q$ closer to $q_{\min }$ we find that extremely low temperatures are required to observe the asymptotic behavior, due to the influence of the tricritical point at $q_{\min }$. In the low-energy theory, one can expect the quartic nonlinearity to vanish at $q_{\min }$ and a marginal sixth-order term should govern the scaling, which leads to a crossover at a temperature $T^{*}\left(q\right)$ between logarithmic tricritical scaling and zero-scale-factor universality, with $T^{*}\left(q\right)\to 0$ when $q\to q_{\min }$.

Figure 1

Examples of VBS configurations of $S=1/2$ spins in one dimension. Each blue ellipse represents a singlet pair: $\left(|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle \right)/\sqrt{2}$. In a real VBS there are also fluctuations in the singlet patterns (except in special cases) but the density of singlets on the bonds is still modulated with periodicity two lattice spacings. (a), (b) Show the two degenerate VBS ground states, (c) illustrates a triplet excitation in which a singlet bond is broken, and (d) illustrates a triplet excitation deconfined into two independently propagating spinons.

Figure 2

Schematic phase diagram of the $J\text{−}Q\text{−}h$ chain defined in Eqs. (2) and (3). The different phases and special points indicated are described in the text.

Figure 3

Magnetization density of the $J\text{−}Q\text{−}h$ chain as a function of the external field for a set of coupling ratios $0\le q\le 1.2$ (from Heisenberg limit to beyond the VBS transition). The system size is $L=96$ and the inverse temperature is $\beta =12$ in all cases. Error bars are smaller than the markers.

Figure 4

Magnetization density of the $J\text{−}Q\text{−}h$ chain at $q=1.2$ as a function of the external field $h$, with the inverse temperature scaled with size as $\beta =L/4$. The system sizes are between $L=8$ and 256 as indicated. The inset shows a zoomed-in view of the paramagnetic regime. The error bars are smaller than the markers in main figure and have been omitted for clarity also in the inset (where they are some times slightly larger than the markers).

Figure 5

Saturation field versus the coupling ratio for the $L=30$ periodic $J\text{−}Q\text{−}h$ chain calculated using the Lanczos method. The dot indicates $q_{\min }$.

Figure 6

The lowest-energy eigenvalue ${\overline{E}}_2(J=1,Q=q,L)$ in the two-magnon sector $\left(m_z=S-2\right)$ in the $J\text{−}Q\text{−}h$ chain for system sizes $L=8,16,32,1024$.

Figure 7

The probability $P\left(r\right)=\langle {\psi }_0\left(r\right)|{\psi }_0\left(r\right)\rangle$ of the particles being separated by distance $r$ in the lowest state in the two-magnon sector $\left(m_z=S-2\right)$ of the $J\text{−}Q\text{−}h$ chain.

Figure 8

Alternating dimer-dimer correlation function, defined in Eq. (10), for several values of the magnetization in chains of length $L=96$ at $\beta =12,q=1.2$.

Figure 9

The binding energy defined in Eq. (13) for a $J_1\text{−}{J}_2$ chain with $j\equiv -J_2/J_1$ and anisotropy parameters $\mathrm{\Delta }=0,\pm 1$. Here, a relatively small system $\left(L=128\right)$ is used, to make it easier to see the crossings. When $\mathrm{\Xi }(j,\mathrm{\Delta })>0$, there is a bound state of two magnons.

Figure 10

Test of zero-factor scaling using the rescaled density (18) of flipped spins near saturation for a $J\text{−}Q\text{−}h$ chain of 96 sites for several different inverse temperatures $\beta$ and values of the coupling ratio $q$ (in different panels as indicated). The results are graphed versus the rescaled magnetic field $t\equiv \beta (h_s-h)$. The black lines are the exact predicted universal function (19) with the bare magnon mass $M=1$.

Figure 11

Finite-size behavior of the zero-factor-scaled magnon density (18) for the $J\text{−}Q\text{−}h$ chain at $t\equiv \beta (h_s-h)=0$ for several different inverse temperatures $\beta$ and values of the coupling ratio $q$ (in different panels as indicated). In all cases, the error bars are smaller than the markers. The black horizontal lines in each panel show the value from the exact universal function (19) with the bare magnon mass $M=1$.

Figure 12

Temperature dependence of the rescaled magnon density (18) for an $L=96$ $J\text{−}Q\text{−}h$ chain at $h=h_s$ and several values of the coupling ratio $q$. Error bars are smaller than the markers. The black dashed line shows the exact asymptotic $\left(T\to 0\right)$ value from the universal function (19), setting the bare magnon mass $M=1$.