Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field

Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the “$\tau$ order metric” of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the $x$-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the “$\tau$ order metric,” even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.

Figure 1

Size dependence of the critical point using various observables as the order parameters for the nearest-neighbor ferromagnetic Ising model: (a) Average magnetization along the $z$-axis, Eq. (3); (b) $\tau$ order metric, Eq. (7); (c) second cumulant of the transverse magnetization, $h_x$, Eq. (2); (d) measure of hyperuniformity, $S_0$, Eq. (6). The variation of the critical value of $\mathrm{\Gamma }$ with $N$ is the least for $h_x$. The value of $h_x$ undergoes a maximum for small $N$, which changes to a point of nondifferentiability as $N$ becomes larger.

Figure 2

Basic observables as functions of $\mathrm{\Gamma }$ for systems of size $N=30$: (a) Energy per site; (b) $\tau$ order metric; (c) second cumulant of the transverse magnetization, $h_x$; (d) degree of hyperuniformity, $S_0$. Black line: $J_r=r^{-2}$. Red line: $J_r=r^{-3}$. Green line: $J_r=r^{-4}$. Blue line: ferromagnetic Ising model. Black markers: $J_r={(-1)}^r{r}^{-2}$. Red markers: $J_r={(-1)}^r{r}^{-3}$. Green markers: $J_r={(-1)}^r{r}^{-4}$. Blue markers: antiferromagnetic Ising model. The degree of hyperuniformity, $S_0$, undergoes a sudden jump for the ferromagnetic cases at the critical point. This jump in $S_0$ is absent in the antiferromagnetic cases.

Figure 3

Basic observables as functions of $\mathrm{\Gamma }$ for systems of size $N=30$ interacting via $J_1\left(r\right)$ (described in Appendix pp1): (a) Energy per site; (b) $\tau$ order metric; (c) second cumulant of the transverse magnetization, $h_x$; (d) degree of hyperuniformity, $S_0$ and $S_1$, Eq. (9). Parameter demonstrates the possibility of a weak first-order QPT.

Figure 4

Ground-state structure factor as a function of the wave vector, $k$, and the strength of the transverse field, $\mathrm{\Gamma }$, for $N=30$ spins interacting via $J_1\left(r\right)$ (described in Appendix pp1).

Figure 5

Basic observables as functions of $\mathrm{\Gamma }$ for systems of size $N=30$ interacting via $J_2\left(r\right)$ (described in Appendix pp1): (a) Energy per site; (b) $\tau$ order metric; (c) second cumulant of the transverse magnetization, $h_x$; (d) degree of hyperuniformity, $S_0$ and $S_1$. Parameter demonstrates the possibility of a first-order QPT indicated by sharp discontinuities in (b), (c), and (d) around $\mathrm{\Gamma }\approx 0.1$.

Figure 6

Structure factor as a function of the wave vector, $k$, and the strength of the transverse field, $\mathrm{\Gamma }$, for $N=30$ spins interacting via $J_2\left(r\right)$ (described in Appendix pp1). We have marked out the critical value of $\mathrm{\Gamma }$ where the phase transition occurs. Between $\mathrm{\Gamma }=0.095$ and 0.10, there is a phase change.

Figure 7

Basic observables as functions of $\mathrm{\Gamma }$ for systems of size $N=30$ interacting via $J_3\left(r\right)$ (described in Appendix pp1): (a) Energy per site; (b) $\tau$ order metric; (c) second cumulant of the transverse magnetization, $h_x$; (d) degree of hyperuniformity, $S_0$ and $S_1$. Parameter demonstrates the possibility of two first-order QPTs indicated by sharp discontinuities in (b), (c), and (d) around $\mathrm{\Gamma }\approx 0.065$ and $\mathrm{\Gamma }\approx 0.125$.

Figure 8

Structure factors for spins interacting via $J_3\left(r\right)$ (described in Appendix pp1). Left: ground states for select intervals of $\mathrm{\Gamma }$ for systems of size $N=30$. Ground state before the first critical point $(\mathrm{\Gamma }<0.06)$ is labeled $|0\rangle$. Right: other low-lying states at $\mathrm{\Gamma }=0$, labeled $|1\rangle$, $|2\rangle$, and $|3\rangle$, respectively.

Figure 9

Structure factor as a function of the wave-vector, $k$, and the strength of the transverse field, $\mathrm{\Gamma }$, for $N=30$ spins interacting via $J_3\left(r\right)$ (described in Appendix pp1). We have marked out critical values of $\mathrm{\Gamma }$ where the phase transitions occur. There are two phase transitions, one at $\mathrm{\Gamma }\approx 0.07$ and one at $\mathrm{\Gamma }\approx 0.125$, after which there is a steady continuous loss of order.

Figure 10

Energy per site with respect to different references.