Quantum Phase Transition in the One-Dimensional Water Chain

The concept of quantum phase transitions (QPTs) plays a central role in the description of condensed matter systems. In this Letter, we perform high-quality wave-function-based simulations to demonstrate the existence of a quantum phase transition in a crucially relevant molecular system, namely, water, forming linear chains of rotating molecules. We determine various critical exponents and reveal the water chain QPT to belong to the ($1+1$)-dimensional Ising universality class. Furthermore, the effect of breaking symmetries is examined, and it is shown that, by breaking the inversion symmetry, the ground state degeneracy of the ordered quantum phase is lifted to yield two many-body states with opposite polarization. The possibility of forming ferroelectric phases together with a thermal stability of the quantum critical regime up to $\sim 10\mathrm{K}$ makes the linear water chain a promising candidate as a platform for quantum devices.

Figure 1

Phase diagram of the one-dimensional water system. The disordered phase is characterized by water molecules delocalized over all orientations at large water-water distances. In contrast, in the ordered phase, the electric dipole moments (black arrows) are aligned, leading to a twofold degenerate ground state which is a superposition of left-and right-polarized states. The solid line represents the Schmidt gap $\mathrm{\Delta }\lambda$ as a function of the water-water distance $R$.

Figure 2

(a) and (b) Energy difference between the ground state and the first and second excited states for chains of full symmetry (FS) and of broken symmetry (BS). (c) and (d) von Neumann entanglement entropy for FS and BS chains. (e) and (f) First ten Schmidt values for FS and BS chains. (g) and (h) Derivative of the nearest-neighbor dipole-dipole correlation function $C(R)=(1/N-1){\sum }_{i=1}^{N-1}⟨{\mu }_i^z{\mu }_{i+1}^z⟩$. In all calculations, a small field of strength is $|\overrightarrow{E}·\overrightarrow{\mu }|={10}^{-7}{E}_h$ applied at the two edge molecules. The system size is $N=300$.

Figure 3

(a) von Neumann entanglement entropy for different numbers of water molecules. (b) Schmidt gap for different numbers of water molecules. The inset in (b) shows the rescaled Schmidt gaps (for $N=200–400$) with fitted critical exponents $\nu =1.004\pm 0.0040$ and $\beta =0.118\pm 0.008$. (c) $z$ component of the total polarization per molecule for different numbers of water molecules. For these calculations, the inversion symmetry was broken by applying a very small electric field ($|\overrightarrow{E}·\overrightarrow{\mu }|={10}^{-7}{E}_h$) at the two edge molecules. The inset in (c) shows the rescaled polarization per bond (for $N=200–400$) with fitted critical exponents $\nu =1.069\pm 0.025$ and $\beta =0.125\pm 0.001$. The critical distance $R_c$ is indicated by the dashed vertical line in (a)–(c). (d) von Neumann entanglement entropy for different bipartitions at the QPT (solid circles). The solid line shows a fit of the entropy points to $(c/3)\ln [(N/\pi )\sin (n\pi /N)]$, with a central charge of $c=0.5028\pm 0.0005$. The two edge points were excluded from the fit. The inset in (d) shows the von Neumann entanglement entropy calculated with open-boundary conditions (OBCs, solid line) and sine-squared deformation (SSD, dashed line). All points are calculated for $N=100$. (e) $z$ component of dipole-dipole correlation between the central water molecule at site $m=51$ and the $j$th water molecule (solid circles). The solid line shows a fit of the correlation function to $j^{-\eta }$ with a critical exponent of $\eta =0.253\pm 0.002$. All points are calculated for $N=101$.

Figure 4

(a) Difference of the von Neumann entanglement entropy of a fully symmetric chain and a chain with broken inversion symmetry. (b) von Neumann entanglement entropy of a fully symmetric chain with a central bond weakened by a factor $ϵ$. The dashed line indicates the value of $\ln (2)$. (c) $z$ component of the electric dipole-dipole correlation of the two central water molecules of a fully symmetric chain with a central bond weakened by a factor $ϵ$. The system size is $N=300$.