Quantum Phase Transitions in Coherent Ising Machines: XY Model for Demonstration

TL;DR

Problem: investigate quantum phase transitions in a 1D XY spin chain using a photonic platform. Approach: establish an exact spectral mapping between the XY model and a network of DOPOs in a coherent Ising machine, deriving parameter correspondences $J = 2 sqrt(J_x J_y)$, $Delta = - h J_s / sqrt(J_x J_y)$, and $D^2 = J_d^2 ( h^2 /(J_x J_y) - 4 )$, and showing $e_g^{D} = - e_g^{XY} + h J_s /(2 sqrt(J_x J_y))$; demonstrate that the DOPO network reproduces the XY QPT across isotropic, anisotropic, and TFI limits. Contributions: explicit spectral mapping, validation of second-order QPTs via ground-state energy derivatives and magnetic susceptibility, and demonstration of an optical simulator for universal quantum critical phenomena. Significance: enables scalable optical simulations of quantum criticality with potential impacts on quantum sensing and photonic quantum information processing.

Abstract

Quantum phase transitions (QPTs) in coherent Ising machines (CIMs) are studied via a spectral mapping between the one-dimensional XY spin model and a network of degenerate optical parametric oscillators (DOPOs). This exact correspondence reveals that the DOPO network faithfully reproduces the quantum critical behavior of the XY model across its anisotropic, isotropic, and transverse-field Ising regimes. The ground-state energy density and its derivatives are analyzed to reveal second-order QPTs characterized by singularities in magnetic susceptibility at critical points. These results show that CIMs do not only serve as powerful platforms for solving combinatorial optimization problems but also provide a versatile optical simulator for studying universal quantum critical phenomena, bridging quantum-spin models and photonic quantum systems.

Figures (3)

• Figure 1: (a) Schematic representation of a DOPO, a key component for constructing a CIM. (b) Simplified phase-space illustration of a DOPO: the vacuum state below the threshold and the coherent state corresponding to phases $0$ and $\pi$ above the threshold.
• Figure 2: Numerical results for three quantum-spin-chain models under an external field $h$: (a) Ground-state energy density $e_g^{\mathrm{XY}}$, (b) magnetization $m_z^{\mathrm{XY}} = -\partial e_g^{\mathrm{XY}}/\partial h$, and (c) magnetic susceptibility $\chi^{\mathrm{XY}} = -\partial^2 e_g^{\mathrm{XY}}/\partial h^2$. Note that sub-figure (c) uses a logarithmic scale. The curves correspond to the anisotropic XY model ($J_x = 2, J_y = 1$, blue solid line), the isotropic XY model ($J_x = J_y = 1$, green dash-dotted line), and the transverse-field Ising model ($J_x = 1, J_y = 0$, red dashed line). The vertical dotted lines mark the calculated critical fields $h_c$ for each model.
• Figure 3: QPTs in a DOPO network mapped from the XY model. Columns show DOPO parameter regimes: (left) $J=2\sqrt{2}$, $\Delta = -3h/\sqrt{2}$; (middle) $J=2$, $\Delta = -2h$ ($D=0$); (right) $J = 0.2$, $\Delta = -10h$. These correspond to anisotropic XY, isotropic XY, and near-TFI spin chains, respectively. From top to bottom, the panels display: (a--c) the ground-state energy density $e_g^{\mathrm{D}}$; (d--f) the longitudinal magnetization $m_z^{\mathrm{D}}$; and (g--i) the magnetic susceptibility $\chi^{\mathrm{D}}$. Each panel features dual horizontal axes: the upper axis shows the DOPO detuning parameter $\Delta$, and the lower axis shows the corresponding transverse field $h$ in the XY model. All energies are scaled by $J_0=1$ to render them dimensionless. Vertical dashed lines indicate the critical points at $\Delta_c = -2J - D$.