Kinematic Emergence of the Page Curve in a Local Transverse-Field Ising Model

Abstract

We present a controllable quantum spin-chain model that reproduces the Page curve (the rise-and-fall of bipartite entanglement expected in black-hole evaporation), using only local interactions and a kinematic reduction of the subsystem size. Two transverse-field Ising chains are coupled to form a pure bipartite state; Hawking-like evaporation is implemented by dynamically shrinking the 'system' chain and enlarging the 'environment' chain, while unitary real-time evolution is simulated with matrix product state (MPS) tensor networks. The characteristic Page curve profile emerges robustly under this controlled subsystem resizing and notably persists even when the explicit Hamiltonian coupling across the boundary is set to zero, demonstrating that shrinking Hilbert-space dimension alone can generate Page curve behaviour. We show that the detailed shape of the curve depends on the internal information dynamics: operation at criticality yields a smooth profile, whereas moving away from criticality distorts entanglement growth and decay. These results position locally interacting spin chains as a realistic platform for probing black-hole-inspired information dynamics on current quantum hardware.

Figures (10)

• Figure 1: A Schematic of the idealised Page curve which depicts the behaviour of the entropy of the environment over time. Time is most intuitively defined as how much Hawking radiation has left the black hole until it eventually evaporates. In the first half of the black holes life, the entropy of the radiation grows and reaches a maximum until the growth is limited by the degrees of freedom present in the black hole Hilbert space. This turning point is the Page time. After this the entropy of the system decreases and information is recovered. This preserves unitarity and information within the system and is a proposed solution to the information paradox Page2.
• Figure 2: A schematic of the full spin chain modelled by the Hamiltonian given in Eq.\ref{['eq.hamiltonian']}. The chain is split, one for the system coloured in red, of size $N$ with interaction strength $J_{sys}$ and one for the environment, in green, with interaction strength $J_{env}$ and has length $M$. The overall size of the system is $L = N+M$ with a Hilbert space of size $\mathcal{H} \sim d^{L}$. The two subsystems are coupled with dynamic coupling $h$. To simulate evaporation we systematically move the dynamic coupling $h$ to the left to grow the environment and decrease the system whilst allowing the system to naturally evolve through time using a TEBD protocol (see Appendix \ref{['sec.method']} for more information) to simulate unitary evolution.
• Figure 3: We simulate time evolution of the time independent Hamiltonian $\hat{H}$ through unitary evolution simulated with a TEBD protocol (see Appendix \ref{['sec.method']} for more information). The x-axis denotes the time evolution passed, simulated in increments of $\tau = 0.1$ with an evaporation time-step $T = 5$, each marker denotes a point where an evaporation event was performed. When $\frac{t}{T} \in \mathbb{Z}$ the von Neumann entropy is measured, recorded and then an evaporation event is performed. The result is the von Neumann behaves in alignment with the Page curve after around half the time has passed (the Page time) the entropy decreases due to being limited by the size of the system since Eq.\ref{['eq.Svn']} holds for the bipartite model. $N$ is denoted in the differing colours and can be seen in the figure. The rest of the parameters are as follows: $J_{sys} = g_{sys} = h =3$, $J_{env} = g_{env} = 1$ and $M=150$.
• Figure 4: We simulate time evolution of the time independent Hamiltonian $\hat{H}$ through unitary evolution simulated with a TEBD protocol (see Appendix \ref{['sec.method']} for more information). The x-axis denotes the time evolution passed, simulated in increments of $\tau = 0.1$ with an evaporation time-step $T = 5$, each marker denotes a point where an evaporation event was performed. When $\frac{t}{T} \in \mathbb{Z}$ the von Neumann entropy is measured, recorded and then an evaporation event is performed. Note when the dynamic coupling between the system and the environment $h=0$, there is still a characteristic rise and fall throughout the entire process when the system is entirely kinematic. Since there is no dynamic coupling between the two subsystem when $h=0$ the Page curve is purely down to the kinematic transfer of the degrees of freedom. We see that having no dynamic coupling has little effect on the entropic behaviour. The variations of $h$ are denoted in the differing colours and can be seen in the figure. The rest of the parameters are as follows: $J_{sys} = g_{sys} =3$, $J_{env} = g_{env} = 1$, $N=15$ and $M=150$.
• Figure 5: We simulate time evolution of the time independent Hamiltonian $\hat{H}$ through unitary evolution simulated with a TEBD protocol (see Appendix \ref{['sec.method']} for more information). In this plot we vary the relationships $\frac{J_{sys}}{g_{sys}}$ and $\frac{J_{env}}{g_{env}}$ to probe how criticality effects the outcome of the entropy profile. The x-axis denotes the time evolution passed, simulated in increments of $\tau = 0.1$ with an evaporation time-step $T = 5$, each marker denotes a point where an evaporation event was performed. We keep $N=15$, $M=150$ and $h = 3$. When $\frac{t}{T} \in \mathbb{Z}$ the von Neumann entropy is measured, recorded and then an evaporation event is performed. We ran this simulation for each permutation of the system and environment subsystems being governed by the Hamiltonian with $\frac{J}{g} > 1$,$\frac{J}{g} < 1$ or $\frac{J}{g} = 1$. These permutations are categorised above into three subplots, the data is grouped via the conditions imposed on the system with (a) being when the system is at criticality, (b) when $J_{sys} > g_{sys}$, and (c), when $J_{sys} < g_{sys}$. We find in figure (b) that when the system has a large $J$ then it significantly effects how entropy behaves within the environment when compared to the system at criticality. We observe a significant suppression in the entropy growth in the first half of the Page curve, which is dependent on the dominance of the $J_{sys}$.