Quantum Ising Model

The transverse field Ising model is defined with the hamiltonian¹:

$$H=-\sum _{i,j}^{}{\sigma }_i^z\otimes {\sigma }_j^z-h\sum _i^{}{\sigma }_i^x$$

where ${\sigma }^z$ and ${\sigma }^x$ are the Pauli Z and X matrics respectively, and $h$ is the transverse magnetic field strength. This model is realized in physical experiments related to quantum computing²³. The state that is observed in these physical experiments is the ground state of $H$, defined as the eigenvector of $H$'s smallest eigenvalue $|{\psi }_{ground}⟩=\sum _b^{2^N}{c}_b|{\psi }_b⟩$, where $|{\psi }_b⟩$ are basis states $|\uparrow \uparrow \uparrow \cdots ⟩$, $|\downarrow \downarrow \downarrow \cdots ⟩$, $|\uparrow \uparrow \downarrow \cdots ⟩$ and so on. Note that for a composite system of $N$ spins, there are in total $2^N$ basis states.

Magnetization is defined as the average Z spin

$$⟨M⟩=\sum _b^{2^N}{p}_b\frac{\sum _i^N{\sigma }_{bi}^z}{N}$$

where $p_b=\overline{c_b}{c}_b$ is the probability at basis state $|{\psi }_b⟩$.

With magnetization as the order parameter, we get the following graph showing the existence of a phase transition at critical field strength $h_c=1$ in the one dimensional case:

This differs from the classical situation where the 1-D Ising model lacks a phase transition. In the quantum model, the fluctuations required by the phase transition is provided by the transverse field ${\sigma }^x$ which contains off-diagonal matrix elements. In the two dimensional case, we get $h_c\simeq 2.5$ which is close to the theoretical infinite lattice size case of $h_c=3$ (Table 3.1 Suzuki⁴, Fig. 4 Hesselmann⁵).

This phase transition can also be seen in the band diagram of the ground and excitation states. We see that as the transverse field approaches $h_c$, the ground and 1st excitation energies diverge from being in a shared degenerate state, to two separate states.

Non-symmetric probability distribution in simulation

The basis state probability $p_b$ used in our simulations is a non-symmetric one, as opposed to the one belonging to the original hamiltonian $p_b^H$. $p_b$ is defined as

$$\begin{array}{c}{p}_b=\{\begin{array}{rlr}{p}_b^H+p_{b^{\prime }}^H,& & b_{\mathrm{\#}up}>b_{\mathrm{\#}down}\\ p_b^H,& & b_{\mathrm{\#}up}=b_{\mathrm{\#}down}\\ 0,& & b_{\mathrm{\#}up}<b_{\mathrm{\#}down}\\ \end{array}\end{array}$$

where $p_{b^{\prime }}^H$ is the probability of the basis state with all the spins in $|{\psi }_b⟩$ flipped, $b_{\mathrm{\#}up}$ and $b_{\mathrm{\#}down}$ are the number of spin ups and downs in basis state $|{\psi }_b⟩$.

In real world experiments, such a non-symmetric distribution is a more realistic description, as it is unlikely to see all $N$ spins flipped down if the initial state is prepared with all up spins, especially when $N$ is very large. However, due to the finite lattice size of simulations, the characteristic time where the system flips from +M to -M state is much smaller than the observation time as seen in the below graph from Fig 3.10, Section 3.4, Binder et al.⁶:

Such a non-symmetric distribution in effect measures the average absolute value of magnetization $⟨\left|M\right|⟩$ as opposed to the raw average magenetization $⟨M⟩$, which in the case of $h\sim 0$ is always 0 for a symmetric hamiltonian like ours. In the classical 2-D Ising model where $T_c=2.269$, we see from the below graphsthat the non-symmetric distribution gives better results (upper graph non-symmetric, lower graph symmetric, from Fig. 14, Fig 15, J. Kotze⁷):

References

Footnotes

1. Quantum Ising Phase Transition, Carsten Bauer, Katharine Hyatt, link ↩

2. Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator, Sepehr Ebadi et al., arXiv:2012.12281 ↩

3. Observation of a Many-Body Dynamical Phase Transition with a 53-Qubit Quantum Simulator, J. Zhang, arXiv:1708.01044 ↩

4. Quantum Ising Phases and Transitions in Transverse Ising Models, 2nd Edition, Sei Suzuki, Jun-ichi Inoue, Bikas K. Chakrabarti ↩

5. Thermal Ising transitions in the vicinity of two-dimensional quantum critical point, S. Hesselmann, S. Wesse, arXiv:1602.02096 ↩

6. Monte Carlo Simulation in Statistical Physics 6th Ed., Binder, K. & Heermann, D. W. ↩

7. Introduction to Monte Carlo methods for an Ising Model of a Ferromagnet, Jacques Kotze. arXiv:0803.0217 ↩