Coherence susceptibility as a probe of quantum phase transitions

We introduce a coherence susceptibility method, based on the fact that it signals quantum fluctuations, for identifying quantum phase transitions, which are induced by quantum fluctuations. This method requires no prior knowledge of order parameter, and there is no need for careful considerations concerning the choice of a bipartition of the system. It can identify different types of quantum phase transition points exactly. At finite temperatures, where quantum criticality is influenced by thermal fluctuations, our method can pinpoint the temperature frame of quantum criticality, which perfectly coincides with recent experiments.

Figure 1

Ising model ground state. Main plot: Coherence of one-site reduced density matrix for the symmetry-broken ground state (dash-dot line) and the thermal ground state (solid line). For the symmetry broken state the singularity occurs exactly at the quantum phase transition point. Inset: Coherence susceptibility for the thermal ground state. This is obtained by numerically carrying out the derivative of the coherence curve in the main plot with respect to $\lambda$, therefore the actual divergence of $\chi$ at the phase transition point manifests itself as a big negative value in the numerical result due to the finite step.

Figure 2

The coherence of two adjacent spins for the $\mathit{XX}$ model in the thermodynamic limit.

Figure 3

Kitaev model ground state. The left inset is the phase diagram of the Kitaev model. We calculate the ground state along the dashed line. The right inset shows the obtained coherence for two sites with an $x$ link. The main figure plots the coherence susceptibility. The singular point occurs exactly at the topological phase transition point $J_x=0.5$.

Figure 4

Ising model thermal states at different temperatures. Main plot: Coherence susceptibility at different temperatures. At finite temperatures the quantum phase transition is smoothed out due to thermal fluctuation. As a result the singularity deforms into maximum points. Left inset: Phase diagram at finite temperatures. Right inset: The locations of the maximum point at different temperatures. Here $k_{B}T$ is in unit of the interaction energy in the system Hamiltonian.

Figure 5

Quantum discord for Ising model thermal states at different temperatures. From top to bottom the temperature increases from $k_{B}T=0$ to 1 in unit of the interaction energy in the system Hamiltonian.

Figure 6

The $\lambda$ location of the maximal quantum discord with respect to temperatures for the Ising model. We show it only up to $k_{B}T=0.7$ because the real maximal quantum discord for higher temperatures is out of the calculation scope of $\lambda$. Here $k_{B}T$ is in unit of the interaction energy in the system Hamiltonian.

Figure 7

A graphic representation of the Kitaev model. There are two sublattices (empty and full circles). Each unit cell (marked by elliptic circle) contains one site of each kind. Three types of bonds are labeled by “$x$ links,” “$y$ links,” and “$z$ links.” We choose the coordinate axes in ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ directions.