Anomalous dynamical phase in quantum spin chains with long-range interactions

The existence or absence of nonanalytic cusps in the Loschmidt-echo return rate is traditionally employed to distinguish between a regular dynamical phase (regular cusps) and a trivial phase (no cusps) in quantum spin chains after a global quench. However, numerical evidence in a recent study (J. C. Halimeh and V. Zauner-Stauber, arXiv:1610.02019) suggests that instead of the trivial phase, a distinct anomalous dynamical phase characterized by a novel type of nonanalytic cusps occurs in the one-dimensional transverse-field Ising model when interactions are sufficiently long range. Using an analytic semiclassical approach and exact diagonalization, we show that this anomalous phase also arises in the fully connected case of infinite-range interactions, and we discuss its defining signature. Our results show that the transition from the regular to the anomalous dynamical phase coincides with ${\mathbb{Z}}_2$-symmetry breaking in the infinite-time limit, thereby showing a connection between two different concepts of dynamical criticality. Our work further expands the dynamical phase diagram of long-range interacting quantum spin chains, and can be tested experimentally in ion-trap setups and ultracold atoms in optical cavities, where interactions are inherently long range.

Figure 1

The dynamical phase diagram of the one-dimensional LR-TFIM (2) after a global quench with initial field strength ${\mathrm{\Gamma }}_{\text{i}}=0$, showing three distinct dynamical phases: regular, anomalous, and trivial (see main text). The dynamical critical line is marked in solid black. The results for the nonintegrable model are obtained using iMPS [18], while the dynamical critical point for the NN-TFIM ($\alpha \to \infty$) is known analytically [6]. The phase diagram for the FC-TFIM ($\alpha =0$) is the main result of this work.

Figure 2

The periodicity of the order parameter (dotted black line) in the FC-TFIM for a quench from ${\mathrm{\Gamma }}_{\text{i}}=0$ to ${\mathrm{\Gamma }}_{\text{f}}$, derived in a semiclassical approach. This periodicity is also that of the nonanalytic cusps arising in the Loschmidt-echo return rate (1). The period diverges at the dynamical critical point ${\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.25$, i.e., at the point where the infinite-time average of the longitudinal magnetization (solid blue line) is nonanalytic as a function of ${\mathrm{\Gamma }}_{\text{f}}$. The critical point ${\mathrm{\Gamma }}_{\text{c}}^{\text{d}}$ also separates the anomalous and regular phases.

Figure 3

Loschmidt-echo rate function and expectation value of the magnetization after a quench from ${\mathrm{\Gamma }}_{\text{i}}=0$ to ${\mathrm{\Gamma }}_{\text{f}}=0.30$ (left), ${\mathrm{\Gamma }}_{\text{f}}=0.40$ (middle), and ${\mathrm{\Gamma }}_{\text{f}}=0.50$ (right). All quenches are in the regular phase (${\mathrm{\Gamma }}_{\text{f}}>{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.25$) (compare Fig. 4 for quenches in the anomalous phase). Each plot shows four different system sizes, $N=200,400,600,800$, from light to dark red, with the latter achieving convergence for the results shown here. The dotted grid indicates the turning points of $\langle s_z\rangle$ in the thermodynamic limit according to (8).

Figure 4

Loschmidt-echo rate function and expectation value of the magnetization after a quench from ${\mathrm{\Gamma }}_{\text{i}}=0$ to ${\mathrm{\Gamma }}_{\text{f}}=0.10$ (left), ${\mathrm{\Gamma }}_{\text{f}}=0.15$ (middle), and ${\mathrm{\Gamma }}_{\text{f}}=0.20$ (right). All quenches are in the anomalous phase (${\mathrm{\Gamma }}_{\text{f}}<{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.25$) (compare Fig. 3 for quenches in the regular phase). Each plot shows four different system sizes, $N=200,400,600,800$, from light to dark red, with the latter achieving convergence for the results shown here. The dotted grid indicates the turning points of $\langle s_z\rangle$ in the thermodynamic limit according to (8).

Figure 5

Loschmidt-echo rate function and expectation value of the magnetization after a quench from ${\mathrm{\Gamma }}_{\text{i}}=0$ to ${\mathrm{\Gamma }}_{\text{f}}=0.23$ (top left), ${\mathrm{\Gamma }}_{\text{f}}=0.24$ (top right), ${\mathrm{\Gamma }}_{\text{f}}=0.26$ (bottom left), and ${\mathrm{\Gamma }}_{\text{f}}=0.27$ (bottom right). Quenches in the top (bottom) panels are in the anomalous (regular) phase ${\mathrm{\Gamma }}_{\text{f}}<{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.25$ (${\mathrm{\Gamma }}_{\text{f}}>{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.25$). Each plot shows four different system sizes, $N=600,800,1000,1200$, from light to dark red. The dotted grid indicates the turning points of $\langle s_z\rangle$ in the thermodynamic limit according to (8).

Figure 6

Loschmidt-echo rate function and expectation value of the magnetization after a quench from ${\mathrm{\Gamma }}_{\text{i}}=0.20$ to ${\mathrm{\Gamma }}_{\text{f}}=0.25$ (top left), ${\mathrm{\Gamma }}_{\text{f}}=0.30$ (top right), ${\mathrm{\Gamma }}_{\text{f}}=0.45$ (bottom left), and ${\mathrm{\Gamma }}_{\text{f}}=0.50$ (bottom right). Quenches in the top (bottom) panels are in the anomalous (regular) phase ${\mathrm{\Gamma }}_{\text{f}}<{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.35$ (${\mathrm{\Gamma }}_{\text{f}}>{\mathrm{\Gamma }}_{\text{c}}^{\text{d}}=0.35$). Each plot shows four different system sizes, $N=600,800,1000,1200$, from light to dark red, with the latter achieving convergence for the results shown here. The dotted grid indicates the turning points of $\langle s_z\rangle$ in the thermodynamic limit according to (8).