Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines

Using the framework of infinite matrix product states, the existence of an anomalous dynamical phase for the transverse-field Ising chain with sufficiently long-range interactions was first reported in J. C. Halimeh and V. Zauner-Stauber [Phys. Rev. B 96, 134427 (2017)], where it was shown that anomalous cusps arise in the Loschmidt-echo return rate for sufficiently small quenches within the ferromagnetic phase. In this work we further probe the nature of the anomalous phase through calculating the corresponding Fisher-zero lines in the complex time plane. We find that these Fisher-zero lines exhibit a qualitative difference in their behavior, where, unlike in the case of the regular phase, some of them terminate before intersecting the imaginary axis, indicating the existence of smooth peaks in the return rate preceding the cusps. Additionally, we discuss in detail the infinite matrix product state time-evolution method used to calculate Fisher zeros and the Loschmidt-echo return rate using the matrix product state transfer matrix. Our work sheds further light on the nature of the anomalous phase in the long-range transverse-field Ising chain, while the numerical treatment presented can be applied to more general quantum spin chains.

Figure 1

Fixed-point relations for the left and right MPO transfer matrices (7) and definition of the effective Hamiltonians $H$ for $A_C^s$ and $K$ for $C$. The symbol $\widehat{=}$ refers to equality up to terms contributing to the energy expectation value per site [35] (for more details, see also Appendix C.2 in Ref. [36]).

Figure 2

Example of how cusps in the return rate $r\left(t\right)$ can be detected through level crossings of the rate-function branches $r_i\left(t\right)$ (11). Shown are the first four $r_i\left(t\right)$ for a quench from $h_{\mathrm{i}}=0$ to $h_{\mathrm{f}}=1.3$ and $\alpha =2.2$, where the lowest branch represents the rate function (9). At $t\approx 2.4$ there are no level crossings; $r\left(t\right)$ is thus smooth. At times $t\approx 4$ and $t\approx 5.5$ there are level crossings in the rate-function branches, which cause cusps in $r\left(t\right)$. The example shows anomalous cusps; however, regular cusps (or in fact any other type thereof) show up due to the same mechanism. The insets show a magnification of these cusps, where branches that are going up are colored red, and branches coming down are colored blue for better visualization.

Figure 3

Fisher-zero lines for $\alpha =2.2$ and $h_{\mathrm{i}}=0$ in the complex $z=R+it$ plane. We show data for quenches to (a) $h_{\mathrm{f}}=1.5<h_{\mathrm{c},\mathrm{z}}^{\mathrm{II}}$ in the anomalous phase and (b) $h_{\mathrm{f}}=2>h_{\mathrm{c},\mathrm{z}}^{\mathrm{II}}$ in the regular phase. It can be seen that in (a) the lowest FZL cutting the imaginary axis ($R=0$) terminates around $R\approx 0.6$ (this process is depicted in Fig. 4). With decreasing $h_{\mathrm{f}}$, this end point (along with the end points of all higher FZLs cutting the imaginary axis) moves to smaller and smaller $R$, until the end points cross the imaginary axis and the cusps in $r\left(t\right)$ vanish one by one, starting at early times $t$. The black dots represent data points; the blue lines are guides for the eye. Numerical data were not sufficient for display in the top left grayed-out areas in both (a) and (b).

Figure 4

Plots of the rate-function branches $r_i\left(z\right)$ with complex argument $z=R+it$ vs real time $t$ for four different (fixed) values of $R=\{0.4,0.6,0.64,0.68\}$, showing the mechanism responsible for the termination of FZLs in the complex plane in the anomalous phase (here, in particular, for the lowest FZL in Fig. 3). It can be seen that the crossing of the two lowest branches present in the top two panels has disappeared with increasing $R$ in the bottom two panels. Notice that this is not due to one branch moving up such that they no longer cross, but by a disconnecting process that roughly leaves the branches in their general position, but turns the crossing more and more into an avoided crossing. This feature is characteristic for the disappearance of anomalous cusps, whereas regular cusps always vanish due to relative movement of the branches only.

Figure 5

Examples of double cusps for quenches within the anomalous phase. Here for $\alpha =2$, the third and fifth cusps are double cusps; for $\alpha =2.2$ the fourth and sixth cusps are double cusps.

Figure 6

Example of rate-function branches $r_i\left(t\right)$ responsible for developing regular cusps (red) coming down and taking over branches responsible for developing anomalous cusps (blue) with increasing $h_{\mathrm{f}}$ (here $h_{\mathrm{c},\mathrm{z}}^{\mathrm{II}}=1.85$).