Boundaries and Unphysical Fixed Points in Dynamical Quantum Phase Transitions

We show that dynamic quantum phase transitions (DQPT) in many situations involve renormalization group (RG) fixed points that are unphysical in the context of thermal phase transitions. In such cases, boundary conditions are shown to become relevant to the extent of even completely suppressing the bulk transitions. We establish these by performing an exact RG analysis of the quantum Ising model on scale-invariant lattices of different dimensions, and by analyzing the zeros of the Loschmidt amplitude. Further corroboration of boundaries affecting the bulk transition comes from the three-state quantum Potts chain, for which we also show that the DQPT corresponds to a pair of period-2 fixed points.

Figure 1

Construction of hierarchical lattices. The sites are represented by squares. Replace each bond by a motif of $b$ branches. (a) $b=1$, (b) $b=2$ (diamondlike motif), and (c) $b=3$. Three generations are shown for $b=2$.

Figure 2

Zeros of $L(y)$ in the complex-$y$ plane, and RG flows. The red circle is the unit circle (UC) for time evolution. For $b=1$, (a) only one zero at $y=-1$ for open BC, while (b) the zeros populate the imaginary axis for periodic or fixed BC. (c) For $b=2$, the zeros meet the UC at four points, $A_p$, $p=1$, 2, 3, 4. Under RG, UC flows to the positive real axis, taking each $A_p$ to $y_c=3.38298\dots$. (d) For $b=3$, the four meeting points are of two types; $A_1$, $A_2$ flow to $y_c=2.05817\dots$, while $K_1$, $K_2$ to $-y_c<0$. See Fig. 3.

Figure 3

Zeros of $L(y)$ as Julia sets in the complex-$y$ plane for (a) $b=2$ and (b) $b=3$. See Fig. 2. The zeros pinch the UC at four points. (c) For $b=3$, zoomed view of the region near $A_1$ of Fig. 2.

Figure 4

Rate function $\mathrm{Re}f(y=e^{i2Jt})$ vs $2Jt(=\theta )$. (a) $b=1$: for a chain of $N=8$ sites with boundary field $h=0.1$. $\mathrm{Re}f(y)$ from a direct time evolution (using matlab) (discs) agrees with Eq. (6) (solid green line). The $N\to \infty$ limit is shown by the blue dash-dot ($f_{+}$) and orange dashed ($f_{-}$) lines, with DQPT at $A_1$, $\theta =\pi /2$, and $A_2$, $\theta =3\pi /2$ [Fig. 2]. (b) Same as (a) but without the boundary field ($h=0$): time evolution data from matlab agree with $f_{+}$ (solid line) and with the values (green crosses) from Eq. (4b) [which fails for $\theta \in (\pi /2,3\pi /2)$]. (c) For $b=3$: $f$ (thin brown line) from Eq. (4b) for $n=8$ with $++$ spins at the two boundary points. The intersections of the horizontal line at $f_c$ (the critical value of $f$, Supplemental Material [21]) with $f$ locates the transition points $A_{1,2}$ and $K_{1,2}$ [Fig. 2]. At these points $d^3(\mathrm{Re}f)/d{t}^3$ diverges (solid blue line). Around $\theta =0$ and $\pi$, $f(y)$ matches with $f_{+}$ (magenta dash-dot) and $f_{-}$ (green dashed lines), respectively.