Julia Set in Quantum Evolution: The case of Dynamical Quantum Phase Transitions

TL;DR

This work develops an exact analytical framework that connects dynamical quantum phase transitions (DQPTs) to complex dynamics via a real-space RG in the complex plane for the one-dimensional transverse-field Ising model. By mapping the RG flow to a rational function $R(y)=(y+1/y)/2$ and identifying its Julia set with the imaginary axis, the authors show that DQPTs correspond to intersections of the unit circle $|y|=1$ with this Julia set, yielding periodic DQPTs at times $t_c=(2n-1)\pi\hbar/2J$ for a periodic chain. The analysis also reveals a striking boundary-condition sensitivity: open chains suppress DQPTs and instead exhibit an orthogonality catastrophe at $t=\pi\hbar/J$, explained via quantum speed limits. In addition, the paper clarifies the role of zeros of the partition function, deriving their density along the imaginary axis and linking them to the Julia set through backward RG iterations. Overall, the work provides a rigorous, topology-driven perspective on non-equilibrium quantum criticality and demonstrates how boundary topology controls dynamical phases in quantum many-body systems.

Abstract

Dynamical quantum phase transitions (DQPTs) are a class of non-equilibrium phase transitions that occur in many-body quantum systems during real-time evolution, rather than through parameter tuning as in conventional phase transitions. This paper presents an exact analytical approach to studying DQPTs by combining complex dynamics with the real-space renormalization group (RG). RG transformations are interpreted as iterated maps on the complex plane, establishing a connection between DQPTs and the Julia set, the boundary separating the basins of attraction of the stable fixed points. This framework is applied to a quantum quench in the one-dimensional transverse field Ising model, where we examine the sensitivity of DQPTs to variations in boundary conditions. We show that altering the topology of the spin chain can suppress DQPTs and provide a qualitative explanation based on quantum speed limits.

Figures (6)

• Figure 1: (a) The complex $y$-plane. The positive x-axis (Re $y>0$) describes the thermal problem while the quantum evolution is along the unit circle ($|y|=1$). The basins of attraction for the RG map in the complex y-plane are colour-coded. (b) Representation on the Riemann sphere for the extended complex plane. The unit circle for quantum dynamics is the equator. The unstable fixed point at infinity is the North pole, N, (red dot), the South pole (S) being the origin. The Julia set is the dashed blue circle through the two poles with the DQPT points marked as blue dots.
• Figure 2: Schematic representation of the RG procedure: (1) coarse-graining by eliminating interior degrees of freedom, (2) adjustment of effective interactions, and (3--4) rescaling with renormalized parameters to preserve macroscopic quantities such as the partition function $Z_n(y|\alpha\beta)$, where $\alpha, \beta$ denote the states of the boundary spins at the nth generation.
• Figure 3: (a) One bond as the starting unit ($n=0$) for an open chain. (b) A triangle as the basic unit ($n=0$) for the periodic chain. The partition functions for the smallest structures are shown above the respective figures.
• Figure 4: The free energy per spin $f(t)$ for the one-dimensional transverse-field Ising model dataset. The periodic chain (red) shows nonanalytic cusps at critical times. The value of $f(t)$ at the transition point is $(\ln 2)/2$. The free energy $f(t)$ for the open chain (blue) remains smooth around transition points. The log divergence corresponds to the orthogonal catastrophe.
• Figure 5: Two geometries: (a) a periodic chain, (b) an open chain. The link connecting site 1 to site $N$ has a coupling $J_b$ while others have $J$. The open chain geometry is achieved by cutting the boundary bond or by setting $J_b=0$.