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arXiv:2509.14923 · Physical review. B./Physical review. B

Julia set in quantum evolution: The case of dynamical quantum phase transitions

Manmeet Kaur, Somendra M. Bhattacharjee

Published Jan 9, 2026• 1 citation
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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.

DOI
10.1103/sbdw-2ly3

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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.
This paper introduces complex dynamics methods to study dynamical quantum phase transitions in the one- and two-dimensional quantum three-state Potts model. The quench involves switching off an infinite transverse field. The time-dependent Loschmidt echo is evaluated by an exact renormalization group (RG) transformation in the complex plane where the thermal Boltzmann factor is along the positive real axis, and the quantum time evolution is along the unit circle. One of the characteristics of the complex dynamics constituted by repeated applications of RG is the Julia set, which determines the phase transitions. We show that special boundary conditions can alter the nature of the transitions, and verify the claim for the one-dimensional system by transfer matrix calculations. In two dimensions, there are alternating symmetry-breaking and restoring transitions, both of which are first order, despite the criticality of the Curie point. In addition, there are finer structures because of the fractal nature of the Julia set. Our approach can be extended to multi-variable problems, higher dimensions, and approximate RG transformations expressed as rational functions.
Dynamical quantum phase transitions (DQPTs) at critical times appear as nonanalyticities during nonequilibrium quantum real-time evolution. Although there is evidence for a close relationship between DQPTs and equilibrium phase transitions, a major challenge is still to connect to fundamental concepts such as scaling and universality. In this work, renormalization group transformations in complex parameter space are formulated for quantum quenches in Ising models showing that the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models. Therefore, these DQPTs obey scaling and universality. On the basis of numerical simulations, signatures of these DQPTs in the dynamical buildup of spin correlations are found with an associated power-law scaling determined solely by the fixed point's universality class. An outlook is given on how to explore this dynamical scaling experimentally in systems of trapped ions.
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.
Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.