🔬 condensed matter

Dynamic quantum phase transitions in the two-leg Creutz ladder with long-range hopping

This paper investigates dynamic quantum phase transitions in a two-leg Creutz ladder with long-range hopping by deriving exact solutions for the dynamical free energy and demonstrating that the density of critical times increases with the hopping range and decreases with the power-law decay exponent.

Original authors: J. A. da Silva, J. C. Xavier

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J. A. da Silva, J. C. Xavier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Physics often seeks to understand how matter organizes itself into distinct states, much like water freezing into ice or a magnet suddenly aligning its internal forces. These changes, known as phase transitions, usually happen when a system is slowly cooled or heated until it reaches a tipping point. For decades, scientists have mastered the rules of these transitions when a system sits quietly in equilibrium, waiting for the temperature to shift. However, a newer and more frantic frontier explores what happens when a system is jolted out of balance. Imagine a calm lake suddenly struck by a massive boulder; the ripples that follow are not just noise, but a complex rearrangement of energy that can reveal hidden properties of the water itself. In the quantum world, where particles behave like both waves and solid objects, these sudden jolts are called quantum quenches. When a parameter of a quantum system is changed instantly, the system does not simply settle into a new state; it oscillates, and at specific moments in time, its behavior can become mathematically sharp and unpredictable. These moments are called dynamical quantum phase transitions, and they represent a new kind of order that exists only in the flow of time, not in the temperature of the room.

Researchers at the Federal University of Uberlândia in Brazil have taken a deep dive into this phenomenon using a specific theoretical model known as the Creutz ladder. This model describes a simplified world where particles move along two parallel tracks, or legs, connected by rungs, all under the influence of a magnetic field. While previous studies looked at how particles jump only to their immediate neighbors, this new work investigates what happens when particles can leap much farther, skipping over several rungs at once. The strength of these long-distance jumps is not constant; it fades as the distance increases, following a specific mathematical rule where the jump probability drops off like a fading echo. The researchers wanted to know if allowing these long-range leaps would fundamentally change how the system reacts to a sudden jolt, specifically whether it would create more of those sharp, unpredictable moments in time.

To answer this, the team first developed a general mathematical framework that could solve any two-track system of this type exactly. They treated the system as a collection of waves moving through momentum space, a way of describing the system based on how its parts move rather than where they sit. By solving the equations for this generic setup, they derived a precise formula for the "Loschmidt amplitude," a quantity that measures how much the system remembers its original state after being jolted. From this, they calculated the dynamical free energy, a value that acts like a thermometer for the system's memory. When this value develops a sharp corner or a cusp, it signals that a dynamical phase transition has occurred. The researchers showed that these sharp moments happen only when specific conditions are met by the system's internal waves, conditions that depend on the strength of the magnetic field and the range of the particle jumps.

Applying this framework to the Creutz ladder, the team discovered that the length of the jump range and the rate at which the jump strength fades are critical factors. When the jumps are short and the strength fades quickly, the system behaves in a familiar way: sharp transitions occur only when the sudden change in the magnetic field crosses a specific equilibrium boundary. However, the story changes dramatically when the particles can jump across long distances and the fading of their strength is slow. In these cases, the researchers found that the system can undergo these sharp transitions even when the magnetic field change does not cross any traditional boundary. The number of these transition moments increases as the jump range grows and the fading slows down. In the most extreme scenarios, where particles can jump very far and the strength fades very slowly, these transition moments become so numerous and close together that they appear almost continuously, creating a dense forest of sharp changes in the system's behavior over time.

The study confirms that long-range interactions do not merely add complexity to the system; they qualitatively alter the relationship between the system's static properties and its dynamic response. The researchers demonstrated that the emergence of these densely packed transition times is not a fluke of a single model but likely a general feature of systems with long-range hopping, echoing similar findings in other theoretical chains. While these results are currently based on exact mathematical simulations rather than physical experiments, the Creutz ladder has already been realized in optical lattices in laboratories. This suggests that the dense, continuous-like behavior predicted by the team could soon be observed in real experiments, offering a new window into how quantum matter behaves when pushed to the edge of stability. The work provides a clear roadmap for understanding how the ability of particles to reach across space reshapes the very nature of time-dependent phase transitions.

Technical Summary: Dynamic Quantum Phase Transitions in the Two-Leg Creutz Ladder with Long-Range Hopping

Problem Statement
This work investigates the effects of long-range hopping on Dynamical Quantum Phase Transitions (DQPTs) within the two-leg Creutz ladder model. While DQPTs—characterized by non-analyticities in the dynamical free energy at critical times following a quantum quench—have been extensively studied in short-range systems, the specific influence of long-range interactions (where hopping amplitudes decay as a power law $1/\ell^\nu$ with range $D$ ) remains less understood. Previous studies on the Su-Schrieffer-Heeger (SSH) model with long-range hopping suggested that increasing the hopping range $D$ and decreasing the decay exponent $\nu$ could lead to a dense set of critical times, potentially causing non-analyticities at almost all times. The authors aim to determine if this phenomenon is a generic feature of long-range hopping systems or specific to the SSH model, using the exactly solvable Creutz model as a testbed.

Methodology
The authors employ an analytical approach based on the exact diagonalization of a generic two-band free-fermion model in momentum space.

Generic Two-Band Framework: They first derive the exact solution for a generic Hamiltonian $H(\theta)$ defined by a $2\times2$ matrix $M(k, \theta)$ . They obtain closed-form expressions for the eigenenergies $\omega_{\pm}^k$ and the eigenoperators.
Loschmidt Amplitude and Free Energy: Using a quantum quench protocol where a parameter $\theta$ changes instantaneously from $\theta$ to $\tilde{\theta}$ , they calculate the Loschmidt amplitude $Z(t) = \langle \psi_0(\theta) | e^{-iH(\tilde{\theta})t} | \psi_0(\theta) \rangle$ . From this, they derive the dynamical free energy $f(t) = -\frac{1}{N} \ln |Z(t)|^2$ .
Yang-Lee-Fisher (YLF) Zeros: The non-analyticities in $f(t)$ are identified by locating the YLF zeros in the complex time plane. The authors show that these zeros reach the real-time axis (signaling a DQPT) only when a specific condition involving the overlap of pre- and post-quench Bloch vectors is met, leading to a nonlinear equation for the critical momenta $k^*$ .
Application to Creutz Model: They apply this framework to the two-leg Creutz ladder with long-range hopping in the presence of a uniform magnetic field. The hopping amplitudes decay as $t_{\ell} \propto \ell^{-\nu}$ up to a finite range $D$ . The magnetic field is incorporated via the Peierls substitution, introducing a phase factor $\theta$ dependent on the magnetic flux.

Key Contributions and Results

Exact Analytical Formulation: The paper provides a general analytical framework for calculating DQPTs in any two-band model, explicitly linking critical times to the eigenenergies and the geometric overlap of the initial and final states.
Critical Time Density: The central finding is that the number of nontrivial critical times $N_s$ $N_{s}$ depends critically on the hopping range $D$ $D$ and the decay exponent $\nu$ $ν$ .
- For the nearest-neighbor case ( $D=1$ ), DQPTs occur primarily when the quench crosses an equilibrium critical point (e.g., $\theta_{eq}^c = 0$ ).
- For long-range hopping ( $D > 1$ ) with small $\nu$ (slow decay), the number of solutions to the critical momentum equation increases significantly.
Emergence of Dense Critical Times: In the regime of large $D$ and small $\nu$ , the critical times become increasingly dense. The authors demonstrate that for specific parameters, this leads to a situation where non-analyticities appear at an increasingly dense set of times, mirroring observations in the long-range SSH model.
Decoupling from Equilibrium Criticality: Unlike the $D=1$ case, the study reveals that for $D > 1$ , DQPTs can occur even when the quench does not cross an equilibrium critical point. The region of parameter space where DQPTs occur does not necessarily coincide with the equilibrium phase transition region.
Finite-Size Considerations: The authors note that while non-analyticities are strictly defined in the thermodynamic limit, finite-size systems may exhibit smeared local maxima that do not correspond to true non-analyticities, highlighting the difficulty of detecting these features experimentally without theoretical guidance.

Significance and Claims
The authors claim that their results demonstrate that the emergence of densely distributed critical times is likely a general characteristic of models with long-range hopping, rather than a feature unique to the SSH model. By showing that long-range hopping can qualitatively modify the relationship between equilibrium and non-equilibrium critical behavior (specifically, allowing DQPTs without crossing equilibrium critical points), the work provides a deeper theoretical understanding of non-equilibrium dynamics in long-range systems.

Given that the Creutz model has recently been realized experimentally in one-dimensional optical lattices, the authors suggest their findings may guide future experimental studies on DQPTs and the effects of long-range hopping. However, they remain modest regarding immediate experimental verification, noting the challenge of resolving true non-analyticities in finite-size experimental setups. The paper concludes that the derived analytical expressions offer a robust tool for predicting and interpreting dynamical critical behavior in these systems.

Technical Summary: Dynamic Quantum Phase Transitions in the Two-Leg Creutz Ladder with Long-Range Hopping

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