[Paper Review] Dynamical quantum phase transitions in strongly correlated two-dimensional spin lattices following a quench

Fredrik Brange, Sebastiano Peotta+2 more|arXiv (Cornell University)|Feb 11, 2022

Quantum many-body systems57 references16 citations

TL;DR

This paper presents a novel application of the cumulant method to detect dynamical quantum phase transitions (DQPTs) in two-dimensional strongly correlated spin lattices after a quench. By mapping the complex zeros of the Loschmidt amplitude using finite-size lattices, the authors identify critical times where DQPTs occur, demonstrating that these transitions emerge even in 2D systems with ferromagnetic or antiferromagnetic couplings, and show that initial energy fluctuations can predict DQPTs in quantum simulators.

ABSTRACT

Dynamical quantum phase transitions are at the forefront of current efforts to understand quantum matter out of equilibrium. Except for a few exactly solvable models, predictions of these critical phenomena typically rely on advanced numerical methods. However, those approaches are mostly restricted to one dimension, making investigations of two-dimensional systems highly challenging. Here, we present evidence of dynamical quantum phase transitions in strongly correlated spin lattices in two dimensions. To this end, we apply our recently developed cumulant method [Phys. Rev. X 11, 041018 (2021)] to determine the zeros of the Loschmidt amplitude in the complex plane of time and predict the crossing points of the thermodynamic lines of zeros with the real-time axis, where dynamical quantum phase transitions occur. We find the critical times of a two-dimensional quantum Ising lattice and the XYZ model with ferromagnetic or antiferromagnetic couplings. We also show how dynamical quantum phase transitions can be predicted by measuring the initial energy fluctuations, for example, in quantum simulators or other engineered quantum systems.

Motivation & Objective

• To extend the study of dynamical quantum phase transitions (DQPTs) beyond one-dimensional systems to two-dimensional strongly correlated spin lattices.
• To overcome the limitations of existing numerical methods that are restricted to one dimension by introducing a scalable approach based on the cumulant method.
• To establish a connection between DQPTs and equilibrium phase transitions by identifying thermodynamic lines of zeros in the complex time plane.
• To demonstrate that DQPTs can be predicted from initial energy fluctuations, enabling experimental detection in quantum simulators.

Proposed method

• Applies the recently developed cumulant method to compute Loschmidt amplitude zeros in the complex time plane for finite-size 2D spin lattices.
• Uses Loschmidt cumulants ⟨⟨Ĥ^n⟩⟩_τ derived from the logarithmic derivative of the Loschmidt amplitude to invert and locate the complex zeros τ_k.
• Employs the Krylov subspace method to efficiently compute high-order cumulants for small lattices (3×3 to 4×4), enabling accurate zero-finding.
• Maps the thermodynamic lines of zeros in the complex time plane and identifies their crossings with the imaginary axis as critical times for DQPTs.
• Relies on the Lee-Yang theory framework, treating time as a complex variable to locate phase transition singularities via zeros of the partition function.
• Validates predictions by comparing with known 1D results and analyzing quenches across equilibrium critical points.

Experimental results

Research questions

• RQ1 Can dynamical quantum phase transitions be reliably detected in two-dimensional strongly correlated spin systems, which are classically intractable?
• RQ2 How do the critical times of DQPTs in 2D spin lattices depend on the initial and final magnetic field strengths and exchange couplings?
• RQ3 To what extent do DQPTs in 2D systems mirror those in 1D systems, particularly in the field-dominated regime?
• RQ4 Can initial energy fluctuations in the post-quench Hamiltonian predict the occurrence of DQPTs in quantum simulators?
• RQ5 What is the relationship between the thermodynamic lines of zeros and the equilibrium phase diagram of the system?

Key findings

• Dynamical quantum phase transitions are observed in 2D quantum Ising and XYZ models with both ferromagnetic and antiferromagnetic couplings after a quench.
• Critical times for DQPTs are identified as crossings of the thermodynamic lines of zeros with the imaginary time axis, with characteristic time scales τ₀ ≈ π/J for interaction-dominated quenches and τ₀ ≈ π/h₂ for field-dominated quenches.
• For field-dominated quenches, the critical times are equidistant and match those of the 1D quantum Ising model, indicating weak dimensionality dependence.
• The method successfully predicts DQPTs using only initial energy fluctuations, enabling experimental detection in quantum simulators.
• The thermodynamic lines of zeros separate distinct dynamical regions, with some lines corresponding to phase boundaries of the associated equilibrium system.
• The approach is robust across small lattices (3×3 to 4×4), allowing accurate prediction of DQPTs without requiring large system sizes or singularities in the rate function.