[Paper Review] Signatures of a quantum phase transition on a single-mode bosonic model

Emmanouil Grigoriou, Carlos Navarrete–Benlloch|arXiv (Cornell University)|Mar 22, 2023

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper demonstrates that a single-mode bosonic Hamiltonian can exhibit a second-order quantum phase transition despite lacking spatial extension, by showing that the thermodynamic limit—defined by an infinite-dimensional Hilbert space—induces non-analytic behavior in the ground state energy. The model exhibits critical exponents and scaling laws identical to those of the quantum Rabi and Lipkin-Meshkov-Glick models, confirming its universality class, while revealing that the thermodynamic and classical limits coincide due to vanishing quantum fluctuations in the large-L limit.

ABSTRACT

Equilibrium phase transitions usually emerge from the microscopic behavior of many-body systems and are associated to interesting phenomena such as the generation of long-range order and spontaneous symmetry breaking. They can be defined through the non-analytic behavior of thermodynamic potentials in the thermodynamic limit. This limit is obtained when the number of available configurations of the system approaches infinity, which is conventionally associated to spatially-extended systems formed by an infinite number of degrees of freedom (infinite number of particles or modes). Taking previous ideas to the extreme, we argue that such a limit can be defined even in non-extended systems, providing a specific example in the simplest form of a single-mode bosonic Hamiltonian. In contrast to previous non-extended models, the simplicity of our model allows us to find approximate analytical expressions that can be confronted with precise numerical simulations in all the parameter space, particularly as close to the thermodynamic limit as we want. We are thus able to show that the system undergoes a change displaying all the characteristics of a second-order phase transition as a function of a control parameter. We derive critical exponents and scaling laws revealing the universality class of the model, which coincide with that of more elaborate non-extended models such as the quantum Rabi or Lipkin-Meshkov-Glick models. Analyzing our model, we are also able to offer insights into the features of this type of phase transitions, by showing that the thermodynamic and classical limits coincide. In other words, quantum fluctuations must be tamed in order for the system to undergo a true phase transition.

Motivation & Objective

• To investigate whether quantum phase transitions can emerge in non-extended systems, such as a single-mode bosonic system, by redefining the thermodynamic limit via infinite-dimensional Hilbert space.
• To determine if such a system displays hallmark signatures of a second-order phase transition, including non-analyticity in the ground state energy and diverging correlation length.
• To derive critical exponents and scaling laws to identify the universality class of the transition in this minimal model.
• To clarify the relationship between the thermodynamic limit and the classical limit in quantum systems, particularly in the absence of spatial extension.

Proposed method

• The study employs a single-mode bosonic Hamiltonian with a control parameter, where the thermodynamic limit is defined not by infinite particle number or system size, but by the Hilbert space dimension growing to infinity as L → ∞.
• The model is analyzed using the positive P representation, which maps the von Neumann equation to a Fokker-Planck equation in a doubled phase space with independent complex variables α and α⁺.
• Stochastic Langevin equations are derived from the Fokker-Planck form, allowing exact simulation of quantum dynamics without approximation.
• The normalized variables β = α/√L and β⁺ = α⁺/√L are introduced to analyze the L → ∞ limit, showing that noise terms vanish and the system becomes deterministic.
• Analytical approximations are derived and compared with precise numerical simulations across the full parameter space, especially near the thermodynamic limit.
• Critical exponents and scaling laws are extracted by analyzing the behavior of the ground state energy and order parameter near the critical point.

Experimental results

Research questions

• RQ1 Can a quantum phase transition occur in a non-extended system, such as a single-mode bosonic model, without spatial extension or infinite particle number?
• RQ2 Does the thermodynamic limit defined by infinite Hilbert space dimension (L → ∞) lead to non-analytic behavior in the ground state energy, signaling a phase transition?
• RQ3 What are the critical exponents and scaling laws of this transition, and how do they compare to those of established models like the quantum Rabi or Lipkin-Meshkov-Glick models?
• RQ4 Is the thermodynamic limit equivalent to the classical limit in this model, and what does this imply for the role of quantum fluctuations?

Key findings

• The single-mode bosonic model exhibits a second-order quantum phase transition as the control parameter is varied, evidenced by a non-analyticity in the ground state energy at a critical point.
• The critical exponents and scaling laws derived match exactly those of the quantum Rabi and Lipkin-Meshkov-Glick models, placing the system in the same universality class.
• In the L → ∞ limit, the stochastic noise terms in the Langevin equations vanish, leading to deterministic evolution and the preservation of coherent states.
• The thermodynamic limit and the classical limit are shown to be equivalent in this model, as quantum fluctuations are tamed in the large-L regime.
• The model allows for precise analytical approximations that are validated against numerical simulations across the entire parameter space, including near the thermodynamic limit.
• The study confirms that the essential features of phase transitions—such as long-range order and critical scaling—can emerge even in minimal, non-extended systems when the Hilbert space dimension is taken to infinity.