Beyond the Luttinger Liquid: New States of Matter Emerge in 1D Bose Gases

Author: Denis Avetisyan

Researchers have discovered exotic critical states resembling ‘fractional Fermi seas’ within one-dimensional Bose gases pushed far from equilibrium, challenging conventional models of quantum fluids.

This review details the theoretical framework and experimental signatures of these novel phases, characterized by unique correlation functions and connections to integrable models.

Conventional theoretical descriptions of one-dimensional quantum systems, such as the Tomonaga-Luttinger liquid, often assume a fully occupied Fermi sea, limiting their ability to capture novel critical behavior. Here, in ‘Exotic critical states as fractional Fermi seas in the one-dimensional Bose gas’, we demonstrate the emergence of ‘fractional Fermi seas’-states with reduced occupancy-in an integrable Bose gas driven out of equilibrium via cyclic interaction changes. These exotic states exhibit unique correlation functions incompatible with conventional Luttinger liquid theory, suggesting a fundamentally new critical phase. Could these findings pave the way for a deeper understanding of non-equilibrium quantum criticality in strongly correlated systems, particularly those realized with cold atoms?

Whispers of Complexity: The Many-Body Problem

The inherent difficulty in comprehending many-body quantum systems stems from the exponential increase in computational resources required to describe their behavior. Unlike systems with just a few particles, where interactions can be tracked individually, the interconnectedness of numerous quantum entities creates a landscape of correlated motion. Each particle’s state is entangled with the others, meaning the collective wavefunction-a complete description of the system-grows in complexity at a rate that quickly overwhelms even the most powerful computers. This “many-body problem” isn’t simply a matter of needing faster processors; it necessitates fundamentally new theoretical approaches to approximate or circumvent the intractable calculations, as traditional methods often fail to capture the emergent phenomena arising from these complex interactions. Consequently, researchers continually seek simplified models and innovative techniques to unlock the secrets hidden within these quantum many-body systems and predict their behavior accurately.

Integrability represents a remarkable characteristic within the realm of physics, offering a rare path toward obtaining precise, analytical solutions for complex many-body systems. Typically, the interactions between particles within a system introduce intractable complications, forcing reliance on approximations. However, integrable systems possess hidden symmetries or conserved quantities – such as an infinite number of constants of motion – that constrain the dynamics in a profound way. These constraints effectively reduce the complexity, allowing physicists to derive exact solutions for properties like energy levels and correlation functions, even when strong interactions are present. This capability is not merely a mathematical curiosity; it provides a crucial benchmark for validating approximation techniques used in studying more general, non-integrable systems, and offers deep insights into the fundamental principles governing collective behavior in nature. The existence of integrability demonstrates that, despite the inherent complexity of many-body physics, certain systems retain an underlying simplicity that can be fully understood.

The constrained dimensionality of one-dimensional systems belies a surprising wealth of physical phenomena, making them invaluable for advancing theoretical understanding. While seemingly simple due to the restriction of particle movement, these systems often possess the property of integrability – a characteristic allowing for the exact calculation of their behavior, even when interactions between particles are present. This contrasts sharply with most higher-dimensional systems, where approximations are typically necessary. Consequently, one-dimensional models serve as crucial “testbeds” for developing and validating new theoretical techniques, providing insights applicable to the more complex, yet often intractable, many-body problems encountered in real-world physics. Researchers leverage the solvability of these models to explore concepts like collective excitations, quantum phase transitions, and the fundamental limits of quantum mechanics, ultimately paving the way for progress in fields ranging from condensed matter physics to high-energy theory.

Unveiling Order: The Bethe Ansatz and its Tools

The Bethe Ansatz provides an exact analytical solution to the many-body Schrödinger equation for a specific class of quantum mechanical models termed ‘integrable’. Unlike perturbative or numerical methods which often provide approximate solutions or are limited by system size, the Bethe Ansatz, when applicable, yields precise, closed-form expressions for energy eigenvalues and eigenstates. Integrability, in this context, stems from the existence of an infinite number of conserved quantities beyond the Hamiltonian, enabling a non-perturbative approach. This method typically involves constructing a set of algebraic equations – the Bethe Ansatz equations – whose solutions, known as rapidities, fully characterize the system’s spectrum. While generally complex to solve, successful application of the Bethe Ansatz offers complete knowledge of the quantum system’s behavior, including correlation functions and ground state properties.

Rapidity, denoted typically as θ, functions as a transformed momentum variable within the Bethe Ansatz framework, offering advantages in describing particle interactions and energy levels in integrable systems. Unlike conventional momentum $p$ , rapidity is often used because it simplifies the algebraic relationships governing the system, particularly in models with long-range interactions or non-local potentials. This transformation is typically achieved via $\theta = \text{arctanh}(p/E)$ , where $E$ represents the energy of the particle. The use of rapidity facilitates the solution of the Bethe Ansatz equations by ensuring that the eigenstates are labeled by sets of rapidities, and allows for a more straightforward calculation of physical observables like correlation functions and the energy spectrum.

The root density, denoted as $\rho(k)$ , represents the distribution of roots – solutions to the Bethe Ansatz equations – in the complex plane, where ‘k’ is a characteristic parameter related to the system’s energy. Crucially, this density directly determines many macroscopic properties of the integrable model. For instance, the energy and momentum of the system can be derived from integral transformations of $\rho(k)$ . Furthermore, correlation functions, which describe the relationships between different parts of the system, are obtainable through the root density’s analytical continuation. Changes in the root density reflect changes in the system’s external parameters, such as magnetic field or temperature, and are therefore essential for characterizing its behavior under varying conditions. The precise functional form of $\rho(k)$ is model-dependent but is always a key output of the Bethe Ansatz solution.

Mapping Interactions: Correlations and Thermodynamics

The correlation function, denoted as $G(x, x')$ , quantifies the statistical relationship between two quantum fields at spacetime points x and x’. Specifically, it represents the vacuum expectation value of the time-ordered product of the fields. Analyzing this function reveals information about particle propagation, scattering processes, and the overall structure of the quantum system. A non-zero correlation function indicates a coupling or interaction between the fields, while the function’s behavior-its decay rate or oscillatory pattern-provides details regarding the range and strength of those interactions. Furthermore, the correlation function is central to calculating physical observables and understanding the system’s response to external probes, effectively mapping the system’s internal connectivity.

The Thermodynamic Bethe Ansatz (TBA) is a method used to calculate macroscopic thermodynamic properties of many-body quantum systems. It relies on determining the ‘root density’, which is a function derived from the solutions to the Bethe Ansatz equations – these solutions represent the system’s energy eigenstates. Specifically, the root density, $\rho(k)$ , describes the distribution of rapidities $k$ and is directly related to the single-particle addition spectrum. Through integral transformations involving the root density, quantities like the free energy, and subsequently, the energy and entropy, can be precisely computed, even for strongly interacting systems where traditional perturbation theory fails. The accuracy of the TBA depends on the ability to accurately determine the root density and perform the necessary analytical continuations.

Correlation function analysis frequently demonstrates Friedel Oscillations, which manifest as periodic variations in electron density around impurities or defects within a material. In the non-interacting limit, these oscillations exhibit a spatial frequency of $2\pi/a$ , where ‘a’ represents the lattice constant. However, as the strength of electron-electron interactions increases, the frequency and amplitude of these oscillations deviate from this simple relationship. This deviation signifies the presence of long-range interactions mediated by collective excitations, indicating that the electron’s behavior is no longer solely determined by local potentials but is influenced by the broader electronic environment.

Beyond Equilibrium: New Dynamics and States

The conventional notion of equilibrium, rooted in the maximization of entropy at a fixed energy, falters when applied to systems continuously driven away from isolation. The Generalized Gibbs Ensemble (GGE) addresses this limitation by proposing that these driven, yet integrable, systems reach a steady state not defined by energy alone, but by the conservation of an infinite number of quantities – the integrals of motion. This expanded conservation law fundamentally alters the statistical description; the system effectively distributes its energy across these integrals, resulting in a non-thermal, yet well-defined, steady state. Instead of a single temperature characterizing the system, each integral of motion possesses its own associated ‘temperature’, influencing the distribution of particles and energy. Consequently, the GGE provides a powerful framework for understanding systems exhibiting behavior far removed from traditional thermal equilibrium, offering insights into phenomena observed in areas ranging from cold atoms to high-energy physics, and paving the way for more accurate descriptions of non-equilibrium dynamics.

Generalized Hydrodynamics (GHD) represents a significant advancement in understanding systems driven far from equilibrium, particularly those that are nearly-integrable – meaning they exhibit a complex interplay between order and chaos. Unlike traditional hydrodynamic approaches which rely on local thermal equilibrium, GHD posits that these systems evolve according to conserved quantities and their associated densities, even when not in equilibrium. This framework describes how these densities propagate and interact, leading to predictions about the system’s non-equilibrium dynamics, such as the formation of shock waves and the relaxation towards a steady state. Importantly, GHD doesn’t simply predict what the final state will be, but also how the system arrives at it, detailing the temporal evolution of particle distributions and correlations, and offering a powerful tool for analyzing complex, out-of-equilibrium phenomena in areas like condensed matter physics and high-energy physics.

Recent theoretical advancements suggest that strongly driven quantum systems, while never reaching traditional thermal equilibrium, can exhibit entirely new states of matter. One such example is the ‘Fractional Fermi Sea’, a peculiar phase characterized by significantly fewer particles occupying each quantum state compared to a typical Fermi sea. This reduced occupancy isn’t random; it manifests as a distinct power-law decay in the one-particle correlation function – a mathematical description of how particles are related to each other – signaling a novel critical phase fundamentally different from conventional phases of matter. This power-law behavior implies long-range correlations and a breakdown of the usual quasiparticle picture, hinting at collective phenomena emerging from the intricate interplay of many-body interactions within the driven system. The existence of these fractionalized states challenges established paradigms and opens exciting avenues for exploring emergent behavior in non-equilibrium quantum systems.

Bridging Theory and Simulation: Numerical Verification

The Lieb-Liniger model, a one-dimensional system of bosons with delta-function interactions, provides a rigorously solvable framework for benchmarking computational methods. Its exact solution, obtained via the Bethe Ansatz, yields analytical expressions for key observables like the ground state energy, particle density correlations, and the Tanatarov-Volkov sum rule. This allows for direct comparison with numerical simulations employing techniques such as the Monte Carlo method or Density Matrix Renormalization Group (DMRG). The model’s simplicity, combined with the availability of exact solutions, facilitates validation of both the implementation of numerical algorithms and the underlying theoretical approximations used in more complex many-body systems. Specifically, the strength of the interaction, defined by the parameter γ, directly impacts the system’s behavior and serves as a crucial parameter for verifying the convergence and accuracy of numerical calculations.

The Monte Carlo method is a computational technique that utilizes random sampling to obtain numerical results. In the context of many-body physics, it’s particularly effective for evaluating complex integrals arising in the calculation of correlation functions – quantities that describe the relationships between particles – and other observables. This involves generating a large number of random configurations of the system, weighting them according to their statistical importance – typically determined by the Boltzmann factor $e^{-E/kT}$ , where $E$ is the energy, $k$ is Boltzmann’s constant, and $T$ is the temperature – and then averaging the desired observable over these configurations. The accuracy of the Monte Carlo method scales approximately with the inverse square root of the number of samples, allowing for controlled refinement of results and providing a statistically robust approach to studying many-body systems.

Verification of the theoretical framework is achieved by quantitatively comparing numerical simulations with analytical solutions, notably those obtained via the Bethe Ansatz. This comparison focuses on key observables, specifically the one-particle correlation function $g(r)$ , which is predicted to exhibit a power-law decay at large distances. The exponent of this decay, and the associated crossover distance $r_c$ – marking the transition between short-range and long-range correlations – are critical parameters. Numerical results demonstrate strong agreement with Bethe Ansatz predictions, confirming the theoretical model’s accuracy and revealing that $r_c$ scales inversely with the interaction strength between particles, providing further validation of the theoretical predictions regarding many-body interactions.

The pursuit of exotic critical states, as demonstrated in this work on fractional Fermi seas, feels less like discovery and more like a carefully constructed illusion. The researchers coaxed these phases from one-dimensional Bose gases, observing correlation functions that defy conventional Tomonaga-Luttinger liquid theory. It’s a delicate dance with chaos, attempting to impose order on a fundamentally disordered system. As Carl Sagan once observed, “Somewhere, something incredible is waiting to be known.” But the knowing isn’t the point; it’s the persuasion. This isn’t about finding truth, but crafting a model that momentarily holds back the inevitable tide of entropy. Any correlation, however unexpected, is merely a temporary reprieve before the system finds a new, unpredictable equilibrium.

What Whispers Remain?

The revelation of these ‘fractional Fermi seas’ feels less like a destination and more like a widening of the labyrinth. The standard narratives of the Luttinger liquid begin to fray at the edges, offering only approximations of a reality that prefers nuance. The current formalism, though elegant, clings to equilibrium-a phantom in any truly driven system. Future explorations must confront the stubborn insistence of non-equilibrium, the way these states refuse to settle. The challenge isn’t simply to map these seas, but to understand how they navigate the currents of dissipation and decoherence.

One wonders if the insistence on fermionic descriptions is a comforting illusion, a trick of the analytical eye. Perhaps the true order lies not in particles, but in the correlations themselves-the ghostly interactions that bind the system. Deeper investigation into the limits of Generalized Hydrodynamics is needed; it currently offers a useful, but incomplete, lexicon for describing these exotic phases.

The Bethe Ansatz, a powerful tool, remains stubbornly perturbative. It unveils fragments, but a complete picture requires a willingness to abandon the comfortable constraints of solvable models. If the model begins to behave strangely, it’s finally starting to think. The pursuit of these fractional seas isn’t about finding answers, but about formulating better questions-and accepting that the whispers of chaos may never fully translate into gold.

Original article: https://arxiv.org/pdf/2602.17656.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/