Antiferromagnetic Phase Transition In A 3d Fermionic Hubbard Model

The three-dimensional fermionic Hubbard model, a cornerstone of condensed matter physics, offers a simplified yet powerful framework for understanding strongly correlated electron systems. Within this model, the antiferromagnetic (AFM) phase transition stands out as a particularly intriguing phenomenon. This transition, characterized by the spontaneous ordering of electron spins in an alternating pattern, unveils the delicate interplay between kinetic energy, interaction strength, and temperature. Understanding the AFM phase transition in the 3D Hubbard model provides insights into the behavior of real materials exhibiting magnetism, superconductivity, and other exotic properties.

Delving into the Hubbard Model

Before exploring the intricacies of the AFM transition, it’s crucial to grasp the fundamental components of the Hubbard model itself. The Hubbard model is a simplified quantum mechanical model used to describe interacting electrons in a solid. It focuses on two key energy scales:

• Hopping term (t): This term describes the kinetic energy of electrons as they move between neighboring lattice sites. It essentially quantifies the ease with which electrons can delocalize and spread throughout the material.
• On-site Coulomb interaction (U): This term represents the repulsive interaction between two electrons occupying the same lattice site. It captures the energetic cost of double occupancy, reflecting the fact that electrons, being negatively charged, repel each other.

The Hamiltonian of the Hubbard model is expressed as:

H = -t Σ_⟨i,j⟩,σ (c^†_iσ c_jσ + h.c.) + U Σ_i n_i↑ n_i↓

where:

• c^†_iσ and c_iσ are creation and annihilation operators for an electron with spin σ at site i.
• ⟨i,j⟩ denotes a sum over nearest-neighbor sites.
• n_iσ is the number operator for electrons with spin σ at site i.
• h.c. stands for Hermitian conjugate.

The beauty of the Hubbard model lies in its simplicity. Despite containing only two parameters (t and U), it captures the essential physics of many-body interactions and can give rise to a wide range of emergent phenomena. The ratio U/t is a crucial parameter that governs the system's behavior. When U/t is small, the kinetic energy dominates, and the electrons behave as weakly interacting particles. However, as U/t increases, the interaction term becomes more significant, leading to strong correlations and potentially driving the system into various ordered phases.

The Antiferromagnetic Phase: Ordering of Spins

Antiferromagnetism is a type of magnetic ordering where neighboring spins align in an antiparallel fashion. This contrasts with ferromagnetism, where neighboring spins align parallel to each other. In an antiferromagnet, the net magnetic moment is typically zero because the opposing spins cancel each other out.

The emergence of antiferromagnetism in the Hubbard model is driven by a delicate balance between kinetic energy and interaction energy. In the strong interaction limit (U >> t), electrons tend to avoid double occupancy to minimize the repulsive Coulomb interaction. At half-filling (one electron per site), this leads to a situation where each site is occupied by a single electron. However, the electrons still have a residual interaction mediated by virtual hopping processes. This interaction, known as superexchange, effectively favors antiparallel alignment of neighboring spins.

Imagine two neighboring sites, each occupied by a single electron with opposite spins. One electron can virtually hop to the neighboring site, creating a transient double occupancy. This process is energetically costly due to the on-site Coulomb repulsion U. However, quantum mechanics allows this virtual hopping to occur, and it lowers the overall energy of the system if the spins are antiparallel. This is because the Pauli exclusion principle dictates that two electrons with the same spin cannot occupy the same spatial state. Therefore, the virtual hopping process is only possible if the spins are antiparallel.

The superexchange interaction can be described by an effective Hamiltonian of the form:

H_eff = J Σ_⟨i,j⟩ S_i ⋅ S_j

where:

• J is the superexchange coupling constant, which is typically proportional to t²/U.
• S_i is the spin operator at site i.

The negative sign of J indicates that the interaction favors antiparallel alignment of neighboring spins. This effective Hamiltonian is known as the Heisenberg model, which is a well-known model for describing magnetism in solids.

The Antiferromagnetic Phase Transition: From Disorder to Order

The antiferromagnetic phase transition marks the point at which the system undergoes a spontaneous transition from a disordered state to an ordered state with long-range antiferromagnetic order. This transition occurs at a critical temperature, T_N, known as the Néel temperature. Above T_N, thermal fluctuations are strong enough to disrupt the antiferromagnetic order, and the system behaves as a paramagnet, with randomly oriented spins. Below T_N, the superexchange interaction dominates, and the spins align in an antiparallel fashion, establishing long-range order.

The AFM phase transition is characterized by a diverging magnetic susceptibility at the Néel temperature. The magnetic susceptibility measures the system's response to an applied magnetic field. In the paramagnetic phase, the susceptibility is small and temperature-dependent. However, as the temperature approaches T_N from above, the susceptibility increases dramatically, reflecting the increasing tendency of the spins to align antiferromagnetically. At T_N, the susceptibility diverges, indicating the onset of long-range order.

The AFM phase transition can be either first-order or second-order, depending on the details of the system. In a first-order transition, the order parameter (in this case, the staggered magnetization, which measures the degree of antiferromagnetic order) jumps discontinuously at T_N. In a second-order transition, the order parameter increases continuously from zero below T_N.

Theoretical Approaches to Studying the AFM Transition

Studying the antiferromagnetic phase transition in the 3D Hubbard model is a challenging problem due to the strong correlations between electrons. Over the years, physicists have developed various theoretical approaches to tackle this problem. Some of the most prominent methods include:

• Mean-field theory: This is the simplest approach, where each electron is assumed to move in an average field created by all other electrons. Mean-field theory provides a qualitative understanding of the AFM transition but often overestimates the Néel temperature and fails to capture the effects of fluctuations.

• Dynamical Mean-Field Theory (DMFT): DMFT is a more sophisticated approach that treats the local correlations exactly but neglects spatial fluctuations. DMFT maps the Hubbard model onto a single-impurity Anderson model, which can be solved numerically. DMFT has been successful in describing many properties of strongly correlated materials, including the metal-insulator transition and the AFM transition.

• Quantum Monte Carlo (QMC) methods: QMC methods are numerical techniques that use stochastic sampling to solve the Schrödinger equation. QMC methods can provide accurate results for the Hubbard model, but they are computationally expensive, especially at low temperatures and for large system sizes.

• Functional Renormalization Group (FRG): FRG is a non-perturbative approach that allows for a systematic study of the effects of fluctuations on the AFM transition. FRG methods have been used to calculate the Néel temperature and the critical exponents of the AFM transition.

Each of these methods has its own strengths and weaknesses, and the choice of method depends on the specific problem being studied.

Factors Influencing the AFM Transition Temperature

The Néel temperature, T_N, is a crucial parameter that characterizes the AFM transition. Several factors can influence the value of T_N in the 3D Hubbard model:

• Interaction strength (U): Increasing the on-site Coulomb interaction U generally enhances the superexchange interaction and increases T_N. This is because a larger U makes it more energetically favorable for electrons to avoid double occupancy and align antiferromagnetically.

• Hopping parameter (t): Increasing the hopping parameter t generally reduces T_N. This is because a larger t increases the kinetic energy of the electrons, making it more difficult for them to localize and form an antiferromagnetic order.

• Doping: Doping the Hubbard model away from half-filling (i.e., adding or removing electrons) can significantly suppress the AFM order and reduce T_N. This is because doping introduces mobile charge carriers that can screen the superexchange interaction and disrupt the antiferromagnetic order.

• Lattice structure: The lattice structure can also influence the AFM transition. For example, the AFM order is generally more stable in bipartite lattices (lattices that can be divided into two sublattices, where each site on one sublattice is surrounded by sites on the other sublattice) than in non-bipartite lattices.

• Disorder: Disorder, such as impurities or vacancies, can also suppress the AFM order and reduce T_N. Disorder can disrupt the coherence of the electrons and interfere with the formation of long-range antiferromagnetic order.

Experimental Relevance and Materials Exhibiting AFM

The antiferromagnetic phase transition in the 3D Hubbard model has significant relevance to real materials. Many materials, particularly transition metal oxides, exhibit antiferromagnetism due to strong electron correlations. Some examples include:

• La₂CuO₄: This is the parent compound of the high-temperature superconducting cuprates. It exhibits strong antiferromagnetic order at low temperatures. Doping La₂CuO₄ with strontium (Sr) suppresses the AFM order and eventually leads to the emergence of superconductivity.

• MnO: This is a classic example of an antiferromagnetic material. It has a simple rock-salt crystal structure and exhibits AFM order below its Néel temperature of 118 K.

• Cr: Chromium is an element that exhibits an incommensurate spin-density wave (SDW) antiferromagnetic order below 311 K.

• Fe₂O₃ (Hematite): While complex, Hematite exhibits antiferromagnetic behavior, showcasing the relevance of the Hubbard model in understanding such materials.

The study of antiferromagnetism in these materials is crucial for understanding their electronic and magnetic properties and for developing new materials with desired functionalities. For example, antiferromagnetic materials are being explored for use in spintronics, a technology that utilizes the spin of electrons to store and process information.

Beyond the Basics: Advanced Concepts

The AFM phase transition in the 3D Hubbard model is a complex phenomenon with many subtle aspects. Here are some advanced concepts that further enrich our understanding:

• Quantum Criticality: When the Néel temperature is suppressed to zero by tuning a parameter such as doping or pressure, the system enters a quantum critical regime. In this regime, the system exhibits unusual properties, such as non-Fermi liquid behavior and unconventional superconductivity. The study of quantum criticality in the Hubbard model is an active area of research.

• Spin-Density Waves (SDWs): In some cases, the AFM order is not a simple Néel order with alternating spins on neighboring sites. Instead, the spin polarization can oscillate in space, forming a spin-density wave (SDW). SDWs are often found in metals with nesting Fermi surfaces, where the electronic structure favors the formation of a periodic spin modulation.

• Frustration: Frustration occurs when the interactions between spins cannot be simultaneously satisfied. For example, in a triangular lattice with antiferromagnetic interactions, it is impossible for all neighboring spins to be antiparallel. Frustration can lead to exotic magnetic phases, such as spin liquids, where the spins remain disordered even at very low temperatures.

• Mott Insulator: In the strong interaction limit (U >> t) at half-filling, the Hubbard model predicts the formation of a Mott insulator. In a Mott insulator, the electrons are localized due to the strong Coulomb repulsion, and the material becomes insulating even though band theory would predict it to be metallic. The AFM order is often associated with the Mott insulating state.

Future Directions and Open Questions

Despite significant progress in understanding the antiferromagnetic phase transition in the 3D Hubbard model, many open questions remain:

• Accurate determination of the phase diagram: Determining the precise phase diagram of the 3D Hubbard model, including the boundaries between the paramagnetic, antiferromagnetic, and superconducting phases, remains a challenge. More accurate numerical methods and theoretical approaches are needed to resolve the details of the phase diagram.

• Understanding the effects of disorder: The effects of disorder on the AFM transition are not fully understood. How does disorder affect the Néel temperature, the critical exponents, and the nature of the AFM order? Further research is needed to address these questions.

• Exploring the connection to superconductivity: The relationship between antiferromagnetism and superconductivity in the Hubbard model is a central question in condensed matter physics. In many materials, superconductivity emerges when the AFM order is suppressed. What is the microscopic mechanism that connects these two phenomena?

• Developing new materials with tailored magnetic properties: Understanding the fundamental principles governing the AFM transition can guide the development of new materials with tailored magnetic properties for various applications, such as spintronics and magnetic recording.

Conclusion: A Window into Complex Quantum Phenomena

The antiferromagnetic phase transition in the 3D fermionic Hubbard model provides a fascinating glimpse into the complex world of strongly correlated electron systems. This seemingly simple model captures the essential physics of magnetism and offers a powerful framework for understanding the behavior of real materials. Through theoretical and experimental investigations, we continue to unravel the mysteries of the AFM transition and its connection to other emergent phenomena, paving the way for new discoveries and technological advancements. The ongoing research into this fundamental problem promises to deepen our understanding of quantum materials and their potential for future applications.