Quantum criticality

Quantum Criticality

Quantum criticality refers to the behavior of systems at a quantum critical point (QCP), which is a zero-temperature phase transition driven by quantum fluctuations. Unlike classical phase transitions, which are driven by thermal fluctuations, quantum phase transitions occur at absolute zero temperature and are driven by non-thermal control parameters such as pressure, magnetic field, or chemical composition. Quantum criticality has profound implications for understanding various exotic phases of matter, including high-temperature superconductors, heavy fermion systems, and quantum magnets.

Quantum Phase Transitions

Quantum phase transitions occur at zero temperature when a non-thermal control parameter, such as pressure or magnetic field, is varied. These transitions are characterized by a quantum critical point, where the system undergoes a continuous phase transition. At the QCP, the system exhibits scale invariance, meaning that its properties do not depend on the length scale being observed. This scale invariance leads to unique physical phenomena that are not present in classical phase transitions.

The study of quantum phase transitions involves understanding the behavior of the system near the QCP. This includes examining the critical exponents, which describe how physical quantities diverge near the transition, and the universality class, which categorizes phase transitions based on their critical behavior.

Quantum Fluctuations

Quantum fluctuations are the temporary changes in energy that occur in a system due to the Heisenberg uncertainty principle. At zero temperature, these fluctuations become the dominant factor driving phase transitions. Unlike thermal fluctuations, which are random and due to thermal energy, quantum fluctuations are intrinsic to the quantum nature of the system.

In the vicinity of a QCP, quantum fluctuations lead to critical phenomena such as non-Fermi liquid behavior, where the traditional description of electrons in metals as Fermi liquids breaks down. This results in unusual properties like anomalous scaling of physical quantities and unconventional superconductivity.

Experimental Realizations

Several experimental systems exhibit quantum criticality, providing valuable insights into the underlying physics. Notable examples include:

**Heavy Fermion Systems**: These materials contain f-electron elements like cerium or ytterbium, which exhibit large effective electron masses. The competition between Kondo screening and Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions leads to quantum critical behavior.
**High-Temperature Superconductors**: In cuprate and iron-based superconductors, the superconducting phase is often found near a QCP, suggesting that quantum criticality plays a role in the mechanism of superconductivity.
**Quantum Magnets**: Systems like the spin-1/2 antiferromagnetic chain can be tuned to a QCP by varying an external magnetic field, leading to a quantum spin liquid state.

Theoretical Framework

The theoretical understanding of quantum criticality involves several approaches, including:

**Renormalization Group (RG) Theory**: RG techniques are used to study the behavior of systems near the QCP by examining how physical quantities change with scale. This approach helps identify the universality class of the transition.
**Dynamical Mean-Field Theory (DMFT)**: DMFT is a non-perturbative approach that treats local quantum fluctuations exactly while approximating the effects of spatial fluctuations. It is particularly useful for studying strongly correlated electron systems.
**Holographic Duality**: Inspired by string theory, holographic duality provides a powerful tool for studying quantum critical systems by mapping them to higher-dimensional gravitational theories.

Quantum Critical Metals

In quantum critical metals, the QCP separates a metallic phase from an ordered phase, such as an antiferromagnet or superconductor. Near the QCP, the electronic properties of the metal are profoundly affected by quantum fluctuations. This leads to non-Fermi liquid behavior, characterized by a breakdown of the quasiparticle description and anomalous temperature dependence of physical quantities like resistivity and specific heat.

The interplay between quantum criticality and superconductivity is of particular interest. In many high-temperature superconductors, the superconducting phase emerges in the vicinity of a QCP, suggesting that quantum critical fluctuations may mediate the pairing mechanism.

Quantum Critical Insulators

Quantum criticality is not limited to metallic systems; it also occurs in insulating systems. For example, in quantum magnets, a QCP can separate different magnetic orders or a magnetic order from a quantum spin liquid state. In these systems, the critical behavior is governed by the interplay between spin interactions and quantum fluctuations.

Quantum critical insulators exhibit unique properties such as fractionalized excitations and topological order. These exotic phases of matter are of great interest for potential applications in quantum computing and other advanced technologies.

Quantum Criticality in Two Dimensions

Two-dimensional (2D) systems provide a fertile ground for studying quantum criticality due to enhanced quantum fluctuations and reduced dimensionality. In 2D systems, the role of quantum fluctuations is more pronounced, leading to rich and complex phase diagrams.

Notable examples of 2D quantum critical systems include:

**Graphene**: The electronic properties of graphene can be tuned to a QCP by varying the carrier density, leading to phenomena such as the quantum Hall effect and Dirac fermions.
**Transition Metal Dichalcogenides (TMDs)**: These materials exhibit a variety of quantum phases, including charge density waves, superconductivity, and Mott insulators, which can be tuned to a QCP by external parameters.

Quantum Criticality and Entanglement

Quantum criticality is closely related to quantum entanglement, a fundamental feature of quantum mechanics where the state of one particle is dependent on the state of another, regardless of the distance between them. Near a QCP, the entanglement entropy, which measures the degree of entanglement between different parts of the system, exhibits scaling behavior that reflects the critical properties of the system.

The study of entanglement in quantum critical systems provides deep insights into the nature of quantum correlations and the structure of the quantum state. It also has implications for quantum information theory and the development of quantum technologies.

Applications and Implications

Understanding quantum criticality has significant implications for various fields of physics and technology. Some potential applications include:

**Quantum Materials**: The design and discovery of new quantum materials with tailored properties for applications in electronics, spintronics, and quantum computing.
**High-Temperature Superconductors**: Insights into the mechanism of high-temperature superconductivity, which could lead to the development of more efficient superconductors for power transmission and other applications.
**Quantum Simulators**: The use of quantum critical systems as simulators for studying complex quantum phenomena that are difficult to investigate experimentally or theoretically.