Quantum critical point
Quantum Critical Point
A **quantum critical point** (QCP) is a point at zero temperature where a continuous phase transition occurs due to quantum fluctuations. Unlike classical critical points, which are driven by thermal fluctuations, quantum critical points are governed by the principles of quantum mechanics. These points are of significant interest in condensed matter physics because they often lead to novel and exotic phases of matter.
Introduction
Quantum critical points arise in systems where the ground state of the system changes as a function of a non-thermal control parameter, such as pressure, magnetic field, or chemical composition. At a QCP, the system undergoes a quantum phase transition (QPT), which is a transition between different quantum phases at absolute zero temperature.
The study of QCPs is crucial for understanding a variety of phenomena in strongly correlated electron systems, including high-temperature superconductivity, non-Fermi liquid behavior, and heavy fermion compounds. The critical behavior near a QCP is characterized by diverging correlation lengths and timescales, leading to scale invariance and universality.
Theoretical Framework
The theoretical understanding of quantum critical points involves several advanced concepts in quantum field theory and statistical mechanics. The key idea is that near a QCP, the system can be described by a quantum field theory with a critical action that captures the essential physics of the phase transition.
Quantum Phase Transitions
Quantum phase transitions are driven by quantum fluctuations rather than thermal fluctuations. These transitions occur at absolute zero temperature and are induced by varying a non-thermal control parameter. The critical behavior near a QPT is described by critical exponents, which characterize how physical quantities diverge as the QCP is approached.
The quantum critical region, which extends to finite temperatures, exhibits unique scaling behavior. In this region, the thermal energy \( k_B T \) is comparable to the energy scale set by the quantum fluctuations, leading to a crossover from quantum to classical critical behavior.
Renormalization Group Approach
The renormalization group (RG) approach is a powerful tool for analyzing the behavior near a QCP. By systematically integrating out short-wavelength fluctuations, the RG approach allows one to derive effective theories that describe the long-wavelength behavior of the system. The fixed points of the RG flow correspond to the critical points, and the critical exponents can be determined from the scaling dimensions of the operators at these fixed points.
In the context of QCPs, the RG approach reveals that the critical behavior is governed by a quantum field theory with a specific set of symmetries and interactions. The universality class of the QCP is determined by these symmetries and interactions, leading to universal scaling laws that apply to a wide range of systems.
Experimental Realizations
Quantum critical points have been observed in a variety of experimental systems, including heavy fermion compounds, high-temperature superconductors, and low-dimensional magnets. These systems provide valuable insights into the nature of quantum criticality and the associated emergent phenomena.
Heavy Fermion Compounds
Heavy fermion compounds, such as CeCu_6 and YbRh_2Si_2, are prototypical systems for studying QCPs. These materials exhibit a rich phase diagram with multiple quantum phase transitions driven by pressure, magnetic field, or chemical doping. Near the QCP, heavy fermion compounds often display non-Fermi liquid behavior, characterized by anomalous temperature dependence of physical properties such as resistivity and specific heat.
High-Temperature Superconductors
High-temperature superconductors, such as the cuprates and iron pnictides, also exhibit quantum critical behavior. The phase diagram of these materials often features a QCP associated with the suppression of antiferromagnetic order or charge density wave order. The proximity to a QCP is believed to play a crucial role in the mechanism of high-temperature superconductivity, leading to enhanced pairing interactions and unconventional superconducting states.
Low-Dimensional Magnets
Low-dimensional magnetic systems, such as spin chains and ladders, provide another class of systems where QCPs can be studied. In these systems, quantum fluctuations are enhanced due to reduced dimensionality, leading to a variety of quantum phase transitions. For example, the transition from a gapped spin liquid state to a gapless critical state in a spin-1/2 chain is a well-known example of a QCP in a low-dimensional magnet.
Critical Behavior and Scaling
The critical behavior near a QCP is characterized by diverging correlation lengths and timescales. The correlation length \(\xi\) diverges as the control parameter \(g\) approaches the critical value \(g_c\), following a power law \(\xi \sim |g - g_c|^{-\nu}\), where \(\nu\) is the correlation length exponent. Similarly, the correlation time \(\tau\) diverges as \(\tau \sim \xi^z\), where \(z\) is the dynamical critical exponent.
The scaling behavior near a QCP can be described by a set of scaling functions that relate different physical quantities. For example, the free energy density \(f\) near a QCP follows a scaling form \(f \sim \xi^{-d} \mathcal{F}(k_B T / \Delta)\), where \(d\) is the spatial dimension, \(\Delta\) is the energy gap, and \(\mathcal{F}\) is a universal scaling function.
Quantum Criticality and Non-Fermi Liquid Behavior
One of the most intriguing aspects of quantum criticality is the emergence of non-Fermi liquid behavior. In conventional Fermi liquids, the low-energy excitations are well-described by quasiparticles with a well-defined Fermi surface. However, near a QCP, the quasiparticle picture often breaks down, leading to non-Fermi liquid behavior characterized by anomalous temperature dependence of physical properties.
For example, the electrical resistivity \(\rho(T)\) in a non-Fermi liquid typically follows a power-law dependence \(\rho(T) \sim T^n\) with \(n \neq 2\), in contrast to the \(T^2\) dependence expected in a Fermi liquid. Similarly, the specific heat \(C(T)\) often exhibits a non-trivial temperature dependence, reflecting the breakdown of the quasiparticle picture.
Emergent Phenomena
Quantum critical points often give rise to emergent phenomena that are not present in the classical phases. These emergent phenomena are a result of the collective behavior of the system near the QCP and can include novel phases of matter, such as unconventional superconductivity, spin liquids, and topological phases.
Unconventional Superconductivity
In many systems, the proximity to a QCP enhances the pairing interactions between electrons, leading to unconventional superconductivity. For example, in the cuprate superconductors, the superconducting transition temperature \(T_c\) is maximized near the QCP associated with the suppression of antiferromagnetic order. The pairing mechanism in these materials is believed to be mediated by spin fluctuations, which are enhanced near the QCP.
Spin Liquids
Spin liquids are exotic phases of matter characterized by the absence of long-range magnetic order and the presence of fractionalized excitations. These phases often arise near QCPs in frustrated magnetic systems, where the competition between different interactions prevents the system from ordering. The study of spin liquids provides valuable insights into the nature of quantum entanglement and topological order.
Topological Phases
Topological phases are characterized by non-trivial topological invariants and robust edge states. These phases can emerge near QCPs in systems with strong spin-orbit coupling and time-reversal symmetry. For example, the transition from a trivial insulator to a topological insulator can be driven by tuning a control parameter such as the strength of spin-orbit coupling. The study of topological phases has important implications for quantum computing and the development of topological quantum bits.
Conclusion
Quantum critical points represent a fascinating and rich area of research in condensed matter physics. The study of QCPs provides deep insights into the nature of quantum phase transitions, the role of quantum fluctuations, and the emergence of novel phases of matter. As experimental techniques continue to advance, the exploration of QCPs in new materials and systems is likely to yield further discoveries and deepen our understanding of quantum criticality.