11.5 Quantum phase transitions
7 min read•august 21, 2024
Quantum phase transitions occur at absolute zero, driven by quantum fluctuations rather than thermal ones. These transitions provide insights into exotic quantum states and critical phenomena in condensed matter systems, crucial for developing novel quantum materials and technologies.
Understanding quantum phase transitions involves studying order parameters, critical exponents, and universality classes. These concepts help explain the scaling behavior and universality of different systems near quantum critical points, where quantum fluctuations dominate and lead to unique phenomena.
Fundamentals of quantum phase transitions
- Quantum phase transitions occur at absolute zero temperature driven by quantum fluctuations rather than thermal fluctuations
- Studying quantum phase transitions provides insights into exotic quantum states of matter and critical phenomena in condensed matter systems
- Understanding quantum phase transitions is crucial for developing novel quantum materials and technologies in condensed matter physics
Definition and characteristics
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- Quantum phase transitions involve abrupt changes in the ground state of a quantum system at zero temperature
- Triggered by varying a non-thermal parameter such as pressure, magnetic field, or chemical composition
- Characterized by long-range quantum correlations and entanglement between particles in the system
- Exhibit quantum critical points where the system becomes scale-invariant
Quantum vs classical phase transitions
- Classical phase transitions driven by thermal fluctuations (temperature changes)
- Quantum phase transitions governed by quantum fluctuations (zero-point motion)
- Quantum transitions can occur even at absolute zero temperature
- Quantum critical behavior extends to finite temperatures in a quantum critical fan
Role of quantum fluctuations
- Quantum fluctuations arise from Heisenberg's uncertainty principle
- Drive the system between different quantum ground states
- Become dominant near the , leading to novel quantum phenomena
- Can induce long-range entanglement and non-local correlations in the system
Order parameters and critical exponents
- Order parameters quantify the degree of order in a system and characterize different phases
- Critical exponents describe how physical quantities scale near the quantum critical point
- These concepts are essential for understanding universality and scaling behavior in quantum phase transitions
Quantum critical points
- Special points in the phase diagram where quantum fluctuations dominate
- Occur at zero temperature when a non-thermal parameter is tuned to a critical value
- Exhibit diverging correlation length and vanishing energy gap
- Lead to the emergence of scale-invariant quantum critical behavior
Universality classes
- Groups of seemingly different systems that share the same critical behavior
- Determined by symmetries, dimensionality, and nature of interactions
- Examples include Ising, XY, and Heisenberg universality classes
- Allow for the classification and prediction of critical phenomena across various quantum systems
Scaling behavior
- Physical quantities near the quantum critical point follow power-law scaling relations
- Characterized by critical exponents that are universal within a given universality class
- Scaling functions describe the behavior of observables in the quantum critical region
- techniques used to derive scaling relations and critical exponents
Types of quantum phase transitions
- Quantum phase transitions can be classified based on their nature and underlying mechanisms
- Understanding different types of transitions is crucial for predicting and controlling quantum critical behavior
- Various quantum phase transitions can lead to the emergence of exotic quantum states of matter
Continuous vs discontinuous transitions
- Continuous transitions (second-order) exhibit smooth changes in
- Discontinuous transitions (first-order) involve abrupt jumps in order parameter
- Continuous transitions characterized by diverging correlation length and critical fluctuations
- Discontinuous transitions may exhibit phase coexistence and hysteresis
Symmetry-breaking transitions
- Involve spontaneous breaking of symmetry in the ground state
- Examples include ferromagnetic transition (breaks rotational symmetry)
- Antiferromagnetic transition (breaks translational symmetry)
- Characterized by the emergence of long-range order and Goldstone modes
Topological transitions
- Involve changes in topological properties of the system without breaking any symmetries
- Examples include quantum Hall transitions and topological insulator transitions
- Characterized by changes in topological invariants (Chern numbers)
- Often associated with the emergence of protected edge or surface states
Quantum critical phenomena
- Quantum critical phenomena emerge in the vicinity of quantum critical points
- Studying these phenomena provides insights into exotic quantum states and non-Fermi liquid behavior
- Understanding quantum critical phenomena is crucial for developing novel quantum materials and technologies
Quantum critical region
- Fan-shaped region in the phase diagram extending from the quantum critical point
- Characterized by strong quantum fluctuations and absence of well-defined quasiparticles
- Exhibits universal scaling behavior and anomalous transport properties
- Bounded by crossover lines separating it from conventional phases
Finite temperature effects
- Quantum critical behavior persists at finite temperatures within the quantum critical fan
- Interplay between thermal and quantum fluctuations leads to novel phenomena
- Temperature acts as a relevant perturbation, introducing a new energy scale
- Quantum-to-classical crossover occurs as temperature increases
Dynamical scaling
- Describes how correlation functions scale in both space and time near the quantum critical point
- Characterized by dynamic critical exponent z relating spatial and temporal correlations
- Leads to the concept of effective dimensionality d_eff = d + z in quantum systems
- Important for understanding transport properties and relaxation dynamics in quantum critical systems
Experimental observations
- Experimental studies of quantum phase transitions provide crucial insights into quantum critical phenomena
- Observations in real materials help validate theoretical predictions and uncover new physics
- Challenges in experimental detection drive the development of novel measurement techniques
Materials exhibiting quantum criticality
- Heavy fermion compounds (CeCu6-xAux) show antiferromagnetic quantum phase transitions
- Cuprate superconductors exhibit quantum critical behavior near optimal doping
- Iron pnictides display magnetic and structural quantum phase transitions
- Quantum magnets (TlCuCl3) demonstrate pressure-induced Bose-Einstein condensation of magnons
Measurement techniques
- probes magnetic order and spin fluctuations
- (ARPES) measures electronic structure
- Nuclear magnetic resonance (NMR) studies local magnetic properties and spin dynamics
- Quantum oscillation measurements reveal Fermi surface properties near quantum critical points
Challenges in detection
- Extremely low temperatures required to access quantum critical regime
- Presence of disorder and impurities can mask quantum critical behavior
- Difficulty in tuning systems precisely to quantum critical points
- Distinguishing quantum critical effects from other low-temperature phenomena
Theoretical approaches
- Theoretical frameworks for studying quantum phase transitions provide tools for predicting and analyzing critical behavior
- Combining different approaches allows for a comprehensive understanding of quantum criticality
- Theoretical predictions guide experimental investigations and material design
Landau theory for quantum transitions
- Extension of classical to quantum systems
- Describes the free energy in terms of an order parameter and its fluctuations
- Incorporates quantum mechanics through imaginary time formalism
- Predicts mean-field critical exponents but fails to capture strong fluctuations near quantum critical points
Renormalization group methods
- Powerful technique for studying scale-invariant behavior near quantum critical points
- Involves systematic coarse-graining of degrees of freedom
- Determines relevant and irrelevant operators governing critical behavior
- Yields non-trivial critical exponents and scaling functions beyond
Numerical simulations
- Quantum Monte Carlo simulations study large-scale quantum systems
- Density matrix renormalization group (DMRG) effective for one-dimensional systems
- Tensor network methods capture entanglement in higher-dimensional quantum systems
- Exact diagonalization techniques applicable to small quantum systems
Applications and implications
- Understanding quantum phase transitions has far-reaching implications for various fields in condensed matter physics
- Applications range from fundamental science to technological innovations
- Quantum criticality plays a crucial role in the development of novel quantum materials and devices
Quantum materials design
- Tailoring materials to exhibit desired quantum critical behavior
- Engineering quantum critical points to enhance specific properties (superconductivity)
- Designing topological materials with protected edge states
- Creating quantum spin liquids and other exotic quantum phases
Quantum computing
- Quantum phase transitions relevant for adiabatic quantum computing algorithms
- Quantum critical systems as potential platforms for quantum information processing
- Topological quantum computation exploiting protected edge states
- Quantum annealing techniques utilizing quantum fluctuations near critical points
High-temperature superconductivity
- Quantum criticality proposed as a mechanism for unconventional superconductivity
- Antiferromagnetic quantum critical points in cuprates and iron-based superconductors
- Quantum fluctuations may enhance electron pairing and increase transition temperatures
- Understanding quantum criticality crucial for designing new
Current research directions
- Ongoing research in quantum phase transitions explores new frontiers and challenges in condensed matter physics
- Emerging topics combine concepts from quantum criticality with other areas of physics
- Current directions aim to push the boundaries of our understanding of quantum many-body systems
Non-equilibrium quantum criticality
- Studying quantum phase transitions in driven and quenched systems
- Investigating dynamical quantum phase transitions following sudden quenches
- Exploring Floquet engineering of quantum critical behavior
- Examining the interplay between quantum criticality and many-body localization
Quantum phase transitions in driven systems
- Investigating periodically driven (Floquet) systems exhibiting novel quantum phases
- Studying dissipative quantum phase transitions in open quantum systems
- Exploring quantum criticality in optically driven materials
- Examining topological phase transitions induced by time-periodic perturbations
Exotic quantum phases
- Investigating quantum spin liquids and fractionalized excitations
- Studying deconfined quantum critical points beyond the Landau paradigm
- Exploring quantum critical behavior in strange metals and non-Fermi liquids
- Investigating emergent gauge theories and duality relations near quantum critical points
Key Terms to Review (18)
Angle-resolved photoemission spectroscopy: Angle-resolved photoemission spectroscopy (ARPES) is a powerful experimental technique used to study the electronic structure of materials by measuring the energy and momentum of electrons emitted from a sample when it is illuminated by ultraviolet or X-ray radiation. This method provides critical information about the band structure, Fermi surface, and other electronic properties of solids, which are essential for understanding phenomena like superconductivity, surface states, and quantum phase transitions.
Bardeen-Cooper-Schrieffer theory: The Bardeen-Cooper-Schrieffer (BCS) theory is a fundamental explanation of superconductivity, describing how electron pairs, known as Cooper pairs, interact to form a condensed state that exhibits zero electrical resistance. This theory revolutionized our understanding of superconducting materials, showing that at low temperatures, attractive interactions between electrons can overcome their natural repulsion, leading to the formation of these pairs and resulting in the macroscopic quantum phenomena observed in superconductors.
First-order quantum phase transition: A first-order quantum phase transition is a type of transition between different phases of matter that occurs at absolute zero temperature and involves a discontinuous change in some thermodynamic quantity, such as entropy or density. This transition can happen when a system's ground state changes due to variations in external parameters, like pressure or magnetic field, leading to abrupt changes in the system's properties. Unlike classical phase transitions that occur at finite temperatures, first-order quantum phase transitions are driven purely by quantum mechanical effects.
Heavy fermion systems: Heavy fermion systems are materials that exhibit unusual electronic properties due to the presence of strongly correlated electron interactions, leading to an effective mass of conduction electrons that is significantly larger than their actual mass. These systems often display phenomena such as magnetism and superconductivity, arising from quantum fluctuations and interactions between localized f-electrons and conduction electrons. The interplay of these features makes heavy fermion systems a rich ground for studying quantum phase transitions.
High-temperature superconductors: High-temperature superconductors are materials that exhibit superconductivity at temperatures significantly higher than traditional superconductors, typically above 77 K. These materials often have complex crystal structures and are primarily composed of copper oxide compounds, which play a critical role in their unique electronic properties and the underlying mechanisms of superconductivity.
K. G. Wilson: K. G. Wilson is a renowned physicist known for his groundbreaking contributions to the understanding of quantum phase transitions and critical phenomena. His work established fundamental ideas that connect statistical mechanics with quantum mechanics, especially in the context of renormalization group theory, which has become an essential tool in studying systems near critical points.
Landau Theory: Landau Theory is a theoretical framework used to describe phase transitions and critical phenomena in physical systems, particularly focusing on the concept of order parameters. It connects thermodynamic properties to microscopic interactions and offers a mathematical approach to understand various states of matter, such as solid, liquid, and gas, as well as magnetic and superconducting states.
Magnetism: Magnetism is a physical phenomenon produced by the motion of electric charges, resulting in attractive or repulsive forces between objects. It is deeply tied to the behavior of electrons in materials and plays a significant role in various condensed matter systems, especially in describing electron interactions within lattices and their influence on phase transitions.
Mean-field theory: Mean-field theory is a theoretical approach used in statistical physics and condensed matter physics to analyze phase transitions and collective behaviors in systems with many interacting components. By simplifying the interactions between particles, this method approximates the effects of all other particles on any given particle as an average or 'mean field', making complex problems more tractable. It helps in understanding phenomena such as symmetry breaking, critical points, and the behavior of systems near phase transitions.
Neutron scattering: Neutron scattering is a powerful experimental technique used to probe the structure and dynamics of materials at the atomic scale by analyzing the scattering of neutrons off atomic nuclei. This method provides valuable insights into various physical properties, such as lattice vibrations, magnetic ordering, and phase transitions, making it an essential tool for understanding complex phenomena in condensed matter physics.
Order Parameter: An order parameter is a measurable quantity that describes the degree of order in a system undergoing a phase transition. It serves as an essential tool in characterizing the transition from one phase to another, as it changes its value at the critical point, helping to understand the system's symmetry and the nature of the phases involved.
Quantum critical point: A quantum critical point is a special point at zero temperature where a continuous phase transition occurs as a function of some parameter, such as pressure or magnetic field. At this point, the system undergoes significant changes in its ground state properties, often leading to novel quantum phenomena. The concept highlights the interplay between quantum fluctuations and thermal effects, allowing for the exploration of exotic states of matter.
Quantum phase transition: A quantum phase transition is a change in the ground state of a system that occurs at absolute zero temperature due to quantum fluctuations rather than thermal fluctuations. This type of transition is driven by changes in external parameters like pressure or magnetic field, leading to distinct phases that exhibit unique properties. Quantum phase transitions are critical for understanding various phenomena in condensed matter physics, such as edge states and the behavior of materials at low temperatures.
Renormalization group: The renormalization group is a powerful mathematical and conceptual framework used in physics to analyze how physical systems behave at different scales. It focuses on the idea that the properties of a system can change when viewed at different length scales, allowing for the classification of systems into universality classes based on shared critical behavior. This framework is essential for understanding phase transitions and quantum phase transitions, highlighting how microscopic interactions lead to macroscopic phenomena.
S. sachdev: S. Sachdev is a prominent physicist known for his contributions to the understanding of quantum phase transitions and critical phenomena. His work has significantly advanced the theoretical framework surrounding these concepts, focusing on how systems undergo transitions between different quantum states as parameters such as temperature and external fields are varied.
Superfluidity: Superfluidity is a phase of matter characterized by the ability of a fluid to flow without viscosity, allowing it to move with zero resistance. This phenomenon occurs at very low temperatures and is closely linked to quantum mechanical effects, where particles behave collectively as a single quantum entity. Superfluidity is significant in understanding systems like liquid helium and plays a crucial role in concepts such as phase transitions and collective excitations.
Symmetry breaking: Symmetry breaking occurs when a system that is initially symmetric ends up in a state that lacks that symmetry due to changes in conditions, such as temperature or external fields. This concept is vital for understanding various physical phenomena, where the ground state of a system can have a different symmetry than the underlying laws governing it, leading to new phases and behaviors. The implications of symmetry breaking can be observed in diverse systems, influencing the emergence of order and collective behaviors.
Topological Order: Topological order is a unique type of quantum order in many-body systems that cannot be described by local order parameters. It provides a global characterization of the system's ground state, which remains robust against local perturbations. This concept is crucial for understanding various phenomena in condensed matter physics, including fractionalization and edge states.