12.4 Quantum field theory and condensed matter systems
6 min read•august 14, 2024
is a powerful tool for understanding complex systems in condensed matter physics. It treats particles as excitations of quantum fields, allowing us to tackle many-body problems and like and .
QFT techniques like , , and methods help us calculate physical quantities in condensed matter systems. These approaches provide insights into the behavior of materials at different energy scales and the emergence of effective low-energy theories.
Quantum Field Theory in Condensed Matter
Application of QFT Methods to Condensed Matter Systems
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- (QFT) provides a framework for describing in condensed matter physics
- Particles are treated as excitations of underlying quantum fields
- QFT methods allow for the systematic treatment of interactions and collective phenomena in condensed matter systems (superconductivity, , magnetism)
- The application of QFT to condensed matter involves the use of second quantization
- Creation and annihilation operators are introduced to describe the creation and destruction of particles in the system
- QFT techniques are used to calculate physical quantities in condensed matter systems
- Feynman diagrams and calculate correlation functions and response functions
- The renormalization group approach, a key concept in QFT, is used to study condensed matter systems
- Examines behavior at different energy scales and understands the emergence of effective low-energy theories
Techniques and Approaches in Condensed Matter QFT
- Second quantization is a key technique in applying QFT to condensed matter
- Introduces creation and annihilation operators to describe the creation and destruction of particles
- Allows for the treatment of many-body systems and the inclusion of interactions
- Feynman diagrams are a graphical tool used to calculate physical quantities in QFT
- Represent the mathematical expressions for correlation functions and response functions
- Enable the systematic treatment of interactions and perturbative expansions
- Perturbation theory is a method used to approximate solutions to quantum mechanical problems
- Expands the Hamiltonian into a sum of a solvable part and a small perturbation
- Allows for the calculation of physical quantities in the presence of weak interactions
- The renormalization group approach is used to study the behavior of condensed matter systems at different energy scales
- Identifies the relevant degrees of freedom and interactions at each scale
- Allows for the derivation of effective low-energy theories that capture the essential physics of the system
Effective Field Theories for Condensed Matter
Concept and Utility of Effective Field Theories
- () are simplified models that capture the essential low-energy physics of a system
- Ignore the high-energy details that are not relevant at the scale of interest
- EFTs are constructed by identifying the relevant degrees of freedom and symmetries that govern the low-energy behavior
- The most general Lagrangian consistent with these symmetries is written down
- The use of EFTs allows for a more tractable description of complex condensed matter systems
- High-energy degrees of freedom are integrated out, leading to a simpler theory with fewer parameters
- EFTs are particularly useful in describing universal properties of condensed matter systems
- near phase transitions are independent of the microscopic details of the system
Examples of Effective Field Theories in Condensed Matter
- The of superconductivity is an example of an EFT
- Describes the superconducting state using a complex field
- Captures the essential physics near the superconducting phase transition
- The Landau Fermi liquid theory is an EFT describing the low-energy properties of interacting fermions
- Treats the system as a gas of weakly interacting with renormalized parameters
- Explains the properties of metals, such as the specific heat and magnetic susceptibility
- The hydrodynamic description of superfluids is an EFT that captures the long-wavelength, low-frequency behavior
- Describes the system in terms of a few macroscopic variables, such as the density and velocity fields
- Accounts for the existence of sound waves and vortices in superfluids
- The nonlinear sigma model is an EFT used to describe the low-energy physics of systems with broken continuous symmetries
- Describes the dynamics of Goldstone bosons, which are the massless excitations associated with the broken symmetry
- Applies to a wide range of systems, including magnets, superfluid helium, and quantum chromodynamics (QCD)
Symmetry Breaking and Phase Transitions in Condensed Matter
Role of Symmetry Breaking in Condensed Matter
- is a fundamental concept in condensed matter physics
- A system in a symmetric phase undergoes a transition to a phase with lower symmetry
- occurs when the ground state of a system has a lower symmetry than the Hamiltonian
- Leads to the emergence of new phases of matter
- Phase transitions in condensed matter systems are often associated with the breaking of certain symmetries
- Breaking of in superconductors
- Breaking of in ferromagnets
- The concept of symmetry breaking is crucial for understanding the properties of various condensed matter systems
- Existence of in systems with continuous symmetry breaking
- that arise from the breaking of discrete symmetries
Ginzburg-Landau Theory and Order Parameters
- The Ginzburg-Landau theory provides a general framework for describing phase transitions and symmetry breaking
- Uses an order parameter to characterize the degree of symmetry breaking
- The order parameter is a physical quantity that is zero in the symmetric phase and non-zero in the broken-symmetry phase
- Examples include the magnetization in ferromagnets and the superconducting gap in superconductors
- The Ginzburg-Landau free energy is a functional of the order parameter and its gradients
- Includes terms that favor the symmetric phase and terms that favor the broken-symmetry phase
- Minimizing the Ginzburg-Landau free energy determines the equilibrium value of the order parameter
- Describes the behavior of the system near the phase transition
- The Ginzburg-Landau theory provides a unified description of various types of phase transitions
- Second-order phase transitions, where the order parameter changes continuously (ferromagnetic transition)
- First-order phase transitions, where the order parameter changes discontinuously (liquid-gas transition)
Examples of Quantum Field Theories in Condensed Matter
Ginzburg-Landau Theory of Superconductivity
- The Ginzburg-Landau theory is a phenomenological quantum field theory used to describe superconductivity
- Introduces a complex order parameter field that describes the superconducting state
- Includes a Lagrangian with terms describing the coupling between the order parameter and electromagnetic fields
- The theory successfully explains various properties of superconductors
- Meissner effect: expulsion of magnetic fields from the interior of a superconductor
- Existence of Type I and Type II superconductors, distinguished by their magnetic properties
- Behavior of superconductors near the critical temperature
- The Ginzburg-Landau equations derived from the theory describe the spatial variation of the order parameter and magnetic fields
- Allow for the calculation of the superconducting coherence length and penetration depth
- The theory has been extended to describe unconventional superconductors, such as high-temperature superconductors
- Incorporates additional order parameter symmetries and coupling terms
Kondo Problem and Renormalization Group Methods
- The describes the behavior of magnetic impurities in metallic hosts
- An example of a strongly correlated quantum system that can be studied using QFT methods
- The describes the interaction between a localized magnetic impurity and the conduction electrons in the host metal
- Leads to the formation of a singlet state at low temperatures, known as the
- The Kondo problem can be treated using renormalization group methods
- Reveal the existence of a characteristic energy scale, the
- Below , the impurity is screened by the conduction electrons, leading to the formation of the Kondo singlet
- The has important consequences for the transport properties of metals with magnetic impurities
- Leads to a logarithmic increase in the electrical resistivity at low temperatures
- Gives rise to a peak in the specific heat and a Curie-like magnetic susceptibility
- The Kondo problem has been studied extensively using various QFT techniques
- : a non-perturbative method that iteratively diagonalizes the Hamiltonian
- : an exact solution technique that exploits the integrability of the Kondo model
- : describes the critical behavior of the Kondo problem at the fixed point
Key Terms to Review (31)
Bethe Ansatz: The Bethe Ansatz is a powerful mathematical technique used to solve certain one-dimensional quantum many-body systems, particularly in the context of integrable models. It provides a way to construct wave functions for these systems by using a set of parameters known as rapidities, which describe the momenta of the particles involved. This method connects closely to both quantum field theory and condensed matter physics, revealing deep insights into the behavior of quantum systems, especially in terms of their energy spectra and correlations.
Collective phenomena: Collective phenomena refer to behaviors or properties that emerge from the interactions of many individual components in a system, resulting in macroscopic effects that cannot be understood by examining the components in isolation. These phenomena are particularly important in systems like condensed matter, where quantum effects and thermal fluctuations can lead to emergent behaviors such as superconductivity, magnetism, and phase transitions.
Conformal Field Theory: Conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations, which are transformations that preserve angles but not necessarily distances. This property makes CFTs particularly useful in studying critical phenomena and phase transitions, where systems exhibit scale invariance. CFTs also play a crucial role in holographic theories, connecting strongly coupled quantum field theories with gravity in higher-dimensional spaces, and have applications in condensed matter physics, revealing insights into quantum phase transitions and emergent phenomena.
Critical Exponents: Critical exponents are numerical values that characterize the behavior of physical quantities near continuous phase transitions, describing how these quantities diverge or vanish as a system approaches the critical point. They provide essential insights into the universality class of the phase transition, linking systems with different microscopic details but similar critical behavior. These exponents are typically denoted by letters like $eta$, $
u$, $eta$, and $ au$, each representing different physical properties such as order parameter, correlation length, and specific heat.
Effective Field Theories: Effective field theories are simplified models that capture the essential physics of a system at a specific energy scale while ignoring the details of high-energy interactions. They allow physicists to focus on relevant degrees of freedom and phenomena without needing a complete understanding of all underlying processes, making them particularly useful in quantum field theory and condensed matter physics, where complex interactions often arise. By using effective field theories, researchers can address important questions while keeping computations manageable and results insightful.
EFTs: Effective Field Theories (EFTs) are simplified models used in quantum field theory that focus on low-energy phenomena by integrating out high-energy degrees of freedom. They provide a way to capture the essential features of a physical system while ignoring irrelevant details, making them particularly useful in condensed matter physics and particle physics.
Feynman diagrams: Feynman diagrams are pictorial representations of the interactions between particles in quantum field theory. They simplify complex calculations in particle physics by visually depicting the paths and interactions of particles, facilitating the understanding of processes like scattering and decay.
Gauge symmetry: Gauge symmetry is a fundamental concept in physics that refers to the invariance of a system under local transformations of certain fields. It plays a crucial role in the formulation of physical theories, particularly in defining interactions between particles and fields without changing the observable outcomes. This principle helps unify forces and leads to the conservation laws that govern particle interactions and their dynamics.
Ginzburg-Landau Theory: Ginzburg-Landau theory is a mathematical framework used to describe phase transitions in condensed matter systems, particularly in superconductors. It introduces a complex order parameter that represents the degree of order within a system and employs a free energy functional to analyze the thermodynamic properties associated with phase transitions. This theory connects quantum field theory concepts with the behavior of macroscopic systems, showcasing how microscopic interactions lead to observable phenomena.
Goldstone Modes: Goldstone modes refer to the massless excitations that arise in a system with spontaneously broken continuous symmetries. When a symmetry is broken, the Goldstone theorem states that there will be corresponding massless scalar particles, known as Nambu-Goldstone bosons, which represent the directions in which the symmetry is broken. These modes play a significant role in various fields, including particle physics and condensed matter systems, highlighting how symmetry breaking can lead to observable physical phenomena.
Kondo Effect: The Kondo Effect is a phenomenon in condensed matter physics where the electrical resistance of a metal containing magnetic impurities decreases at low temperatures, leading to an increase in conductivity. This effect arises from the interactions between the conduction electrons and the localized magnetic moments of the impurities, resulting in a complex scattering process that can be analyzed through quantum field theory approaches.
Kondo Hamiltonian: The Kondo Hamiltonian is a theoretical model used to describe the interaction between localized magnetic moments, typically from impurities, and conduction electrons in a metal. This model captures the essential physics of the Kondo effect, where the resistance of a metal increases as temperature decreases due to the scattering of conduction electrons off these localized spins. The Kondo Hamiltonian provides insight into phenomena such as the temperature-dependent resistivity in metals with magnetic impurities and has significant implications in quantum field theory and condensed matter systems.
Kondo Problem: The Kondo problem describes the behavior of a magnetic impurity in a metal and its interaction with conduction electrons, leading to non-Fermi liquid behavior at low temperatures. It is a key concept in condensed matter physics that helps explain phenomena such as the resistance minimum in metals with magnetic impurities and the development of local magnetic moments due to electron screening.
Kondo Singlet: A Kondo singlet is a quantum state formed when a localized spin (such as that from a magnetic impurity) couples with the conduction electrons in a metal, leading to a collective behavior that screens the localized spin. This phenomenon is crucial in understanding the low-temperature properties of metals with magnetic impurities and illustrates how quantum field theory can describe interactions in condensed matter systems.
Kondo Temperature: The Kondo temperature is a characteristic energy scale associated with the Kondo effect, where the scattering of conduction electrons by localized magnetic impurities leads to an increase in electrical resistivity at low temperatures. This phenomenon highlights the interactions between localized spins and conduction electrons, playing a significant role in the behavior of strongly correlated electron systems, particularly in metals and alloys.
Magnetism: Magnetism is a physical phenomenon produced by the motion of electric charge, which results in attractive and repulsive forces between objects. It is closely linked to electricity and can be understood through classical physics, but its manifestations become richer and more complex when analyzed through the lens of quantum field theory, especially in condensed matter systems where collective behavior of particles gives rise to magnetic properties.
Many-body systems: Many-body systems refer to physical systems that consist of a large number of interacting particles, where the collective behavior emerges from the complex interactions among these particles. In the context of quantum field theory and condensed matter systems, understanding many-body systems is crucial because they exhibit phenomena like superconductivity and magnetism that cannot be explained by studying individual particles alone.
Numerical renormalization group: The numerical renormalization group (NRG) is a computational technique used to study quantum systems at critical points by systematically removing high-energy degrees of freedom. This method allows researchers to analyze the behavior of many-body systems, particularly in condensed matter physics, where quantum field theory concepts can be applied. By focusing on low-energy excitations, NRG provides insights into phenomena such as quantum phase transitions and impurity problems in interacting systems.
Order parameter: An order parameter is a measurable quantity that characterizes the degree of order in a system, typically used to describe phase transitions. It provides insight into the symmetry of the system and can indicate the presence of a particular phase, such as ordered or disordered states. In both quantum field theory and condensed matter systems, the order parameter plays a crucial role in understanding how systems transition from one phase to another and how symmetry breaking occurs.
Perturbation theory: Perturbation theory is a mathematical technique used in quantum mechanics and quantum field theory to approximate the behavior of a system that is subject to small disturbances or interactions. It allows for the calculation of physical quantities by treating the interaction as a small perturbation of a solvable system, providing a powerful method to understand complex systems and their dynamics.
Quantum Field Theory: Quantum Field Theory (QFT) is a fundamental framework in theoretical physics that combines classical field theory, special relativity, and quantum mechanics to describe the behavior of subatomic particles and their interactions. This approach allows for the incorporation of particle creation and annihilation processes, making it essential for understanding the dynamics of fundamental forces and particles in a unified way.
Quantum Field Theory (QFT): Quantum Field Theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics to describe the fundamental interactions of particles and fields in a unified way. It treats particles as excitations in underlying fields, enabling the description of particle creation and annihilation, which is essential for understanding interactions in high-energy physics and condensed matter systems.
Quasiparticles: Quasiparticles are collective excitations that arise in many-body systems, behaving like particles while not being actual individual particles. They represent emergent phenomena that simplify the analysis of complex interactions in condensed matter physics, enabling a better understanding of properties such as conductivity and magnetism.
Renormalization Group: The renormalization group is a collection of techniques used to study the changes in physical systems as one varies the energy scale or length scale. It helps deal with infinities that arise in quantum field theories by systematically relating the parameters of a theory at different scales, allowing for the prediction of physical phenomena in a consistent manner. This concept connects deeply with divergences in calculations, running couplings, effective field theories, critical phenomena, and condensed matter systems.
Rotational Symmetry: Rotational symmetry refers to a property of a shape or object that looks the same after being rotated about a central point by a certain angle. This concept is crucial in understanding how physical systems behave under transformations, particularly in quantum field theory and condensed matter systems, where symmetry principles often dictate the fundamental laws governing particle interactions and material properties.
Second quantization: Second quantization is a formalism in quantum mechanics that extends the concept of quantizing fields, allowing for the description of many-particle systems. This approach introduces operators that create or annihilate particles in a quantum field, facilitating the analysis of phenomena where particle numbers can change, such as in interactions. It provides a framework to merge the principles of quantum mechanics with the principles of special relativity, enabling applications in various areas like particle physics and condensed matter.
Spontaneous Symmetry Breaking: Spontaneous symmetry breaking occurs when a system that is symmetric under a certain transformation chooses a specific configuration that does not exhibit that symmetry. This phenomenon is crucial in various fields, leading to the emergence of distinct states and particles, and it helps explain many physical processes, including mass generation and phase transitions.
Superconductivity: Superconductivity is a phenomenon where a material exhibits zero electrical resistance and expels magnetic fields when cooled below a certain critical temperature. This behavior is significant in understanding phase transitions and broken symmetries, leading to the emergence of Nambu-Goldstone bosons and various condensed matter theories.
Superfluidity: Superfluidity is a phase of matter characterized by the ability of a fluid to flow without viscosity, allowing it to move through small openings and around obstacles with no energy loss. This unique behavior is often observed in liquid helium at very low temperatures and connects deeply with quantum mechanics, as it arises from the collective quantum state of particles in the fluid. Superfluidity challenges our classical understanding of fluids and plays a significant role in the study of condensed matter systems.
Symmetry Breaking: Symmetry breaking occurs when a system that is symmetric under a certain transformation loses that symmetry due to a change in its conditions or configuration. This phenomenon is pivotal in various areas of physics, as it helps explain how certain fundamental forces arise and why certain particles acquire mass while others do not.
Topological Defects: Topological defects are irregularities or discontinuities that occur in ordered systems, arising when a system undergoes a phase transition. They can manifest in various forms, such as vortices, monopoles, and domain walls, and are crucial in understanding the behavior of both condensed matter systems and quantum field theory. These defects reflect the underlying symmetries of the system and can have significant implications for its physical properties.