🔬Condensed Matter Physics Unit 11 – Many-Body Physics & Correlated Systems

Key Concepts and Foundations

• Many-body physics studies systems with a large number of interacting particles, such as electrons in solids or atoms in quantum gases
• Focuses on emergent collective phenomena that arise from interactions between particles, leading to properties different from individual constituents
• Quantum mechanics provides the fundamental framework for describing many-body systems, accounting for wave-particle duality, quantum entanglement, and Pauli exclusion principle
• Statistical mechanics connects microscopic properties of many-body systems to macroscopic observables, using concepts like partition functions, ensembles, and thermodynamic potentials
• Symmetry and conservation laws play a crucial role in understanding many-body systems, including translational, rotational, and gauge symmetries, as well as conservation of energy, momentum, and particle number
• Quasi-particles are elementary excitations in many-body systems that behave like particles with renormalized properties (phonons in solids, magnons in magnets, or Cooper pairs in superconductors)
• Phase transitions occur when many-body systems undergo qualitative changes in their properties, often accompanied by symmetry breaking or the emergence of long-range order (ferromagnetism, superconductivity, or Bose-Einstein condensation)

Many-Body Quantum Mechanics

• Many-body quantum mechanics extends single-particle quantum mechanics to systems with multiple interacting particles, describing their collective behavior and emergent properties
• The many-body wavefunction depends on the coordinates of all particles in the system, making the Schrödinger equation high-dimensional and challenging to solve exactly
• Indistinguishability of identical particles leads to quantum statistics (Fermi-Dirac for fermions, Bose-Einstein for bosons) and the symmetrization or antisymmetrization of the many-body wavefunction
• The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state, resulting in the formation of Fermi seas and the concept of Fermi energy
• Exchange interactions arise from the antisymmetry of the fermionic wavefunction, leading to effective interactions between particles (exchange energy in atoms or exchange coupling in magnetic systems)
• Variational principles, such as the Rayleigh-Ritz method, provide a powerful tool for approximating the ground state and excited states of many-body systems by minimizing the energy functional
• Perturbation theory allows for systematic corrections to the non-interacting picture, accounting for weak interactions between particles (Møller-Plesset perturbation theory or the GW approximation)

Second Quantization

• Second quantization is a formalism that describes many-body quantum systems using creation and annihilation operators, providing a convenient and compact representation of the many-body Hilbert space
• Creation operators $a^\dagger_i$ add a particle to a specific quantum state $i$, while annihilation operators $a_i$ remove a particle from state $i$, satisfying commutation relations for bosons and anticommutation relations for fermions
• The occupation number representation uses Fock states $|n_1, n_2, ...\rangle$ to describe the number of particles in each quantum state, with the vacuum state $|0\rangle$ representing the absence of particles
• The many-body Hamiltonian can be expressed in terms of creation and annihilation operators, capturing single-particle terms (kinetic energy and external potentials) and two-particle interactions
• Wick's theorem allows for the evaluation of expectation values and correlation functions of products of creation and annihilation operators, expressing them in terms of contractions and normal-ordered products
• The Heisenberg picture of quantum mechanics describes the time evolution of operators, with creation and annihilation operators acquiring time dependence, while the state vector remains constant
• Field operators $\psi^\dagger(\mathbf{r})$ and $\psi(\mathbf{r})$ create and annihilate particles at position $\mathbf{r}$, providing a continuous representation of the many-body system and enabling the formulation of quantum field theories

Correlation Functions and Green's Functions

• Correlation functions describe the statistical relationships between physical quantities in many-body systems, capturing the effects of interactions and collective behavior
• The one-particle Green's function $G(1,1')$ represents the probability amplitude for a particle to propagate from space-time point $1$ to $1'$, containing information about single-particle excitations and spectral properties
• The two-particle Green's function $G(1,2;1',2')$ describes the propagation of two particles and their interaction, relevant for studying particle-particle scattering, screening, and collective modes
• The spectral function $A(\mathbf{k},\omega)$ is the imaginary part of the one-particle Green's function, providing information about the single-particle excitation spectrum and the density of states
• The self-energy $\Sigma(1,1')$ captures the effects of interactions on the single-particle Green's function, leading to renormalization of the particle's properties (effective mass, lifetime, and spectral weight)
• Dyson's equation relates the interacting Green's function to the non-interacting Green's function and the self-energy, allowing for a perturbative treatment of interactions
• The fluctuation-dissipation theorem connects the correlation functions to the linear response of the system to external perturbations, relating equilibrium fluctuations to dissipative properties (conductivity or susceptibility)

Fermi Liquid Theory

• Fermi liquid theory describes the low-energy properties of interacting fermionic systems, such as electrons in metals or liquid $^3$He, in terms of long-lived quasi-particles with renormalized properties
• Landau's adiabatic continuity principle states that the low-energy excitations of an interacting Fermi system are in one-to-one correspondence with those of a non-interacting Fermi gas, with interactions leading to renormalization of quasi-particle properties
• Quasi-particles in a Fermi liquid have a well-defined momentum and energy, with a lifetime that diverges as $(\varepsilon-\varepsilon_F)^{-2}$ near the Fermi energy $\varepsilon_F$, ensuring the stability of the Fermi liquid state
• The Landau interaction function $f(\mathbf{k},\mathbf{k}')$ parametrizes the effective interaction between quasi-particles, determining the renormalization of their properties and the stability of the Fermi liquid
• Fermi liquid parameters, such as the effective mass $m^*$, the Landau parameter $F_0^s$, and the Wilson ratio, characterize the renormalization of quasi-particle properties and the response functions of the system
• The Fermi surface, defined by the locus of points in momentum space with quasi-particle energy equal to the Fermi energy, plays a central role in determining the low-energy properties of the system
• Collective modes in Fermi liquids, such as zero sound and spin waves, arise from the coherent oscillations of the Fermi surface, with their dispersion relations determined by the Landau interaction function

Strongly Correlated Systems

• Strongly correlated systems are materials in which the interactions between particles are comparable to or larger than their kinetic energy, leading to the breakdown of conventional perturbative approaches and the emergence of novel quantum phases
• Mott insulators are materials that should be metallic based on band theory but are insulating due to strong electron-electron interactions, with the Mott transition occurring as a function of the interaction strength or band filling
• Hubbard models capture the essential physics of strongly correlated systems, describing the competition between the kinetic energy of particles hopping between lattice sites and the on-site Coulomb repulsion
• Antiferromagnetism often emerges in strongly correlated systems due to the superexchange mechanism, where virtual hopping of electrons between neighboring sites leads to an effective antiferromagnetic coupling
• Heavy fermion materials exhibit large effective masses and enhanced electronic correlations due to the hybridization between localized $f$-electrons and conduction electrons, leading to unconventional superconductivity and quantum criticality
• Frustrated magnetism arises when the geometry of the lattice or the presence of competing interactions prevents the system from achieving a unique ground state, leading to highly degenerate manifolds and exotic spin liquid phases
• Unconventional superconductivity, such as high-temperature cuprate superconductors or iron-based superconductors, emerges in strongly correlated systems, often in proximity to other ordered phases (antiferromagnetism or charge order)

Computational Methods

• Computational methods play a crucial role in understanding many-body physics, providing numerical solutions to problems that are intractable by analytical means
• Exact diagonalization directly solves the many-body Schrödinger equation by representing the Hamiltonian matrix in a truncated Hilbert space, yielding numerically exact results for small systems
• Quantum Monte Carlo methods, such as variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC), use stochastic sampling to efficiently explore the high-dimensional configuration space and compute expectation values of observables
• Tensor networks, including matrix product states (MPS) and projected entangled pair states (PEPS), provide a compact representation of quantum states by exploiting the local entanglement structure, enabling efficient simulations of 1D and 2D systems
• Dynamical mean-field theory (DMFT) maps a lattice problem onto a self-consistent impurity problem, capturing local quantum fluctuations while treating non-local correlations in a mean-field manner
• The density matrix renormalization group (DMRG) is a variational method that optimizes the MPS representation of the ground state by iteratively truncating the Hilbert space, providing accurate results for 1D and quasi-1D systems
• Quantum embedding methods, such as the self-energy embedding theory (SEET) and the dynamical vertex approximation (DΓA), combine local and non-local correlations by embedding a correlated subsystem into a larger environment
• Machine learning techniques, including neural networks and kernel methods, are being increasingly applied to many-body physics for representing quantum states, solving optimization problems, and identifying phase transitions

Applications and Experimental Techniques

• Many-body physics finds applications in a wide range of physical systems, from condensed matter to cold atomic gases and quantum information processing
• Quantum materials, such as topological insulators, Weyl semimetals, and quantum spin liquids, exhibit novel properties arising from the interplay of topology, symmetry, and strong correlations
• Cold atomic gases provide a versatile platform for simulating many-body physics, with the ability to control interactions, dimensionality, and lattice geometry, enabling the study of quantum phase transitions and non-equilibrium dynamics
• Quantum simulators, built using cold atoms, trapped ions, or superconducting qubits, aim to solve specific many-body problems by engineering the Hamiltonian of interest and measuring relevant observables
• Spectroscopic techniques, such as angle-resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM), and neutron scattering, probe the electronic structure, local density of states, and magnetic excitations in many-body systems
• Transport measurements, including electrical conductivity, thermal conductivity, and Hall effect, provide information about the charge and heat transport properties of many-body systems, revealing the nature of quasi-particles and collective modes
• Thermodynamic measurements, such as specific heat, magnetic susceptibility, and compressibility, probe the response of many-body systems to external fields and temperature changes, allowing for the identification of phase transitions and critical behavior
• Ultrafast spectroscopy, using femtosecond laser pulses, enables the study of non-equilibrium dynamics and the real-time evolution of many-body systems, revealing the timescales of electronic and structural processes