INTRODUCTION In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field!), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → −∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0; due to zero point motions which ‘smear out’ the particles over some region around these lattice points.
INTRODUCTION In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field!), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → −∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0; due to zero point motions which ‘smear out’ the particles over some region around these lattice points.
INTRODUCTION In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field!), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → – ∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0; due to zero point motions which ‘smear out’ the particles over some region around these lattice points.
Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).
In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties.
Commentaries.- A Survey by the Editors.- I. Thermodynamic Limit for Coulomb Systems.- I. 1 Existence of Thermodynamics for Real Matter with Coulomb Forces.- I. 2 The Constitution of Matter: Existence of Thermodynamics for Systems Composed of Electrons and Nuclei.- II. Hard Sphere Virial Coefficients.- II. 1 Suppression at High Temperature of Effects Due to Statistics in the Second Virial Coefficient of a Real Gas.- II. 2 Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals.- III. Zeros of Partition Functions.- III. 1 Monomers and Dimers.- III. 2 Theory of Monomer-Dimer Systems.- III. 3 A Property of Zeros of the Partition Function for Ising Spin Systems.- III. 4 A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets.- IV. Reflection Positivity.- IV. 1 Existence of Phase Transitions for Anisotropic Heisenberg Models.- IV. 2 Phase Transitions in Anisotropic Lattice Spin Systems.- IV. 3 Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions.- IV. 4 Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models.- IV. 5 Phase Transitions and Reflection Positivity. II. Lattice Systems with Short-Range and Coulomb Interactions.- IV. 6 Lattice Models for Liquid Crystals.- IV. 7 Existence of Neel Order in Some Spin-1/2 Heisenberg Antiferromagnets.- IV. 8 The XY Model Has Long-Range Order for all Spins and all Dimensions Greater than One.- V. Classical Thermodynamics.- V. 1 The Third Law of Thermodynamics and the Degeneracy of the Ground State for Lattice Systems.- V. 2 A Guide to Entropy and the Second Law of Thermodynamics.- V. 3 A Fresh Look at Entropy and the Second Law of Thermodynamics.- VI. Lattice Systems.- VI. 1 Properties of a Harmonic Crystal in a Stationary Nonequilibrium State.- VI. 2 The Statistical Mechanics of Anharmonic Lattices.- VI. 3 Time Evolution of Infinite Anharmonic Systems.- VI. 4 Lattice Systems with a Continuous Symmetry III. Low Temperature Asymptotic Expansion for the Plane Rotator Model.- VII. Miscellaneous.- VII. 1 The Finite Group Velocity of Quantum Spin Systems.- VII. 2 The Classical Limit of Quantum Spin Systems.- VII. 3 A Refinement of Simon's Correlation Inequality.- VII. 4 Fluxes, Laplacians, and Kasteleyn's Theorem.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.
Variational quantum algorithms offer a promising framework for solving eigenvalue problems on near-term quantum hardware, yet their applicability beyond electronic structure calculations remains relatively unexplored. In this work, we investigate the quantum computing of lattice vibrational and thermodynamical properties by applying the variational quantum eigensolver and variational quantum deflation to phonon Hamiltonians derived from first-principles force constants obtained using density functional theory. The mass-weighted dynamical matrix is mapped onto a qubit-encoded Hermitian operator, enabling computation of the full set of acoustic and optical phonon branches of crystalline silicon and graphene using a reduced qubit register and direct benchmarking against classical diagonalization. The quantum-computed phonon spectrum is further used to evaluate vibrational entropy, constant-volume specific heat, and thermal expansion coefficient, reproducing expected low-temperature quantum behavior and the high-temperature Dulong-Petit limit. We further demonstrate that combined error mitigation strategies help recover phonon dispersions and thermodynamic behavior consistent with expected trends on near-term quantum hardware. Although classical phonon methods remain computationally superior, our results establish phonon-based thermodynamics as a stringent and physically transparent benchmark for assessing variational quantum algorithms on near-term quantum devices.
Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo techniques.