scholarly journals ORDER, DISORDER AND PHASE TRANSITIONS IN QUANTUM MANY BODY SYSTEMS

2020 ◽  
Author(s):  
Alessandro Giuliani
Keyword(s):  
Quantum Statistical Mechanics ◽  
Lattice Models ◽  
Theoretical Physics ◽  
Electron Interaction ◽  
Many Body ◽  
Many Body Theory ◽  
Decay Properties ◽  
Condensed Matter Theory ◽  
Interacting Electrons ◽  
Many Body Systems

In this paper, I give an overview of some selected results in quantum many body theory, lying at the interface between mathematical quantum statistical mechanics and condensed matter theory. In particular, I discuss some recent results on the universality of transport coecients in lattice models of interacting electrons, with specic focus on the in- dependence of the quantum Hall conductivity from the electron-electron interaction. In this context, the exchange of ideas between mathematical and theoretical physics proved particu- larly fruitful, and helped in clarifying the role played by quantum conservation laws (Ward Identities), together with the decay properties of the Euclidean current-current correlation functions, on the interaction-independence of the conductivity. 

scholarly journals Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

2016 ◽  
Vol 113 (47) ◽  
pp. 13278-13282 ◽  
Author(s):  
Ramis Movassagh ◽  
Peter W. Shor
Keyword(s):  
Physical Models ◽  
Matching Theory ◽  
Many Body ◽  
Quantum Matter ◽  
Condensed Matter Theory ◽  
Simple Quantum ◽  
Matter Theory ◽  
Many Body Systems ◽  
Classical Computer ◽  
Area Law

Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.

PHILIPPE NOZIÈRES: FEENBERG MEDALIST 2001: MICROSCOPIC AND PHENOMENOLOGICAL FOUNDATIONS OF THE THEORY OF QUANTUM MANY-BODY SYSTEMS PHILIPPE NOZIÈRES REPLIES

2003 ◽  
Vol 17 (28) ◽  
pp. 4947-4952
Author(s):  
A. J. LEGGETT ◽  
E. KROTSCHECK ◽  
J. W. NEGELE
Keyword(s):  
Fermi Liquid ◽  
Single Channel ◽  
Landau Theory ◽  
Liquid Mixtures ◽  
Many Body ◽  
Many Body Theory ◽  
Temperature Solution ◽  
Many Body Systems ◽  
The Many ◽  
Solid Liquid

The Eighth Eugene Feenberg Medal is awarded to Philippe Nozières in recognition of his many pathbreaking contributions to many-body theory, including • His definitive work on the properties of the free electron gas, in particular in the region of realistic metallic densities, • his rigorous development of the theory of a normal Fermi liquid, which provided a firm microscopic foundation for the Landau theory, • his analysis of the nonequilibrium thermodynamics of 3-He solid-liquid mixtures, • his exact solution to the X-ray edge problem, • his elegant formulation of the low-temperature solution to the single-channel Kondo problem in the language of Fermi-liquid theory, • his introduction of the many-channel problem as a new class of quantum impurity systems, and • his innovative work on the static and dynamic behavior of the liquid-solid interface.

Quantum Ensembles

2019 ◽  
pp. 285-296
Author(s):  
Robert H. Swendsen
Keyword(s):  
Statistical Mechanics ◽  
Probability Distributions ◽  
Quantum Statistical Mechanics ◽  
Stationary States ◽  
Mechanical State ◽  
Many Body ◽  
Model Probability ◽  
Exact Position ◽  
Quantum Statistical ◽  
Many Body Systems

The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.

scholarly journals Quantum scrambling and state dependence of the butterfly velocity

2019 ◽  
Vol 7 (4) ◽  
Author(s):  
Xizhi Han ◽  
Sean Hartnoll
Keyword(s):  
High Temperature ◽  
Energy Gap ◽  
Lattice Models ◽  
State Dependence ◽  
Many Body ◽  
Ising Spin ◽  
Local Quantum ◽  
Local Lattice ◽  
Many Body Systems ◽  
Mixed Field

Operator growth in spatially local quantum many-body systems defines a scrambling velocity. We prove that this scrambling velocity bounds the state dependence of the out-of-time-ordered correlator in local lattice models. We verify this bound in simulations of the thermal mixed-field Ising spin chain. For scrambling operators, the butterfly velocity shows a crossover from a microscopic high temperature value to a distinct value at temperatures below the energy gap.

scholarly journals Universal quantum Hamiltonians

2018 ◽  
Vol 115 (38) ◽  
pp. 9497-9502 ◽  
Author(s):  
Toby S. Cubitt ◽  
Ashley Montanaro ◽  
Stephen Piddock
Keyword(s):  
Lattice Models ◽  
Body System ◽  
Many Body ◽  
Diverse Range ◽  
Spin Lattice ◽  
Universal Quantum ◽  
Hamiltonian Simulation ◽  
Many Body Systems ◽  
Physical Phenomena ◽  
Practical Field

Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. However, we prove that the entire physics of any quantum many-body system can be replicated by certain simple, “universal” spin-lattice models. We first characterize precisely what it means for one quantum system to simulate the entire physics of another. We then fully classify the simulation power of all two-qubit interactions, thereby proving that certain simple models can simulate all others, and hence are universal. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a step toward justifying why error correction may not be required for this application of quantum information technology.

scholarly journals SCALING IN MANY-BODY SYSTEMS AND PROTON STRUCTURE FUNCTION

2003 ◽  
Vol 17 (28) ◽  
pp. 5139-5149 ◽  
Author(s):  
OMAR BENHAR
Keyword(s):  
Scaling Function ◽  
Careful Consideration ◽  
Proton Structure Function ◽  
Large Momentum ◽  
Inelastic Electron ◽  
Many Body ◽  
Many Body Theory ◽  
Proton Structure ◽  
Rich Information ◽  
Many Body Systems

The observation of scaling in processes in which a weakly interacting probe delivers large momentum q to a many-body system simply reflects the dominance of incoherent scattering off target constituents. While a suitably defined scaling function may provide rich information on the internal dynamics of the target, in general its extraction from the measured cross section requires careful consideration of the nature of the interaction driving the scattering process. The analysis of deep inelastic electron-proton scattering in the target rest frame within standard many-body theory naturally leads to the emergence of a scaling function that, unlike the commonly used structure functions F1 and F2, can be directly identified with the intrinsic proton response.

scholarly journals The importance of being integrable: Out of the paper, into the lab

2014 ◽  
Vol 28 (18) ◽  
pp. 1430010 ◽  
Author(s):  
Murray T. Batchelor
Keyword(s):  
Quantum Physics ◽  
Three Dimensional ◽  
Integrable Model ◽  
Theoretical Physics ◽  
Many Body ◽  
One Dimensional ◽  
S Matrix ◽  
Many Body Systems ◽  
Low Dimensional ◽  
Traditional Setting

The scattering matrix (S-matrix), relating the initial and final states of a physical system undergoing a scattering process, is a fundamental object in quantum mechanics and quantum field theory. The study of factorized S-matrices, in which many-body scattering factorizes into a product of two-body terms to yield an integrable model, has long been considered the domain of mathematical physics. Many beautiful results have been obtained over several decades for integrable models of this kind, with far reaching implications in both mathematics and theoretical physics. The viewpoint that these were only toy models changed dramatically with brilliant experimental advances in realizing low-dimensional quantum many-body systems in the lab. These recent experiments involve both the traditional setting of condensed matter physics and the trapping and cooling of atoms in optical lattices to engineer and study quasi-one-dimensional systems. In some cases the quantum physics of one-dimensional systems is arguably more interesting than their three-dimensional counterparts, because the effect of interactions is more pronounced when atoms are confined to one dimension. This article provides a brief overview of these ongoing developments, which highlight the fundamental importance of integrability.

scholarly journals Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

2018 ◽  
Author(s):  
Johannes Hauschild ◽  
Frank Pollmann
Keyword(s):  
Matrix Product ◽  
Many Body ◽  
Density Matrix Renormalization ◽  
Tensor Networks ◽  
Condensed Matter Theory ◽  
Tensor Network ◽  
Matter Theory ◽  
Particle Number Conservation ◽  
Many Body Systems ◽  
The One

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

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