We develop a general framework to calculate the many-body density of states (DOS) of isolated and interacting quantum systems. Based on the generalized coherent state formalism and the Simon-Lieb bounds for a quantum partition function, our method provides a general method of calculation for the DOS in high-dimensional irreducible sectors. This framework further provides rigorous bounds for the ground state energy in each sector and enables the calculation of microcanonical observables across the entire spectrum. Using the Lipkin-Meshkov-Glick (LMG) model as a test bed, we validate our framework by successfully identifying quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs) across spin sectors. Unlike existing model-specific numerical or analytical techniques, our formalism relies on general underlying symmetries, making it broadly applicable. Applying our method to the ferromagnetic transverse field Ising chain with power law interactions, we demonstrate that its highest-spin-sector DOS is qualitatively identical to that of LMG-type Hamiltonians. Our work establishes a versatile and computationally efficient bridge between algebraic structure and many-body thermodynamics.
We develop a general framework to calculate the many-body density of states (DOS) of isolated and interacting quantum systems. Based on the generalized coherent state formalism and the Simon-Lieb bounds for a quantum partition function, our method provides a general method of calculation for the DOS in high-dimensional irreducible sectors. This framework further provides rigorous bounds for the ground state energy in each sector and enables the calculation of microcanonical observables across the entire spectrum. Using the Lipkin-Meshkov-Glick (LMG) model as a test bed, we validate our framework by successfully identifying quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs) across spin sectors. Unlike existing model-specific numerical or analytical techniques, our formalism relies on general underlying symmetries, making it broadly applicable. Applying our method to the ferromagnetic transverse field Ising chain with power law interactions, we demonstrate that its highest-spin-sector DOS is qualitatively identical to that of LMG-type Hamiltonians. Our work establishes a versatile and computationally efficient bridge between algebraic structure and many-body thermodynamics.
Abstract Part I: Quantum Collective Spin Systems We use a coherent spin state formalism in order to study collective spin models, which have many applications in physics. In particular, the Lipkin-Meshkov-Glick (LMG) has been analyzed in the thermodynamic limit. The method developed during this work can, in principle, be used for more general Hamiltonians written as a function of the su(2) algebra generators. We have derived exact expressions for the integrated density of states for this model. The nature of the density of states singularities has been detailed. The first order finite size corrections as well as observables expectation values were also computed. The standard Bohr-Sommerfeld quantization approach, adapted to the spin case, is no longer valid near the spectral singularities. In order to treat these cases, we have developed a new approach that permits to describe the spectrum in the neighbourhood of the critical points. Part II: Adiabatic Quantum Computation We have proposed a simple model which highlights the relations between quantum phase transitions and adiabatic (quantum) computation. This model puts in evidence the importance of the choice of the initial Hamiltonian and of the adiabatic path in the parameter space; it should be helpful as a toy model for more realistic cases. We have also studied the state populations dynamics when a quantum phase transition point is crossed, for the LMG model studied in the first part. Numerical simulations show that the dynamics of the populations is very sensitive to the presence of exceptional points in the spectrum, which a simplified model for the quantum evolution already suggests.
Quantum many body physics is an exciting research area, involving novel phases of matter with fundamentally new properties, but is also notoriously hard due to complexity of interacting quantum systems. Some popular approaches involve approximation techniques and numerical simulations, which are known to fail in several important cases. In contrast, rigorous mathematical tools, such as exact solutions and operator inequalities, have a narrower range of applicability, but provide guaranteed results and insights into the underlying physical mechanisms. This research aims to develop new techniques in this direction and use them to explore novel phases of matter. My first direction is to construct toy models where exact solutions are possible. Such models are important as they prove that certain physical phenomena are theoretically possible in nature, and often lead to the discovery of new phases. Such insights are provided by three exactly solvable models I have discovered: (1) a family of 1D quantum spin models hosting free parastatistical quasiparticles (an exotic type of identical particles beyond fermions, bosons, and anyons), proving for the first time that parastatistics is theoretically possible as an emergent phenomenon; (2) a 3D classical Ising model whose phases are characterized by topological features of certain loop observables, suggesting existence of previously unknown classical phases and phase transitions with topological order parameters; and (3) a family of models with exact p-wave superconducting ground states demonstrating the existence of Majorana quasiparticles and non-Abelian statistics in particle number-conserving systems. My second direction is to derive rigorous bounds and exact constraints on physical observables, which are applicable to large families of quantum many-body systems. I present three directions of progress: (1) a method that dramatically improves the upper bounds on the speed of information propagation in locally-interacting systems, which significantly extends the scope of these bounds and enables new applications; (2) bounds on finite-size errors in numerical simulations of lattice systems, including quench dynamics and gapped ground states; and (3) a locality bound on gapped ground states of power-law interacting systems, which leads to a generalization of the aforementioned error bounds to such systems. These error bounds have important theoretical implications such as proving the existence of the thermodynamic limit and stability of phases, and are practically useful in determining the validity of finite-size numerical simulations.
Excited-state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-21 system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.
Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.
Abstract In this paper, we employ the bootstrap method, a technique that relies on consistency relations instead of direct diagonalization, to determine the expectation values in quantum many-body systems. We then use these values to assess the entanglement content of the system. Our work extends the bootstrap approach to quantum many-body systems, rather than single-body or few-body systems, concentrating on the well-known Lipkin–Meshkov–Glick (LMG) model with both transverse and longitudinal external magnetic fields. In the bootstrap method we solve the LMG model with up to 16 sites. Unlike previous studies that have focused mainly on ground-state properties, our methodology allows for the calculation of a broad range of properties, including energy spectrum, angular momentum, concurrence, tangle, residual tangle, and quantum Fisher information (QFI), for all eigenstates or a particular sector of the eigenstates, without referring to the explicit wavefunctions of these states. We show that this approach offers not only a new computational methodology but also a comprehensive view of both bipartite and multipartite entanglement properties across the entire spectrum of eigenstates. Specifically, we demonstrate that states typically found in the central region of the spectrum exhibit greater multipartite entanglement, as indicated by larger QFI values, compared to states at the edges of the spectrum. In contrast, concurrence displays the opposite trend. This observed behavior is in line with the monogamy principle governing quantum entanglement.
The thermal or equilibrium ensemble is one of the most ubiquitous states of matter.For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed-matter physics, high-energy physics, quantum chemistry and quantum computing, among others.We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the fact that interactions in the Hamiltonian are short ranged at its core.We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory.These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms.We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.