In this diploma thesis precursors of quantum phase transitions in finite many-body systems are studied. The main attention is paid to the mechanism, how nonanalytic behaviour of the ground state is generated for certain critical values of real control parameters. It is shown that nonanalytic behaviour of energy levels and eigenstates is closely connected with exceptional points of the hamiltonian, which are points in control parameter space extended into a complex domain where at least two eigenvalues and corresponding eigenvectors coincide. Differences in the distribution of exceptional points in the complex plane of control parameter for the first and second order phase transitions and also evolutions of the position of exceptional points with increasing particle number are discussed.
In this diploma thesis precursors of quantum phase transitions in finite many-body systems are studied. The main attention is paid to the mechanism, how nonanalytic behaviour of the ground state is generated for certain critical values of real control parameters. It is shown that nonanalytic behaviour of energy levels and eigenstates is closely connected with exceptional points of the hamiltonian, which are points in control parameter space extended into a complex domain where at least two eigenvalues and corresponding eigenvectors coincide. Differences in the distribution of exceptional points in the complex plane of control parameter for the first and second order phase transitions and also evolutions of the position of exceptional points with increasing particle number are discussed.
We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.
What is a quantum phase transition? Consider a Hamiltonian, H ( g ), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling, g . Let us follow the evolution of the ground state energy of H ( g ) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e. H ( g ) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.
What Is a Quantum Phase Transition? Consider a Hamiltonian, H(g) , whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g . Let us follow the evolution of the ground state energy of H(g) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = g c (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = g c in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.
A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points (EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we use the Lipkin-Meshkov-Glick (LMG) model as a test bed to explore the strong connection between EPs and the onset of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies (EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linked with the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The values of the complex control parameter $\ensuremath{\alpha}$ that lead to the EPs approach the real axis as the system size $N\ensuremath{\rightarrow}\ensuremath{\infty}$. This happens for both the EPs that are close to the separatrix that marks the ESQPT and also for those that are far away, although in the latter case, the convergence rate is smaller. With the method of Pad\'e approximants, we can extract the critical value of $\ensuremath{\alpha}$.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
The aim of this work is to develop a technique for identifying quantum phase transitions which does not rely on the existence of a thermodynamic limit, for studies in finite systems. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-particle Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine critical ground-state couplings. By considering the behaviour of certain ground-state correlation functions, these may be identified as quantum phase transitions in the many-body integrable system with finite particle number. We study two particular examples of bosonic Hamiltonians which admit second-order transitions, and discuss further applications.