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.
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.
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.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.
Quantum phase transitions are sudden changes in the ground-state wavefunction of a many-body system that can occur as a control parameter such as a concentration or a field strength is varied. They are driven purely by the competition between quantum fluctuations and mutual interactions among constituents of the system, not by thermal fluctuations; hence they can occur even at zero temperature. Examples of quantum phase transitions in many-body physics may be found in systems ranging from high-temperature superconductors to topological insulators. A quantum phase transition usually can be characterized by nonanalyticity/discontinuity in certain order parameters or divergence of the ground state energy eigenvalue and/or its derivatives with respect to certain physical quantities. Here in a circular one-dimensional spin model with Heisenberg XY interaction and no magnetic field, we observe critical phenomena for the n0=1/N→0 Mott insulator caused by a qualitative change of the boundary condition. We demonstrate in the vicinity of the transition point a sudden change in ground-state properties accompanied by an avoided level-crossing between the ground and the first excited states. Notably, our result links conventional quantum phase transitions to microscopic boundary conditions, with significant implications for quantum information, quantum control, and quantum computing.
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.
Quantum phase transitions, being in charge of the property of matter, are normally induced by varying the Hamiltonian and are characterized by an avoided level crossing between the ground and an excited state. Recently, a quantum phase anomaly induced by a boundary is predicted theoretically as the number of sites N goes to infinity in a one-dimensional Heisenberg spin XY model without an external magnetic field. The infinite-size lattice apparently is inaccessible in experiment, which prevents the experimental observation of the theoretical prediction. In this paper, we find that the finite-size odd-sited lattice can also give the infinite-size lattice's ground-state behavior, and successfully observe such a critical anomaly between two phases having topologically different boundary conditions for the Mott insulator in a finite-size photonic lattice.
Abstract We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H = K + gV , where K and V are two non commuting operators acting on the space of states F , we may always write F = F cond ⊕ F norm where F cond is the subspace spanned by the eigenstates of V with minimal eigenvalue and F norm = F cond ⊥ . If, in the thermodynamic limit, M cond / M → 0, where M and M cond are, respectively, the dimensions of F and F cond , the above decomposition of F becomes effective, in the sense that the ground state energy per particle of the system, ϵ , coincides with the smaller between ϵ cond and ϵ norm , the ground state energies per particle of the system restricted to the subspaces F cond and F norm , respectively: ϵ = min{ ϵ cond , ϵ norm }. It may then happen that, as a function of the parameter g , the energies ϵ cond and ϵ norm cross at g = g c . In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace F