The Simplest Density-Driven Transition The simplest metal–insulator (MI) transition is that due to band filling (or emptying) in a system of non-interacting electrons. This is the fermionic equivalent of Bose–Einstein condensation in a system of non-interacting bosons. Let us consider a system of non-interacting electrons described by a tight-binding band in a hyper-cubic lattice in d -dimensional space. As the number of electrons per site n , for a given spin direction, approaches the value n c = 1 the system evolves from a metallic to an insulating state. This transition is a quantum critical phenomenon with which we can associate critical exponents. We can introduce a characteristic length χ , which can be identified with a screening length which diverges at the metal–insulator transition. The natural variable to describe this zero-temperature transition is the distance from the chemical potential or Fermi level μ , to the bottom, E b = 0, or to the top of the band E t , i.e. g = μ − E b,t . In terms of this variable, ξ ∝ | g | − ν and the ground state free energy density has a singular part that behaves as, f ∝ | g | 2− α , close to the transition. This expression defines the critical exponent α . The critical exponents ν and α associated with this zero temperature transition are related through the quantum hyperscaling relation 2 − α = ν ( d + z ). For this simple MI transition it is possible to determine its universality class for any dimension (Continentino, 1995a). One starts by noticing that the gap for excitations is actually a linear function of the distance g = μ − E t . This fixes the gap exponent at the value νz = 1. Furthermore, for a hypercubic lattice the density of states close to the band edges varies as, such that, the number where μ is the chemical potential.
The Simplest Density-Driven Transition The simplest metal–insulator (MI) transition is that due to band filling (or emptying) in a system of non-interacting electrons. This is the fermionic equivalent of Bose–Einstein condensation in a system of non-interacting bosons. Let us consider a system of non-interacting electrons described by a tight-binding band in a hyper-cubic lattice in d -dimensional space. As the number of electrons per site n , for a given spin direction, approaches the value n c = 1 the system evolves from a metallic to an insulating state. This transition is a quantum critical phenomenon with which we can associate critical exponents. We can introduce a characteristic length χ , which can be identified with a screening length which diverges at the metal–insulator transition. The natural variable to describe this zero-temperature transition is the distance from the chemical potential or Fermi level μ , to the bottom, E b = 0, or to the top of the band E t , i.e. g = μ − E b,t . In terms of this variable, ξ ∝ | g | − ν and the ground state free energy density has a singular part that behaves as, f ∝ | g | 2− α , close to the transition. This expression defines the critical exponent α . The critical exponents ν and α associated with this zero temperature transition are related through the quantum hyperscaling relation 2 − α = ν ( d + z ). For this simple MI transition it is possible to determine its universality class for any dimension (Continentino, 1995a). One starts by noticing that the gap for excitations is actually a linear function of the distance g = μ − E t . This fixes the gap exponent at the value νz = 1. Furthermore, for a hypercubic lattice the density of states close to the band edges varies as, such that, the number where μ is the chemical potential.
Conductivity and Charge Stiffness In this chapter, we will consider initially metal–insulator (MI) transitions which occur in pure systems without disorder. In the final section we consider a superfluid–insulator transition induced by some special type of disorder. We will show that the quantities which characterise the phase transitions in both problems obey similar scaling laws and are governed by similar exponents. Differently from the magnetic transitions treated before metal–insulator (MI) transitions have no obvious order parameter to distinguish the metallic from the insulating phase. This precludes a power expansion of the free energy in terms of a small quantity as for magnetic phase transitions. In spite of that, the concepts of phase transitions and critical phenomena turn out to be very useful to describe metal–insulator transitions. In case this problem is approached using the renormalisation group, the localisation transition is associated with an unstable fixed point and the flow of the RG equations to the different attractors is sufficient to distinguish the nature of the phases. Then, in general, we can associate a set of critical exponents with the MI transitions. Besides, these exponents are not independent but obey scaling relations. Since we are concerned with zero temperature instabilities, the quantum hyperscaling law, Eq. 1.18, plays a crucial role in this problem. We will distinguish here two different types of metal–insulator transitions. Those due to a competition between parameters of the relevant Hamiltonian and metal–insulator transitions arising by varying the number of electrons or chemical potential. An example of the former is the Mott transition due to the competition between kinetic energy and the local Coulomb repulsion as described by the half-filled band Hubbard model. The latter will be referred as density-driven metal– insulator transitions in analogy with the superfluid–insulator transition in bosonic systems with varying particle number (Fisher et al ., 1989). These two types of transitions may occur in the same system or Hamiltonian. They in general belong to different universality classes and consequently have different critical exponents.
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
We analyze the transformation from insulator to metal induced by thermal fluctuations within the Falicov-Kimball model. Using the dynamic mean field theory (DMFT) formalism on the Bethe lattice we find rigorously the temperature dependent density of states (DOS) at half filling in the limit of high dimensions. At zero temperature (T=0) the system is ordered to form the checkerboard pattern and the DOS has the gap $\ensuremath{\Delta}$ at the Fermi level ${\ensuremath{\varepsilon}}_{F}=0$, which is proportional to the interaction constant U. With an increase of T the DOS evolves in various ways that depend on U. For U>Ucr the gap persists for any T (then $\ensuremath{\Delta}>0$), so the system is always an insulator. However, if U<Ucr, two additional subbands develop inside the gap. They become wider with increasing T and at a certain U-dependent temperature TMI they join with each other at ${\ensuremath{\varepsilon}}_{F}$. Since above TMI the DOS is positive at ${\ensuremath{\varepsilon}}_{F}$, we interpret TMI as the transformation temperature from insulator to metal. It appears that TMI approaches the order-disorder phase transition temperature ${T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}$ when U is close to 0 or Ucr, but TMI is substantially lower than ${T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}$ for intermediate values of U. Moreover, using an analytical formula we show that TMI=0 at U=2<Ucr, so we prove that the quantum critical point exists for the ordered metal at (T=0,U=2). Having calculated the temperature dependent DOS we study thermodynamic properties of the system starting from its free energy F. Then we find how the order parameter d and the gap $\ensuremath{\Delta}$ change with T and we construct the phase diagram in the variables T and U, where we display regions of stability of four different phases: ordered insulator, ordered metal, disordered insulator, and disordered metal. Finally, we use a low temperature expansion to demonstrate the existence of a nonzero DOS at a characteristic value of U on a general bipartite lattice.
Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in (V,Cr)2O3 and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.
FINDING: Topological insulators exhibit a metal-insulator transition with critical exponent ν ≈ 2.7, driven by disorder, linking to universal scaling behavior. | MATH: Critical exponent ν ~ 2.7 from scaling analysis of localization length; no deviation from metal-ordinary insulator transitions in systems of orthogonal symmetry class. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, etc.) or base-60 links; however, the critical exponent ν ~ 2.7 is near 2.618 (φ², where φ = 1.618), suggesting a possible resonance with golden ratio scaling in critical phenomena. Lattice structure implicit in topological band theory (e.g., honeycomb, square lattices) but not specified in findings. | DEPTH: 6 — The critical exponent is a precise mathematical constant from scaling theory, but the connection to geometric harmony is speculative; the finding advances understanding of universality classes in disordered topological phases, not fundamental geometric structure. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We investigate how strongly interacting bosons, described by the Bose-Hubbard Hamiltonian, transition from the Mott insulator to the superfluid phase and, ultimately, to a regime of vanishing interaction where the system smoothly transitions to an ensemble of free bosons undergoing Bose-Einstein condensation. Our study focuses on bosons residing on a three-dimensional simple cubic lattice, with tight-binding interactions characterized by the kinetic energy t between neighboring lattice sites and on-site repulsion U. To achieve this, we develop a formalism based on mapping the original quantum mechanical Hamiltonian onto an effective quantum spherical model. This approach is nonperturbative in interaction energy and spans the entire range of the interaction parameter U. Our findings are presented through a series of multiparameter phase diagrams that include temperature, interaction energy, chemical potential, and particle density, highlighting the corresponding critical surfaces and lines. In this context, we address the issue of quantum critical points at fixed densities, which appear as endpoints in the temperature-interaction phase diagram. We also compare our theoretical results, where applicable, with other approaches, including quantum Monte Carlo simulations.