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Journal of the Physical Society of Japan

Scaling Theory of Transitions between the Mott Insulator and Quantum Fluids

Masatoshi Imada

Published Dec 15, 1994• 8 citations
Publisher

Abstract

Critical properties of the transition between the Mott insulator and quantum fluids are investigated by using the scaling theory. The Mott insulating phase and the quantum fluid phase of bi-component systems are categorized into several types depending on the existence of order and the type of excitation spectra. The critical exponents of the transition in multicomponent systems are discussed for various combinations of metal-insulator and superfluid-insulator transitions. For the generic transition between a genuinely multicomponent metal and the Mott insulator in d dimensions, the hyperscaling is satisfied with the correlation exponent ν=1/2 d , the dynamical exponent z =2 d , and the Drude weight exponent \(\zeta =\frac32-\frac1d\). It is argued that the upper critical dimension d c satisfies d c ≥2. The above exponents are compared with ν=1/2, z =2, and \(\zeta =\frac{d}2\) for the transition to the band insulator. Critical properties of the energy gap, the compressibility and other quantities are also discussed.

DOI
10.1143/jpsj.63.4294

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Critical properties of the transition between the Mott insulator and quantum fluids are investigated by using the scaling theory. The Mott insulating phase and the quantum fluid phase of bi-component systems are categorized into several types depending on the existence of order and the type of excitation spectra. The critical exponents of the transition in multicomponent systems are discussed for various combinations of metal-insulator and superfluid-insulator transitions. For the generic transition between a genuinely multicomponent metal and the Mott insulator in d dimensions, the hyperscaling is satisfied with the correlation exponent ν=1/2 d , the dynamical exponent z =2 d , and the Drude weight exponent (\zeta =\frac32-\frac1d). It is argued that the upper critical dimension d c satisfies d c ≥2. The above exponents are compared with ν=1/2, z =2, and (\zeta =\frac{d}2) for the transition to the band insulator. Critical properties of the energy gap, the compressibility and other quantities are also discussed.
Transitions between the Mott insulator and metals in clean systems are analyzed with the scaling theory in terms of quantum critical phenomena. In the case of generic transitions controlled by the filling, the scaling theory is established from analyses of the Drude weight and the compressibility based on hyperscaling. In the transition to the Mott insulator, a new universality class with the correlation length exponent ν<1/2 and the dynamical exponent z >2 is derived, which is in contrast to the transition to the band insulator characterized by ν=1/2 and z =2. The unusual exponents lead to various anomalous characters of metals near the Mott insulator such as strong suppression of the degeneracy temperature, and nonlinear dependence of the Drude weight on the doping concentration. Remarkable properties are also found in the specific heat, the compressibility and spin correlations. A mechanism of high temperature superconductivity is discussed in terms of the release from the suppressed coherence.
Critical properties of transitions between a variety of quantum fluids and the Mott insulator are examined in terms of the transition to an incompressible state. In one dimension, they show universally common properties each other irrespective of the statistics, components and the interaction of particles. In two dimensions, they show qualitative differences originated from the differences in the nature of the quantum fluid. It is useful to categorize the singularity at the Mott transition into two types, namely, one characterized by the charge mass divergence and the other by the vanishment of the carrier number. This difference originates from the number of components. Its variety is further explored by examining various cases such as strongly correlated fermions, hard core bosons and boson t - J models. Critical properties to other incompressible states such as spin gap states and the fractional quantum Hall state are also discussed from the same viewpoint. Relevance of our analyses to the understanding of normal state properties in high- T c cuprates and other transition metal oxides is discussed.
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.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
We study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as κ∝∣μ−μc∣−0.58±0.08κ\propto |μ- μ_c|^{-0.58\pm0.08}κ∝∣μ−μc​∣−0.58±0.08 where μcμ_cμc​ is the critical chemical potential. In the insulating phase, the localization length follows ξl∝∣μ−μc∣−νlξ_l \propto |μ- μ_c|^{-ν_l}ξl​∝∣μ−μc​∣−νl​ with νl=0.26±0.05ν_l = 0.26 \pm 0.05νl​=0.26±0.05. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent νκ=0.21±0.04ν_κ= 0.21 \pm 0.04νκ​=0.21±0.04. Our results show that the exponents νκν_κνκ​ and νlν_lνl​ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent ν=1/4ν= 1/4ν=1/4 and dynamical exponent z=4z = 4z=4. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents ν=1/2ν= 1/2ν=1/2 and z=2z = 2z=2.