The types of Mott insulator Dung-Hai Leea,b and Steven A. Kivelsonc (a)Department of Physics,University of California, Berkeley, CA 94720, USA (b)Center for Advanced Study, Tsinghua University, Beijing 100084, China (c)Physics Department,University of California, Los Angeles, CA 90095-9547 USA There are two classes of Mott insulators in nature, distinguished by their responses to weak doping. With increasing chemical potential, Type I Mott insulators undergo a first order phase transition from the undoped to the doped phase. In the presence of long-range Coulomb interactions, this leads to an inhomogeneous state exhibiting “micro-phase separation.” In contrast, in Type II Mott insulators charges go in continuously above a critical chemical potential. We show that if the insulating state has a broken symmetry, this increases the likelihood that it will be Type I. There exists a close analogy between these two types of Mott insulators and the familiar Type I and Type II superconductors. The study of Mott insulators and doped Mott insula- fractionalization[6, 7, 8, 9]. To this list, we add a “type” tors constitutes one of the central focuses of condensed index that classifies Mott insulators into two categories matter physics of the past two decades. At a simple level, depending on their response to doping. a Mott insulator is a system in which the strong repul- Generally speaking, states with charge gaps (this in- sion between particles impedes their flow; the simplest cludes both band and Mott insulators) occur in crys- cartoon of this is a system with a classical ground-state talline systems at isolated rational “occupation num- in which there is one particle on each site of a crystalline bers,” ν = ν∗, where ν is the number of particles per lattice and such a large repulsion between two particles chemical unit cell. By “doping” we mean a process which on the same site that fluctuations involving the motion causes the occupation number to shift away from ν∗. of a particle from one site to the next are suppressed. When lattice translation symmetry is not spontaneously The Mott insulator is thus essentially classical in nature broken, ν∗ is typically an integer for bosons, consistent (and so accessible to particles of any[1] statistics.) In with the simple classical cartoon described above, and an contrast a band insulator[2], including all familiar semi- even integer for spin 1/2 fermions (e.g. electrons). The conductors, is a state produced by subtle quantum inter- significance of this latter statement is that a necessary ference effects which arise from the fact that electrons are condition for an electronic system to be adiabatically con- fermions. Nevertheless, one generally considers band in- nected to a weakly interacting band-insulator is that ν∗ sulators to be “simple” because the band theory of solids be an even integer. Insulating states can also occur when successfully accounts for their properties; although the ν∗ is a fraction (for fermions this includes odd-integer physical origin of the insulating character of a Mott in- ν∗). Usually when that happens, translational symme- sulator is understandable to any child, other properties, try is spontaneously broken so that the new unit cell has especially the response to doping, are still only at best integral occupation number (or even integral occupation partially understood. number for fermions)[10, 11]. For instance, electronic There is even debate concerning the definition of a Mott insulators with ν∗ = 1 often exhibit antiferromag- Mott insulator. In many of the most familiar cases[3] netic N`eel long-range order, which doubles the unit cell involving electrons, one can, in theory, continuously vary leading to an effective ν∗ = 2. Nevertheless, we will the strength of the repulsion between electrons, such that eff mention [6, 7, 8] model bosonic systems for which the in the limit of vanishing strength interactions the system Mott state can be shown to have no broken symmetries can be well understood on the basis of band theory, while for ν∗ = 1/2, and fermionic systems in which ν∗ = 1. in the strong interaction limit Mott physics better ac- (Currently, no laboratory system has been found which counts for the insulating properties; if there is no phase unambiguously exhibits this exotic behavior.) transition between these two limits, it is not clear at what interaction strength one should start thinking of the sys- The purpose of this paper is to address the response tem as a Mott insulator. However, recently an interest- of Mott insulators to light doping (i.e. ν → ν∗ − δ). Our arXiv:cond-mat/0208536v1 [cond-mat.str-el] 27 Aug 2002 ing Bose Mott insulator has been produced by trapping central observation is that there are two types of Mott in- bosonic neutral atoms in an optical lattice[4]; there is no sulators (henceforth referred to as Type I and Type II). In non-interacting limit in which bosons are insulating, so a Type I Mott insulator an increasing chemical potential no such ambiguity arises here. induces a first order phase transition from an undoped Mott states, in addition to being insulating, can be state to a charge-rich state so that changes occur discon- characterized by the presence or absence of a spon- tinuously. In other words, there is a range of “forbidden taneously broken symmetry (e.g. spin antiferromag- charge density,” and two-phase coexistence. In a Type II netism), by the nature of the low energy excitation Mott insulator, charges go in continuously above a criti- spectrum (e.g. gapped or gapless)[5], and by the cal chemical potential. Since doping of a band insulator presence or absence of topological order and charge is continuous, a doped Type II Mott insulator may be 2 adiabatically connected to a band insulator, but a doped have antiferromagnetic order when undoped, there ex- Type I Mott insulator is a thermodynamically distinct ists considerable evidence that doping induces spa- state of matter. tial inhomogeneity[20, 21, 22, 23, 24]. Conversely, In the presence of long-range Coulomb forces macro- Sr1−xLaxTiO3, which is also an antiferromagnet when scopic charge inhomogeneity is impossible. However, so undoped[25], is generally believed[26] to be Type II on long as the long-range forces are not too strong, there can the basis of the observation of conventional Fermi liquid exist a range of low doping in which the doped charges behavior[27] for doping as low as x = 5%, with an effec- tend to cluster[12]. We still refer to this kind of system tive mass that shows a tendency to diverge with decreas- as a Type I Mott insulator[13]. For instance, if a low ing x; taken at face value[28], this evidence of a quantum density of doped charges form puddles or stripes[14], we critical precursor to a continuous metal-insulator transi- consider this a form of charge clustering, while a Wigner tion indeed suggests that this material is Type II. crystal of doped charges is deemed homogeneous. Another purpose of this paper is to elucidate the What is the situation in real materials? That simple analogy between Type I and Type II Mott insulators semiconductors can be continuously doped, and hence and the familiar types of superconductors. In partic- are Type II, is well known, although it is not clear they ular, when the constituent particles are bosons there are profitably thought of as Mott insulators, even when exists a mathematically precise mapping, the so-called correlation effects significantly renormalize the gap mag- “duality transformation”[10, 29], that relates the zero- nitude. There are many materials in which the insulating temperature properties of the doped Mott insulator in state is also antiferromagnetically ordered. It has been two spatial dimensions, D=2, to the finite-temperature argued[15, 16, 17] that doping of this class of Mott insu- response of a 3D superconductor to a magnetic field. Ta- lators generically[18] leads to phase separation, i.e. that ble I summarizes the correspondence between the two. they are Type I. We will show below that the presence of This same mapping was exploited previously by Balents, a broken symmetry always increases the tendency of an Fisher, and Nayak[30] in the context of a theory of a insulator to be Type I, although the conjectured strict proposed “nodal liquid” Mott insulating state; in this re- connection has neither been proven[19] nor disproved. gards, the present paper simply underlines the generality For the cuprates and manganates, both of which of the analogy. T =0 properties of 2D Bose Mott insulators T > 0 properties of 3D Superconductors Doping Applying magnetic field Chemical potential µ Applied magnetic field H Induced particle density ρ Magnetic induction B World line of doped particles Flux tubes Quantum delocalization of doped particles Thermal meandering of flux tubes Type I Mott insulator Type I superconductor Mott gap Hc Effective attraction between doped particles Positive N-S interface energy Type II Mott insulator Type II superconductor Effective repulsion between doped particles Negative N-S interface energy Mott gap Hc1 Wigner crystal of doped particles Abrikosov flux lattice Superfluid state Entangled vortex fluid Critical µ at which Wigner crystal melts Hc2 I. DOPING A TRANSLATIONALLY experiments in Ref.[4]. Consider the following Hamilto- INVARIANT BOSE MOTT INSULATOR nian To begin with, we avoid the interesting complications which arise from spontaneous symmetry breaking or from t + U + + H = − (a aj + h.c.)+ (a ai)(a ai − 1) the fermionic character of electrons by focusing on the 2 i 2 i i <ij>X Xi simplest kind of Mott insulators formed by spin zero 1 point bosons on a lattice at ν = 1.