arXiv:cond-mat/0208536v1 [cond-mat.str-el] 27 Aug 2002 pcrm(..gpe rgpes[] n ythe charge by and and order excitation topological gapless)[5], energy of or low absence gapped the or spon- of antiferromag- presence a (e.g. nature spin of the spectrum by absence (e.g. or symmetry netism), presence broken the taneously by characterized so here. insulating, arises are ambiguity no bosons such is which no there in lattice[4]; trapping optical limit by an non-interacting produced in interest- been atoms an neutral has recently bosonic insulator However, Mott Bose insulator. sys- ing Mott the a of as thinking what start tem phase at should clear no one not strength is is ac- it interaction there limits, better two if these physics properties; between transition Mott insulating the limit while for interaction theory, counts band strong of basis the the in system on the understood interactions well strength be vanishing can of that such cases[3] limit electrons, the familiar between in repulsion most the vary of continuously the theory, strength the in of can, many one electrons, In involving insulator. best Mott at only still are doping, understood. properties, to partially in- other response Mott child, the a any especially of to character understandable the is insulating sulator although the of properties; origin solids their of physical theory for band accounts the because successfully in- “simple” band be considers to generally sulators are electrons one that Nevertheless, fact the inter- fermions. from quantum arise In subtle which by effects semi- produced ference familiar statistics.) state all a any[1] is including of conductors, suppressed. insulator[2], particles band are nature a to next contrast in accessible the classical essentially so to thus (and site is motion insulator one the Mott from involving The particle fluctuations particles a that two site of between crystalline same repulsion a the of large site on a ground-state each such classical on and particle a one lattice with is simplest system there repul- the a which strong in is flow; the this their which of impedes in cartoon particles system between a level, simple is sion a condensed insulator At of Mott decades. two focuses a past central the of the physics of matter one constitutes tors otsae,i diint en nuaig a be can insulating, being to addition in states, Mott a of definition the concerning debate even is There insula- Mott doped and insulators Mott of study The xssacoeaaoybtenteetotpso otinsula Mott of types l two the chem these superconductors. increases between II critical this analogy symmetry, close a long-r broken a above of exists a continuously has presence state the in insulating se In go “micro-phase charges exhibiting phase. state insulators doped inhomogeneous an the insulators to to Mott leads undoped I the Type from potential, chemical increasing With hr r w lse fMt nuaosi aue distingu nature, in insulators Mott of classes two are There ( c ) ( hsc eatetUiest fClfri,LsAngeles Los California, of Department,University Physics a ( ) b eateto hsc,nvriyo aiona Berkeley California, of Physics,University of Department ) etrfrAvne td,Tigu nvriy ejn 10 Beijing University, Tsinghua Study, Advanced for Center ugHiLee Dung-Hai h ye fMt insulator Mott of types The a,b n tvnA Kivelson A. Steven and aln ytm tioae ainl“cuainnum- crys- “occupation in rational occur isolated insulators) bers,” at Mott systems and talline band both cludes doping. categories to two response “type” into their a insulators on add Mott depending we list, classifies this that To index 9]. 8, 7, fractionalization[6, assteocpto ubrt hf wyfrom away shift to number which process occupation a mean the we “doping” causes By cell. unit chemical otsaecnb hw ohv oboe symmetries the broken which no have for to systems shown for bosonic be can model state 8] Mott 7, [6, mention ubrfrfrin)1,1] o ntne electronic instance, For with insulators 11]. Mott fermions)[10, occupation has for integral cell even unit number (or new number the that occupation so integral broken spontaneously is try ν ea vnitgr nuaigsae a loocrwhen occur also can states Insulating ν integer. even an be etdt ekyitrcigbn-nuao sthat is band-insulator interacting necessary weakly a con- a adiabatically to that be nected to The is system electronic statement electrons). an for latter condition (e.g. this fermions of 1/2 significance spin an and for above, integer described even cartoon classical simple the with hnltietasainsmer sntspontaneously not is broken, symmetry translation lattice When ei `e ogrneodr hc obe h ntcell effective unit an the to doubles leading which order, N`eel long-range netic fMt nuaost ih oig(i.e. doping light to insulators Mott of which behavior.) found exotic been this has exhibits unambiguously system laboratory no (Currently, scniuu,adpdTp IMt nuao a be may insulator Mott II insulator Type band doped a a of continuous, doping criti- Since is a above potential. continuously chemical in cal II go Type “forbidden charges a of insulator, In range Mott coexistence. a two-phase is and discon- density,” there occur charge undoped words, changes other an that In so from tinuously. state transition charge-rich a phase to order potential state chemical first increasing a an insulator In induces II). Mott Type I and Type I in- a Type Mott as of to types referred two (henceforth are sulators there that is observation central ∗ ∗ eeal paig ttswt hregp ti in- (this gaps charge with states speaking, Generally h ups fti ae st drs h response the address to is paper this of purpose The .Uulywe hthpes rnltoa symme- translational happens, that when Usually ). safato frfrin hsicue odd-integer includes this fermions (for fraction a is ν ∗ ν 1 = aain”I otat nTp IMott II Type in contrast, In paration.” se yterrsosst ekdoping. weak to responses their by ished klho hti ilb yeI There I. Type be will it that ikelihood ν osadtefmla yeIadType and I Type familiar the and tors neg rtodrpaetransition phase order first a undergo clptnil eso hti the if that show We potential. ical = ∗ / stpclya nee o oos consistent bosons, for integer an typically is ν ,adfrincssesi which in systems fermionic and 2, ∗ neCuobitrcin,this interactions, Coulomb ange where , A90594 USA 90095-9547 CA , A970 USA 94720, CA , c ν 04 China 0084, ∗ ν ν eff fe xii antiferromag- exhibit often 1 = ∗ stenme fprilsper particles of number the is .Nvrhls,w will we Nevertheless, 2. = ν → ν ∗ − ν δ ∗ .Our ). 1. = ν ν ∗ ∗ . 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 X Xi simplest kind of Mott insulators formed by spin zero 1 point bosons on a lattice at ν = 1. This is not a purely + V a+a a+a − µ a+a , (1) 2 ij i i j j i i academic exercise[31], as it applies rather directly to the Xi,j Xi 3 where i, j label the lattice sites on a D-dimensional hy- N is the total number of lattice sites.) For Jz < Jxy + percubic lattice, and ai creates a boson at site i. The (i.e.V Hubbard model, and has been extensively is Type II, and the doped system is a uniform superfluid studied from various perspectives[32]. for all 1 >ν> 0. Let us focus on the U/t → ∞ limit. In this “hardcore” Conversely, for Jz > Jxy (i.e. V >t) the model is limit, doubly occupied sites cost too much energy, hence effectively an Ising ferromagnet with a fully polarized are excluded. In this case for large positive µ there is an ground state. In this case Mz exhibits a discontinuity unique ground state ∆Mz = N as hz is varied through zero. If we consider a constrained groundstate, with |Mz| < N/2, the state + |Mott >= aj |0 >, (2) exhibits phase separation into two oppositely polarized Yj domains. In terms of the bosons, these two domains are Mott insulating (ν = 1) and empty (ν = 0) respectively. in which each site is occupied by one and only one boson On the boundary between the two behaviors (Jz = Jxy), and hence ν = 1. Since this state is separated from all the system is equivalent to a Heisenberg ferromagnet in other states with the same particle number by an energy a magnetic field. In the absence of h the ground state is gap of order U, clearly we have an insulator. (For µ large z N+1 fold degenerate correspond to Mz = −N/2, ..., N/2. and negative, the groundstate is the empty lattice, |0 >.) In the presence of h the state with M = ±N/2 (± de- The behavior of the system upon varying µ (i.e. dop- z z pends on the sign of hz) has the lowest energy. For this ing) depends on the values of Vij . For Vij = 0 the doped system there is again a first order transition at hz = 0. holes (i.e the empty sites) interact only through the hard- However, precisely at the transition point the boson in- core exclusion. Since the kinetic energy favors a uni- verse compressibility κ−1 diverges, and a uniform ground form density state with delocalized holes, this limit cor- state exists for any ν; at this critical point, only, the sys- responds to a Type II Mott insulator. Indeed, a small tem is neither truly Type I nor Type II. concentration of doped holes, δ ≡ 1 − ν << 1, is equiva- lent to a system of dilute bosons (defined relative to the vacuum state |Mott >) with short-range interactions, a problem whose solution has long been known[33, 34]. The II. DUALITY TRANSFORMATION FOR THE BOSE MOTT INSULATOR IN D=2 energy is clearly minimized by delocalizing the holes so the ground state in D = 2 is a uniform superfluid with superfluid stiffness proportional to tδ. Conversely, if Vij There is a particularly convenient way of thinking is attractive (negative), it is clear that the holes will clus- about Bose Mott insulators that is specific to D = 2. It turns out that for the class of model given by Eq. (1), ter when | i Vij | ≫ t. Thus depending the nature of Vij Eq. (1) canP describe either a Type I or Type II Mott in- there is an exact mapping, the “duality” transformation, sulator. Note that in this example, the Mott state does that provides us an alternative view of the physics of not break any symmetry, so any inhomogeneity of the Eq. (1) in terms of the vortices of the boson field[10, 29]. the doped state cannot be attributed to order parameter It is this mapping that enables us to establish a precise competition. The Type I behavior discussed here is more connection between the two types of Mott insulators with general[35] than that which occurs in the bicritical point Type I and Type II superconductors[10, 29, 30]. In the scenario in the Landau theory of two competing order following we discuss the physical content of the duality parameters[36]. transformation without going into its technical details of To be still more explicit, consider the case in which the transformation[10]. Vij = −V for (ij) nearest-neighbor sites, and Vij = 0 A vortex is a topological defect in the Bose field. When otherwise. Technically in this limit Eq. (1) is equivalent a boson is adiabatically transported around a vortex, to the S=1/2 ferromagnetic XXZ model in a z-direction the boson wavefunction acquires a phase factor - the magnetic field Aharonov-Bohm phase θ = 2π. In the dual picture, the role of the vortices and the particles are interchanged: x x y y z z z H = −Jxy (Si Sj +Si Sj )−Jz Si Sj −hz Si . the vortices are the dual particles (they turn out to have X X Xi Bose statistics as well), and when a dual particle is trans- (3) ported around an original boson, the wave function ac- The mapping between these two models relates Jxy to t, quires the same Aharonov-Bohm phase[10, 29] discussed Jz to V , hz to µ + V c/2 (c = the coordination number). above. The fact that a boson and a vortex acquire a phase The z component of the magnetization is related to the when they go around one another implies that bosons boson density according to Mz = (ν − 1/2)N. (Here and vortices can not Bose condense simultaneously. As 4 is well known, in the Bose superfluid phase the vortices The properties of classical 3D superconductors in a must be absent or localized, i.e. form an Abrikosov lat- magnetic field are generally well known, so most of the tice. Conversely, in the dual phase, where the vortices comparisons in Table I are self-evident. We therefore form a superfluid[37], the boson density (and hence the confine ourselves to commenting on a few of the subtleties dual magnetic flux) must be frozen; the vortex super- of this comparison: fluid phase is the Mott insulating phase of the original Although a Type I supercondunductor typically expels bosons[10, 29]. any applied field of magnitude H , < jk >, < kl >, and < li > are all nearest neighbor and 0 otherwise. For sufficiently pairs of nearest-neighbors. strong V and µ = 0 the ground state breaks translation On a square lattice, all zero temperature Mott insu- symmetry and bosons form a checkerboard lattice and lating phases with ν = 1/2 derived from this Hamil- Mott insulate. In this two-fold degenerate ground state tonian are believed to break translational symmetry, at the unit cell is doubled. Is this Mott insulator Type I or least doubling the unit cell size, so that the effective ν Type II? is integral[7]. However, on a triangular lattice, it has With the above specific choice of U and Vij Eq. (1) is now been established[7] that for a range of parameters equivalent to Eq. (3) with Jxy = −t, Jz = V and h = near t ∼ V , there is a dimer liquid phase, with exponen- µ − V c/2. For this choice of parameters it is known that tially falling correlation functions and no broken symme- as a function of hz Eq. (3) exhibits a “spin flop” transition tries. From the dual viewpoint, this phase can be viewed from the antiferromagnetic Ising (Sz) ordered phase into as a superfluid of vortex pairs[6]. As mentioned previ- the ferromagnetic XY ordered phase. Translate this into ously, there are observable topological consequences of the boson language it implies that it is Type I. the vortex pairing, including certain predictable ground- There is, in fact, a general reason to expect broken state degeneracies on closed surfaces. symmetry to increase the tendency of a Mott insulator This same model can serve as a paradigmatic example to be Type I. If there is a broken symmetry in the in- of a Mott insulating state of spinfull fermions at an odd sulating state, then there generally is a corresponding integer ν = 1. To see this, we merely need to define † local order parameter (ψ) (for the above example ψ is fermion creation operators, cσ,ij , for an electron with spin the two-sublattice density wave order parameter). We polarization σ on bond . In terms of this, we can now consider the effect of this order parameter on the express the dimer creation operators as interface energy between the insulating and the metal- † † † lic (doped) state. Typically, ψ will have a reduced value bij = c↑,ij c↓,ij (6) (perhaps 0) in the doped state. Thus, at the interface between doped and undoped region a spatially varying ψ † † and an additional constraint, σ cσ,ij cσ,ij = 0 or 2, is necessary. The spatial gradient energy which can be thought of as a consequenceP of a strong attractive interaction between two electrons on the same ddx|∇ψ|2 (4) bond. Z Is this sort of fractional Mott insulator Type I or will thus make a positive contribution to the interface Type II? The intensive study of what happens when energy, increasing the tendency of the Mott insulator to one dopes such a “spin-liquid” state was initiated by be Type I. the proposal[43] that doping such a liquid would lead inevitably to high temperature superconductivity. The basis for this proposal was the thought that pairing cor- IV. AN EXAMPLE OF TRANSLATIONALLY relations (e.g. a spingap[9]) might be present already INVARIANT BOSE MOTT INSULATOR AT in the insulating state, and these would evolve smoothly ν = 1/2 into superconducting pairing upon doping. Central to this proposal is the assumption that the spin-liquid is a A recently discovered system which exhibits a Bose Type II Mott insulator. Indeed, an early triumph of this Mott insulating state at a fractional ν is the quantum idea was the observation by Ioffe and Larkin[8, 44] that, hardcore dimer model on a triangular lattice. The model for at least a range of parameters, doped holes in the is defined in terms of bosons (called dimers) which live quantum dimer model do not phase separate. Moreover, reversing the logic of the previous section, the fact that on the nearest neighbor bonds of a lattice, i.e. b† creates ij this state has no broken symmetry increases the likeli- a boson on the bond connecting sites i and j. We impose hood that it is Type II. a hardcore constraint - this can be thought of as deriving from the U → ∞ limit of a contact interaction, which forbids two dimers to occupy the bonds emanating from † † V. CONCLUSIONS a single site, i.e. bij bij bjkbjk = 0 for any triplet of nearest neighbor sites, < ijk >. Thus, since each bond touches exactly two sites, it is possible on any regular lattice with To summarize, in this work we introduce the concept that there are two types of Mott insulator. In particular, one site per unit cell to satisfy this constraint if and only if ν ≤ 1/2. The simplest (shortest range) Hamiltonian[8] there exist a whole class of Mott insulators, the type I Mott insulators, that become inhomogeneous upon one can construct for these dimers is doping. The type I behavior does not have to originate † † from order parameter competition. We give a familiar H = −t [b b bilbjk + h.c.] (5) (ijkl) ij kl lattice boson example where the Mott state exhibits P † † † † +V (ijkl)[bij bklbklbij + bilbkj bkj bil] no symmetry breaking, however depending on the sign P 6 of certain interaction, it can be either type I or type insulator?” “What is the ultimate low energy behavior II after doping. Through duality these two types of of these inhomogeneities?” insulating states have a close tie to Type I and Type II superconducting states.[30] Finally after addressing the “zeroth order” physics of doped Mott insulator, we point Acknowledgements: We wish to thank C.Nayak for out that many important “first order” issues remain useful discussions. This work was supported in part by unanswered. Among them are “what determines the NSF grants # DMR 99-71503 (DHL) and DMR - 0110329 structure of the inhomogeneity in a doped Type I Mott (SAK).

[1] In two space dimensions hardcore particles with quan- D.Andelman, Science 267, 476-483 (1995). tum statistics between that of fermions and bosons are [15] V.J.Emery, S.A.Kivelson and H-Q.Lin Phys. Rev. Lett. allowed. See, “Fractional Statistics and Anyon Super- 64, 475-478 (1990); S.A.Kivelson and V.J.Emery, in conductivity” edited by Frank Wilczek, World Scientific, Strongly Correlated Electronic Materials: The Los 1990. Alamos Symposium 1993, Eds. K. S. Bedell, Z. Wang, D. [2] A band insulator insulates because of Pauli exclusion E. Meltzer, A. V. Balatsky, and E. Abrahams (Addison- principle. In the non-interacting prototype of this type Wesley, Reading Massachusetts, 1994) pp. 619-656. of insulator all available single-particle orbitals are oc- [16] E.L.Nagaev JETP Lett 6, 18 (1967) and 16, 394 (1972); cupied. This type of insulator usually possesses a even P.B.Visscher Phys. Rev. B 10, 943 (1974). number of electrons per crystalline unit cell. [17] D.I. Khomskii and K.I. Kugel, Europhysics Letters 55, [3] For instance, a two leg Hubbard ladder with suitable 208 (2001). choice of the rung hopping evolves adiabatically from a [18] There are, in each case, various additional conditions con- band insulator, for small U, to a so-called Haldane gap cerning the nature of the antiferromagnet specified in the system, one of the canonical examples of a Mott insulat- various conjectural relations between antiferromagnetism ing state of electrons without a broken symmetry, in the and Type I behavior. Before gleefully constructing pro- large U limit. posed counterexamples, such as a doped Ising antiferro- [4] M. Greiner et al, Nature, 415, 39 (2002). magnet with additional strong, short-range repulsive in- [5] A Mott insulator necessarily has a charge gap, however teractions, the reader should consult the original papers. it is possible for it to have gapless neutral excitations. [19] It was shown by L. Pryadko, D.Hone, and S.A.Kivelson [6] T. Senthil and M.P.A. Fisher, Phys. Rev. B 63, 134521 [Phys. Rev. Lett. 80, 5651-5654 (1998)] that a long-range (2001); C. Nayak and K.Shtengel, Phys. Rev. B 64, uniformly attractive interaction between static charges is 064422 (2001); X-G Wen, cond-mat/0107071; L. Balents, induced by spin-wave fluctuations in a Heisenberg quan- M. P. A. Fisher, S. M. Girvin, cond-mat/0110005. tum antiferrmagnet - these forces are sufficient to prove [7] R. Moessner and S.L. Sondhi, Phys. Rev. Lett. 86, 1881 that phase separation is generic in doped antiferromag- (2001); R.Moessner, S.L.Sondhi, and E.Fradkin, Phys. nets up to a few nasty caveats. Rev. B 65, 24504 (2002). [20] For a review of stripes in the cuprates see V. J. Emery, [8] D.S. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, S. A. Kivelson, J. M. Tranquada Proc. Natl. Acad. Sci. 2376-9 (1988). USA 96, 8814 (1999). [9] S.A.Kivelson, D.S.Rokhsar, and J.P.Sethna, Phys. Rev. [21] C. Howald, P. Fournier, and A. Kapitulnik, Phys. Rev. B 35, 8865 (1987); S.A.Kivelson, Phys. Rev. B 39, 259- B 64, 100504 (2001); S. H. Pan et al. Nature, 413, 282 264 (1989). (2001). [10] D-H Lee and R. Shankar, Phys. Rev. Lett., 65, 1490 [22] K.M. Lang et al, Nature 415, 412 (2002). (1990). [23] For a review of the possible role of intrinsic electronic [11] M. Oshikawa, Phys. Rev. Lett. 84, 1535 (2000). inhomogeneity in the manganates see, e.g., E. Dagotto, [12] U. L¨ow et al, Phys. Rev. Lett. 72, 1918 (1994); T. Hotta, A. Moreo, Physics Report 344, 1 (2001), and V.J.Emery and S.A.Kivelson, Physica C 209, 597-621 Rev. Mod. Phys., in press. (1993). [24] For a more general experimental review of stripe forma- ∗ [13] Strictly speaking, in the limit δ ≡ ν − ν → 0, since tion in doped antiferromagnets, see J.M. Tranquada, J. the Madulung energy per hole diverges, the groundstate Phys. Chem. Solids, 59, 2150 (1988). in the presence of long-range Coulomb interactions is al- [25] B. Keimer et al, Phys. Rev. Lett. 85, 3946 (2000). ways a Wigner crystal of holes, independently of whether [26] M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70, in the absence of such interactions the state would be 1039 (1998). Type I or Type II. However, where the Coulomb interac- [27] Y. Tokura et al, Phys. Rev. Lett. 70, 2126 (1993). tions are weak, due to the presence of a large dielectric [28] Although the cited evidence suggests that LaTiO3 is a constant, clustering occurs in a Type I Mott insulator Type II Mott insulator, and thus a counterexample to the for δ > δc, where δc is a parametrically small (in powers general conjecture that antiferromagnetic insulators are of ǫ−1 ) critical doping concentration. In the absence of generically Type I, it seems to us that more work needs to long-range forces and for small enough δ , the charges in be done to establish this fact. The apparent divergence of a Type II Mott insulator always form a uniform fluid. the effective mass with decreasing x reported in [27], and [14] For an extremely pedagogical review of the consequences the good Fermi liquid character of the metallic state even of Coulomb frustrated phase separation, see M.Seul and at the smallest reported x, certainly argue in favor of a 7

continuous transition to the insulating state. However, [33] V. N. Popov, Theor. Math. Phys. 11, 565 (1972). indirect NMR evidence, Y.Furukawa et al, Phys. Rev. [34] D.S. Fisher and P.C. Hohenberg, Phys. Rev. B 37, 4936 B 59, 10550-10558 (1999), apparently implies that there (1988). is little or no antiferromagnet character to the magnetic [35] M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. fluctuations in the metallic phase; a continuous transi- A 4, 1071 (1971) tion between a paramagnetic metal and an antiferromag- [36] For a discussion of the bicritical point scenario in the netically ordered insulator would be expected to exhibit context of the cuprate superconductors, see S.-C. Zhang diverging (quantum critical) antiferromagnetic fluctua- Science 275, 1089 (1997). tions as the critical point is approached. The failure to [37] We note that the vortex superfluid discussed here differs see such fluctuations is most naturally understood if the from the generic boson superfluid because the net vortic- transition is ultimately first order. The issue is further ity is zero. The field theory that describes such ”particle- complicated by the fact that the correlation effects are hole symmetric” condensate has space and time on equal relatively weaker in this material than in the cuprates footing (i.e. relativistic). This fact makes the later anal- or manganates; the charge gap in the insulating state is ogy with thermal fluctuating 3D superconductors com- only about 0.1eV. This might well mean that the range pletely appropriate. of doping over which the actual physics of the “doped an- [38] S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 1999 tiferromagnet” is dominant may be rather small, and in (1983). fact the experiments seem never to access the important [39] The field strength Fµν (µ = 0, 1, 2) of a 2+1-dimensional range of doping with x < 5%. It would be very interest- EM field has three independent components. The corre- ing to look more closely at the low doping regime of this spondence of the these three components with the bo- and related materials, and especially to study the mag- son three-current (ρ, j) is as follows: F12 ↔ ρ, F01 ↔ netic structure by neutron scattering in this regime, to j2, F02 ↔ j1. unambiguously determine whether they are truly Type [40] In the correspondence between a quantum superconduc- II. tor at T = 0 and a thermally fluctuating superconductor [29] M.P.A. Fisher and D-H Lee, Phys. Rev. B 39, 2756 at finite temperature the magnetic field (F12) and electric (1989). field (F01, F02) in the 2+1 dimensional quantum theory [30] L. Balents, M.P.A. Fisher and C. Nayak, Int. J. Mod. are mapped on to Bz and (By, −Bx) respectively. Phys. B 12, 1033 (1998). [41] M. Tinkham “Introduction to superconductivity”, [31] The system considered in this Ref.[4] can be “doped” by McGraw-Hill, New York (1975); P.-G. de Gennes, “Su- varying the period of the trap lattice. perconductivity of metals and alloys”, Benjamin, New [32] For an introduction to the bose Hubbard model, see, York (1966). for example, R.T.Scalettar et al, Phys. Rev. B 51, 8467 [42] D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988). (1995) and especially, G. G. Batrouni and R. T. Scalet- [43] P.W.Anderson, Science 235, 1196-8 (1987). tar Phys. Rev. Lett. 84, 1599-1602 (2000) where issues [44] L.Ioffe and A.Larkin, Phys. Rev. B 40, 6941-7 (1989). of phase separation are explicitly addressed.