AJO-DO 1972 Sep (296-309): The six keys to normal occlusion - AndrewsThe six keys to normal occlusionLawrence F. Andrews, D.D.S.San Diego, Calif.This article will discuss six significant characteristics observed in a study of 120 casts ofnonorthodontic patients with normal occlusion. These constants will be referred to as **the** "sixkeys to normal occlusion." The article will also discuss **the** importance of **the** six keys,individually and collectively, in successful orthodontic treatment.Orthodontists have **the** advantage of a classic guideline in orthodontic diagnosis, that is, **the**concept given to **the** specialty a half-century ago by Angle that, as a sine qua non of properocclusion, **the** cusp of **the** upper first permanent molar must occlude in **the** groove between **the**mesial and middle buccal cusps of **the** lower first permanent molar.But Angle, of course, had not contended that this factor alone was enough. Clinicalexperience and observations of treatment exhibits at national meetings and elsewhere hadincreasingly pointed to a corollary fact— that even with respect to **the** molar relationship itself,**the** positioning of that critical mesiobuccal cusp within that specified space could be inadequate.Too many models displaying that vital cusp-embrasure relationship had, even after orthodontictreatment, obvious inadequacies, despite **the** acceptable molar relationship as described byAngle.Recognizing conditions in treated cases that were obviously less than ideal was not difficult,but nei**the**r was it sufficient, for it was subjective, impressionistic, and merely negative. Areversal of approach seemed indicated: a deliberate seeking, first, of data about what wassignificantly characteristic in models which, by professional judgment, needed no orthodontictreatment. Such data, if systematically reduced to ordered, coherent paradigms, could constitutea group of referents, that is, basic standards against which deviations could be recognized andmeasured. The concept was, in brief, that if one knew what constituted "right," he could **the**ndirectly, consistently, and methodically identify and quantify what was wrong.A ga**the**ring of data was begun, and during a period of four years (1960 to 1964), 120nonorthodontic normal models were acquired with **the** cooperation of local dentists,orthodontists, and a major university. Models selected were of teeth which (1) had never hadorthodontic treatment, (2) were straight and pleasing in appearance, (3) had a bite which lookedgenerally correct, and (4) in my judgment would not benefit from orthodontic treatment.The crowns of this multisource collection were **the**n studied intensively to ascertain whichcharacteristics, if any, would be found consistently in all **the** models. Some **the**ories took formbut soon had to be discarded; o**the**rs required modification and **the**n survived. Angle's molarcusp groove concept was validated still again. But **the**re was growing realization that **the** molarrelationship in **the**se healthy normal models exhibited two qualities when viewed buccally, notjust **the** classic one, and that **the** second was equally important.O**the**r findings emerged. Angulation (mesiodistal tip) and inclination (labiolingual orArticle Text 1

PRL 107, 026401 (2011) PHYSICAL REVIEW LETTERSweek ending8 JULY 2011solution, which is formally exact in **the** limit of largecoordination. Here **the** Hubbard model maps onto aneffective Anderson impurity model supplemented by aself-consistency condition [7]. To solve **the** DMFT equationswe use **the** iterated perturbation **the**ory (IPT) [7] andcross-check our results with numerically exact continuoustime quantum Monte Carlo (CTQMC) calculations [8,9].We find, in agreement with previous work [10], that afterappropriate energy rescaling (see below), **the** two methodsproduce qualitatively and even quantitatively identical resultsin **the** incoherent crossover region that we examine.It is well known that at very low temperatures TT c , however, different crossover regimes have beententatively identified [7,11], but **the**y have not been studiedin any appreciable detail. The fact that **the** first-ordercoexistence region is restricted to such very low temperaturesprovides strong motivation to examine **the** high temperaturecrossover region from **the** perspective of ‘‘hiddenquantum criticality.’’ In o**the**r words, **the** presence of acoexistence dome at TT c , along this instability line. Consequently, as inRefs. [10,13,14], our problem is recast as an eigenvalueanalysis of **the** corresponding free energy functionalF ½Gðiw n ÞŠ for which **the** DMFT Green’s function solutionG DMFT ðiw n Þ represents a local extremum and can be regardedas a vector in an appropriate Hilbert space.The free energy **near** such an extremum can be written asF ½Gðiw n ÞŠ ¼ F 0 þ Tt 2P m;nGði! m ÞM mn Gði! n Þþ,whereM mn ¼ 1 @ 2 F ½GŠ2Tt 2(2)@Gði! m Þ@Gði! n Þ G¼GDMFTand Gði! n ÞGði! n Þ G DMFT ði! n Þ. The curvature of**the** free energy functional is determined by **the** lowesteigenvalue of **the** fluctuation matrix M. As explained inRef. [15], can be obtained from **the** iterative solution ofDMFT equations. The difference of **the** Green’s functionsin iterations n and n þ 1 of **the** DMFT self-consistencyloop is given byG ðnþ1Þ ði! n Þ G ðnÞ ði! n Þ¼e n G ð0Þ ði! n Þ; (3)and **the**refore determines **the** rate of convergence of **the**Green function to its solution.An example of our calculations is shown in **the** inset inFig. 2, where **the** eigenvalues at several temperatures areplotted as a function of interaction U=U c . The minima of**the**se curves define **the** locus of **the** instability trajectoryU ? ðTÞ, which terminates at **the** critical end point (U c , T c ),as shown in Fig. 2. Note that **the** immediate vicinity T T cof **the** critical end point has been carefully studied **the**oretically[10] and even observed in experiments [2],revealing classical Ising scaling (since one has a finitetemperature critical point) of transport in this regime. Inour study, we examine **the** crossover behavior at muchhigher temperatures T T c , displaying very differentbehavior: precisely what is expected in presence of quantumcriticality.Resistivity calculation.—To reveal quantum criticalscaling, we calculate **the** temperature dependence of **the**resistivity along a set of trajectories parallel to our instabilitytrajectory [fixed U ¼ U U ? ðTÞ]. Resistivity wascalculated by using standard DMFT procedures [7], with026401-2

PRL 107, 026401 (2011) PHYSICAL REVIEW LETTERSweek ending8 JULY 2011**the** maximum entropy method [16] utilized to analyticallycontinue CTQMC data to **the** real axis. The resistivityresults are shown in Fig. 1, where in panel (a) IPT resistivitydata for U ¼ 0; 0:025; 0:05; 0:1; 0:15; 0:2in **the** temperature range of T 0:07–0:2 are presented(CTQMC data are not shown for **the** sake of clarity of **the**figure). The resistivity is given in units of **Mott** , maximalresistivity according to **the** Boltzmann quasiclassical**the**ory of transport [17]. The family of resistivity curvesabove (U > 0) **the** ‘‘separatrix’’ c ðTÞ (dashed line, correspondingto U ¼ 0) has an insulatinglike behavior,while metallic dependence is obtained for U < 0.Scaling analysis.—According to what is generally expectedfor quantum criticality, our family of curves shouldsatisfy **the** following scaling relation:ðT;UÞ ¼ c ðTÞfðT=T 0 ðUÞÞ: (4)We thus first divide each resistivity curve by **the** separatrix c ðTÞ ¼ðT;U ¼ 0Þ and **the**n rescale **the** temperature,for each curve, with an appropriately chosen parameterT 0 ðUÞ to collapse our data onto two branches [Fig. 1(b)].Note that this unbiased analysis does not assume anyspecific form of T 0 ðUÞ: It is determined for each curvesimply to obtain optimum collapse of **the** data [18].This puts us in a position to perform a stringent test ofour scaling hypo**the**sis: True quantum criticality correspondsto T 0 ðUÞ, which vanishes at U ¼ 0 and displayspower-law scaling with **the** same exponents for both scalingbranches. As seen in Fig. 3(a), T 0 falls sharply asU ¼ U ? is approached, consistent with **the** QC scenariobut opposite to what is expected in a ‘‘classical’’ phasetransition. The inset in Fig. 3(a) with log-log scaleshows clearly a power-law behavior of T 0 ¼ cjUj z , with**the** estimated power ðzÞ IPTU0¼ 0:57 0:01 for an insulatingbranch.We also find [Fig. 3(b)] a very unusual form of our criticalresistivity c ðTÞ, corresponding to **the** instability trajectory.Its values largely exceeds **the** **Mott** limit, yet it displaysmetalliclike but non-Fermi liquidlike temperature dependence c ðTÞT. Such puzzling behavior, while inconsistentwith any conventional transport mechanism, has beenobserved in several strongly correlated materials close to**the** **Mott** transition [17,20]. Our results thus suggest that itrepresents a generic feature of **Mott** quantum criticality. function and mirror symmetry of scaled curves.—Tospecify **the** scaling behavior even more precisely, we compute**the** corresponding function [4] ðgÞ ¼ d lngd lnT , withg ¼ c = being **the** inverse resistivity scaling function.Remarkably [Fig. 4(a)], it displays a **near**ly li**near** dependenceon lng and is continuous through U ¼ 0 indicatingprecisely **the** same form of **the** scaling function on both sidesof **the** transition—ano**the**r feature exactly of **the** form expectedfor genuine quantum criticality. This functional formis very natural for **the** insulating transport, as it is obtainedeven for simple activated behavior ðTÞe Eg=T . The factthat **the** same functional form persists well into **the** metallicside is a surprise, especially since it covers almost an orderof magnitude for **the** resistivity ratio. Such a behavior hasbeen interpreted [4] to reflect **the** ‘‘strong coupling’’ natureof **the** critical point, which presumably is governed by **the**same physical processes that dominate **the** insulator. Thispoints to **the** fact that our QC behavior has a strong coupling,i.e., nonperturbative character.The fact that **the** function assumes this logarithmicform on both sides of **the** transition is ma**the**maticallyequivalent [4] to stating that **the** two branches of **the**corresponding scaling functions display ‘‘mirror symmetry’’over **the** same resistivity range. Indeed, we find thattransport in this QC region exhibits a surprisingly developedreflection symmetry [dashed vertical lines of Fig. 4(a)mark its boundaries]. Such a symmetry is clearly seen inFig. 4(b), where **the** resistivity = c (for U > 0) andconductivity = c ¼ c = (U < 0) can be mappedonto each o**the**r by reflection with ðUÞ ð UÞ c¼ c[21].0.15T 0=cδU zν(a)1815(b)1β = 0.00889 + 1.76 Ln(ρ c /ρ)zν=0.567(a)4(b)T 00.10.050IPTCTQMCT 00.10.01δU>0δU 0); vertical dashed lines indicate **the**region where mirror symmetry of curves is found. (b)Reflection symmetry of scaled curves close to **the** transition.

PRL 107, 026401 (2011) PHYSICAL REVIEW LETTERSweek ending8 JULY 2011Note that T=T 0 ¼ 1 sets **the** boundary of **the** quantumcritical region, over which **the** reflection symmetry ofscaled curves is observed. It is depicted by dash-dottedcrossover lines T 0 in **the** phase diagram of Fig. 2 [15].These remarkable features of **the** function, and associatedreflection symmetry, have been observed earlier inexperimental [1,21] and **the**oretical [4] studies, whichtentatively associated this with disorder-dominated MITs.Speculation that lng reveals disorder as **the** fundamentaldriving force for MIT presumably reflects **the** fact that,historically, it has first been recognized for Anderson transitions[5]. Our work, however, shows that such behaviorcan be found even in **the** absence of disorder—ininteraction-driven MITs. This finding calls for rethinkingof basic physical processes that can drive **the** MIT.Conclusions.—We have presented a careful and detailedstudy of incoherent transport in **the** high temperature crossoverregime above **the** critical end point T c of a single-bandHubbard model. Our analysis revealed a so-far overlookedscaling behavior of **the** resistivity curves, which we interpretedas evidence of hidden **Mott** quantum criticality.Precisely locating **the** proposed QC point in our model ishindered by presence of **the** low temperature coexistencedome, which limits our quantum critical scaling to **the**region well above T c . Regarding **the** nature of transportin **the** QC regime, we found that **the** critical resistivity wellexceeds **the** **Mott** limit, and yet it—surprisingly—assumesa metallic form, in dramatic contrast to conventionalMIT scenarios. These features, toge**the**r with largeamounts of entropy characterizing this entire regime[22], prove surprisingly reminiscent of **the** ‘‘holographicduality’’ scenario [23,24] for a yet-unspecified QC point.Interestingly, **the** holographic duality picture has—so far—been discussed mostly in **the** context of quantum criticalityin correlated metals (e.g., T ¼ 0 magnetic transitions inheavy fermion compounds). Ours is **the** first work proposingthat **the** same physical picture could apply to quantumcriticality found at **the** MIT.We believe that our results provide a significant newperspective on QC around **the** **Mott** transition and a deeperunderstanding of an apparent universality in **the** high temperaturecrossover regime. Our method traces a clear avenuefor fur**the**r searches for QC scaling, which are likely tobe found in many o**the**r regimes and models.In particular, it would be interesting to study a correspondingcritical regime by going beyond **the** single-siteDMFT analysis. It was shown in Ref. [19] that inclusion ofspatial fluctuations does not significantly modify **the** hightemperature crossover region in **the** half filled Hubbardmodel. Consequently, we expect our main findings topersist. An even more stringent test of our ideas shouldbe provided in models where **the** critical end point T c canbe significantly reduced. This may include studies of **the****Mott** transition away from half filling [25] or in systemswith frustrations [6,26]. In such situations **the** proposedscaling regime should extend to much lower temperatures,perhaps revealing more direct evidence of **the**—so far—hidden **Mott** QC point. Our ideas should also be testedby performing more detailed transport experiments in **the**relevant incoherent regime, a task that may be easilyaccessible in various organic **Mott** systems [3], where T cis sufficiently below room temperature.The authors thank K. Haule for **the** usage of his CTQMCcode and M. Jarrell for **the** use of his maximum entropycode for analytical continuation of **the** CTQMC data. Thiswork was supported by **the** National High Magnetic FieldLaboratory and **the** NSF through Grants No. DMR-0542026and No. DMR-1005751 (H. T. and V. D.) and SerbianMinistry of Science under Project No. ON171017 (J. V.and D. T.). D. T. acknowledges support from **the** NATOScience for Peace and Security Programme GrantNo. EAP.RIG.983235. Numerical simulations were run on**the** AEGIS e-Infrastructure, supported in part by FP7projects EGI-InSPIRE, PRACE-1IP, and HP-SEE.[1] E. Abrahams et al., Rev. Mod. Phys. 73, 251 (2001).[2] P. Limelette et al., Science 302, 89 (2003).[3] F. Kagawa et al., Nature (London) 436, 534 (2005).[4] V. Dobrosavljević et al., Phys. Rev. Lett. 79, 455 (1997).[5] E. Abrahams et al., Phys. Rev. Lett. 42, 673 (1979).[6] A. Camjayi et al., Nature Phys. 4, 932 (2008).[7] A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).[8] P. Werner et al., Phys. Rev. Lett. 97, 076405 (2006).[9] K. Haule, Phys. Rev. B 75, 155113 (2007).[10] G. Kotliar et al., Phys. Rev. Lett. 84, 5180 (2000).[11] M. J. Rozenberg et al., Phys. Rev. Lett. 75, 105 (1995).[12] J. H. Mooij, Phys. Status Solidi A 17, 521 (1973).[13] G. Moeller, V. Dobrosavljevic, and A. E. Ruckenstein,Phys. Rev. B 59, 6846 (1999).[14] M. J. Case and V. Dobrosavljević, Phys. Rev. Lett. 99,147204 (2007).[15] See supplemental material at http://link.aps.org/supplemental/10.1103/PhysRevLett.107.026401 for eigenvalueanalysis and scaling procedure.[16] M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133 (1996).[17] N. E. Hussey et al., Philos. Mag. 84, 2847 (2004).[18] O**the**r criteria have also been employed to identify a MITcrossover line. Reference [19] used **the** flatness of **the** lowestMatsubara points of **the** self-energy; Refs. [7,11] used **the**inflection point of **the** resistivity curves. We have not revealany apparent scaling behavior following **the**se trajectories.[19] H. Park, K. Haule, and G. Kotliar, Phys. Rev. Lett. 101,186403 (2008).[20] B. J. Powell and R. H. McKenzie, J. Phys. Condens. Matter18, R827 (2006).[21] D. Simonian, S. V. Kravchenko, and M. P. Sarachik, Phys.Rev. B 55, R13 421 (1997).[22] L. De Leo et al., Phys. Rev. A 83, 023606 (2011).[23] S. Sachdev, Phys. Rev. Lett. 105, 151602 (2010).[24] J. McGreevy, Physics 3, 83 (2010).[25] G. Sordi, A. Amaricci, and M. J. Rozenberg, Phys. Rev. B80, 035129 (2009).[26] M. Eckstein et al., Phys. Rev. B 75, 125103 (2007).026401-4