Quantum Critical Transport near the Mott Transition


Quantum Critical Transport near the Mott Transition

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, theconcept 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 themesial 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 neither 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 thendirectly, consistently, and methodically identify and quantify what was wrong.A gathering 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 then studied intensively to ascertain whichcharacteristics, if any, would be found consistently in all the models. Some theories took formbut soon had to be discarded; others required modification and then survived. Angle's molarcusp groove concept was validated still again. But there was growing realization that the molarrelationship in these healthy normal models exhibited two qualities when viewed buccally, notjust the classic one, and that the second was equally important.Other 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 theory (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 they 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 other 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 ofthe 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 therefore determines the rate of convergence of theGreen 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 ofthese 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 theoretically[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 theresistivity 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 2011the 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 thefigure). The resistivity is given in units of Mott , maximalresistivity according to the Boltzmann quasiclassicaltheory 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 then 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 hypothesis: 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 , withthe 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 tothe 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 computethe corresponding function [4] ðgÞ ¼ d lngd lnT , withg ¼ c = being the inverse resistivity scaling function.Remarkably [Fig. 4(a)], it displays a nearly linear dependenceon lng and is continuous through U ¼ 0 indicatingprecisely the same form of the scaling function on both sidesof the transition—another 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 thesame 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 mathematicallyequivalent [4] to stating that the two branches of thecorresponding 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 other 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 theregion 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 theoretical [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 theregion 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, together 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 further searches for QC scaling, which are likely tobe found in many other 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 theMott 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 therelevant 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 onthe 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. 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