Universality of liquid-gas Mott transitions at finite temperatures
Stefanos Papanikolaou, Rafael Monteiro Fernandes, Eduardo Fradkin, Philip W. Phillips, Joerg Schmalian, Rastko Sknepnek
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Universality of liquid-gas Mott transitions at finite temperatures
Stefanos Papanikolaou, Rafael Monteiro Fernandes,
EduardoFradkin, Philip W. Phillips, Joerg Schmalian, and Rastko Sknepnek Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801-3080 Ames Laboratory and Department of Physics and Astronomy, Iowa State University Ames, IA 50011, USA Instituto de F´ısica ”Gleb Wataghin”, Universidade Estadual de Campinas,and Laborat´orio Nacional de Luz S´ıncrotron,Campinas, SP, Brazil (Dated: November 10, 2018)We explain in a consistent manner the set of seemingly conflicting experiments on the finite temperatureMott critical point, and demonstrate that the Mott transition is in the Ising universality class. We show that,even though the thermodynamic behavior of the system near such critical point is described by an Ising orderparameter, the global conductivity can depend on other singular observables and, in particular, on the energydensity. Finally, we show that in the presence of weak disorder the dimensionality of the system has crucialeffects on the size of the critical region that is probed experimentally.
Although band theory predicts that a system of electrons ina solid with one electron per site (unit cell) should be metal-lic, such a system ultimately insulates [1, 2] once the localelectron repulsive interactions exceeds a critical value. Theonset of the insulating state, the Mott transition, arises fromthe relative energy cost of the on-site Coulomb repulsion U between two electrons on the same lattice site, and the kineticenergy, represented by the band width W . Then, the transi-tion is governed solely by the ratio of U/W . At T = 0 , itis often the case that symmetries of the microscopic system,associated with charge, orbital or spin order, may be broken inthe Mott insulating state. However, at sufficiently high tem-peratures T , or in strongly frustrated systems, no symmetry isbroken at the finite- T Mott transition. Then, the transition ischaracterized by paramagnetic insulating and metallic phases,whose coexistence terminates at a second-order critical point,depicted in Fig. 1(a). In this paper, we are concerned withthe universal properties of this classical critical point [3], asrevealed by a series of apparently conflicting experiments on (Cr − x V x ) O [4] and organic salts of the κ − ET family [5].Since no symmetry is broken at the finite- T Mott transi-tion, in a strict sense there is no order parameter. Nonetheless,experimental [4, 5], as well as theoretical evidence [6, 7] sug-gest that the transition is in the Ising universality class, sim-ilar to the liquid-vapor transition. For example, Castellani etal. [6] constructed an effective Hamiltonian for this problem,and proposed that double occupancy should play the role ofan order parameter for the Mott transition. On the insulatingside, doubly occupied sites are effectively localized, but inthe metal, they proliferate. A Landau-Ginzburg analysis [7]provided further evidence for a non-analyticity in the doubleoccupancy at a critical value of
U/W that defines a Mott tran-sition. Ising universality follows immediately because doubleoccupancy, h n i ↑ n i ↓ i , is a scalar local density field.Experimentally, the universality of the Mott critical pointis typically probed by some external parameter, such as pres-sure, which can tune the ratio W/U . Measurements of theconductivity, Σ , on (Cr − x V x ) O [4] found that away from the critical point, the exponents defined through ∆Σ ( t, h = 0) = Σ( t, h = 0) − Σ c ∝ | t | β σ , ∆Σ ( t = 0 , h ) ∝ | h | /δ σ ,∂ Σ( t, h ) /∂h | h =0 ∝ | t | − γ σ , (1)have mean-field Ising values, β σ ≃ / , γ σ ≃ and δ σ ≃ .Here, t = ( T − T c ) /T c and h = ( P − P c ) /P c , with (Σ c , T c , P c ) denoting the corresponding values at the criticalendpoint. Close to the critical region, Limelette et al. [4] ob-served a drift to the critical exponents of the 3D Ising univer-sality class. Mean field behavior is also seen in NiS [8].However, similar pressure measurements [5] on the quasi-2D organic salts of the κ -ET family appear to challenge theview that the Mott transition is in the Ising universality class.In this material, Kagawa et al. [5] found that their data is de-scribed by the exponents β σ ≃ , γ σ ≃ , and δ σ ≃ ,which do not seem to be consistent with the known expo-nents of the 2D Ising model whose exponents are [9] β = , γ = and δ = 15 . Since the exponents obey the scaling law γ σ = β σ ( δ σ − , it was proposed that the Mott transition isin a new, as yet unknown universality class. The situation isfurther complicated by thermal expansion measurements [10]that claim to measure the heat capacity exponent α and find . < α < . . This result is not only in sharp contrast to theexpectation for an Ising transition (where α = 0 for d = 2 ),it also strongly violates the scaling law α + 2 β σ + γ σ = 2 , ifone uses the exponents of Ref. [5].In this paper, we present a unified phenomenological de-scription of all of these experimental facts within an Ising-type model, and resolve the issue of the universality classof the Mott transition. A complete description of these ex-periments requires to take into account that the conductivitydepends on all possible singular observables of the associ-ated critical system, and not just on the thermodynamic or-der parameter associated with the phase transition. Similarconsiderations were made in magnetic systems near the Curietemperature [11, 12, 13], to explain the critical exponent ofthe conductivity along the coexistence curve. In that case, asymmetry of the microscopic definition of the conductance P T AFI PI PM (a) (b) FIG. 1: (a) Typical phase diagram of Mott transitions as a functionof pressure P and temperature T . At low T and P , a N´eel antiferro-magnetic insulator (AFI) appears; it becomes a paramagnetic insula-tor (PI) if T increases, or a paramagnetic metal (PM) if P increases.Dashed line: first-order transition ending at a liquid-gas critical point.Full lines: continuous phase transition to the ordered state. Coloredregions: critical (dark) and mean-field (light) regimes of the criticalpoint. (b) An Ising configuration on the triangular lattice. Up (down)spins correspond to conducting (insulating) grains of linear size ofthe order of the system’s dephasing length. prevented any coupling of the global conductivity to odd mo-ments of the order parameter, along the coexistence line. Eventhough similar in spirit, the situation here is much different.Starting from an effective microscopic model near an Isingcritical point, we show that: 1) the conductivity typically de-pends on all possible singular thermodynamic observables ofthe system, namely the order parameter and energy densityof the Ising model; 2) when the coupling to the energy den-sity dominates, there exists a large regime around the criticalpoint, where the critical exponents for the conductivity are ( β σ , γ σ , δ σ ) = (1 , , ) , that agree (within the error bars)with the findings of Kagawa et al. [5], and the correspond-ing mean-field exponents are ( β MFσ , γ
MFσ , δ
MFσ ) = (1 , , ) ;3) a crossover to Ising exponents is obtained in the order pa-rameter dominated regime as seen in Refs. [4, 8]; 4) in thepresence of disorder the Mott critical point ultimately belongsto the random-field Ising model universality class, and there-fore the dimensionality of the system under study is even moreimportant for specifying its critical properties.In order to resolve the discrepancies raised by these ex-periments, we consider the behavior of the conductivity ofthe system near the Mott critical point, assuming that it be-longs to the 2D Ising universality class. Rather than startingfrom a microscopic picture, e.g. a Hubbard model, we con-sider a coarse-grained model with the correct symmetries inwhich the physics of the relevant transport degrees of free-dom is captured. In this picture, one defines coarse-grainedregions, of linear size of the order of the dephasing length l φ of the system, which are either insulating or conducting.Along these lines, we consider an Ising model on a 2D lattice(cf. Fig. 1(b)). Near the critical point, where the correlationlength for density fluctuations ξ diverges, it is expected thatthe relevant degrees of freedom behave classically. The Isingvariables S i on each lattice site represent the fluctuating den-sity of mobile carriers on microscopic “grains” of linear sizeof the order of the dephasing length l φ , which are conducting ( S i = +1 ), or insulating ( S i = − ). The Hamiltonian is βH = − T X h ij i S i S j + hT X i S i , (2)where T is the temperature, P and P c are the pressure and thecritical pressure, respectively, and h ∝ P − P c plays the roleof the Ising magnetic field. This model is expected to describethe physics near the critical point, where ξ ≫ l φ . In this limit,all other interactions beyond nearest-neighbor are irrelevant.Near the critical point, the most singular effect of the pressureis described by a coupling to the order parameter.To relate the order parameter fluctuations to the transportproperties we will define an associated resistor network forthis model, an approach that has been successfully used inother strongly correlated systems [14, 15]. Let σ C and σ I bethe local conductivities of the conducting and insulating re-gions, respectively. We define the bond conductance of thenetwork model simply by adding these two conductivities inseries. The bond conductance has three possible values, de-pending on the state of each grain, which can both be conduct-ing, both insulating, one conducting and the other insulating.Thus, the conductance of the bond ( i, j ) has the form σ ij = σ (1 + g m ( S i + S j ) + g ǫ S i S j ) . (3)Even in this toy model, the microscopic conductivity, σ ij ,couples both to the order parameter, S i , and to the energydensity, S i S j , of the Ising model with naturally large cou-plings, g m and g ǫ , defined in Eq.(3). More specifically, wefind that σ = ( σ C + σ I ) + σ C σ I σ C + σ I , g m = σ C − σ I σ , and g ǫ = ( σ C − σ I ) σ ( σ C + σ I ) . At high contrast, σ C ≫ σ I , we get g m ≃ g ǫ ≃ , whereas, at low contrast, | σ I − σ C | ≪ σ C ,we get g ǫ < g m → .The conductivity of the 2D Ising model we described isa non-trivial quantity to compute. As it was shown in thesimpler case of the random resistor network (RRN) [16], net-works of bonds with conductance σ C ( σ I ) chosen randomly with probability p and − p , the global conductivity becomesnon-zero as soon as an infinite percolating and conductingcluster emerges in the system. When σ I = 0 , the criticalexponent β σ of the conductivity is non-trivially related to thefractal properties of the incipient infinite conducting cluster.This exponent is larger than unity for random uncorrelatednetworks and larger than the exponent of the order parame-ter, because dangling bonds of the infinite cluster do not con-tribute to the conductivity. On the other hand, it becomesmuch less than unity for correlated networks, and typicallyvery close to the exponent of the order parameter, since the in-finite cluster is efficiently connected with few dangling bonds.On the other hand, when σ I > , a conducting cluster isless distinguishable from that of an insulating one, and thecomplex effects coming from the fractal cluster boundaries aresmeared out. In the context of RRN, the percolation transitionis not seen in the behavior of the conductivity, which seemsto show just a crossover. If the contrast is low, σ I ≃ σ C ,
10 20 50 100 200 L Σ T c
10 100 g m = g ε = 0.999999Σ T c =3.9 L − 0.20
10 20 50 100 300 L Σ T c − Σ
10 100 g m = 0.001, g ε = 0.01 Σ T c − Σ = 0.02 L − + 0.012 Σ T c − Σ = − + 0.011 FIG. 2: Crossover behavior of the conductivity at finite contrast: theenergy density (order parameter) dominates at short (long) lengthscales. Inset: Fractal scaling at large contrast. the actual conductivity of a single bond between sites i, j , Σ ,should depend only on local observables, and we can formallyexpand it in powers of g m and g ǫ [17], Σ = σ + g m h ( S i + S j ) i + g ǫ h S i S j i + . . . , (4)where the ellipsis represents more complex products of localspin operators (weighed by rapidly decaying functions) [17].Near the Ising critical point, the most singular contributionof the expectation values of multi-spin operators in Eq. 4 isgiven by the expectation value of the most singular, “primary”,operators of the Ising critical point, the order parameter m andthe energy density ǫ . Thus, the most singular term of multi-spin operators with odd (even) number of spins is proportionalto the order parameter (energy density). Therefore, within therange of convergence of this expansion, Σ = Σ ( g m , g ǫ ) + f m ( g m , g ǫ ) h m i + f ǫ ( g m , g ǫ ) h ǫ i , (5)where Σ , f m , f ǫ are non-universal regular polynomials in g m and g ǫ . Provided that the critical behavior is still controlled bythe fixed point theory of the Ising model, the total conductivityshould have the structure of Eq. (5). Thus, at finite contrast,Eq. (5) predicts that the actual conductivity is the sum of evenand odd components, under the Ising symmetry transforma-tion, Σ = Σ +Σ even +Σ odd , and it should exhibit a crossover from an energy density dominated behavior at short distancesto an order parameter dominated behavior at long distances.The crossover scale is controlled by the relative size of thefunctions f m and f ǫ (cf. Fig. 2). This behavior breaks downat high contrast where there is multi-fractal behavior (cf. Insetin Fig.2 and Ref. [18]).We can understand the experiments of Refs. [4, 5, 8], ifwe assume that Eq. (5) applies. The results of Refs. [4, 8]follow by assuming that f m ( g m , g ǫ ) > f ǫ ( g m , g ǫ ) , and theconductivity scales as the order parameter. Conversely, theresults of Ref. [5] follow if f ǫ ( g m , g ǫ ) ≫ f m ( g m , g ǫ ) , and theconductivity, for an extended regime near the critical point,scales as the energy density of the Ising model. In this case
10 20 50 100 300 L Σ oddT c
10 1000.01 g m = g ε = 0.01 Σ oddT c = 0.017 L −
10 20 50 100 300 L Σ evenT c
10 100 g m = g ε = 0.01 Σ evenT c = 0.016 L − + 0.011 FIG. 3: Monte Carlo data which verify the expected behavior of theconductivity when g m , g ǫ ≪ . Σ even scales as the energy density(see text). Inset: Σ odd scales as the order parameter. holds ∆Σ ∝ | m | θ , where θ = (1 − α ) /β . Then, it followsthat β σ = θβ , δ σ = δ/θ and γ σ = γ + β (1 − θ ) . Theresulting critical exponents are ( β σ , γ σ , δ σ ) = (1 , , ) , veryclose to the experimental values. These exponents obey γ σ = β σ ( δ σ − , (6)if γ = β ( δ − , i.e. , the conductivity exponents obey a scal-ing relation identical to the Ising exponents, in agreementwith the experimental verification of this scaling relation inRef. [5]. In addition, the scaling function obtained by Kagawa et al. [5] only depends on βδ = β σ δ σ , as in our theory.In order to verify the theoretical picture presented above,we performed Monte Carlo simulations of the 2D Ising modelon square and triangular lattices, using the Wolff cluster algo-rithm [19]. For the calculation of the conductivity, for eachIsing configuration we used the Franck - Lobb algorithm [20],or explicitly solved Kirchhoff equations. As expected, wefound that at the Ising critical point, for g m , g ǫ ≪ , the evencomponent of the conductivity Σ even scales as the energy den-sity, while the odd component Σ odd scales as the order param-eter (cf. Fig. 3). As g m , g ǫ approach unity, a slow crossoverexists to a fractal regime of the Ising clusters, which is cru-cial for specifying the critical exponent of the conductivity,consistent with the results of Ref. [18] (cf. Inset in Fig.2.)Refs. [4, 8] report 3D mean-field Ising behavior and a smallcritical region in (Cr − x V x ) O and NiS under pressure re-spectively, in contrast to the extended critical region with 2DIsing exponents of Ref. [5]. We can understand these exper-iments by considering the effects of quenched disorder on anIsing critical point. The difference between a quasi-2D and a3D material is a strongly anisotropic Ising interaction alongthe direction perpendicular to the planes. Disorder that lo-cally favors the localized over the delocalized state or viceversa, corresponds to a random magnetic field , and couples tothe order parameter of the Ising transition. This induces den-sity fluctuations. The relevant model for this discussion is theanisotropic 3D random-field Ising model (RFIM), H = − J xy X { ij } xy S i S j − J z X { kl } z S k S l + X i h i S i , (7)where h i is a random field with variance ∆ . For d = 3 , thereis a continuous phase transition in the D random-field Isingmodel (3DRFIM) universality class [21] for any anisotropy J xy /J z , an irrelevant operator at the 3D RFIM fixed point.However, for large anisotropy and weak disorder, relevant tothe quasi-2D organics, there is a large dimensional crossoverregime from 2D RFIM behavior, with an exponentially longcorrelation length, to the narrower 3D RFIM criticality [22].What changes between the 3D isotropic materials and thequasi-2D organics is not the universality class, but wherethe planar correlations become critical. For weak disorder ∆ ≪ J xy and strong anisotropy J z /J xy ≪ , the planesare essentially decoupled and 2D-RFIM behavior holds with ξ xy ≫ in a large region away from the transition point.When J z ≃ J xy , the critical region is narrow, and controlledby the 3D RFIM fixed point.With regards to the thermal expansion measurements thatclaim to measure the heat capacity exponent α , we argue thatthe authors of Ref. [10] misinterpret their results. The volumechange is proportional to the Ising order parameter of the Motttransition, i.e. ∆ l ∝ m , yielding l − dl/dT ∝ t β − . Thethermal expansion diverges with exponent − β = 0 . ,consistent with Ref. [10] who find it in the range . − . .Some major predictions can be drawn from our picture.Firstly, all thermodynamic observables near the Mott criti-cal point should have Ising critical exponents. Secondly, re-garding the critical behavior in quasi-2D organic salts [5], inthe clean system, the conductivity along the coexistence lineshould have the same critical exponent ( β σ = 1 ) in both mean-field and true-critical regimes. This means that the conductiv-ity jump ∆Σ J ≡ Σ( T, h = 0 + ) − Σ( T, h = 0 − ) along thecoexistence line, which should be proportional to the order pa-rameter, should have distinct mean-field and critical regimes,where β ∆Σ J = 1 / and β ∆Σ J = 1 / respectively. Also,the first-order Mott transition is expected to be broadened bydisorder [23, 24]. Thus, instead of a sharp jump in the con-ductance one should see a continuous change which wouldbecome more abrupt for clean systems. The net effect is tomake the system spatially inhomogeneous, as in charge or-dered phases, stripes and electron nematics [25] and in themanganites [26], which tend to round their phase transitionsand replace the first-order transition by an inhomogeneousphase. Thus, hysteretic glassy-like aging effects [27], a prob-lem that has been studied only recently in strongly correlatedsystems [14, 28], are also expected at this Mott transition.In conclusion, we have explained under a consistent phe-nomenological framework the series of experiments that wereperformed during the last few years on Mott criticality. Weshowed that the conductivity of a system near a critical pointdepends on all possible local singular observables of the sys-tem, which in the case of interest, are the order parameter and the energy density of the effective Ising model. This descrip-tion holds when the contrast between conducting and insulat-ing regions is small. Should this not hold, fractal behavior ofthe incipient infinite clusters at the critical point is crucial forspecifying the critical exponents of the conductivity. Finally,disorder affects the effective dimensionality of the system. Inparticular, we showed that for quasi-2D materials, such as theorganic salts in the κ -ET family, critical fluctuations are ex-pected to be much larger than in a 3D material in an extendedregime in the ( P, T ) plane around the Mott critical point. Acknowledgements
We thank K. Kanoda for discussions.This work was supported in part by the National ScienceFoundation grants DMR 0442537 (EF) and DMR 0605769(PP) at UIUC, by the U.S. Department of Energy, Division ofMaterials Sciences under Award DE-FG02-07ER46453 (EF),through the Frederick Seitz Materials Research Laboratory atUIUC, and the Ames Laboratory, operated for the U.S. De-partment of Energy by Iowa State University under ContractNo. DE-AC02-07CH11358 (JS), and by CAPES and CNPq(Brazil) (RF). [1] N. F. Mott, Proc. Phys. Soc. A , 416 (1949).[2] W. F. Brinkman and T. M. Rice, Phys. Rev. B , 4302 (1970).[3] This should not be confused with the putative quantum criticalscenario of M. Imada, Phys. Rev. B, , 075113 (2005).[4] P. Limelette et al. , Science , 89 (2003).[5] F. Kagawa, K. Miyagawa, and K. Kanoda, Nature , 534(2005).[6] C. Castellani et al. , Phys. Rev. Lett. , 1957 (1979).[7] G. Kotliar, E. Lange, and M. J. Rozenberg, Phys. Rev. Lett. ,5180 (2000).[8] N. Takeshita et al. , arXiv:0704.0591 (2007), H. Takagi, privatecommunication.[9] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, 1973).[10] M. de Souza et al. , arXiv:cond-mat/0610576v1 (2006).[11] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. , 435(1977).[12] M. E. Fisher and J. S. Langer, Phys. Rev. Lett. , 665 (1968).[13] I. Mannari, Phys. Lett. A , 134 (1968).[14] E. W. Carlson et al. , Phys. Rev. Lett. , 097003 (2006).[15] J. Burgy et al. , Phys. Rev. Lett. , 277202 (2001).[16] D. Stauffer and A. Aharony, Introduction to Percolation Theory (CRC Press, 1991).[17] J. A. Blackman, J. Phys. C: Solid State Phys. , 2049 (1976).[18] P. J. M. Bastiaansen and H. J. F. Knops, J. Phys. A: Math. Gen. , 1791 (1997).[19] U. Wolff, Phys. Rev. Lett. , 361 (1989).[20] D. J. Franck and C. J. Lobb, Phys. Rev. B , 302 (1988).[21] T. Nattermann, in Spin Glasses and Random Fields , edited byA. P. Young (World Scientific, Singapore, 1998).[22] O. Zachar and I. Zaliznyak, Phys. Rev. Lett. , 036401 (2003).[23] Y. Imry and M. Wortis, Phys. Rev. Lett. , 3580 (1979).[24] M. Aizenman and J. Wehr, Phys. Rev. Lett. , 2503 (1989).[25] S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature , 550(1998).[26] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. , 1 (2001).[27] J. P. Sethna, K. A. Dahmen, and O. Perkovi´c, in Science of Hys- teresis , edited by I. D. Mayergoyz and G. Bertotti (AcademicPress, London, 2004). [28] J. Schmalian and P. Wolynes, Phys. Rev. Lett.85