Single-particle excitations under coexisting electron correlation and disorder: a numerical study of the Anderson-Hubbard model
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Full Paper
Single-particle excitations under coexisting electron correlation and disorder: anumerical study of the Anderson-Hubbard model
Hiroshi
SHINAOKA ∗ and Masatoshi IMADA , Department of Applied Physics, The University of Tokyo, Tokyo 113-8656 CREST, JST, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656 (Received October 31, 2018)
Interplay of electron correlation and randomness is studied by using the Anderson-Hubbardmodel within the Hartree-Fock approximation. Under the coexistence of short-range interactionand diagonal disorder, we obtain the ground-state phase diagram in three dimensions, whichincludes an antiferromagnetic insulator, an antiferromagnetic metal, a paramagnetic insulator(Anderson-localized insulator) and a paramagnetic metal. Although only the short-range in-teraction is present in this model, we find unconventional soft gaps in the insulating phasesirrespective of electron filling, spatial dimensions and long-range order, where the single-particledensity of states (DOS) vanishes with a power-law scaling in one dimension (1D) or even fasterin two dimensions (2D) and three dimensions (3D) toward the Fermi energy. We call it softHubbard gap . Moreover, exact-diagonalization results in 1D support the formation of the softHubbard gap beyond the mean-field level. The formation of the soft Hubbard gap cannot beattributed to a conventional theory by Efros and Shklovskii (ES) owing the emergence of softgaps to the long-range Coulomb interaction. Indeed, based on a picture of multivalley energylandscape, we propose a phenomenological scaling theory, which predicts a scaling of the DOS, A in energy E as A ( E ) ∝ exp[ − ( − γ log | E − E F | ) d ]. Here, d is the spatial dimension, E F is theFermi energy and γ is a non-universal constant. This scaling is in perfect agreement with thenumerical results. We further discuss a correction of the scaling of the DOS by the long-rangepart of the Coulomb interaction, which modifies the scaling of Efros and Shklovskii. Further-more, explicit formulae for the temperature dependence of the DC resistivity via variable-rangehopping under the influence of the soft gaps are derived. Finally, we compare the present theorywith experimental results of SrRu − x Ti x O . KEYWORDS: electron correlation, disorder, Anderson-Hubbard model, single-particle density of states,soft gap, variable-range hopping
1. Introduction
Strongly-correlated electron systems continue to be achallenging issue of the condensed matter physics. Es-pecially, metal-insulator transitions have been attractingmuch attention, because of various phases found in theirvicinities. When the electron correlation becomes dom-inant compared to the kinetic energy, the ground stateundergoes a transition from a metal into a correlation-induced insulator. The Mott transition at specific elec-tron density is a typical example. In the Mott insula-tor, the electrons are localized on the individual atomicorbitals to avoid the strong on-site Coulomb repulsion,leading to the opening of a gap in the single-particle DOSas illustrated in Fig. 1 (a). There only the spin and or-bital degrees of freedom remain at low energies. The ab-sence of the single-particle excitations at low energies isa common feature of correlation-induced insulators, suchas antiferromagnetic insulators and charge-ordered insu-lators.Another source of the localization in the strongly-correlated electron systems is disorder (or randomness),which is inevitably present in real materials. The disor-der drives the metal-insulator transition as the Andersontransition.
There, the insulators are characterized not ∗ E-mail: [email protected] Fig. 1. (Color online) Schematic of insulators with different typesof gaps in the single-particle density of states: (a) Insulator in-duced by interaction, (b) Anderson insulator, (c) Disorder depen-dence of a single-particle gap and (d) Insulator with soft gap. by the vanishing carrier number but by a vanishing relax-ation time accompanying the quantum localization of thewave functions by the impurity scattering. In contrast to
Full Paper H. Shinaoka et al. the Mott insulator, the Anderson insulators without elec-tron correlations exhibit no gap as illustrated in Fig. 1(b), indicating that the gapless single-particle excitationsare essential in determining their physical properties atlow energies. This makes the Anderson insulators com-pletely different from the Mott insulators.Since the electron correlation and randomness in-evitably coexist in real materials, clarification of thesingle-particle excitations under the coexistence is impor-tant for understanding their physical properties. How-ever, since the Mott and Anderson insulators have qual-itatively different low-energy excitations, the DOS atlow energies under their coexistence is highly-non-trivial,which is the main topic of this paper. Let us consider howa single-particle gap behaves when fluctuating randompotentials are introduced as disorder into the pure
Mottinsulator. When the disorder strength is weak enough,the single-particle gap might survive despite appearanceof localized impurity levels induced within the gap asillustrated in Fig. 1 (c). With the increase of disorderstrength W , width of distribution of random potentials,however, the gap amplitude gradually decreases and fi-nally the gap collapses at W = W GAP . One might thinkthat for
W > W
GAP , the gap completely closes and theDOS at E F becomes nonzero. Namely, the ground stateundergoes a transition to a simple Anderson insulator,where the gapless single-particle excitations dominatethe low-energy physics. Unfortunately, this naive expec-tation is not correct.In their seminal work, Efros and Shklovskii (ES) con-sidered an amorphous or a doped crystalline semiconduc-tor, where the Coulomb interaction is nearly unscreened.They showed that assuming nonzero A ( E F ) in the groundstate in the presence of the long-range Coulomb inter-action, the ground state is unstable against a particle-hole excitation between localized states at E F , whichis not a single-particle excitation but a multiple exci-tation from the ground state. In other words, this meansthat the supposed ground state with nonzero A ( E F ) isrelaxed by electron-hole excitations to reduce A ( E F ) un-der the influence of the long-range Coulomb interaction.As a result of this excitonic effect, soft Coulomb gap in-deed opens and the DOS at E F remains zero even for W > W
GAP as illustrated in Fig. 1 (d), where the DOSis scaled near E F as A ( E ) ∝ | E − E F | d − . (1.1)Here d is the spatial dimension. Note that although wefocus on half filling in Fig. 1, the soft Coulomb gap isnot restricted to half filling.It is important to emphasize that this power-law scal-ing is not restricted to the critical point which sepa-rates the correlation-induced and Anderson insulators.Instead, according to the ES theory, this power-law scal-ing generically dominates the whole insulating phase,which indicates that the insulator region is always criti-cal. Indeed, the existence of the Coulomb gap was con-firmed by numerical simulations and in electron tun-neling experiments later. When the screening gets stronger e.g., near metal-insulator transitions with divergence of the dielectricconstant, the effects of the long-range part of theCoulomb interaction are restricted to lower and lowerenergies. In this case, the soft Coulomb gap arising fromthe ES mechanism is expected to shrink to extremelylow energies and the effect of the short-range part prac-tically determines electronic structures in the experimen-tally accessible energy scale. However, within the ES the-ory, short-range interactions do not generate soft gaps,because the excitonic effect is negligible for the short-range interaction. Therefore, according to the ES theory,one could speculate that the soft gap vanishes in the in-sulator near the metal-insulator transition.Although the coexistence of the interaction and ran-domness is common in strongly-correlated materials, par-ticularly perovskite-type compounds with B-site sub-stitution AB − x B ′ x O offer a suitable stage for the in-vestigation of the combined effects. SrRu − x Ti x O is apromising candidate for this purpose. One of the endcomponent SrRuO is a correlated ferromagnetic metal( T C = 165 K); the other end component SrTiO is aband insulator with a wide band gap ( ≃ . d Ru band is located at the Fermi energy, while the Ti 3 d band is well separated from the Ru 4 d band around theFermi energy. Thus the Ti atoms act as impurities forthe itinerant Ru 4 d electrons, and the potential heightof the introduced disorder is as much as of the orderof 1 eV. As a consequence, around 0 . < x < .
5, thematerial undergoes a metal-insulator transition into acorrelated Anderson insulator. In the vicinity of themetal-insulator transition, indeed, recent photoemissionresults of SrRu − x Ti x O indicates breakdown of theES scaling in 3D. Although the power law with the ex-ponent α = 2 is expected from the ES theory for 3D,the exponent obtained by the fitting of the experimentaldata ( ≃ .
2) is clearly different from α = 2. The break-down of the ES theory has been observed in other mate-rials such as LaNi − x Mn x O , indicating the ubiquityof unconventional soft gaps in the vicinities of the metal-insulator transitions. The deviation from the ES theorynear the metal-insulator transition was also observed in anumerical study with the unrestricted Hartree-Fock ap-proximation. Despite long-time efforts to go beyond the ES theory,however, effects of short-range interaction have not beenfully understood yet because of the difficulties in han-dling quantum effects, which is important for short-rangeinteraction. The Anderson-Hubbard model is one of theminimal models of real strongly-correlated materials un-der the coexistence of the interaction and randomness.The Anderson-Hubbard Hamiltonian is defined by H = − t X h i,j i ,σ c † iσ c jσ + U X i n i ↑ n i ↓ + X i,σ ( V i − µ ) n iσ , (1.2)with N s sites and N e electrons, where t is a hopping in-tegral, U is the on-site repulsion, c † iσ ( c iσ ) is the creation(annihilation) operator for an electron with spin σ on thesite i . The number operator is defined by n iσ = c † iσ c iσ . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. and µ is the chemical potential. In addition to the usualHubbard Hamiltonian representing the itinerancy andthe short-range interaction (namely, the first and the sec-ond terms of eq. (1.2)), disorder is represented by thespatially uncorrelated spin-independent random poten-tial V i .Many numerical techniques have been applied to theAnderson-Hubbard model. For example, the quantumMote Carlo (QMC) method was applied in 1D,
19, 20 and 3D. Furthermore, the dynamical mean-field the-ory was extended to disordered systems. At infinite di-mensions, Dobrosavljevi´c and Kotliar formulated a vari-ant of the dynamical mean-field theory (DMFT),
23, 24 so-called statistical DMFT, which is exact in the limit of theinfinite coordination number or in the non-interactinglimit. By using the site-dependent bath functions, sta-tistical DMFT can partially treat spatial correlations.Subsequently, Dobrosavljevi´c et al. derived a mean-fieldtheory, which is simpler but ignores the spatial cor-relations. Another approach is the (unrestricted) site-dependent Hartree-Fock (HF) approximation. Althoughit treats the electron correlation in the mean-field level,it is certainly beyond the mean-field theory of the typethat allows only spatially uniform mean fields. In fact, itcan describe the inhomogeneity of the electronic struc-tures by using site-dependent mean-fields. Tusch et al. obtained the ground-state phase diagram including bothof the magnetic and charge degrees of freedom in 3D. Recently, several numerical studies have reported sup-pression of the DOS at the Fermi energy even for short-range interaction. For example, soft gaps were reportedin a HF study in 3D. Although they claimed a power-law scaling of the DOS with α ≃ .
5, the origin of the softgap has not been clarified at all. Recent numerical studiesin 2D by using the exact diagonalization showed a dip ofthe DOS near E F . These strongly suggest the presenceof an unconventional mechanism which suppresses theDOS even with short-range interaction. However, a nu-merical study with statistical DMFT claimed nonzero A ( E F ) in the insulating phase, and even the divergenceof A ( E F ) toward the metal-insulator transition from themetallic side, which completely disagrees with the HFresults. On the other hand, in several mean-field studieswhich ignores the spatial correlations, no singularity wasreported in the DOS.
25, 30
We clearly need further stud-ies for comprehensive understanding of the short-rangecase.In this paper, through numerical analyses of the 3DAnderson-Hubbard model, we show that there existsa soft gap even though only short-range interaction ispresent in the Anderson-Hubbard model. We call thisunconventional soft gap soft Hubbard gap . We show nu-merical evidences of the soft Hubbard gaps within theHF approximation in 1D, 2D and 3D. Further supportby the exact diagonalization in 1D is given. In order toclarify the origin of the soft Hubbard gap, we propose aphenomenological theory based on a picture of multival-ley energy landscape , which corresponds to emergence ofmany excited states degenerated with the ground state.Our scaling theory predicts an unconventional scalingof the density of states A ( E ) in energy E as A ( E ) ∝ exp[ − ( − γ log | E − E F | ) d ]. Here γ is a non-universal con-stant. We show that this predicted scaling is consistentwith the numerically-observed scaling. A part of the nu-merical evidences for the soft Hubbard gap and the scal-ing theory have already been given in a letter briefly. Inthis paper, however, we analyze the ground-state phasediagram of the 3D Anderson-Hubbard model in greaterdetail. Especially, the criticality of the metal-insulatortransitions is clarified from a viewpoint of the formationof the soft Hubbard gap. Furthermore, by extending ourscaling theory, we clarify effects of the long-range part ofthe Coulomb interaction responsible for low-energy exci-tations. Especially, we show that the ES theory is seri-ously modified in the presence of the long-range Coulombinteraction, when we consider the multiply-excited statesby extending our scaling theory originally constructedfor the short-range interaction. In order to inspire fu-ture experimental efforts to examine the validity of thepresent fundamental proposal, we study temperature de-pendence of the DC resistivity in the presence of thesoft gap. We further compare the experimental resultsfor SrRu − x Ti x O with the present theory.This paper is organized as follows; In §
2, we intro-duce the Anderson-Hubbard model and numerical meth-ods employed. In §
3, we show numerical results ofthe Anderson-Hubbard model in 3D and 1D within theHartree-Fock approximation as well as by the exact di-agonalization. Section 4 is devoted to the scaling theoryof the soft gap and its extension to discrete distribu-tion functions of random potentials and the long-rangeCoulomb interaction. In §
5, we derive transport prop-erties in the presence of the soft gap. Comparisons withexperimental results are also given. The summary anddiscussion are given in §
2. Model and Method
In this paper, we analyze the Anderson-Hubbardmodel, whose Hamiltonian is defined by eq. (1.2). Weemploy a cubic lattice for d = 3, a square lattice for d = 2 and a chain lattice for d = 1. We take the latticespacing as the length unit. The spin-independent randompotential V i representing randomness is assumed to fol-low two models of the distribution P V ( V i ): the box typeof width 2 W , P V ( V i ) = (cid:26) W ( | V i | < W )0 (otherwise) (2.1)with the average h V i i = 0, and the Gaussian type, P V ( V i ) = 1 √ πσ exp (cid:18) − V i σ (cid:19) , (2.2)where σ = W /
12 with the average h V i i = 0. Because W is proportional to the standard deviation of the dis-tribution for the both models, W is the parameter tocontrol the strength of disorder. For both the two distri-butions, µ = U/ We first employ the Hartree-Fock (HF) approxima-tion, where the wave function is approximated by a sin-
J. Phys. Soc. Jpn.
Full Paper H. Shinaoka et al. gle Slater determinant consisting of a set of orthonor-mal single-particle orbitals { φ n } ( n is an orbital index).Within this trial wave function, the variational principleleads to the HF equation as {H + U X i ( h n i ↓ i n i ↑ + h n i ↑ i n i ↓ ) } φ n = ǫ n φ n , (2.3)where H is the one-body part of the Hamiltonian and weneglect h c † i ↑ c i ↓ i . Here h n iσ i are the site-dependent meanfields. Later, we will show that the inhomogeneity of theelectronic structures is necessary for the formation of thesoft Hubbard gap. To find a site-dependent mean-fieldsolution h n iσ i for the HF equations, we employ the iter-ative scheme starting from an appropriate initial guessuntil the convergence condition, | ∆ n iσ | < − ( ∀ iσ ) issatisfied. Here ∆ n iσ denotes a change in the mean fieldat the site i with spin σ , n iσ before and after an iteration.Initial guesses of mean fields employed in our calculationswill be described later in each case. In order to acceleratethe convergence of the iteration, we employ the Ander-son mixing, which is easily applicable to the 3D casebecause its computational cost scales as O ( N s ) and it isnot heavy.In general, many stable mean-field solutions may co-exist for a given realization of the random potentials.Therefore, after repeating the calculations for several dif-ferent initial guesses of the mean fields as the groundstate, we employ the solution that has the lowest energy. In this paper, we further employ the exact diagonal-ization method in 1D. Because the exact diagonalizationmethod takes into account quantum fluctuations ignoredby the Hartree-Fock approximation, it is suitable for theexamination of robustness of our Hartree-Fock resultsagainst the quantum fluctuations. By diagonalizing thefull Hamiltonian matrix by using LAPACK routines, we obtain all the eigenstates. Then the single-particledensity of states is calculated by A ( E ) = P n P iσ |h n ; N e + 1 | c † iσ | N e i| × δ ( E − E + ( n )) ( E > E F ) P n P iσ |h n ; N e − | c † iσ | N e i| × δ ( E − E − ( n )) ( E < E F ) , (2.4)where N e is the number of electrons in the ground state, | n ; N i is the n -th eigenstates with N electrons ( n ≥ | N e i is the ground state. The single-particle exci-tation energies E ± ( n ) are defined as E ± ( n ) = E − E n ( N e ±
1) + E ( N e ) − E F . (2.5)Here E n ( N ) is the energy of the n -th eigenstates with N electrons, | N e i . In order to reduce the computationalcost, we divide the full Hilbert space into subspaces witha fixed set of ( N ↑ , N ↓ ), where N ↑ and N ↓ denote the num-ber of up-spin electrons and that of down-spin electrons,respectively. The maximum dimension of the subspacesis 400 ( N s = 6).
3. Numerical Results
In this section, we analyze the Anderson-Hubbardmodel in 1D and 3D numerically. Our main new results are the following:(1) Determination of the ground-state phase diagram ofthe Anderson-Hubbard model in 3D within the HFapproximation,(2) Discovery of the unconventional soft gap driven bythe short-range interaction regardless of electron fill-ing and spatial dimensionality,(3) Clarification of the unconventional scaling of the softgap; the DOS decays faster than any power law in3D, while it follows a power law in 1D,(4) Clarification of the criticality of the metal-insulatortransitions.We first analyze the Anderson-Hubbard model in 3Dwithin the HF approximation. Further numerical evi-dences of the soft gap in 1D are given within the Hartree-Fock approximation. We also show exact-diagonalizationresults in 1D beyond the mean-field level. Detailed scal-ing analyses within the Hartree-Fock approximation in2D will be given in the next section § in this paper, we show more detailed numericalanalyses which were omitted in the previous letter, e.g. the criticality of the metal-insulator transitions. We study the ground state of the 3D Anderson-Hubbard model with the Gaussian distribution of P V in detail. A part of the results have already been re-ported briefly. In our 3D study, we take the hoppingintegral t as the energy unit. Figure 2 shows the cal-culated ground-state phase diagram within the HF ap-proximation. At U = 0, the Anderson-Hubbard modelundergoes a metal-insulator transition (Anderson transi-tion) from the paramagnetic metal (PM) to the param-agnetic insulator (PI) at a finite strength of the disorder, W c = 21 . ± . On the other hand, at W = 0,since the system is half-filled with a perfect nesting, theground state is the antiferromagnetic insulator (AFI) forany nonzero value of U . Here, we discuss the ground-state phase diagram for U, W >
0. First, we focus on thespin degree of freedom. For
W >
0, the ground state isparamagnetic near U = 0. With increasing the interac-tion, the ground states undergoes an antiferromagnetictransition at a critical point U c ( > U c monotonically increases asthe disorder strength W increases. Next, we focus on thecharge degree of freedom. The ground state is insulatingfor U, W ≫
1, which contains AFI as well as paramag-netic insulator (PI) (PI is usually identified as Andersoninsulator). Metallic phases are restricted to a dome-likeregion (
U <
W < . Phys. Soc. Jpn.
Full Paper H. Shinaoka et al. P V . Abbreviations are: AFI, antiferromagnetic insula-tor; AFM, antiferromagnetic metal; PI, paramagnetic insula-tor (Anderson insulator); PM, paramagnetic metal. The dot-ted line denotes the asymptotic scaling of the antiferromagnetictransition for U ≫ t and W ≫ t . The critical value of themetal-insulator transition in the non-interacting limit is givenby W c = 21 . ± . We first discuss the antiferromagnetic transition. Theantiferromagnetic order parameter m ( Q ) is defined by m ( Q ) = 1 N s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i h S zi i e iQr i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.1)where Q = ( π, π, π ). At W = 0, since the system is half-filled with the perfect nesting condition satisfied, the an-tiferromagnetic order vanishes in an essentially-singularway toward U = 0 as m ( Q ) = tU exp (cid:18) − πtU (cid:19) . (3.2)While the amplitude of the antiferromagnetic gap givenby U m ( Q ) also follows essentially-singular scaling and issmall near U = 0, U c increases sharply from zero withthe increasing disorder W .We consider the antiferromagnetic transition in theatomic limit ( U, W ≫ t ). In this limit, there are two typesof sites: singly occupied sites for | V i | < U/
2, and doublyoccupied or empty sites for V i < − U/ V i > U/
2, re-spectively. Thus the Anderson-Hubbard model reducesto a site-diluted antiferromagnet, where the antifer-romagnetic transition coincides with percolation of thesingly occupied sites within the mean-field level, becauseof the antiferromagnetic interaction between neighboringsingly occupied sites. Thus U c is determined by Z + U c − U c P ( V i ) dV i = p c , (3.3)where p c ≃ . U c /W ≃ . Fig. 3. (Color online) Square of antiferromagnetic order parame-ter m ( Q ) at U = 3, W = 20 and W = 30 for (a), (b) and (c),respectively. The solid lines are the fit by the mean-field critical-ity β = 1 /
2. Results for 4 × × × × in the atomic limit. This asymptotic scaling is plotted asthe dotted line in Fig. 2. As we will show below, the phaseboundary numerically obtained starts following this scal-ing with increasing W and U .In order to determine the antiferromagnetic transitionline in the intermediate region, we calculate the anti-ferromagnetic order parameter with cubic unit cells of L × L × L . We employ two boundary conditions: closed-shell and open-shell boundary conditions. At U = 0 and W = 0, the periodic boundary condition correspondsto open-shell configurations for L = 4 n and closed-shell J. Phys. Soc. Jpn.
Full Paper H. Shinaoka et al. configurations for L = 4 n + 2, where n is a positive inte-ger. In contrast, the antiperiodic boundary condition in3D corresponds to closed-shell configurations for L = 4 n and open-shell configurations for L = 4 n + 2. Thereforewe define the open-shell boundary condition as the pe-riodic boundary condition for L = 4 n and the antiperi-odic boundary condition for L = 4 n + 2. The closed-shellboundary condition is defined as the opposite cases. Forfinite-size lattices, the antiferromagnetic order parame-ter calculated with the open-shell boundary condition islarger than that with the closed-shell boundary condi-tion, because of an enhancement of the electron correla-tion for the open-shell boundary condition.We employ four kinds of initial guesses of mean fields:(1) Uniform and antiferromagnetically-ordered states,(2) Ground states in the limit of t → t , which consists of locally-antiferromagnetically-ordered clusters parted bydoublons and holons,(3) Paramagnetic states with uniform charge distribu-tion,(4) States with random charge-spin distributions.All these initial guesses are further perturbed by dif-ferent choices of random noise in both of the spin andcharge sectors before putting into the self-consistent it-eration. We typically need several tens of initial guessesfor convergence of the antiferromagnetic order param-eters. Figure 3 (a) shows the calculated antiferromag-netic order parameter at U = 3 with the open-shell andclosed-shell boundary conditions, respectively. Error barssmaller than the symbols are dropped for the simplicityof the figure. For each boundary condition, as the sys-tem size increases, the antiferromagnetic order param-eter well converges to the mean-field critical behaviorof β = 1 /
2. We estimate the upper and lower limits ofthe critical point by fitting of the data with the open-shell boundary condition and those with the closed-shellboundary condition, respectively.At
W/t = 20 ,
30, we show the antiferromagnetic orderparameter calculated only with the open-shell bound-ary condition, because difference in the calculated an-tiferromagnetic order parameter was found to be negli-gible between the two boundary conditions. As shownin Fig. 3 (b) and (c), the antiferromagnetic order pa-rameter again well converges to the mean-field criticalbehavior, as the system size becomes larger. The esti-mated antiferromagnetic transition points of U c /W =0 . ± . , . ± .
003 for
W/t = 20 ,
30 are closeto the atomic-limit value ( U c /W ≃ . × ×
12 for W = 20, 10 × ×
10 for W = 30,respectively. This correct asymptotic behavior of U c wasnot reproduced in the previous HF study. On the other hand, K. Byczuk et al. found a non-monotonic behavior of U c for the antiferromagnetic tran-sition as a function of the increasing disorder strength W within DMFT with spin degrees of freedom. Namely,
Fig. 4. (Color online) Inverse of spin structure factor S ( Q ) mul-tiplied by the number of sites N s as a function of U at W = 20. they found that U c becomes zero as a function of W at the Anderson transition point in the non-interactinglimit. In contrast, we found a monotonic increase of U c in our 3D study, which we believe more reasonable. Weshow a further numerical evidence at W = 20. We definespin structure factor S ( q ) as S ( q ) = 1 N s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i,j h S i · S j i e iq · ( r i − r j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.4)In Fig. 4, we plot spin structure factor at Q = ( π, π, π )for W = 20. The quantity N s S ( Q ) diverges toward theantiferromagnetic transition point approaching from thelimit of U = 0, which further supports the existence ofthe antiferromagnetic transition around U/t ≃ Next, we discuss the metal-insulator transitions. Weidentify insulating phases by extrapolation of the local-ization lengths ξ to the bulk limit. The localization length ξ is defined by the asymptotic behavior of single-particleorbitals near E F at long distances as φ n ∝ exp( − r/ξ ) , (3.5)where r is the distance from the center of the orbital.In order to observe the asymptotic behavior easily, weemploy pseudo-1D unit cells of L × L × M , where M ≫ L as has been employed in Refs. 36–38. In fact, we adopt M = 1000 , , ,
100 for L = 4 , , ,
10, respectively.We use single-particle orbitals within an energy windowof the width 0 .
01 around the Fermi energy, namely, | E − E F | < .
01 to calculate the localization lengths.Figure 5 shows the extrapolation of the calculated lo-calization lengths for U = 2 , , , U < W from0, metals appear from AFI as in the 2D result, withfurther reentrant transition to insulators (AFI or PI) atlarger W . Even in antiferromagnetically-ordered phases,there exist metal-like regions, where the antiferromag-netic order parameter is locally reduced due to relativelylarge fluctuations of V i . When W is small enough, the . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. system remains insulating even though the DOS is gap-less, because these metal-like regions are small in size andisolated from each other. With increasing W , however,they grow in size and finally percolates, correspondingto the first metal-insulator transition into metals (AFMor PM). Further increase of the disorder i.e. , W ≫ t ,results in the reentranst metal-insulator transition intoinsulators (AFI or PI). It should be noted that becausethe Gaussian-type distribution is unbounded, the gaplessregions appear and the antiferromagnetic gap closes assoon as the disorder strength W becomes nonzero. Thisindicates W GAP = 0. Thus there is no real gap for
W > P V . How-ever, this is not a generic feature for P V bounded in afinite region.In the previous DMFT study, K. Byczuk et al. founda first-order transition between the Mott insulator andthe metal. In contrast, the double occupancy in our3D study exhibits a jump neither at the metal-insulatortransitions nor at the antiferromagnetic transitions asshown in Fig. 6, indicating that they are not of the first-order but continuous. Now, we discuss the criticality ofthe metal-insulator transitions. In the non-interactinglimit, the normalized localization lengths Λ L ≡ ξ L /L forthe different system sizes cross at the Anderson transi-tion. Namely, Λ L is constant with respect to the widthof the 1D bar L when L is large enough. Furthermore,the critical parameter Λ c = Λ L ( W c ) was reported tobe universal, not depending on the distribution of therandom potentials. In Fig. 7, we show scaling plots ofthe localization lengths at U = 2, U = 4 and U = 5.Even for the interacting cases, the normalized localiza-tion lengths indeed seem to cross at the metal-insulatortransitions with a universal critical parameter Λ c ≃ . c = 0 . ± . ), indicating thatthe metal-insulator transitions for U > U = 0. Further support for this unconventionalcriticality will be given in § ξ at t = 1, U = 4 and W = 30 as well as at t = 1, U = 4 and W = 30. Although the localization lengthis nearly independent of the energy near the Fermi en-ergy at U = 0, it depends on the energy in a morecomplex manner at U = 4. Around the Fermi energy,namely | E − E F | < .
5, the inverse localization length1 /ξ is extrapolated to a finite value, being consistentwith the fact that the ground state is insulating. Alsofor | E − E F | >
5, the extrapolated localization length isfinite and increases toward high energies. In the inter-mediate region, namely 0 . < | E − E F | <
5, however,1 /ξ is extrapolated to zero, indicating the existence ofextended states. This reentrant transition in the imme-diate vicinity of the Fermi energy is clearly beyond theconventional picture of the Anderson transitions withoutelectron correlations, where there is no reentrant transi- Fig. 5. (Color online) Quasi-1D localization lengths as a functionof inverse length 1 /L . tion with respect to the energy. In order to reinforce the results obtained by theanalyses of the localization lengths, we further analyzeanother order parameter for the Anderson transition,the geometrically-averaged DOS, The geometrically-averaged DOS, A G ( E ) is defined as A G ( E ) = exp N s N s X i =1 log( A i ( E )) ! , (3.6)where the local DOS at the site i is given by A ( E ) = − π Im G i,i ( E + iη ) , (3.7)which is exact in the limit of η = +0. Here η ( > J. Phys. Soc. Jpn.
Full Paper H. Shinaoka et al.
Fig. 6. (Color online) Double occupancy with the open-shellboundary condition as functions of U . The system sizes are8 × × W = 5 and 20, 10 × ×
10 for the rest. Thedouble occupancy increases as W increases for a fixed value of U . This is consistent with the fact that the double occupancyhas its maximum value 0 . W = + ∞ , where only anequal number of doubly occupied sites and empty sites exist inthe ground state. It should be mentioned that the double occu-pancy is 0 .
25 at U = 0 and W = 0, and 0 in the limit U = + ∞ .
0) is a broadening parameter. In the bulk limit, thegeometrically-averaged DOS at the Fermi energy is an-other order parameter for the Anderson transitions, be-ing nonzero when states at the Fermi energy are extendedand zero when they become localized. This is becausewhen wave functions at a given energy become local-ized in real space, the local DOS at that energy becomesdiscrete: localized wave functions at the same energy arewell separated in position so the local DOS at that energygoes to zero far from the wave function centers. In Fig. 8(b), we show the geometrically-averaged DOS extrapo-lated to the bulk limit and further to the limit of η = +0.At low but nonzero energies, the geometrically-averagedDOS is extrapolated to a nonzero value, indicating theexistence of extended states. This is consistent with theresults obtained by the extrapolation of the localizationlength. We will discuss the implication of this unusualfeature in the next subsection. Next, we discuss the DOS. Figure 9(a) shows theDOS for typical parameters. Naively one might expect A ( E F ) > W >
0. Indeed, there exists no soft gap inthe DOS at the parameter A depicted in Fig. 2 which cor-responds to the non-interacting Anderson insulator. TheDOS A ( E F ) is also nonzero for t = 0 as is seen in the caseD in Fig. 2. However, we find a soft gap over the entire in-sulating phases in the case of U > § § Fig. 7. (Color online) Finite-size scaling plot of localizationlengths at U = 2, U = 4 and U = 5. The metal-insulator transi-tion points are denoted by the centers of the circles. tional soft gap soft Hubbard gap , since it is driven by theshort-range interaction U . The significance of the softHubbard gap is clear because the soft gap is establishedirrespective of the spatial dimension and electron filling.It should be mentioned that the soft Hubbard gap is notrestricted to half filling as well, as we will see in 1D. Al-though the soft gap is observed generically, its formationcertainly has to satisfy minimal requirement: Not onlythe coexistence of the interaction and randomness butalso the itinerancy is required for their formation. In-deed, the soft Hubbard gap vanishes at the parameter Dby switching off the transfer. All of the three terms in theAnderson-Hubbard Hamiltonian (eq. (1.2)), namely itin-erancy, interaction and randomness are imperative forthe formation of the soft gap. . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. ξ at t = 0, U = 4 and W =30. The shade denotes the energy region where the states areextended (1 /ξ = 0). The dotted lines are guides for the eyes. (b)Geometrically-averaged DOS extrapolated to the limit of η = +0and the bulk limit. The shade denotes the energy region wherethe states are extended ( A G = 0) This soft gap moreover bears generically an uncon-ventional nature: Although a power law scaling A ( E ) ∝| E − E F | α with exponents 0 . < α < | E − E F | > . a closer look for | E − E F | < . E F as shown in Fig. 9 (b). Thedeviation from a simple power law strongly suggests thatthe soft Hubbard gap originates from a novel mechanism.The formation of the soft Hubbard gap in the insulat-ing side further supports an unconventional universalityclass of the metal-insulator transitions for the interact-ing case U > § β is defined by A ( E F ) = | W − W c | β (Metallic side) . (3.8)For U = 0, the critical exponent β is zero because thedensity of states remains nonzero in the Anderson insula-tor. However, the critical exponent β should be nonzerofor U >
U > U = 0. The formation of the softHubbard gap in the insulating phases may be respon-sible for the increase of the critical parameter Λ c fromthe non-interacting value (Λ c = 0 . ± . ≃ . U > U = 0. This prevents the localized statesfrom being hybridized with each other and percolating.Thus the single-particle wave functions at the Fermi en-ergy remain localized in the bulk limit for U >
0, even ifthe quantity Λ L = ξ L /L has reached the non-interactingcritical parameter Λ c .Furthermore, the formation of the soft Hubbard is re-sponsible for the low-lying extended excited states in theinsulating phases (see Fig. 8). Because of the formationof the soft Hubbard gap at the Fermi energy, and theweight excluded and transferred from the low-energy re-gion around the Fermi energy, the DOS forms a peakright outside the soft gap as seen in the DOS at U = 4and W = 30 (Fig. 9). Since the mean distance between lo-calized states in an energy window of the width ∆ E cen-tered at the energy E is proportional to ( A ( E )∆ E ) − /d ,the mean distance between localized states is shortest atthe peak position. Thus, with decreasing W , the local-ized states at the peak position become extended beforethose at other energies, leading to the appearance of thelow-lying extended excited states in the vicinity of themetal-insulator transition. Therefore the reentrant local-ization and the low-lying extended excited states may beubiquitous in insulators with a soft gap. In order to clarify how the unconventional scaling ob-served in 3D depends on the spatial dimensionality andis modified in lower dimensions, we further numericallyanalyze the Anderson-Hubbard model in 1D with the boxdistribution of P V in detail here. We find the unconventional soft gaps also in 1D re-gardless of electron filling within the HF approximation.Figure 10 (a) shows the DOS for the hole-doped caseas well as for the half-filled case. We employ the peri-odic boundary condition. Here, holes are doped with thechemical potential µ shifted by − / A ( E ) ∝ | E − E F | α even at low energies as shown inFig. 10 (b) regardless of electron filling. Although theEfros-Shklovskii theory predicts A ( E F ) > A ( E ) ∝ E / | E − E | ) . (3.9)However, this conventional theory cannot explain the ob-served power law, which clearly indicates the existence ofthe unconventional mechanism of the soft gap. Althoughthe gaps again shrink with the decreasing energy scalewhen t or U becomes smaller, the power law still holdswhen t and U are nonzero as shown in Fig. 11. Full Paper H. Shinaoka et al.
Fig. 9. (Color online) DOS with system size 8 × × t =1 , U = 0 , W = 30), B ( t = 1 , U = 4 , W = 30), C ( t = 1 , U =6 , W = 5), D ( t = 0 , U = 4 , W = 30). (a) Linear plot. (b)Double logarithmic plot. The solid lines denote the fitting linesby a power law. In order to investigate effects of quantum fluctuationsnot taken into account in the HF approximation, we fur-ther analyze the Anderson-Hubbard model by using theexact diagonalization in 1D. In Fig. 12, we further showthe calculated DOS at t = 1, U = 10 and W = 10.Because we need all the eigenstates to calculate the ex-citation spectra of each ensemble with a fixed configu-ration of randomness and the ensemble average over anextensive number of random configurations is required,we restrict the system size N s to N s ≤
6. We find adip in the DOS at E F , where the DOS decreases as thesystem size N s increases. This indicates the formationof a soft Hubbard gap beyond the mean-field level. Inorder to clarify the scaling of the DOS, we employ afinite-size scaling analysis. We assume a scaling func-tion, A ( ǫ, N − ) = N − β s f ( ǫN β/α s ) = ǫ α g ( N − β/α s ǫ − ) cor-responding to A ( ǫ, N − = 0) ∝ ǫ α and A ( ǫ = 0 , N − ) ∝ N − β s ( ǫ = | E − E F | ). As shown in Fig. 12, the DOS wellconverges to this scaling function with α = 0 .
075 and β = 0 . et al. found only a dip Fig. 10. (Color online) DOS by HF in 1D at t = 1, U = 10 / W = 20 / N s = 14) in the linear (a) and the logarithmic (b)plots. Electron filling is half-filled and hole-doped for µ = U/ µ = U/ − /
3, respectively. The straight lines in (b) arethe scaling plots. of the DOS in 2D with exact diagonalization. Namely,they found that the DOS is almost system-size indepen-dent within the energy resolution of the order of 0 . t ,which appears to disagree with the present result. How-ever, the pseudogap observed in their study may wellcorrespond to high-energy part of a soft gap in our anal-yses. Indeed, the soft gap is restricted to very low en-ergies e.g. , | E − E F | < . t in our 1D study. At higherenergies, the DOS is almost system-size independent andlooks like a pseudogap if the energy resolution becomespoor ( e.g. , the energy resolution ∆ E = 0 . t ) as shownin Fig. 13 being consistent with their observation. Thuswe believe that further analyses at lower energies withhigher energy resolution will reveal soft gaps also in 2D.
4. Theory of Soft Hubbard Gap
Our numerical observation of an unconventional softgap urges us to explore an unconventional mechanism forits origin and an unprecedented low-energy excitationswhen all of the interaction, randomness and quantum ef-fects are combined. In § . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. t (panel (a)), U (panel (b)) in 1D within the HF approximation ( N s = 14). Thestraight lines are the fit by power laws: (a) U = 10 / W = 20 / t = 1, W = 20 / theory has been reported briefly in the previous letter, we discuss in more detail for the benefit of its further ex-tension: Although a continuous distribution of randompotentials is assumed for the moment for the simplicityof discussion, we extend our scaling theory to discretedistributions of random potentials in § § Fig. 12. (Color online) (a) DOS in 1D with the exact diago-nalization (open boundary condition): t = 1, U = 10, W =10, µ = U/
2, ( N s = 2 , , , , . × realizations of disorder for N s = 6. (b) Scaling plotby A ( ǫ, N − ) = N − β s f ( ǫN β/α s ) ( α = 0 . , β = 0 . . t . In this section, we propose the origin of the soft Hub-bard gap and construct its scaling theory. For simplic-ity without loss of generality, we restrict ourselves to asingle-particle excitation for the electron side, namely,
E > E F . We consider the case of r int ≪ ξ , where r int is the range of the interaction in the model; r int = 0 for Full Paper H. Shinaoka et al. the Anderson-Hubbard model. Even for r int = 0, when t becomes nonzero, virtual hopping of electrons generatesintersite effective interaction U ij though the zero pointfluctuation, which exponentially decreases with the mu-tual distance | r i − r j | as U ij ∝ exp( − b | r i − r j | ) , (4.1)where b is proportional to the inverse of the localiza-tion length. This effect is not considered in the ES the-ory because it regards electrons as classical particles.Within the HF approximation, the interaction energy(the second term of the Anderson-Hubbard Hamiltonian,eq. (1.2)) is decoupled as D U X i n i ↑ n i ↓ E = X n,m n U X i ( | φ m ( r i ; ↑ ) | | φ n ( r i ; ↓ ) | + | φ m ( r i ; ↓ ) | | φ n ( r i ; ↑ ) | ) o , (4.2)where the orbital indices n and m run over all the occu-pied single-particle wave functions. Thus, from eq. (3.5), b is given by b = 2 /ξ within the HF approximation.By assuming the self-averaging of the DOS, the DOSaveraged over the sites is obtained as A ( E ) = DZ ∞−∞ P V ( V ) A ( E, V )d V E { V } , (4.3)where the symbol { V } denotes a set of random poten-tials V i except for V . Note that A ( E, V ) is the localDOS projected on the site 1 under the condition of thefixed V at the site 1. This local DOS A ( E, V ) implic-itly depends on { V } . Here we decompose the averageover the random potential into the two steps; namely,the average over V as described by R P V ( V ) dV at afixed configuration of { V } and the subsequent averagewith respect to { V } . In the following, we first examine V -dependence of the single-particle excitation energy,namely A ( E, V ) and obtain the V -averaged local DOS.Then we average the V -averaged local DOS over { V } to obtain the scaling of the soft Hubbard gap.First, we discuss V -dependence of A ( E, V ) for fixed { V } . When V decreases, the ground-state occupationof the site 1 changes from 0 to 1 and then from 1 to2 at V c and V c , respectively. A possible and typicalground state | φ i at V > V c is illustrated in Fig. 14(a),where the site 1 is empty. Here the total particle num-ber is N e = N a and the energy E ( V ). Near V c butfor V > V c , a single-particle excited state | φ i with N e = N a + 1 and the energy E ( V ) is defined by theelectron configuration fixed to be the same as | φ i ex-cept for an addition of an electron at the site 1, as isillustrated in Fig. 14(b). In the interacting case, because c † iσ | φ i is not necessarily an eigenstate, there may be sev-eral eigenstates that have nonzero matrix elements with c † iσ | φ i , namely, single-particle excited states. However,for simplicity, we discuss the lowest single-particle ex-cited state, | φ i , which dominates at low energies. Onemight think that | φ i becomes the ground state below V c , where N e = N a +1. In this case, however, the single-particle excitation gap E − E vanishes at V c leadingto the absence of the gap in the V -averaged DOS. In-deed, this is what happens in the Anderson insulator at Fig. 14. (Color online) Schematic illustration of (a) the groundstate, (b) a single-particle excited state, (c) a nearly-degeneratestate with the ground state and (d) a multiply-excited state. (e)Schematic of V dependence of excitation energies. U = 0, where the DOS exhibits no soft gap. Thus thenumerical evidences of the soft gaps indicate that | φ i asa single-particle excited state is excluded by the electroncorrelation.In the ES theory, single-particle excited states are ex-cluded from low energies by the ground-state stabilitycondition against an electron-hole excitation. This mech-anism, however, cannot be attributed to the formationof the soft Hubbard gap, because this excitonic effect isnegligible in the case of the short-range interaction. Wenow consider the ground-state stability against excita-tions of more complex form. We assume a multivalleyenergy landscape, which may be characteristic to ran-dom systems. Namely, we assume that there exist manyarbitrarily-low-energy excited states whose occupationsare the same with those of | φ i at the site 1 but whoseconfigurations are globally different on other sites. InFig. 14(c), we illustrate a state | φ ′ i at a local mini-mum E ′ nearly degenerate with | φ i . Here the config-urations of | φ ′ i are relaxed from | φ i not only at theoccupied site n nearest to the site 1 at the distance R but also at farther sites ( > R ). Figure 14(d) shows asingle-particle excited state | φ ′ i from | φ ′ i with the en-ergy E ′ and the site-1 occupancy identical with | φ i .Here, the two nearly-degenerate states, | φ i and | φ ′ i areseparated by a barrier, where multi-particle relaxation isrequired to reach from one to the other. Now E is givenby ( V − E F ) + P i U i + E , where U i is the interactionenergy between electrons on the site 1 and those on thesite i . Note that the interaction energy | U i | is nonzeroonly when the site i satisfies R ≤ | i − | . R + ξ becauseof the localization (Keep in mind that R is the distance . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. to the nearest neighbor electrons from the site 1). On theother hand, E ′ is given by E ′ = ( V − E F )+ P i U ′ i + E ′ ,where U ′ i is again the interaction energy between elec-trons on the site 1 and those on the site i in the state | φ ′ i . Because E ≃ E ′ and the configurations of | φ ′ i onthese sites are different from those of | φ i , E ′ is differ-ent from E typically by the amount as much as | U n | .Now, out of many possible | φ ′ i s, one can choose | φ ′ i sothat the energy E ′ is lower than E by the amount | U n | .Then | φ ′ i is indeed a state that is obtained by a multi-ple excitation from | φ i . This means that h φ ′ | c † iσ | φ i = 0and | φ ′ i is orthogonal to the single-particle excitations.Now we assume linear dependence of the excitationenergies, E ( V ) and E ′ ( V ) as functions of V . Then E ′ ( V ) and E ( V ) crosses at V = V ′ c and for V < V ′ c the ground state becomes | φ ′ i as illustrated in Fig. 14(e). Note that the excitation energy E ′ − E is negativein the vicinity of V = V ′ c . For V > V ′ c , the state | φ ′ i is not counted in the DOS, because this state is not asingle-particle excitation of | φ i , but rather a multiply-excited state. As a result, the single-particle excitationenergy E − E has the lower bound at V = V ′ c . In otherwords, the energy difference ∆ = | E ( V ′ c ) − E ′ ( V ′ c ) | should be the lowest energy of single-particle excitationscounted in A near V = V ′ c in the region V > V c .Therefore the V -averaged local DOS has a gap of ∆ incontrast to the non-interacting case as Z ∞ V ′ c P V ( V ) A ( E, V )d V ∝ H s ( E − E F − ∆) , (4.4)where H s is the Heaviside step function. The same argu-ment applies around V = V c .Here we make an additional comment. One mightthink that, as in the ES theory, it could be possible tolower the energy of | φ i from E to E ′ by relaxing local electronic configurations only near the site n . It, however,always increases the energy of the electrons on the sitesother than the site 1, because they have already been op-timized in the ground state. Thus a global reconstructionis required to lower the energy.Next, in order to derive the scaling of the density ofstates, we discuss the distribution function of ∆ withrespect to { V } . From the above discussion, it is reason-able to assume that ∆ depends on { V } only through R .Under this assumption, from eq. (4.1), ∆ scales as∆( R ) = a exp( − bR ) , (4.5)where a and b ( ∝ ξ − ) are non-universal positive con-stants. It is clear that a is proportional to U when U issmall, and b diverges for t →
0. Hereafter we neglect log-arithmic corrections. The localization lengths may fluc-tuate between the site 1 and the site n . However, thisfluctuation of the localization lengths does not affect theasymptotic behavior of ∆( R ), eq. (4.5), because of theself averaging of the localization length between the site1 and the site n in the limit of R → + ∞ .On the other hand, the probability distribution of R follows P ( R ) = a ′ R d − exp( − b ′ R d ) , (4.6) at long distances, where a ′ and b ′ are non-universal posi-tive constants again. Equation (4.6) means that the prob-ability of formation of a large void of electrons aroundthe site 1 is exponentially rare. Equations (4.5) and (4.6)lead to Q (∆) = P ( R (∆)) (cid:12)(cid:12)(cid:12)(cid:12) dRd ∆ (cid:12)(cid:12)(cid:12)(cid:12) ∝ ( − log ∆) d − ∆ − exp (cid:18) − b ′ b d ( − log ∆) d (cid:19) ≃ ∆ − exp (cid:18) − b ′ b d ( − log ∆) d (cid:19) , (4.7)where Q (∆) are the distribution function of ∆. Here weneglect the logarithmic correction term ( − log ∆) d − . Be-cause the DOS at the energy E is proportional to theprobability of ∆ ≤ | E − E F | , we obtain the scaling of theDOS from Eqs (4.4) and (4.7) as A ( E ) ∝ Z | E − E F | d∆ Q (∆) ∝ exp (cid:18) − b ′ b d ( − log | E − E F | ) d (cid:19) . (4.8)For d = 1, because the exponential and the logarithmfunctions cancel each other, this scaling reduces to apower law with a non-universal exponent, which is consis-tent with the observed power-law scaling in 1D: A ( E ) ∝| E − E F | b ′ /b . Non-universal power-law distributions ofenergies are common in Griffith phases. Equation (4.5)indicates that ∆, namely the gap vanishes as t or U van-ishes because of a → b → ∞ . Furthermore, theexponent α = b ′ /b is expected to decrease as t becomessmaller because of the reduction of ξ . These predictionsare consistent with our HF results in 1D as shown inFig. 11 (a). It should be mentioned that the power lawfor 1D in the present theory is in sharp contrast with theES theory, because the ES theory predicts the logarith-mic scaling of the DOS as eq. (3.9). For d >
1, our scalingpredicts that the DOS vanishes faster than any powerlaw, being consistent again with our HF study in 3D. InFig. 15 (b), we show a scaling plot of the DOS in 3D byour scaling given by eq. (4.8). Indeed, the DOS fits wellwith our scaling in 3D at low energies e.g., | E − E F | < . P V . Fig-ure 15 (b) shows the DOS with the system size 10 ×
10 andthe periodic-boundary condition at t = 1, U = 6, W = 5.We averaged the DOS over as many as 4 . × realiza-tions of random potentials to obtain the high-resolutiondata. As shown in Fig. 15 (b) the DOS fits well with ourscaling obtained from eq. (4.8) rather than a power lawfor | E − E F | < . R )and P ( R ) calculated by the following procedure: First, Full Paper H. Shinaoka et al.
Fig. 15. (Color online) (a) Scaling plot of DOS in 3D at param-eters B and C depicted in Fig. 2: B ( t = 1 , U = 4 , W = 30),C ( t = 1 , U = 6 , W = 5). We employ Lorentz broadeningwith a broadening factor 1 . × − and 6 . × − for A andB, respectively. The broken lines denote | E − E F | = 10 − and10 − . The DOS fits well with A ( E ) ∝ exp( − ( − γ log | E − E F | ) )shown by the fitting lines for 10 − < | E − E F | < − . (b)Scaling plot of DOS in 2D: t = 1 , U = 6 , W = 5 with theGaussian distribution of P V . The inset is a linear plot of thesame data. The black solid curves are fit by the predicted scal-ing, A ( E ) ∝ exp( − ( γ log | E − E F | ) ). If A ( E ) followed A ( E ) ∝ exp( − ( γ log | E − E F | ) ), the data in the panel (b) would followa straight line. The broken lines denotes | E − E F | = 0 . , . we obtain the ground state for each realization of ran-dom potentials. We construct the lowest single-particleexcited state by adding one electron to the lowest unoc-cupied orbital. Next we optimize the mean fields by theiterative scheme starting from those of the single-particleexcited state with N e fixed. Then ∆ is obtained as thedifference of these two excitation energies. We calculate R as the distance between the center of the lowest un-occupied orbital, r and those of the occupied orbitalsnearest to r in the ground state. We define the centerof the orbital as the site that has the maximum weight.Fitting by eqs. (4.5) and (4.6) gives b ′ = 1 . ± .
01 and b = 1 . ± .
01. Estimated exponent of b ′ /b = 0 . ± . α = 0 . ± .
07 obtained di-rectly from the DOS as shown in Fig. 16 (c). This is anumerical evidence for the quantitative validity of ourtheory.In the above discussion, under the assumption of amultivalley energy landscape, we have successfully con-
Fig. 16. (Color online) Numerical estimates of P ( R ) and ∆( R )at t = 1, U = 10 / W = 20 / µ = U/ − /
3. Fittings byeqs. (4.5) and (4.6) as are shown in (a) and (b) by dotted linesgive b ′ = 1 . ± .
01 and b = 1 . ± .
01. The ratio b ′ /b (dashedlines in (c)) well agrees with α obtained by the fitting (solid line)of the DOS in (c). structed the scaling theory which is consistent with allthe numerical results. In the presence of a multivalleyenergy landscape, there are many excited states whoseconfigurations are globally different from those of theground state, which is a non-trivial combined effect ofthe electron correlation and randomness. Indeed, in thefield of spin glasses, it is known that a multivalley en-ergy landscape emerges concurrently with replica sym-metry breaking (RSB) within a spin-glass phase. Exten-sive studies on replica symmetry breaking (RSB) in fi-nite dimensions have been carried out especially in clas-sical Ising models (Ising Edward-Anderson model). Onthe other hand, our Hartree-Fock results indicate that amultivalley energy landscape exists over the whole in-sulating phases for d = 1, 2 and 3. Furthermore, al- . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. though the exact-diagonalization results are restricted tothe strongly-localized region in 1D because of the severelimitation of system sizes, the exact-diagonalization re-sults indicate robustness of the multivalley energy land-scape against quantum fluctuations beyond the mean-field level. Further theoretical studies along this line areintriguing future subjects.Our conclusion disagrees with the DMFT results aswell as several other mean-field studies
25, 30 which sup-port absence of the soft gaps. This may be because theyignore spatial correlations. The latter ignores inhomo-geneity of the electronic structures. Indeed, a DMFTstudy improved by partially taking account of the in-tersite self-energy retrieves the suppression of the DOSnear E F to some extent. In contrast to our scaling, the power law was proposedto interpret the photoemission experiments.
15, 16
How-ever, as shown in Figs. 9 and 15, the asymptotic be-havior of the DOS is restricted to low energies, namely | E − E F | < . t and high-energy part of the DOS seemsas if it approximately follows a power law with a non-universal exponent in our 3D HF study. Because hop-ping integrals between d orbitals on nearest-neighbor Ruatoms are on the order of 0 . , theasymptotic behavior of the soft Hubbard gap may berestricted to the energy region lower than 10 meV. Thusphotoemission experiments with the energy resolution onthe order of 1 meV are desired to observe the presentasymptotic behavior clearly. Although this level of reso-lution has become possible recently, such high resolutionhas not been utilized so far for the present purpose in theliterature.
15, 16
We believe that our paper provides incen-tive for such high-resolution photoemission experiments.In addition to the high-resolution photoemission, otherexperiments accessible to low energies are highly desiredfor experimental confirmation of the present theory. Forexample, the DC transport measurement is suitable forinvestigating the density of states in insulating phases atlow energies i.e.
T <
300 K ( k B T .
30 meV). Actually,in §
5, we discuss the temperature dependence of the DCresistivity in the presence of the soft Hubbard gap.
Although we assume a continuous distribution P V inthe previous sections, the obtained scaling, eq. (4.8) doesnot depend on the distribution. Even for a discrete dis-tribution of a binary alloy form, one obtains the samescaling as that for continuous distributions by modifyingthe above discussion slightly. In the above discussion,we first average the single-particle excitation spectrumover V at fixed configurations of { V } , which is a setof random potentials V i except for V . For a continuousdistribution, as described in eq. (4.4), the single-particleexcitation energy distributes continuously above a lowerbound ∆( R ) depending on { V } through R . Here R isthe minimum distance from the site 1 to the occupiedsites in the ground state. This continuous distribution isa key point for obtaining our scaling. For a discrete dis-tribution, however, the single-particle excitation energydistributes discretely above ∆( R ), because V is discrete. Thus we need an extension of our consideration here. Wefurther average the single-particle excitation spectrumover { V } with a certain common value of R . Then thedistribution of the single-particle excitation energy be-comes continuous above ∆( R ), because of fluctuations ofthe electronic structures at the distance R from the site1 in the ground states. After a further average over R ,one obtains the same scaling as that for the continuousdistribution.We show a numerical evidence in 1D within theHartree-Fock approximation in Fig. 17. We employ thebimodal distribution: V i = ± W with the symmetricprobability P ( W ) = 1 / P ( − W ) = 1 /
2. Althoughthe density of states has a complex peak structure due tothe discreteness of the bimodal distribution, the densityof states clearly exhibits a soft gap near the Fermi energyas shown in Fig. 17 (a). Indeed, as shown in Fig. 17 (b),the soft gap follows a power law, which is consistent withthe above discussion.
Fig. 17. (Color online) Density of states in 1D within the Hartree-Fock approximation with 24 sites: t = 1 , U = 2 , W = 2. (a)Linear plot. (b) Double logarithmic plot. The solid line in (b) isthe fit with a power law. Even near metal-insulator transitions where thescreening is strong, the Coulomb interaction remains longranged in insulating phases, though its amplitude may besmall. Thus the scaling of the DOS deviates from eq. (4.8)
Full Paper H. Shinaoka et al. in real materials, as the long-range part becomes domi-nant at low energies. Note that eq. (4.8) is the asymp-totic scaling for the model with short-range interactiononly. In this section, we discuss effects of the remaininglong-range part of the Coulomb interaction.Owing to the long-range part of the Coulomb inter-action, Efros et al. showed that a ground-state stabilityagainst an exciton excludes low-energy single-particle ex-citations and generates power-law soft gaps as A ( E ) ∝| E − E F | d − (see eq. (1.1)). However, since they consid-ered the stability condition only against electron-hole ex-citations, the DOS may vanish faster than the ES scalingat lower energies because of stability conditions againstfurther many-particle excitations. Indeed, Efros obtaineda correction of the scaling in 3D by considering multi-electron-hole excitations as A ( E ) ∝ exp( − ( ǫ /ǫ ) / ) , (4.9)where ǫ = | E − E F | and ǫ is a constant. For d = 1 ,
2, heclaimed that there is no further correction in ES scaling.However, since our scaling even for the short-range in-teraction shows faster decay of the DOS than any powerlaw in the presence of a multivalley structure, it is nat-ural to infer that the DOS should decay even faster inthe presence of the long-range interaction and certainlybeyond the ES theory at low energies instead of eq. (4.9).Thus we extend our scaling theory to the case of the long-range Coulomb interaction. For the long-range Coulombinteraction, eqs. (4.5) and (4.6) are modified as∆( R ) = aR − , (4.10) P ( R ) = a ′ R d − exp( − b ′ R d ) . (4.11)From eqs. (4.10) and (4.11), we obtain Q (∆) = P ( R ) (cid:12)(cid:12)(cid:12)(cid:12) dRd ∆ (cid:12)(cid:12)(cid:12)(cid:12) = a d a ′ ∆ − d − exp( − a d b ′ ∆ − d ) . (4.12)In a manner similar to the short-range interaction(eq. (4.8)), the scaling of the DOS is given by A ( E ) ∝ Z | ǫ | Q (∆) d ∆ ∝ Z | ǫ | ∆ − d − exp( − a d b ′ ∆ − d ) d ∆= 1 a d b ′ d exp( − a d b ′ ǫ − d ) ∝ exp( − a d b ′ ǫ − d ) . (4.13)Indeed, this scaling shows faster decay of the DOS thaneq. (4.9) proposed by Efros for 3D. Moreover, eq. (4.13)reveals that the ES scaling must be modified even for d = 1 , In Table I, we summarize the scaling laws of the DOSin four kinds of models: (A)/(B) short-range interactionwithout/with a multivalley energy landscape, (C)/(D)long-range interaction without/with a multivalley energylandscape. Now we consider how the scaling of the soft gap depends on the energy for the case (D), which cor-responds to realistic materials. At energies higher thanthe energy scale of the long-range part, the formationprocess of the soft gap is dominated by the short-rangepart. Thus, in this energy region, the soft gap follows thescaling of the soft Hubbard gap (eq. (4.8)). As the long-range part of the Coulomb interaction becomes dominat-ing at low energies, the scaling of the DOS crosses overfrom eq. (4.8) to the ES scaling eq. (1.1) and further toeq. (4.13). It should be mentioned that the intermediatescaling laws may not always be observed clearly.
5. Transport Properties
In this chapter, we discuss the temperature depen-dence of the DC resistivity in the presence of the softHubbard gap or the soft Coulomb gap. Although thetransport coefficient is determined by the two-particle(electron-hole) correlations and is not necessarily identi-cal with the single-particle excitations measured in theDOS, the DC transport measurement is a useful andgood tool for investigating the DOS at low energies withhigh resolution in insulating phases if the transport isdominated by independent single-particle excitations.
When a gap is still open for weak disorder, thermally-excited carriers dominate the conduction and the tem-perature dependence of the DC resistivity follows theArrhenius law as ρ = ρ exp (cid:18) T T (cid:19) , (5.1)where T is a constant. If a gap is closed for stronger disorder, localized statesnear E F dominates the conduction. Mott showed thatat sufficiently low temperatures, conduction results fromelectron hopping between localized state within a nar-row band near E F , which is called variable-range hop-ping (VRH). He also showed that provided there is anon-vanishing DOS at E F , the temperature dependenceof the DC resistivity exhibits universal behavior as ρ = ρ exp "(cid:18) T T (cid:19) / ( d +1) . (5.2) Because VRH explicitly depends on the DOS near E F ,soft gaps modify the temperature dependence of the re-sistivity qualitatively. Now we discuss scaling of the DCresistivity in the presence of a soft gap. As discussed in § ρ also crossovers as a function of the temper-ature. In the following, we discuss scaling of the DC re-sistivity for each energy region separately. (1) Energy regions (D.2) and (D.3) in Table II . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. Model Scaling of DOS Energy (A) Short-range interaction ≃ A > without multivalley energy landscape(B) Short-range interaction exp (cid:0) − γ ( − log ǫ ) d (cid:1) eq. (4.8) with multivalley energy landscape(C) Long-range Coulomb interaction ǫ d − eq. (1.1) (HEs) without multivalley energy landscape exp( − ( ǫ /ǫ ) / ) (3D) eq. (4.9) (LEs)(D.1) exp (cid:0) − γ ( − log ǫ ) d (cid:1) eq. (4.8) (HEs) (D) Long-range Coulomb interaction (D.2) ǫ d − eq. (1.1) ↓ with multivalley energy landscape (D.3) exp( − ( ǫ /ǫ ) / ) (3D) eq. (4.9) ↓ (D.4) exp( − βǫ − d ) eq. (4.13) (LEs) Table I. Summary of scaling laws of DOS four kinds of models: (A)/(B) short-range interaction without/with a multivalley energylandscape, (C)/(D) long-range interaction without/with a multivalley energy landscape. Shaded part denotes the novel scaling lawsobtained in this paper. The scaling of the case (C.2) is expected to appear in energies lower than that of the case (C.1). The case (D.1)corresponds to the energy scale where the short-range part of the Coulomb interaction is dominant, while the cases (D.2-4) correspondto that where the long-range part is dominant. In the cases of the long-range interaction, scaling crossovers are expected as a functionof the energy. Abbreviations are: High energies (HEs); Low energies (LEs).
Efros et al. showed that in the presence of the softCoulomb gap with the power law A ( E ) ∝ | E − E F | d − (eq. (1.1)), the temperature dependence of the DC resis-tivity is modified regardless of the spatial dimension d as ρ = ρ exp "(cid:18) T T (cid:19) / . (5.3)Even in the case (D.3) of Table II where the DOS is scaledas exp( − ( ǫ /ǫ ) / ), Efros concluded that the DC resis-tivity follows eq. (5.3), because the excitation spectrumof the particle screened by excitons that determines theDC transport still follows the power law | E − E F | d − . (2) Energy region (D.1) in Table II On the other hand, at high energies where the short-range part of the Coulomb interaction is dominant, theDOS follows the scaling of the soft Hubbard gap eq. (4.8).Here we discuss the DC transport in the presence ofthe soft Hubbard gap without considering the long-rangeCoulomb interaction. First, we assume the DOS in theform A ( ǫ ) = α | ǫ | β exp( − ( γ | log ǫ | ) d ) , (5.4)where ǫ = | E − E F | . The power-law correction term | ǫ | β isa slight generalization from eq. (4.8). Under this assump-tion, we obtain the resistivity up to the leading order atlow temperatures via variable-range hopping as ρ = ρ exp (cid:18) c exp( − c | log( k B T ) | /d ) k B T (cid:19) , (5.5)for d > c = 1+ a − ( αβ +1 ) − /d , a = ξ/ c >
0. Fordetails of the derivation of eq. (5.5), readers are referredto Appendix A.1. (3) Energy region (D.4) in Table II
We next discuss the modification of the DC resistivityat energies lower than those justified by the ES scal-ing, by starting from eq. (4.13). We derive the temper-ature dependence of the DC resistivity in a way similarto the case of the soft Hubbard gap. Following a pro-cedure similar to the case of the soft Hubbard gap (see Appendix A.2), we obtain the resistivity up to the lead-ing order as ρ = ρ exp (cid:18) ( β/d ) /d | log( k B T ) | − /d k B T (cid:19) . (5.6)The scaling laws of the DC resistivity obtained in theabove discussion are summarized in Table II. In Fig. 18(a), we compare the obtained DC resistivity for the mod-ified ES scaling, namely eq. (5.6) with that for the ESscaling, namely eq. (5.3). Indeed, the DC resistivity forthe modified ES scaling diverges at low temperaturesfaster than that for the ES scaling. However, as shown inFig. 18 (b), the DC resistivity for the modified ES scalingis almost indistinguishable from the Arrhenius law justby tuning the activation energy.Here we propose a procedure to distinguish the modi-fied ES scaling from the Arrhenius law clearly:(1) One estimates ρ in the scaling for the modified ESscaling (eq. (5.6)): From eq. (5.6), one obtainslog( ρ ) = log( ρ ) + ( β/d ) /d | log( k B T ) | − /d k B T → log( ρ ) (1 /T → +0) . (5.7)Thus one can estimate ρ easily by a linear extrap-olation of log( ρ ) to the limit of 1 /T → +0 withrespect to 1 /T .(2) Next, one plots the logarithm of normalized re-sistivity multiplied by the temperature T log( ρ/ρ )against | log( T ) | − /d as shown in Fig. 18 (c). In thisplot, log( ρ/ρ ) is proportional to | log( T ) | − /d forthe modified ES scaling, while the normalized resis-tivity log( ρ/ρ ) is constant when the experimentaldata follows the Arrhenius law. These two functionscan be distinguished clearly from each other.We believe that the validity of the present theory can betested against experiments by using this plot providedthat ρ does not contain other extrinsic factors of tem-perature dependence out of the present consideration,such as the volume expansion. Full Paper H. Shinaoka et al.
Fig. 18. (Color online) Comparison of the DC resistivity for themodified ES scaling, eq. (5.6) with (a) that for the ES scaling,and (b) the Arrhenius law (the dashed and dotted lines). Wenote that the modified ES scaling shown by the solid line ishardly distinguishable from the Arrhenius law (the dotted line)by fitting the activation energy. In eq. (5.6), we take k B = 1and ( β/d ) /d = 1. (c) In this plot, all the data which followthe Arrhenius law converge to a constant, namely unity irre-spective of the activation energy, while log( ρ/ρ ) is proportionalto | log( T ) | − /d for the modified ES scaling. This distinguishesthese two scaling laws clearly. In this section, we compare our formulae for the DCresistivity, eq. (5.5) and eq. (5.6) with transport proper-ties of SrRu − x Ti x O , where the breakdown of the ESscaling was indicated in photoemission experiments.
15, 16
In Fig. 19, we show the experimental data of the DC re-sistivity of SrRu − x Ti x O at x = 0 .
6, where the groundstate is insulating. The experimental data fit well withthe ES scaling for k B T < .
005 eV ( T .
60 K), indicatingthat the ground state is an Anderson insulator. At highertemperatures (0 .
005 eV < k B T < .
011 eV ,
60 K . T .
130 K), however, they fit well with the scaling obtained inthe presence of the soft Hubbard gap for 3D (eq. (A · Fig. 19. (Color online) Scaling analyses of the DC resistivity ofSrRu − x Ti x O (the experimental data illustrated as filled circlesare from Ref. 14). (a) The logarithmic plot. The panels (b), (c)and (d) show scaling plots for the ES scaling ( T < < T < < T ), respectively. The vertical dotted lines denote T =55 K ,
135 K.. Phys. Soc. Jpn.
Full Paper H. Shinaoka et al. are as follows: log( ρ ) = 9 . ± .
2, log( c ) = 0 . ± . c = 2 . ± .
4. On the other hand, for higher temper-atures (0 .
011 eV < k B T ), the experimental data deviatefrom the scaling for the soft Hubbard gap, being con-sistent with the our 3D HF study where the asymptoticbehavior of the soft Hubbard gap is restricted to low en-ergies ( < . t ) as shown in Figs. 9 and 15. On the otherhand, we cannot find the region where the modified ESscaling predicted by eq. (5.6) is satisfied even in the low-est temperature available in the experiment ( ≃
30 K).We believe that further experiments at lower tempera-ture will reveal the existence of the correction to the ESscaling.Another interpretation of the experimental data for0 .
005 eV < k B T < .
011 eV may be that the ES scal-ing ( k B T < .
005 eV) and the Arrhenius law ( k B T > .
011 eV) cross over there; the Arrhenius law arises fromthe hopping conduction between nearest-neighbor local-ized states, which dominates the DC resistivity at hightemperatures instead of the variable-range hopping. In-deed, for 0 .
011 eV < k B T < .
026 eV (130 K . T .
300 K), they seem to follow the Arrhenius law with asmall activation energy 0 . ± .
001 eV. In order to ex-clude or support the possibility of this crossover, furtherexperiments as functions of composition x in the vicinityof the metal-insulator transition are desired.
6. Summary and discussion
In summary, we have examined the ground state andsingle-particle excitations of the 3D Anderson-Hubbardmodel within the Hartree-Fock approximation. In § soft Hubbard gap . Further numericalevidences to support this unconventional soft gap hasbeen given within the Hartree-Fock approximation, andfurther with the exact diagonalization in 1D. In contrastto the 3D case, a power-law scaling is satisfied in 1D.In §
4, based on the picture of a multivalley energylandscape, we have constructed a phenomenological the-ory, being qualitatively as well as quantitatively consis-tent with the numerically-obtained scaling of the softHubbard gap. There, further support for the scaling the-ory has been given in 2D within the Hartree-Fock ap-proximation. Moreover, by considering effects of the long-range part of the Coulomb interaction remaining un-screened at low energies, we have obtained a novel scalingof the soft gap beyond the Efros-Shklovskii theory. Thisnovel scaling is caused by a multivalley energy landscape.Finally, we have proposed scaling crossovers of the DOSas the long-range part becomes dominant at low tem-peratures. Scaling laws and their expected crossovers aresummarized in Table II.In §
5, we have derived the temperature dependence ofthe DC resistivity in the presence of the soft gaps. Pos-sible experiments to verify the present theory has beenproposed. The scaling laws of the density of states and the DC resistivity are summarized in Table II. Finally, wehave shown that the present theory is indeed consistentwith the experimental data for SrRu − x Ti x O .In order to readdress the essence of the present theoryand stimulate further theoretical studies, we now discusspossible extension of the present theory. In §
4, we haveshown that a gapless collective mode accompanying amultivalley energy landscape causes relaxations from asingle-particle excited states to a multiply-excited statesnot counted in the single-particle DOS, which directlyleads to the formation of the unconventional soft gap. Inother words, the single-particle excited states are elim-inated from low energies by the ground-state stabilityagainst the collective excitations at low energies. How-ever, there are actually many other kinds of gapless col-lective modes. A typical example is a spin wave accom-panying magnetic ordering with spontaneous symmetrybreaking, which is not included in the present Hartree-Fock approximation. If these collective modes also causethe relaxation from a single-particle excited states to amultiply-excited states, the present scaling of the den-sity of states will equally be valid, which extends theapplicable scope of the present theory. Whether or notthe present theory can be extended to general collectivemodes is left for a future challenge.Recently magnetic hard gaps have been observed in alarge number of disordered materials such as In-dopedCdMnTe,
48, 49 amorphous GeCr-films, doped Si andamorphous Si − x Mn x -films by measuring the temper-ature dependence of the DC resistivity. These gaps wereidentified to be of magnetic origin by reduction of the gapsizes by an increased external magnetic field. Because theDOS vanishes toward the Fermi energy even faster thana power law in the presence of the soft Hubbard gap, thesoft Hubbard gap causes the rapid divergence of the DCresistivity slightly slower than the Arrhenius law at lowtemperatures as summarized in Table II. Comparisons ofthe present theory with these experiments are interestingissues. Acknowledgments
We thank B. Shklovskii and T. Ohtsuki for fruitful dis-cussions. M. I. thanks Aspen Center for Physics for thehospitality. Numerical calculation was partly carried outat the Supercomputer Center, Institute for Solid StatePhysics, Univ. of Tokyo. This work is financially sup-ported by MEXT under the grant numbers 16076212,17071003 and 17064004. H. S. thanks JSPS for the fi-nancial support.
Appendix: Derivation of temperature depen-dence of DC resistivity
In this appendix, we derive the temperature depen-dence of the DC resistivity in the presence of thesoft Hubbard gap or the soft Coulomb gap within thevariable-range hopping.
A.1 Soft Hubbard gap: Energy region (D.1) in Table II
First, we derive the DC transport in the presence ofthe soft Hubbard gap. We assume the DOS in the form A ( E ) = α | ǫ | β exp( − ( γ | log ǫ | ) d ) , (A · Full Paper H. Shinaoka et al.
Model Scaling of DOS DC resistivity TemperatureClean Mott insulator = 0 (hard gap) exp( T /T ) (A) Short-range interaction ≃ A > T /T ) / ( d +1) ) without multivalley energy landscape(B) Short-range interaction exp (cid:0) − γ ( − log ǫ ) d (cid:1) exp (cid:16) c − c | log( k B T ) | /d ) k B T (cid:17) with multivalley energy landscape · · · eq. (4.8) · · · eq. (5.5) (C) Long-range Coulomb interaction (C.1) ǫ d − exp (cid:0) ( T /T ) / (cid:1) (HTs) without multivalley energy landscape (C.2) exp( − ( ǫ /ǫ ) / ) (3D) exp (cid:0) ( T /T ) / (cid:1) (LTs)(D.1) exp (cid:0) − γ ( − log ǫ ) d (cid:1) exp (cid:16) c − c | log( k B T ) | /d ) k B T (cid:17) (HTs) · · · eq. (4.8) · · · eq. (5.5) ↓ (D) Long-range Coulomb interaction (D.2) ǫ d − exp (cid:0) ( T /T ) / (cid:1) ↓ with multivalley energy landscape (D.3) exp( − ( ǫ /ǫ ) / ) (3D) exp (cid:0) ( T /T ) / (cid:1) ↓ (D.4) exp( − βǫ − d ) exp (cid:16) c | log( k B T ) | − /d k B T (cid:17) (LTs) · · · eq. (4.13) · · · eq. (5.6) Table II. Summary of scaling laws of DOS and DC resistivity for four kinds of models: (A)/(B) short-range interaction without/witha multivalley energy landscape, (C)/(D) long-range interaction without/with a multivalley energy landscape. Shaded part denotes thenovel scaling laws obtained in this paper. The scaling of the case (C.2) is expected to appear in energies lower than that of the case(C.1). The case (D.1) corresponds to the energy scale where the short-range part of the Coulomb interaction is dominant, while thecases (D.2-4) correspond to that where the long-range part is dominant. In the cases of the long-range interaction, scaling crossoversare expected as a function of the temperature. Abbreviations are: High temperatures (HTs); Low temperatures (LTs). where ǫ = | E − E F | . The power-law correction term | ǫ | β isa slight generalization from eq. (4.8). In the following, wediscuss ρ for the soft Hubbard gap without consideringthe long-range Coulomb interaction.In order to determine the energy window ǫ dominat-ing the transport for a given temperature T , we minimizethe resistivity ρ = ρ exp (cid:18) N /d ( ǫ ) a + ǫ k B T (cid:19) , (A · ǫ , where a = ξ/
2. We define thenumber of electrons within the energy window, N ( ǫ ) as N ( ǫ ) = Z ǫ + E F − ǫ + E F A ( E ) dE = 2 α Z + ∞ c exp( − γ d x d − ( β + 1) x ) dx ≃ αβ + 1 exp( − γ d c d − ( β + 1) c )= 2 αβ + 1 ǫ β +10 exp( − γ d | log ǫ | d ) , (A · x = − log( ǫ ), c = − log( ǫ ). Here we employ anapproximation Z + ∞ c exp( − ax d − x ) dx = Z ∞ c ∞ X n =0 n ! ( − a ) n ( x d ) n e − x dx = ∞ X n =0 n ! ( − a ) n Γ(1 + dn, c ) ≃ ∞ X n =0 n ! ( − a ) n c dn e − c = exp( − ac d − c ) , (A ·
4) which is justified for c ≫
1. In the fourth line, we expandthe incomplete gamma function Γ( z, p ) in p asΓ( z, p ) = p z − e − p × ( ∞ X n =1 p n ( z − z − · · · ( z − n ) ) . (A · ·
2) with respect to ǫ , we ob-tain dρdǫ = − (cid:26) A ( ǫ ) adN /d +1 ( ǫ ) − k B T (cid:27) ρ ( ǫ ) . (A · A ( ǫ ) adN /d +1 ( ǫ ) = 1 k B T . (A · ·
7) and sub-stituting eqs. (A ·
1) and (A · γ d | log ǫ | d + ( β + d + 1) | log ǫ | = d | log( k B T ) | , (A · x , a x + b x /d = c (A · x = c a − − b c /d a − − /d + O ( c − /d ) . (A · d >
1, we obtain the solution of eq. (A ·
8) upto the next leading term in the low-temperature limit as γ d | log ǫ | d = dX − ( β + d + 1) d /d γ − X /d + O ( X − /d ) , (A · ǫ = exp (cid:16) − d /d γ − X /d +( β + d + 1) d − /d γ − X − /d + O (cid:0) X − /d (cid:1)(cid:17) , (A · . Phys. Soc. Jpn. Full Paper H. Shinaoka et al. where X represents | log( k B T ) | . By substituting theseequations into eq. (A · N ( ǫ ) = 2 αβ + 1 ( k B T ) d exp( d /d +1 γ − X /d ) . (A · ·
2) up to the leading term is given by1 N ( ǫ ) /d a = exp( − d /d γ − X /d ) a (cid:16) αβ +1 (cid:17) /d k B T . (A · ·
2) is given by ǫ k B T = exp( − d /d γ − X /d ) k B T . (A · ρ = ρ exp (cid:18) c exp( − c | log( k B T ) | /d ) k B T (cid:19) , (A · d > c = 1 + a − ( αβ +1 ) − /d and c >
0. Equa-tion (5.5) gives the scaling of the resistivity for our softHubbard gap (given by eq. (4.8)) obtained for the modelwith the short-range interaction only.
A.2 Modified ES scaling: Energy region (D.4) in Ta-ble II
We next discuss the modification of the DC resistivityat energies lower than those justified by the ES scaling,by starting from eq. (4.13). We derive the temperaturedependence of the DC resistivity in a way similar to thecase of the soft Hubbard gap.We integrate the DOS with respect to energy using theexpansion of the incomplete gamma function and obtain N ( ǫ ) = Z ǫ + E F − ǫ + E F A ( E ) dE = 2 αβ − d − Γ( − d , βǫ − d ) ≃ αβ − d − ǫ d +10 exp( − β | ǫ | − d ) . (A · · βd − ǫ − d + (1 + 1 /d ) log( ǫ − d ) = | log( k B T ) | . (A · x in the limit of c → + ∞ , a x + b log( x ) = c (A · x = c a − − a − b log( c ) + O ( c ) . (A · ·
18) up to the nextleading term in the low-temperature limit as ǫ − d = dβ − X − dβ − (1 + 1 /d ) log( X )+ O ( X ) , (A · ǫ = ( β/d ) /d X − /d +( β/d ) /d d − (1 + 1 /d ) X − (1+1 /d ) log( X )+ O ( X − (1+1 /d ) ) , (A ·
22) where X represents | log( k B T ) | again. Thus the first termin the exponential function in eq. (A ·
2) up to the leadingterm is given by1 N ( ǫ ) /d a = ( C/a ) | log( k B T ) | − (1+1 /d ) k B T , (A · C is a constant. On the other hand, the secondterm in the exponential function up to the leading termin eq. (A ·
2) is given by ǫ k B T = ( β/d ) /d | log( k B T ) | − /d k B T . (A · ρ = ρ exp (cid:18) ( β/d ) /d | log( k B T ) | − /d k B T (cid:19) . (A ·
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