Half metals at intermediate energy scales in Anderson insulators
HHalf metals at intermediate energy scales in Anderson insulators
Kyung-Yong Park, Hyun-Jung Lee, and Ki-Seok Kim
1, 3 Department of Physics, POSTECH, Pohang, Gyeongbuk 37673, Korea Seoul Science High School, Seoul 03066, Korea Asia Pacific Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk 37673, Korea (Dated: October 28, 2020)Although quantum phase transitions involved with Anderson localization had been investigatedfor more than a half century, the role of spin polarization in these metal-insulator transitions has notbeen clearly addressed as a function of both the range of interactions and energy scales. Based on theAnderson-Hartree-Fock study, we reveal that the spin polarization has nothing to do with Andersonmetal-insulator transitions in three dimensions as far as effective interactions between electrons arelong-ranged Coulomb type. On the other hand, we find that metal-insulator transitions appearwith magnetism in the case of Hubbard-type local interactions. One of the most fascinating andrather unexpected results is the appearance of half metals at intermediate energy scales in Andersoninsulators of the Fermi energy, that is, only spin ↑ electrons are delocalized while spin ↓ electronsare Anderson localized. Introduction:
Since the interplay between Andersonlocalization and electron correlation had broken downthe common wisdom of the absence of metals in two di-mensions, there have appeared lots of interests on two-dimensional metal-insulator transitions not only for ex-perimental aspects of semiconducting devices but alsofor theoretical perspectives of both field theoretical andnumerical studies [1]. On the other hand, the elec-tronic structure near Anderson metal-insulator transi-tions has not been characterized in three dimensionsboth intensively and extensively, compared to the caseof two-dimensional metal-insulator transitions. Recentexperiments revealed that eigenfunction multifractality[2] still survives in the presence of Coulomb interac-tions, where the measured multifractal exponent is en-hanced and the nature of multifractality becomes weak-ened [3]. These experiments promoted theoretical inves-tigations such as nonlinear σ − model field-theory studyfor multifractal exponents [4] and metal-insulator transi-tions based on the Hartree-Fock approximation for inter-actions [5] and in hybridization with density functionaltheory [6, 7]. Recently, two of the authors showed thattwo types of mobility edges exist in the presence of long-ranged Coulomb interactions [8]. The multifractal analy-sis within the Hartree-Fock approximation confirms thatthe high-energy mobility edge is connected to that of theAnderson transition without interactions while the low-energy mobility edge near the Fermi energy occurs fromthe interplay between Coulomb correlation and Andersonlocalization. In particular, it turns out that each mobil-ity edge is described by two different multifractal spectralfunctions.Although the interplay between Anderson localizationand electron correlation has been shown to result in anovel electronic structure for three-dimensional metal-insulator transitions, there still remains an essential ques-tion, that is, the role of spin degrees of freedom in thesemetal-insulator transitions. Resorting to the multifrac- tal analysis within the Anderson-Hartree-Fock study, weexamine how local magnetization affects these metal-insulator transitions as a function of both the range ofelectron correlations and energy scales. First, we revealthat the local spin polarization has nothing to do withAnderson metal-insulator transitions in three dimensionswhen effective interactions between electrons are long-ranged Coulomb type. More precisely, both magnetismand local fluctuations occur deep inside an insulatingphase. Second, on the other hand, we find that metal-insulator transitions appear with magnetism in the caseof Hubbard-type local interactions. One of the most fas-cinating and rather unexpected results is the appearanceof half metals at intermediate energy scales in Andersoninsulators of the Fermi energy, that is, only spin ↑ elec-trons are delocalized while spin ↓ electrons are Andersonlocalized. Model Hamiltonian and numerical analysis:
Weconsider an effective model Hamiltonian on a three-dimensional cubic lattice, where the quadratic part is H = − t (cid:88) (cid:104) ij (cid:105) (cid:88) σ c † iσ c jσ − h.c. + (cid:88) i (cid:88) σ ( (cid:15) i − µ ) c † iσ c iσ , (1)and the interaction term is H I = 12 (cid:88) i (cid:88) j (cid:88) σσ (cid:48) c † iσ c iσ U σσ (cid:48) ij c † jσ (cid:48) c jσ (cid:48) . (2)Here, t is the hopping integral between nearest neigh-bor sites (cid:104) ij (cid:105) , and µ is the chemical potential. In thisstudy we set t = 1 as the unit of energy and focus onthe case of half filling. (cid:15) i is a random potential, uni-formly distributed in [ − W, W ]. It is well known thatthe Anderson transition occurs at W c = 8 .
25 withoutelectron correlations [9]. Here, we set W = 7 < W c .The interaction coefficient is given by U σσ (cid:48) ij = U σσ (cid:48) r ij withthe strength U σσ (cid:48) ≡ e κ σσ (cid:48) for the case of Coulomb and U σσ (cid:48) ij = U δ − σσ (cid:48) δ ij for the case of Hubbard, respectively. a r X i v : . [ c ond - m a t . d i s - nn ] O c t Resorting to the Hartree-Fock approximation for in-teractions, we perform the self-consistent analysis in realspace for both (i) spin-density and charge-density orderparameters of the Hartree channel and (ii) exchange hop-ping order parameters of the Fock channel, and find cor-responding eigenfunctions and eigen energies for a givendisorder configuration. Then, we take averaging for vari-ous disorder configurations: Twenty samples for disor-der realizations in the long-ranged interacting system(LRIS) and fifty ones in the short-ranged interacting sys-tem (SRIS). We would like to emphasize that the Ewald-summation technique [10–12] has been utilized, whichallows us to deal with long-ranged Coulomb interactionsquite accurately, where the long-ranged part of interac-tions is taken in the momentum space while the short-ranged interaction is considered in the real space [13].See Ref. [8] for more details. In addition, we performour simulations in the three dimensional cubic lattice,varying the system size L with L = 12, 16, and 20 forthe multifractal analysis to identify the mobility edge. Density of states and magnetization:
Disorder aver-aged density of states (DOS) ρ ( ω ) are shown in Fig. 1(LRIS) and Fig. 2 (SRIS). First of all, it turns out thatthe spin-resolved DOS remains essentially the same asthat of the spinless case [8] in the LRIS. The dip featureof the DOS at the Fermi energy is already pronounced inthe interaction range of 0 . ≤ U ≤ .
2, identified with theEfros-Shklovskii pseudogap [14] in the log-log plot [15]and not shown here but carefully discussed in the spin-less case [8]. According to the multifractal analysis in thespinless case, the metal-insulator transition at the Fermienergy occurs around U ≈ .
3, confirmed by the presentmultifractal analysis below. On the other hand, the DOSevolution of the SRIS shows strong spin resolution in con-trast with that of the LRIS. Besides the dip feature at theFermi energy, where the Altshuler-Aronov correction be-havior [16] is mixed with the Efros-Shklovskii pseudogapfeature [14] in the log-log plot and not shown here, thereappears additional suppression of the DOS at an interme-diate energy scale, identified with ω U . This suppressedDOS is transferred to enhance the DOS at the other en-ergy scale of ω U + U . The opposite-spin DOS shows theopposite behavior, that is, enhancement of the DOS at ω U but its suppression at ω U + U . This spin-resolvedDOS suppression and enhancement at each intermediateenergy scale is one of the main discoveries in this study.Figure 3 shows disorder averaged magnetization as afunction of the interaction strength in both LRIS andSRIS. Magnetization appears at 1 . < U < . . (a) (b) (c) (d) FIG. 1: Disorder averaged density of states (DOS) ρ ( ω )for various interaction parameters in the long-rangedinteracting system (LRIS). This spin-resolved DOS doesnot show much difference in this interaction range,compared with that of the spinless case [8]. Accordingto the multifractal analysis in the spinless case, themetal-insulator transition at the Fermi energy occursaround U ≈ .
10, given by the below multifractal analysis. In ad-dition, we observed strong spatial fluctuations of the lo-cal magnetization in the metal-insulator transition. Thiscomparison study leads us to conclude that the spin dy-namics has nothing to do with the metal-insulator transi-tion in the LRIS while local magnetic fluctuations wouldaffect the Anderson transition in the SRIS.
Multifractal Analysis:
Strong fluctuations of eigen-functions are one of the fingerprints near the Andersonmetal-insulator transition, responsible for the nature ofeigenfunction multifractlity [17]. This eigenfunction mul-tifractality can be characterized by the fractal dimensionof each moment of an eigenfunction, given by a set ofinverse participation ratios and their disorder averages[2]. To identify the mobility edge as a function of theinteraction strength, we obtain the multifractal scalingexponent from the fractal dimension by differentiating itwith respect to the number of the eigenfunction moment[8]. We follow the standard procedure for actual numeri-cal calculations [18, 19], well known in this research andnot shown here. An essential feature of the multifractalscaling exponent is that it is defined at each eigenenergyand becomes enhanced in metals as the size of the systemis enlarged. On the other hand, it decreases in insulatorsas the volume is expanded. As a result, we determinethe mobility edge, which remains unchanged regardlessof the system size [20].We show the spin-resolved multifractal scaling expo- (a) (b) (c) (d)
FIG. 2: Disorder averaged density of states (DOS) ρ ( ω )for various interaction parameters in the short-rangedinteracting system (SRIS). The DOS evolution of theSRIS shows strong spin resolution in contrast with thatof the LRIS. Besides the dip feature at the Fermi energy,there appears additional suppression of the DOS at anintermediate energy scale, identified with ω U , here.This suppressed DOS is transferred to enhance the DOSat the other energy scale of ω U + U . The opposite-spinDOS shows the opposite behavior, that is, enhancementof the DOS at ω U but its suppression at ω U + U .nent α for three system sizes L = 12, 16, and 20 inthe LRIS, given by Fig. 4. As shown in the DOS ofFig. 1, the multifractal scaling exponent remains un-changed from that of the spinless case, where α ↑ and α ↓ are almost identical. Here, we marked size-independenttwo crossing points for scale-invariant multifractal expo-nents [20], which identify two types of mobility edges.It turns out that the high-energy mobility edge is con-nected to that of the Anderson transition without inter-actions while the low-energy mobility edge near the Fermienergy occurs from the interplay between Coulomb cor-relation and Anderson localization [8]. The low-energycrossing point close to the Fermi energy starts to ap-pear from around U = 0 .
3, and the multifractal scal-ing exponent α saturates to the value of the spinlesscase around U = 0 .
8, from which it does not dependon the interaction strength, consistent with the previ-ous study [8]. This leads us to conclude that the criticalstrength for the Anderson metal-insulator transition oc-curs at 0 . ≤ U MIT c < .
5. We recall the critical strength1 . < U FM c < . U MITc (cid:28) U F Mc confirmsthat the local ferromagnetism has nothing to do withthe Anderson metal-insulator transition in the LRIS.On the other hand, the multifractal scaling exponent α for three system sizes L = 12, 16, and 20 in the SRIS U=0.5 U=1.2U=0.8 (a)
U=5 U=10 U=20 (b)
FIG. 3: Disorder averaged magnetization as a functionof the interaction strength in both LRIS and SRIS.Magnetization appears at 1 . < U < . .
3. On the other hand, ferromagnetism already occursnear the metal-insulator transition in the SRIS, thecritical interaction strength of which is 5 < U < U = 5 results in the metallic phase at the Fermi energy,where only the high-energy mobility edge is observed tocause an insulating state for the band edge. Here, thedashed line represents the multifractal scaling exponentof the opposite spin flavor. Increasing the interactionstrength up to U = 10, there appears a crossing pointnear the Fermi energy, identified with an Anderson insu-lating phase. One unexpected feature is that the high-energy mobility edge of spin ↓ electrons in the right paneldiffers from that of spin ↑ electrons in the left panel. As aresult, only spin ↑ electrons become metallic in the energy Extended Localized (a) (b) (c)
LocalizedExtended (d)
LocalizedExtended
FIG. 4: Spin-resolved multifractal scaling exponent α for three system sizes L = 12, 16, and 20 in the LRIS.As shown in the DOS of Fig. 1, the multifractal scalingexponent remains unchanged from that of the spinlesscase [8], where α ↑ and α ↓ are almost identical. Here, wemarked two types of mobility edges, where thehigh-energy mobility edge is connected to that of theAnderson transition without interactions while thelow-energy mobility edge near the Fermi energy occursfrom the interplay between Coulomb correlation andAnderson localization.range of 4 < E < ↓ electrons are Andersonlocalized. Increasing the interaction strength further upto U = 15, spin ↓ electrons are all localized while spin ↑ electrons remain delocalized in 5 < E <
10, resulting ina half metallic phase.
Half metals at intermediate energy scales in Ander-son insulators:
Figure 6 shows magnetization densityof states m ( ω ) = ρ ↑ ( ω ) − ρ ↓ ( ω ) in the SRIS. In theseinteraction-strength ranges, both spin ↑ and ↓ electronsnear the Fermi energy are Anderson localized. On theother hand, only spin ↑ ( ↓ ) electrons are delocalized nearthe positive (negative) frequency region of the magneti-zation peak (dip).Based on the multifractal analysis Fig. 5 with the DOSanalysis Fig. 2 and the magnetization analysis Fig. 6,we obtain a phase diagram in the plane of energy andHubbard interaction strength, shown in Fig. 7. Here,red dots represent the mobility edge for spin ↑ electrons,and blue ones do that of spin ↓ electrons. Dashed linesare shown to clarify phase boundaries. When the Hub-bard interaction is weak, most energy regions are metallicexcept for the band edge, given by the high-energy mo-bility edge. Increasing the interaction strength, the DOSfor spin ↓ electrons is suppressed at intermediate energyscales while that of spin ↑ electrons is enhanced at thecorresponding energy scales, which results from the roleof the Hubbard-type local interactions. As a result, only ↑ Extended (a) ↓ Extended (b) ↑ Extended (c) ↓ Extended Localized (d) ↑ ExtendedLocalized Localized (e) ↓ Localized (f)
FIG. 5: Spin-resolved multifractal scaling exponent α for three system sizes L = 12, 16, and 20 in the SRIS.Here, the left panel shows the multifractal scalingexponent for spin ↑ electrons, and the right panelrepresents that for spin ↓ electrons. Increasing theinteraction strength, we find that the low-energymobility edge arises near the Fermi energy, identifiedwith an Anderson insulating phase. On the other hand,the high-energy mobility edge shows drastic spinresolution in this multifractal scaling exponent, whichgives rise to a half metallic phase in an intermediateenergy scale, where only spin ↑ electrons are delocalized.Inset figures show the spin-resolved local DOS of Fig. 2. ↑ ↓ ↑ ↓↓ ↑ (a) LOC LOCEXT EXT ↓ LOC ↑ ↑ ↑ ↓ EXT ↓ EXTLOCLOC LOC (b)(b)
FIG. 6: Magnetization density of states m ( ω ) = ρ ↑ ( ω ) − ρ ↓ ( ω ) in the SRIS. In theseinteraction-strength ranges, both spin ↑ and ↓ electronsnear the Fermi energy are Anderson localized. On theother hand, only spin ↑ ( ↓ ) electrons are delocalizednear the positive (negative) frequency region of themagnetization peak (dip). ↑ ↓ ↑ ↓ ↑ ↓ ExtendedLocalized No states
ExtendedLocalized
UVIR
FIG. 7: Phase diagram in the plane of energy andinteraction strength in the SRIS. Here, we focus on thepositive energy side. Red (blue) dots represent the(high-energy = UV and low-energy = IR) mobility edgeof spin ↑ ( ↓ ) electrons, the above of which correspondsto an Anderson localized phase. There exists a halfmetallic phase at intermediate energy scales in theAnderson localized state of the Fermi energy.spin ↑ electrons are delocalized while spin ↓ electrons re-main to be Anderson localized. Increasing the interactionstrength further, the whole band becomes Anderson lo-calized. The emergence of the half metallic state is oneof the main results in the present study. Conclusion:
In this study, we revealed that the roleof local magnetic fluctuations in the Anderson metal-insulator transition differs completely between the caseof long-ranged Coulomb interactions and that of localHubbard-type interactions in three dimensions. It turnsout that the spin dynamics has nothing to do with themetal-insulator transition in the former. On the otherhand, the spin dynamics is essential for the Andersontransition in the latter, where the evolution of the multi-fractal scaling exponent shows strong spin resolution forboth high- and low-energy mobility edges. In particular,we found half metals at intermediate energy scales in theAnderson localized phase of the Fermi energy. We believethat the emergence of the half metal phase in Andersoninsulators can be verified experimentally by the pump-probe technique [21].Before closing, we would like to point out that theterm of Anderson insulator at the Fermi energy shouldbe understood in caution. Our recent simulation resultsbased on the multifractal scaling analysis are confirm-ing the existence of a metal-insulator transition at finitetemperatures, where the insulating phase persists up toa certain critical temperature. This implies that the cor-responding insulating state has to be called a many-bodylocalized phase [22, 23]. Our entanglement-entropy cal-culation will clarify the nature of this insulating phasesooner or later. This study was supported by the Ministry of Educa-tion, Science, and Technology (No. 2011-0030046) of theNational Research Foundation of Korea (NRF). [1] B. Spivak, S. V. Kravchenko, S. A. Kivelson, and X. P.A. Gao, Rev. Mod. Phys. , 1743 (2010).[2] F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008).[3] A. Richardella, P. Roushan, S. Mack, B. Zhou, D. A.Huse, D. D. Awschalom, and A. Yazdani, Science ,665 (2010).[4] I. S. Burmistrov, I. V. Gornyi, and A. D. Mirlin, Phys.Rev. Lett. , 066601 (2013).[5] M. Amini, V. E. Kravtsov, and M. Muller, New J. Phys. , 015022 (2014).[6] Y. Harashima and K. Slevin, Phys. Rev. B , 205108(2014).[7] Edoardo G. Carnio, Nicholas D. M. Hine, and Rudolf A.Romer, arXiv:1710.01742 [cond-mat.dis-nn].[8] Hyun-Jung Lee and Ki-Seok Kim, Phys. Rev. B ,155105 (2018).[9] K. Slevin and T. Ohtsuki, Phys. Rev. Lett. , 382(1999).[10] P. P. Ewald, Ann. Phys. (Leipzig) , 253 (1921)[11] S. W. de Leeuw, J. W. Perram and E. R. Smith, Proc.Roy. Soc. Lond. A , 27-56 (1980); ibid. Proc. Roy.Soc. Lond. A , 57 (1980).[12] H. Lee and W. Cai, Ewald summation for Coulomb inter-actions in a periodic supercell , (Lecture Notes, StanfordUniversity, 2009).[13] In order to clarify the role of long-range interactionsin the Hartree-Fock approximation, we implement theEwald summation technique, where the Hartree term issplit into two parts: a real-space portion based on a short-range interaction potential whose pairwise sum convergesquickly and a long-range portion based on a slowly-varying interaction potential whose pairwise sum con-verges relatively quickly in a reciprocal space [10–12]. Theoptimal implementation of the Ewald technique allows usto resolve the long-standing issue of the ill-convergence ofthe long-range potential and the multiplicity of Hartree-Fock solutions near the Anderson-Mott transition, whichhas been also reported in a recent Hartree-Fock numeri-cal study [5].[14] B. I. Shklovskii and A. L. Efros,
Electronic properties ofdoped semiconductors (Springer Science & BusinessMedia, 2013).[15] W. L. McMillan, Phys. Rev. B , 2739 (1981).[16] B. L. Altshuer, A. G. Aronov, A. L. Efros, and M. Pol-lak, Electron-electron Interactions in Disordered Systems ,(Elsevier, Amsterdam, 1985).[17] Multifractality implies the existence of infinitely manyrelevant operators, which cannot be the case for conven-tional continuous phase transitions [2]. This peculiar fea-ture may be involved with the fact that the upper criticaldimension of the Anderson metal-insulator transition isinfinite, where the conventional dimensional regulariza-tion technique does not work for the problem of Andersonlocalization.[18] A. Chhabra, R. V. Jensen, Phys. Rev. Lett. , 1327 (1989).[19] M. Janssen, Int. J. Mod. Phys. B , 943 (1994).[20] A. Rodriguez, Louella J. Vasquez, K. Slevin, and R. A.Roemer, Phys. Rev. B , 134209 (2011); ibid. Phys.Rev. Lett. , 046403 (2010).[21] M. C. Fischer, J. W. Wilson, F. E. Robles, and W. S. Warren, Review of Scientific Instruments , 031101(2016).[22] D. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. , 1126 (2006).[23] I. Gornyi, A. Mirlin, and D. Polyakov, Phys. Rev. Lett.95