We study the interplay of disorder and correlation in the one-dimensional
hole-doped Hubbard-model with disorder (Anderson-Hubbard model) by using the
density-matrix renormalization group method. Concentrating on the doped-hole
density profile, we find in a large $U/t$ regime that the clean system exhibits
a simple fluid-like behavior whereas finite disorders create locally Mott
regions which expand their area with increasing the disorder strength contrary
to the ordinary sense. We propose that such an anomalous Mott phase formation
assisted by disorder is observable in atomic Fermi gases by setup of the box
shape trap

M. MACHIDA

M. OKUMURA

Nobuhiko TANIGUCHI

S. YAMADA

谷口 伸彦

English

arXiv

M. Okumura

S. Yamada

N. Taniguchi

M. Machida

Crossref

Hole Localization in the One-Dimensional Doped Anderson-Hubbard Model

TANIGUCHI Nobuhiko

0000000608

OKUMURA M.

YAMADA S.

MACHIDA M.

Tsukuba Repository  (Tulips-R)

Hole Localization in the One-Dimensional Doped Anderson-Hubbard ModelM. Okumura,1,2,* S. Yamada,1,2,† N. Taniguchi,3,‡ and M. Machida1,2,x1CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan2CREST (JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan3Institute of Physics, University of Tsukuba, Tennodai, Tsukuba 305-8571, Japan(Received 13 March 2008; published 3 July 2008)We study an interplay of disorder and correlation in the one-dimensional hole-doped Hubbard-modelwith disorder (Anderson-Hubbard model) by using the density-matrix renormalization group method.Concentrating on the doped-hole-density profile, we find in a large U=t regime that the clean systemexhibits a simple fluidlike behavior whereas finite disorders create locally Mott regions which expandtheir area with increasing the disorder strength contrary to the conventional sense. We propose that such ananomalous Mott phase formation assisted by disorder is easily observable in atomic Fermi gases by settingup the box-shape trap.DOI: 10.1103/PhysRevLett.101.016407 PACS numbers: 71.10.Fd, 03.75.Ss, 71.10.Pm, 71.23.kRecently, atomic Fermi gas loaded on an optical lattice(FGOL) has attracted a lot of attention, since FGOL isexpected to be an excellent testbed to resolve controversialissues in condensed matter physics [1]. One of the advan-tages of FGOL is a tunability of the interaction betweentwo atoms associated with the Feshbach resonance [2],which opens up a pathway to systematically study notonly BCS-BEC crossover but also strongly correlated be-haviors. Another advantage is the flexibility in makingplaygrounds such as the periodical lattice, which providesvarious stages including disorder effects for many-bodyinteracting systems [1].Among a huge number of proposals on FGOL, one of theunique challenges is a study of interplay between random-ness and strong correlation [1]. This is one of the mostdifficult but important problems in real solids becausehigh-Tc superconductor is a typical reality. In high-Tcsuperconductors, their common mother phase is the Mottinsulator showing antiferromagnetism [3]. The carrier isdoped by chemical substitution, which inevitably brings arandom potential. However, the disorder effects in stronglycorrelated systems have been too complicated issues tostudy theoretically and experimentally in condensed mat-ters. Thus, its interplay has remained as an unsolved issue.On the other hand, FGOL is a very good experimentalreality in systematically examining such a complex issuedue to the wide tunability and flexibility.In this Letter, we study the doped Mott insulator withdisorder in a form of the Anderson-Hubbard model [seeEq. (1) below], and predict experimental results on FGOLby means of the density-matrix renormalization group(DMRG) method [4,5]. Consequently, we find that thedisorder does not destroy the Mott insulator but help thegrowth of the Mott phase domains contrary to our naiveexpectation. Such a nontrivial feature is kept and amplifieduntil the disorder amplitude fully exceeds over the repul-sive interaction strength.So far, the harmonic trapped FGOL has been consideredin theoretical studies. For instance, Gao et al. have reportedtheir DMRG calculation results for the 1D Anderson-Hubbard model [6,7] with the harmonic trap potential[8]. However, the harmonic trap induces spatially inhomo-geneous filling which is different from usual situations insolid state matters. On the other hand, the box trap withdisorder provides almost equivalent stages. Thus, our tar-get reality is one-dimensional (1D) FGOL confined in abox-shaped trap with lattice including randomness. In itsexperimental setup, see Ref. [9] for the box-shape trap andRef. [1] for the random potential.The Hamiltonian of the 1D Anderson-Hubbard model isgiven by HAH  tXhi;ji;cyicj Xi;ini; XiUni"ni#; (1)where hi; ji refers to the nearest neighbors i and j  i 1,t is the hopping parameter between the nearest neighborlattice sites, U is the on-site repulsion, ci (cyi) is theannihilation (creation) operator with spin index  "; # ,ni; cyic is the site density operator, and the randomon-site potential i is chosen by a box probability distri-bution P i  W=2 jij=W, where x is the stepfunction and the parameter W controls the disorderstrength. In all DMRG calculations, we employ the openboundary condition for the box-shape trap, and focus onthe site density of fermions as ni  ni;"  ni;#.The 1D Anderson-Hubbard model has been intensivelyinvestigated in the context of the transition between theAnderson and Mott insulators [10–12]. To our knowledge,however, this model has not been fully studied in a range ofslight to under doping. Although the model may looksimple, the interplay among the disorder, the interaction,and the doped holes requires more accurate and moresystematic studies. This Letter provides the first systematicresults of doped-hole profiles under a full accuracy.In order to calculate the ground state of the Hamiltonian(1), we use the DMRG method [4,5]. The validity of ourDMRG results was verified by results of the exact diago-PRL 101, 016407 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 20080031-9007=08=101(1)=016407(4) 016407-1 © 2008 The American Physical Societynalization in a case of (8 " , 8 # ) with 18 sites (L  18) atU=t  W=t  30. Both the results give a good agreement.The size dependence in terms of doped-hole profiles ischecked for L  50, 100, 150, and 200, which reveal thatL  50 is enough to characterize hole profiles. In thefollowing, we present results of 1D system with the lengthL  50 in three fillings, n PLi1 ni=L  0:96, 0.88, and0.52. In the use of DMRG, the number of states kept m isset maximum 256 for several cases, and m  100 is con-firmed to be enough to obtain convergent results becausethe largest deviation of the local density between them isbelow 104.First, let us show DMRG results of density profiles when2 holes ( " , # ) are doped and the filling n  0:96.Figures 1(a)–1(d) display typical profiles obtained byvarying W in a fixed U=t  30 under a certain randomconfiguration as depicted at the bottom. In the clean limit(W  0), one finds a fluidlike density profile as Fig. 1(a),which belongs to the Tomonaga-Luttinger liquid with openboundary conditions. With switching on W, flat densityregions whose site density equals to a unit emerge with twoclear dips as Fig. 1(b). The locations of the dips do not shiftby choice of the boundary condition, i.e., open or periodicboundary condition. We also confirm the fact by using theexact diagonalization in L  18 sites. Here, we name theflat density region and the dip ‘‘Mott plateau’’ and ‘‘holevalley’’, respectively. We note that the number of observedhole valleys equals to that of doped holes. This implies thatholes tend not to collect but to localize separately.Furthermore, one finds from Figs. 1(b) and 1(c) that theedges of the Mott plateaus become sharper and the holevalleys become deeper as the randomness strength W in-creases from W=t  2 to 18 [see also Figs. 3(b) and 3(c)for another doping case]. This clearly indicates that therandomness assists the formation of Mott plateaus ratherthan breaking the structure. This is quite nontrivial becausedisorder is usually believed to break flat homogeneousdistribution. Since the clean system exhibits Tomonaga-Luttinger liquid except for the half-filling, these resultsclearly illustrate that the randomness is essential for thelocal development of the Mott state, where the stronginteraction and the randomness cooperatively form theMott plateaus with localizing the doped holes. This issuggestive for the fact that the insulator phase in high-Tcsuperconductors survives tiny doping.When the randomnessW increases and approachesW U, the Mott plateaus are disturbed partly by disorder [seeFig. 1(d)]. By further increase of W above U, the Mottplateau with localized holes breaks down. Figures 1(e)–1(h) show typical density profiles in a weaker interactionU=t  10. Similar to Figs. 1(b) and 1(c), the structure ofthe Mott plateau with localized holes is seen forW <U, asFigs. 1(f) and 1(g). The structure is, however, completelydestroyed by the strong random potentialW >U as seen inFig. 1(h). Under the strong randomness, the local densitytakes values close to 2 (the double occupation) at severalsites, and the Mott plateaus are too small to be identified asa region.The W dependent density profiles seen in Fig. 1 suggestthat there is an optimumW (U) for a fixedU (W) in makingthe Mott plateaus as wide as possible and the hole valleysas sharp as possible. To evaluate an optimum set of pa-rameters (U, W), we introduce a function MU;W in thetwo variable space to characterize the extent of the Mottplateau in the total density profile, which is defined by asummation of ‘‘closeness’’ of the local density to unitexpressed as MU;WXLi1expni;U;W1222 nL; (2)where ni;U;W  ni;"  ni;# means the local site densityunder a realization of the random potential at a certain setof U, W and a local potential , and h	i is the algebraicaverage for multiple random realizations.  characterizesthe peak width of the function, for which we fix   0:05.We note that the factor 1= nL is the normalization con-stant which is adjusted to make MU;W ’ 1 when the ex-tent of the Mott plateaus is the maximum, e.g., MU;W ’1 when ni  0 at two lattice points and ni  1 at other 48lattice points if two holes are doped in L  50.Figure 2 shows a contour plot of MU;W obtained bythe arithmetical average over ten realizations of randomFIG. 1 (color online). The randomness W dependent densityprofiles ni for 2-holes doped case ( n  0:96) at two fixedinteraction strengths, (a)–(d) for U=t  30, and (e)–(h) forU=t  10, under a random potential depicted in the bottom ofeach figure in arbitrary unit (gray dashed line).PRL 101, 016407 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008016407-2potentials for n  0:96 (2 holes doped) case. Along a fixedW=t line, one finds that the value of MU;W increasesmonotonically with increasing U=t. This is consistent withwhat one sees in Fig. 1; i.e., the width of the Mott plateausin theU=t  30 case [Fig. 1(c)] is wider than that inU=t 10 case [Fig. 1(h)] on the same randomness strengthW=t  18. On the other hand, along a fixed U line, thevalue of MU;W initially increases with increasing W,reaches the maximum values close to W  2tU=2 line,and then decreases after crossing the maximum value.Thus, one finds in the slightly doped case that holes areconfined as compactly as possible when W is about a halfof U so that the width of the hole valley is almost a unitlattice constant. It indicates that doped holes almost losetheir quantum delocalizing character.Here, we turn our attention to the cases in which moreholes are doped. Figures 3(a)–3(d) show the total densitydistributions of fermions in n  0:88 (6 holes) case withU=t  20, where we can confirm almost the same behav-iors as n  0:96 (2 holes) case. Namely, the introduction ofdisorder results in the formation of both the Mott plateausand hole valleys, and further increase of W makes the Mottplateaus wider and hole valleys deeper [compare Fig. 3(b)(W=t  2) with (c) (W=t  14)]. One also finds that thenumber of valleys is exactly the same as that of doped holesat W=t  2 case while the number becomes smaller thanthat of doped holes at W=t  14 and W=t  20. Thetendency to create the Mott plateau still remains even forfar away from the half-filling. Figures 3(e)–3(h) show thedensity profile for n  0:52 (24 holes, i.e., close to thequarter filling) at U=t  10. Although the number of val-leys is no longer the same as that of doped holes in all cases(W=t  2, 12, and 20) and the number of fermions is notenough to simply form the Mott plateau at this filling, themaximum density is nearly a unit [see Figs. 3(g) and 3(h)].This suggests even in such a high doping level that both theinteraction and randomness cooperatively work.Figures 4(a) and 4(b) show contour plots ofMU;W forn  0:88 (6 holes) and n  0:52 (24 holes) cases, respec-tively. These essentially show the same tendency as n 0:96 (2 holes) case. The value of MU;W increases withFIG. 2 (color online). A contour plot of MU;W in 2 holesdoped case ( n  0:96) as a function of U=t and V=t. Arithmeticaverage over ten realizations of random potentials is taken. Thestep values are 2 for both U=t and W=t axes.FIG. 4 (color online). Contour plots of the value ofMU;W in(a) n  0:88 (6 holes) and (b) n  0:52 (24 holes) cases with thearithmetic average of ten realizations of random potentials. Thestep value for both U=t and W=t is 2.FIG. 3 (color online). The randomness W dependent densityprofiles ni of (a)–(d) 6 ( n  0:88) and (e)–(h) 24 ( n  0:52)holes doped cases under a random potential shown in the bottomof each figure in arbitrary unit (gray dashed line).PRL 101, 016407 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008016407-3increasing U=t at a fixed W=t, and the optimum value ofMU;W exists for a fixed U=t. On the other hand, onefinds in more details that the maximum of MU;W shiftsto W  2t > U=2 side with increasing the number ofdoped holes. This is because, in high hole-density cases,the random potential magnitude required to push the localdensity up to the unit becomes larger. In addition, highhole-density causes the reduction of the maximum value ofMU;W, because it suppresses the correlation effect andworks unfavorably in forming the Mott plateaus.We also examine fluctuations of MU;W on differentrandomness by evaluating the standard deviation per aver-age DU;W hMU;W=MU;W  12ip, whereMU;W PLi1 exp ni;U;W  12=22= nL.As expected, one finds in Fig. 5(a) (2 holes doped case) thatDU;W becomes relatively small when MU;W showslarge values, and vice versa. This clearly demonstrates thatthe formation of Mott plateaus does not depend on therealization of random potential and MU;W is a goodmeasure to know it. We also obtain almost the samebehavior of DU;W in the 24 holes case as shown inFig. 5(b).In conclusion, we calculated the density profiles offermions in the 1D Anderson-Hubbard model by varyinginteraction strengths, random potential amplitudes, andfillings below the half-filling by means of DMRG. Wefound a clear signature that the presence of disorder assiststhe local formation of the Mott state in the weak disorderregion whereas the Mott phase is destroyed by strongdisorder. As a characterization of the width of the Mottphase, we calculated the function MU;W from the den-sity profiles, and found that the increase of the doping rateshifts the maximum of MU;W from W  2t  U=2 lineto W  2t > U=2 side. These nontrivial behaviors ofdoped holes like the present DMRG works can be system-atically examined by FGOL with the box trap. The experi-mental confirmation will give a crucial contribution tostudies for doped Mott insulators.The authors wish to thank H. Aoki, T. Deguchi, N.Hayashi, K. Iida, T. Koyama, H. Matusmoto, N. Nakai,Y. Ohashi, T. Oka, S. Tsuchiya, and Y. Yanase for illumi-nating discussion. The work was partially supported byGrant-in-Aid for Scientific Research (GrantNo. 18500033) and one on Priority Area ‘‘Physics of newquantum phases in superclean materials’’ (GrantNo. 18043022) from the Ministry of Education, Culture,Sports, Science and Technology of Japan, and a JAEAinternal project represented by M. Kato. One of authors(M. M.) is supported by JSPS Core-to-Core Program-Strategic Research Networks, ‘‘Nanoscience andEngineering in Superconductivity (NES)’’.*okumura.masahiko@jaea.go.jp†yamada.susumu@jaea.go.jp‡taniguch@sakura.cc.tsukuba.ac.jpxmachida.masahiko@jaea.go.jp[1] For a review, see, e.g., M. Lewenstein, A. Sanpera, V.Ahufinger, B. Damski, A. Sen(De), and U. Sen, Adv. Phys.56, 243 (2007), and references therein.[2] See for a review, E. Timmermans, P. Tommasini, M.Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999),and references therein.[3] For a review, see, e.g., P. A. Lee, N. Nagaosa, and X.-G.Wen, Rev. Mod. Phys. 78, 17 (2006), and referencestherein.[4] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev.B 48, 10 345 (1993).[5] For recent reviews, see, e.g., U. Schollwöck, Rev. Mod.Phys. 77, 259 (2005); K. A. Hallberg, Adv. Phys. 55, 477(2006), and references therein.[6] P. W. Anderson, Phys. Rev. 109, 1492 (1958).[7] J. Hubbard, Proc. R. Soc. A 240, 539 (1957); 243, 336(1958).[8] Gao Xianlong, M. Polini, B. Tanatar, and M. P. Tosi, Phys.Rev. B 73, 161103 (2006).[9] T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, andM. G. Raizen, Phys. Rev. A 71, 041604 (2005).[10] M. Ma, Phys. Rev. B 26, 5097 (1982); A. W. Sandvik, D. J.Scalapino, and P. Henelius, ibid. 50, 10474 (1994); R. V.Pai, A. Punnoose, and R. A. Römer, arXiv:cond-mat/9704027; Y. Otsuka, Y. Morita, and Y. Hatsugai, Phys.Rev. B 58, 15 314 (1998).[11] For reviews, see, e.g., P. A. Lee and T. V. Ramakrishnan,Rev. Mod. Phys. 57, 287 (1985); B. Kramer and A.McKinnon, Rep. Prog. Phys. 56, 1469 (1993); D. Beltzand T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994); F.Evers and A. D. Mirlin, arXiv:0707.4378 [Rev. Mod. Phys.(to be published)] and references therein.[12] Another context is the persistent current in mesoscopicrings, which is not cited in this Letter. For its recentprogress, see, e.g., E. Gambetti, Phys. Rev. B 72,165338 (2005), and references therein.FIG. 5 (color online). Contour plots of DU;W in (a) 2 holesdoped case ( n  0:96) and (b) 6 holes doped case ( n  0:88) asa function of U=t and V=t. Arithmetic average over ten realiza-tions of random potentials is taken. The step values are 2 for bothU=t and W=t axes.PRL 101, 016407 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008016407-4

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Hole Localization in One-Dimensional Doped Anderson-Hubbard Model

Hole Localization in One-Dimensional Doped Anderson-Hubbard Model

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