Dimensional tuning of electronic states under strong and frustrated interactions
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Dimensional tuning of electronic states under strong and frustrated interactions
Chisa Hotta and Frank Pollmann Kyoto Sangyo University, Department of Physics, Faculty of Science, Kyoto 603-8555, Japan Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany (Dated: November 30, 2018)We study a model of strongly interacting spinless fermions on an anisotropic triangular lattice. At half-filling and the limit of strong repulsive nearest-neighbor interactions, the fermions align in stripes and form aninsulating state. When a particle is doped, it either follows a one-dimensional free motion along the stripes orfractionalizes perpendicular to the stripes. The two propagations yield a dimensional tuning of the electronicstate. We study the stability of this phase and derive an effective model to describe the low-energy excitations.Spectral functions are presented which can be used to experimentally detect signatures of the charge excitations.
PACS numbers: 71.10.Hf, 71.27.+a, 71.10.-w
Electronic properties of metals are essentially understoodon the basis of band theory which straightforwardly reflectsthe actual crystal geometry. In certain strongly correlated ma-terials, however, the electronic interactions can significantlymodify the spacial geometry of the wave functions (metallic-ity). For example, in high- T c cuprates, the charges on thesquare lattice may align in stripes and the metallic proper-ties become one dimensional (1D) [1]. Also in manganites,the strong correlations of spin and orbital degrees of freedomlead to orbital orders in which the metallicity is confined toa particular spatial direction [2]. Here the density of statesreflects correlation effects and causes an effective modulationof the lattice geometry and thus leads to a change of the in-trinsic thermodynamical and transport properties. A recentlyproposed way to tune the effective geometry of a lattice is tointroduce frustrations in the electronic interactions. This isseen in a model of spinless fermions (i.e., fully spin polarizedelectrons) on a triangular lattice with strong nearest neighborrepulsion which is referred as a ”pinball liquid”. Part of thefermions form a Wigner crystal on one of the three sublattices.The other fermions form a metallic state on the remaining sitesalong a honeycomb lattice [3].In this Letter, we investigate a tuning of the effective di-mensionality of the metallicity due to partially frustrated andstrong interactions . We focus on an anisotropic triangularlattice system and start from an insulator at half-filling withstriped charge order. Doping of particles yields a 1D metal-lic state as in the case of the square lattices in cuprates. Inaddition, we find another non-trivial propagation in which thedoped charges fractionalize and change the stripe geometry.We focus only on the charge degrees of freedom and startfrom a basic t - V model Hamiltonian which reads H t − V = X h i,j i t ij (cid:16) c † i c j + H . c . (cid:17) + X h i,j i V ij n i n j . (1)Here c † j ( c j ) are creation (annihilation) operators of fermionsand n j (= c † j c j ) are number operators. The interactions actonly between neighboring sites h ij i . Anisotropies of thehopping amplitudes and repulsion strengths are given by ( t ij , V ij ) = ( t ′ , V ′ ) > (0 , for vertical bonds (along the y -axis in, e.g., Fig. 1) and ( t, V ) > (0 , for the remainingbond directions. We start from the limit of strong anisotropicinteractions, i.e., t/V = 0 , t/V ′ = 0 , t/ | V ′ − V | = 0 , andassume for simplicity that t ′ = t > . The ground state athalf-filling has an insulating charge order which minimizesthe repulsive nearest-neighbor interactions [4]. We considerthe effect of doping either a particle or a hole into the insulat-ing state. Depending on the anisotropy of the interactions,we find two different scenarios. The case V ′ < V is de-picted schematically in Fig. 1(a). The ground state at half-filling has a vertical stripe-order. An added particle can movefreely along a line of empty sites without changing the po-tential energy of V . All processes of moving orthogonal tothe stripes are prevented by an increase of potential energy by V − V ′ ) . This implies that the added particle is confined toa 1D motion along the y -axis with a dispersion − t cos( k y ) for wave-number k y . Doping of one hole decreases the po-tential energy by V ′ and yields a spectrum identical to thatof the added particle. For the case V ′ > V , the ground stateat half filling has a high degeneracy. The ground states areformed by isolated configurations which have stripes perpen-dicular to the anisotropic bonds with a N x -fold degeneracy[4]. Figure 1(b) shows the horizontal stripe configuration asone example of degenerate ground state configurations. Anadded particle can propagate along the stripes as in the for-mer case. However, an additional propagation can take placenow. The doped particle interacts with the stripes in the verti-cal direction by two V ′ -bonds. These two bonds can separateorthogonally to the stripes without increasing the potential en-ergy. Each of them is carrying a charge of − e/ . These localdefects are in the following regarded as fractional charges. Aslong as we remain in the limit of strong interactions, i.e., as-sume t/V = 0 , the fractional charges are deconfined. When aparticle is removed, a similar picture arises from the fraction-alization of a hole.In the above discussion, we have neglected any quantumfluctuation in t/V . Next, we will investigate how quantumfluctuations affect the ground state and calculate the low-lying excitations. We numerically diagonalize the Hamilto-nian in Eq. (1) exactly on finite size clusters and derive astrong-coupling effective model by a perturbative approach.We start by calculating the local density of states (DOS), D ± ( ω ) = P k A ± ( k, ω ) of the full Hamiltonian in Eq. (1)by means of the Lanczos continued fraction method [5]. Here, A ± ( k , ω ) is the spectral function from adding ( + ) and remov-ing ( − ) a particle. We use a N = N x × N y = 4 × site clusterwith periodic boundary conditions. Figs. 2(a) and 2(b) showthe DOS for V ′ < V and V ′ > V in the strong coupling re-gion with V + V ′ ≡ V = 300 . The spectrum of the firstcase ( V ′ < V ) in Fig. 2(a) has a width of t and correspondsexactly to a 1D dispersion in a single-particle picture. Forthe latter case ( V ′ > V ), we find instead an incoherent broadstructure in Fig. 2 (b). The bandwidth is ≈ t and thus twiceas big as in the former case. This indicates the existence of acollective excitation of fractionalized charges.In the following, we focus on the interesting case in whichfractionalization of charge occurs. We begin by estimating theparameter range for which the fractional-charge phase sus-tains. In this phase, the ground-state energy depends on theinteraction in a specific manner, i.e., we find E ∼ V N − (2 V + 4 t ) and V N + (2 V + 2 V ′ − t ) for one hole and oneparticle doping, respectively. The first term originates fromthe stripes and the second term is the interaction energy of thedoped charges. On the other hand, a pinball liquid phase isfound for V ′ ∼ V (i.e., V ∼ V ) which also shows linear V -dependence in energy as discussed in Ref. [4]. We distinguishthe phases by their different linear slopes in the energy E ( V ) .The phase boundaries are then identified for up to two parti-cle/hole doping ( ∼ doping) as shown in Fig. 2(c). Fromthese results we can expect that the fractional-charge phase re-mains stable for a large range of anisotropic interactions evenif the doping level is increased. Note that if we increase thesystem size towards the bulk limit, the phase boundary for thedoped particle number of O (1) will asymptotically approachthe solely t -dependent line at V − V = − t . We thus con-clude that the physics we discuss is relevant for a large rangeof interaction strength and different doping levels.Next, in order to understand in detail how the fluctua-tions affect the low-energy excitation, we perform a pertur-bative treatment in the limit of strong couplings. We be-gin with the half-filled case which has a degenerate ground-state manifold of stripe configuration and include the pertur-bative processes in t/V , t/V ′ and t/ | V − V ′ | up to fourthorder. The first contribution which lifts the degeneracy are thefourth-order exchange processes around plaquettes as shownin Fig. 3(a). We distinguish between (A) plaquettes with oddnumber of particles (i.e., either one or three) and (B) pla-quettes with two particles. These processes give energy cor-rections of − ǫ and + ǫ per plaquette, respectively, where ǫ = 2 t / (cid:0) (2 V ′ − V ) V ′ (cid:1) > . The different signs originatefrom an exchange of fermions. The conversion of one plaque-tte from (A) to (B) increases the energy by ∆ = ǫ . Then,at half-filling, the afore mentioned degeneracy of stripes at t/V ′ = 0 is lifted and the horizontal stripe filled with (A) pla-quettes becomes a unique ground state. The lowest excitationenergy is N y ∆ which results from a collective rearrangementof charges to another stripe [4]. In the fractional-charge phase (a) vertical stripe+1electron(b) horizontal stripe+1electron xy VV N y N x FIG. 1: (Color online) Representative ground-state configurations ofthe Hamiltonian in Eq.(1) on the anisotropic triangular lattice in thelimit of strong correlations ( t/V = 0 ) with one particle doped athalf-filling. Panel (a) show the case on a vertical stripe order at V ′ < V where an added particle can move only along the stripesof empty sites. Panel (b) shows the horizontal stripe at V ′ > V ,where the fractionalization is illustrated by the separation of the twobold vertical bonds in the y -direction.
10 20 30 40 50fractional charge-50 pinball
V-V ( )/ t (c) (b) D( ) w (a) D( ) w w /t -305 -300 -295 295 300 305-205 -200 -195 195 200 205 projection model / t low doping bulk limit FIG. 2: (Color online) Numerical results on a N x × N y = 4 × -site cluster. DOS of Eq. (1) with t = t ′ = 1 for (a) ( V, V ′ ) =(200 , and (b) ( V, V ′ ) = (100 , . Panel (b) compares theDOS of Eq. (1) for V ′ = 200 , V = 100 (bold line) with the effectivemodel (shaded region) of Eq. (2) on top of each other. (c) showsthe ground state phase diagram. The phase boundary is shown fordifferent doping levels. In the bulk system the phase boundary at lowdoping limit approaches the broken line, where we have fractional-charge phase at V − V < − t . near half-filling, those plaquette-exchange processes lead toa confinement. This can be seen directly in Fig. 3(b); theseparation of fractionalized charges converts (A) plaquettesto (B) plaquettes. The confinement potential Λ c ( | y − y | ) ,which sums up all the possible perturbative processes up tofourth order (excluding a constant shift), is shown in Fig. 3(c). (c) |y -y | L c (0,1)( )= (0.5,1)(1, 0.5)(1,1) (A) (B) (b) horizontal stripe+1electron e =+ /plaquette 5 10-101 -5 D D (A) (B) - e e
32 + - e (a) plaquette processes FIG. 3: (Color online) (a) Fourth order ring exchange pro-cesses along the plaquettes; (A)/(B) lowers/raises the energy by( − ǫ )/( + ǫ ) (both the clockwise and anti-clockwise processes arecounted). (b) The case when one particle is added. The shadedplaquettes between the two vertical bonds are the plaquettes re-placed from (A) to (B) when the bonds are separated by the distance | y − y | . (c) Energy correction Λ c including all the perturbationprocesses up to fourth order as a function of | y − y | for severalparameter choices. The gradient is
2∆ = 5 ǫ , which comes fromthe conversion of (A) to (B) plaquettes. Here y and y are the y -components of the locations of thetwo fractions. The main feature is a linear contribution of per two lattice spacings (for | y − y | > ) which arisesfrom part of the fourth-order processes where the conversionof (A) to (B) plaquettes takes place. The above findings al-low us to introduce the following effective Hamiltonian [14]to describe the low energy excitations of the doped system for V ′ ≫ V ≫ t, t ′ : H pr = X h i,j i P (cid:16) t ij c † i c j + H . c . (cid:17) P + ∆( N ( B ) − N ( A ) ) . (2)The projector P projects onto the manifold of allowed con-figurations which minimize the interaction term of Eq. (1).The projected configurations are connected by first order in t ij . The second term takes into account the confinement ef-fects which result from the conversions of (A) to (B) plaque-ttes. The operators N ( A ) and N ( B ) count the number of cor-responding plaquettes. We validate the effective Hamiltonianby comparing the obtained electron doped DOS with the oneobtained from the Full Hamiltonian Eq. (1). As presented inFig. 2(b), the DOS of Eq. (1) for ( V, V ′ ) = (100 , insolid line agrees perfectly well with the one from the effec-tive model on the same cluster presented as a shaded region.The fact that we have to consider only the manifold of stripe-based configurations reduces the size of the Hilbert space of H pr drastically as compared to the full Hilbert space. Thuswe can perform a careful numerical analysis of the effects ofthe confinement potential.Let us first focus on the case, ∆ ≈ . When a particle isadded, it moves either along the stripes freely by t or frac-tionalizes vertically to the stripes by t ′ . This is observed in (b) -4 -2 0 2 4 ( ) /t w -4-2024 G X M Y G M8 12 N= (a) D =10 -6 G XMY k x k y D( ) + w ( t, t (0,1) (1,0)(1,0.5)(1,1)(0.5,1) N y =4240 -4 -2 0 2 4 ( ) /t w (c) =10 0.1 1 10 D -6 FIG. 4: (Color online) Panel (a) shows the energy dispersion ofthe effective Hamiltonian Eq. (2) for a small confining potential
4∆ = 10 − on a N = 8 × cluster. Panel (b) and (c) show thelocal density of states D ( ω ) for different confining potentials and fordifferent anisotropies in the hopping amplitudes, respectively. Weuse for the spectrums the N = 4 × or × cluster which arelarge enough to give qualitatively the same results (we checked thefinite size effect up to N y = 42 ). Arrows in panel (b) indicate thethree intrinsic peak positions. The energy is shifted by a constant C (∆) for a clear presentation. Inset: Schematic illustration of thevariation of the spectrums. Arrows denote the (non-fractionalized)van Hove singularity. the dispersions within the finite cluster. Figure 4(a) showsthe two-dimensional dispersion for t = t ′ = 1 . The disper-sions along Γ − X are nearly flat since the x -direction mo-tion is sacrificed by the increase of coherence of fractionalcharges in y -direction. There are dispersive degenerate modeswith a width of t along M - Y . These modes originates fromthe non-fractionalized portion which can only propagate in x -direction. They will be shown later to contribute to the weak-ened van Hove singularity. The feature of the fractionalizationin the y -direction is detected along the X - M line. We finddensely packed branches which are following a cosine disper-sion. The bandwidth is ≈ t ′ and is almost independent of N y while the number of modes is N y / . Thus, the dispersions ex-trapolate towards a continuum in the bulk limit. The existenceof continuum has already been discussed in terms of domainwall excitations in 1D Wigner lattices and is visible in the ex-act solution [6]. The two fractions share the total momentum k with arbitrary ratio so that at each k -point the energy cantake almost any values of energy within a certain range.The effect of the confinement potential becomes clear bycomparing the DOS obtained for different choices of ∆ (seeFig. 4(b)). If the confinement is small, we find a three peakstructure which results from the superposition of the 1D vanhove singularities at ω = ± t and a ω = 0 -centered broadsingle peak structure. The former singularity is a result ofthe non-fractionalized propagation in the x -direction. The lat-ter single peak is due to the fractionalization of charge andis consistent with the previous 1D study [6]. The weight ofthe single peak is suppressed by a very large confining poten-tial ∆ . However, even for confinement potentials as large as ∆ ∼ t ′ , t , we still find broad incoherent structure expandingover the energy range of | ω | > t . This is a clear indica-tion that although the two fractions cannot separate to infinitedistance under confinement, the coherence due to t ′ is stilllarge and they form bound pairs with large spatial extent (usu-ally of order t ′ / ∆ in units of the lattice constant). A simi-lar transformation of the DOS is observed when we considerthe case without confinement potential but change instead theanisotropic hopping parameters by hand. Figure 4(c) showsthe electron doped spectrum for different ratios of t ′ /t . For t = 0 , t ′ = 0 , the fractional charges form a broad singlepeak structure. When introducing a finite t , the two 1D van-Hove singularity peaks appear, and it finally becomes a pure1D DOS of free fermion along the x -direction at t ′ = 0 , t = 0 .In summary, we find a state with exotic quasiparticles de-scribed by a t - V model on the triangular lattice in the presenceof strong and anisotropic interactions ( V ′ > V ≫ t, t ′ ). Anadded particle or hole decays into bound pairs of fractionalcharges with large spatial extent. Not only the added particlebut also the other particles move cooperatively and show col-lective excitation behavior in the anisotropic direction. Eventhough the system is originally two dimensional, the strongand anisotropic interactions first lead to a 1D like charge or-dering at half-filling. The dispersion of the quasiparticle is thecombination of the 1D free ( t ) and 1D collective ( t ′ ) propa-gations along and perpendicular to stripes, respectively. Theeffective dimension is tuned from one to two-dimension de-pending on the anisotropy of t ′ /t , on the perturbative interac-tions and on the original interaction strength. It is remarkablethat such delicate and non-trivial dimensional tuning occurs insuch a simple model.Fermions and bosons on certain frustrated lattices haveshown to behave in a similar manner, e.g., a pinball liquidfound previously in the present model [3] and a supersolidityin a hard core bosonic model [7] can be compared. We findthat in the present system, a similar hard-core bosonic modelhas a diagonal stripe ground state at half-filling instead of thehorizontal stripe and it shows similar fractionally charged ex-citations. These fractionalized excitations are bound by a con-finement potential of one-half the magnitude of that of thefermionic model. The different details originate simply fromthe different statistics, i.e., the absence of a fermionic signwhen exchanging two hard-core bosons.Finally we note that such dimensional tuning is not limitedto the anisotropic triangular lattice. We find similar effectsalso on the anisotropic kagome lattice at around 2/3-filling.Such picture in between the classical order and the quantumliquid might more generally be expanded to several other frus-trated lattices as well.The two-dimensional charge ordered stripe state originatingfrom strong Coulomb interactions is already an establishedphenomenon in organic solids [8]. To search for its further interesting possibilities, optics become a powerful tool whichaim for the photo-induced phase transitions as in EDO-TTF[9], θ -ET X. [10] and α -ET I [11]. We expect the highlyconducting photo-induced state based on the horizontal stripein the latter two materials to be a good candidate to realizethe scenario presented in this letter. Fractional charge itselfhas been discussed in many theoretical articles [6, 12, 13].Although the present quantum fractionalization due to frus-tration is not exactly the one found in the classical particlepicture, it is certainly exotic and is the strong indications for acollective many body effect. Its experimental proof should beawaited.We thank Peter Fulde and Karlo Penc for helpful discus-sions. This work is supported by the Max-Planck-Institut f¨urPhysik komplexer Systeme. [1] V. J. Emery, S. A. Kivelson, J. M. Tranquada, Proc. Natl. Acad.Sci. USA 96,8814-8817 (1999)[2] K. I. Kugel and D. I. Khomskii, Sov. Phys. 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511 (2005).[12] P. Fulde, K. Penc, and N. Shannon, Annalen der Physik N x -degenerate stripe configurations. When the twofractions separate over the distance N yy