Mott Transitions and Staggered Orders in the Three-component Fermionic System: Variational Cluster Approach
Takumi Hasunuma, Tatsuya Kaneko, Shohei Miyakoshi, Yukinori Ohta
JJournal of the Physical Society of Japan
FULL PAPERS
Mott Transitions and Staggered Orders in the Three-componentFermionic System: Variational Cluster Approach
Takumi Hasunuma, Tatsuya Kaneko, Shohei Miyakoshi, and Yukinori Ohta ∗ Department of Physics, Chiba University, Chiba 263-8522, Japan
The variational cluster approximation is used to study the ground-state properties and single-particlespectra of the three-component fermionic Hubbard model defined on the two-dimensional square lattice athalf filling. First, we show that either a paired Mott state or color-selective Mott state is realized in theparamagnetic system, depending on the anisotropy in the interaction strengths, except around the SU(3)symmetric point, where a paramagnetic metallic state is maintained. Then, by introducing Weiss fields toobserve spontaneous symmetry breakings, we show that either a color-density-wave state or color-selectiveantiferromagnetic state is realized depending on the interaction anisotropy and that the first-order phasetransition between these two states occurs at the SU(3) point. We moreover show that these staggered ordersoriginate from the gain in potential energy (or Slater mechanism) near the SU(3) point but originate fromthe gain in kinetic energy (or Mott mechanism) when the interaction anisotropy is strong. The staggeredorders near the SU(3) point disappear when the next-nearest-neighbor hopping parameters are introduced,indicating that these orders are fragile, protected only by the Fermi surface nesting.
1. Introduction
The many-particle physics of multicomponentfermions in correlated electron systems, such as orbitalorderings, orbital-selective Mott transitions, andorbital-spin liquids, has recently been one of the majorthemes of condensed matter physics. The ground-stateproperties of the N -component model with N > N is the number of internal degrees of freedomof a fermion, are conjectured to depend strongly on N ,and therefore the occurrence of novel quantum phasesat N > In particular, theSU( N ) symmetric model has been of great interest forsimulating systems of correlated fermionic ultracoldatoms in optical lattices. Thus, the possible experi-mental realization of systems with
N >
Yb,
10, 11) the SU(6) × SU(2)symmetry for a mixture of
Yb and
Yb, theSU(10) symmetry for Sr,
13, 14) and a three-componentfermionic system, although not exactly SU(3), for Li.
15, 16)
The three-component fermionic Hubbard model,which is a natural extension of the two-componentone for correlated electrons, has recently been studiedto clarify its metal-insulator transition, translational-symmetry-broken staggered orders, and superfluidity.Namely, dynamical mean field theory (DMFT) calcula-tions have shown that the metal-insulator transition inthe paramagnetic state of the N = 3 Hubbard model athalf filling (or 3/2 fermions per site) does not occur at theSU(3) symmetric point but it occurs when the interac-tion strengths between components are anisotropic. Moreover, the DMFT calculations assuming transla-tional symmetry breaking have shown that two types ofstaggered orders appear, depending on the anisotropy in ∗ [email protected] the interaction strengths, and the first-order phase tran-sition occurs between the two at the SU(3) symmetricpoint. Superfluidity has also been reported to oc-cur, even in the repulsive Hubbard model near half filling,when the interaction strengths are anisotropic, suggest-ing the presence of an exotic pairing mechanism,
19, 23–25) where the fluctuations of the staggered orders may causethe pairing. We may therefore point out that the effectsof the lattice geometry and Fermi surface nesting, whichare important in the formation of the staggered orders,should be examined carefully in the low-dimensional sys-tems.In this paper, motivated by the above developments inthe field, we study the ground-state properties and ex-citation spectra of the three-component fermionic Hub-bard model defined on the two-dimensional square latticeat half filling by means of the variational cluster approx-imation (VCA). The VCA can treat the short-range spa-tial correlations precisely in the thermodynamic limit oflow-dimensional systems and therefore has a major ad-vantage that the effects of the lattice geometry and Fermisurface topology can be examined, which DMFT studiescannot tackle. We thereby hope that some new insightsinto the physics of three-component fermionic systemswill be obtained.First, we calculate the single-particle spectrum, den-sity of states, and single-particle gap in the paramagneticstate to study the Mott transitions of the model, andwe draw its ground-state phase diagram, which includesthree distinct Mott phases. Then, we introduce the Weissfields to study the translational symmetry breakings andshow that two types of staggered orders appear in theground state of the model. We moreover examine theorigin of the long-range orderings in terms of the gainsin kinetic and potential energies, addressing the Slaterversus Mott mechanisms. We also examine the stabilityof the staggered orders by introducing the next-nearest- a r X i v : . [ c ond - m a t . s t r- e l ] M a y . Phys. Soc. Jpn. FULL PAPERS neighbor hopping parameters to destroy the Fermi sur-face nesting. We thereby discuss the characteristic prop-erties of the three-component fermionic system definedon the two-dimensional square lattice.This paper is organized as follows. In Sect. 2, we in-troduce the three-component Hubbard model and brieflysummarize the method of VCA. In Sect. 3, we presentcalculated results for the paramagnetic and staggered or-dered states and give some discussion. A summary of thepaper is given in Sect. 4.
2. Model and Method
The three-component Hubbard model may be definedby the Hamiltonian H = − t (cid:88) (cid:104) i,j (cid:105) (cid:88) α c † iα c jα − (cid:88) i (cid:88) α µ α n iα + 12 (cid:88) i (cid:88) α (cid:54) = β U αβ n iα n iβ , (1)where c † iα ( c iα ) denotes the creation (annihilation) op-erator of a fermion with color α (= a, b, c ) at site i and n iα = c † iα c iα . t is the hopping integral between the neigh-boring sites, which is taken as the unit of energy, and U αβ (= U βα ) is the on-site interaction between two fermionswith colors α and β . Throughout the paper, we assumethe color-dependent interactions and set U ab = U ( > U bc = U ca = U (cid:48) ( >
0) for simplicity. We also assumethe filling of n = (cid:80) α n α = 3 / n α ≡ (cid:80) i (cid:104) n iα (cid:105) /L = 1 / α fermions in a system of size L . We set thechemical potential as µ α = (cid:80) β (cid:54) = α U αβ / n = 3 / U = U (cid:48) corresponds to the SU(3)Hubbard model, using which studies have been carriedout on superfluidity in the presence of attractive interac-tions as well as on the metal-insulator transition in thepresence of repulsive interactions. In particular, ithas been confirmed that this model at half filling doesnot show the Mott transition in the paramagnetic state,maintaining the metallic state irrespective of the inter-action strength, unlike in the two-component Hubbardmodel. At U (cid:48) = 0, on the other hand, the two interact-ing components undergo the Mott transition, leaving thenoninteracting component metallic. Thus, the anisotropyin the interaction strengths ( U (cid:54) = U (cid:48) ) causes the Motttransition: at U (cid:29) U (cid:48) , the color-selective Mott (CSM)state is realized, where the two components are localizedand one component is itinerant, and at U (cid:28) U (cid:48) , thepaired Mott (PM) state is realized, where the two com-ponents are paired in the same sites and one componentis localized in the other sites.
18, 19, 24)
In the presence of long-range staggered orders, it isknown that fermions are arranged alternately in thelattice and either the color-selective antiferromagnetic(CSAF) state corresponding to the CSM state at U (cid:29) U (cid:48) or the color-density-wave (CDW) state corresponding tothe PM state at U (cid:28) U (cid:48) is realized. It has beenpointed out that in the SU(3) symmetric U = U (cid:48) Hubbard model, the CDW state is realized at half fillingif the perfect Fermi surface nesting of the nesting vec-tor Q = ( π, π ) exists in the two-dimensional square lat-tice. It has also been pointed out
19, 20) that in the SU(3)symmetric model at half filling, the CSAF and CDWstates are energetically degenerate at zero temperatureand that the CSAF state is realized at
U > U (cid:48) and theCDW state is realized at
U < U (cid:48) . DMFT calculationshave suggested the presence of an s -wave superfluid stateat U (cid:48) > U >
0, which is, however, higher in energy thanthe CDW state and is not realized as the ground state ofthe system.
21, 24)
To accomplish the calculations in the thermodynamiclimit, we use the VCA
32, 33) based on self-energy func-tional theory (SFT), which is the variational princi-ple for the grand potential as a functional of the self-energy.
Unlike in DMFT, we can thereby pre-cisely take into account the effects of short-range spa-tial fermionic correlations in low-dimensional systems.In fact, successful explanations were given for the anti-ferromagnetism and superconductivity, as well as forthe pseudogap behaviors, in the two-dimensional Hub-bard model for high- T c cuprate materials. The trial self-energy for the variational method is generated from theexact self-energy of the disconnected finite-size clusters,which act as a reference system. To investigate the spon-taneous symmetry breaking in the VCA, we introducethe Weiss fields in the system as variational parameters.The Weiss fields of the CDW and CSAF states are de-fined as H (cid:48) CDW = M (cid:48) CDW (cid:88) i e i Q · r i ( n ia + n ib − n ic ) (2) H (cid:48) CSAF = M (cid:48) CSAF (cid:88) i e i Q · r i ( n ia − n ib ) , (3)respectively, where M (cid:48) CDW and M (cid:48) CSAF are the strengthsof the Weiss fields of the CDW and CSAF states, re-spectively, which are taken as the variational parameters.Then, the Hamiltonian of the reference system is givenby H (cid:48) = H + H (cid:48) CDW + H (cid:48) CSAF . Within SFT, the grandpotential at zero temperature is given byΩ = Ω (cid:48) − N s (cid:73) C d z πi (cid:88) K ,α ln det [ I − V α ( K ) G (cid:48) α ( z )] , (4)where Ω (cid:48) is the grand potential of the reference sys-tem, N s is the number of clusters in the system, I isthe unit matrix, V α is the hopping parameter betweenthe adjacent clusters, and G (cid:48) α is the exact Green’s func-tion of the reference system calculated by the Lanczosexact-diagonalization method. The K -summation is per-formed in the reduced Brillouin zone of the superlat-tice and the contour C of the frequency integral en-closes the negative real axis. The variational parame-ters are optimized on the basis of the variational prin-ciple, i.e., ∂ Ω /∂M (cid:48) CDW = 0 for the CDW state and ∂ Ω /∂M (cid:48) CSAF = 0 for the CSAF state. The solutions with M (cid:48) CDW (cid:54) = 0 and M (cid:48) CSAF (cid:54) = 0 correspond to the CDWand CSAF states, respectively. In our VCA calculations,
2. Phys. Soc. Jpn.
FULL PAPERS we assume the two-dimensional square lattice and themodulation vector is given by Q = ( π, π ). We use an L c = 2 × which proceeds by tiling the lattice intoidentical, finite-size clusters, solving many-body prob-lems in these clusters exactly, and treating the interclus-ter hopping terms at the first order in strong-couplingperturbation theory. This theory is exact in both thestrong and weak correlation limits, and provides a goodapproximation to the spectral function at any wave vec-tor. In CPT, the Green’s function of color- α fermions isgiven by G cpt α ( k , ω ) = 1 L c L c (cid:88) i,j =1 G ij,α ( k , ω ) e − i k · ( r i − r j ) , (5)where G α ( K , ω ) = (cid:2) G (cid:48)− α ( ω ) − V α ( K ) (cid:3) − . Using theCPT Green’s function G cpt α , the single-particle spectralfunction of the color- α fermions is defined as A α ( k , ω ) = − π Im G cpt α ( k , ω + iη ) , (6)where η gives the artificial Lorentzian broadening of thespectrum. We also calculate the DOS of the color- α fermions defined as ρ α ( ω ) = 1 L (cid:88) k A α ( k , ω ) , (7)where L = N s L c is the total number of lattice sites inthe system.
3. Results of Calculations
First, let us discuss the Mott metal-insulator transi-tions in the paramagnetic state (or in the absence of theWeiss fields). Using the CPT, we calculate the single-particle spectra and DOS, the results of which are shownin Fig. 1 as a function of
U/t at U (cid:48) /t = 4. At U/t = 0,where the PM state is realized [see Fig. 2(a)], we findthat the gap opens in all components of the spectra atthe Fermi level; in particular, the gap in the c componentis about twice as large as the gap in the a and b com-ponents. This may be understood as follows: if a color a (or b ) fermion is added to the system, it is placed onthe site occupied by the color c fermion due to the Pauliprinciple, which increases the energy of the system by U (cid:48) , but if a color c fermion is added to the system, it isplaced on the site occupied doubly by the color a and b fermions, which increases the energy of the system by2 U (cid:48) . Thus, a gap of size U (cid:48) (2 U (cid:48) ) opens in the a and b components ( c component) of the spectra.With increasing U/t , we find that the gap in the a and b components decreases and closes at a certain U/t value, but the gap in the c component remains open up toa larger U/t value. Then, at
U/t = U (cid:48) /t = 4, where ourmodel is SU(3) symmetric, the spectra become equivalentin all the components and the system becomes metallicas shown in Fig. 1(g).Increasing the U/t value further, we find that a gap opens again in the a and b components, but the c com-ponent remains metallic and the DOS curve resemblesthat of the noninteracting band in the two-dimensionalsquare lattice. This situation occurs because the color a and b fermions are localized owing to the large U/t val-ues, whereas the color c fermions hop freely because theinteraction strengths are the same, U bc = U ca = U (cid:48) . Inthis CSM state, a gap of size U opens in the a and b com-ponents of the spectra because, if a color a ( b ) fermion isadded to the system, it is placed on the site occupied bythe color b ( a ) fermion due to the Pauli principle, whichincreases the energy of the system by U .Note that the size of the single-particle gap cannotbe estimated accurately in Fig. 1 because the spectraare broadened artificially by η [see Eq. (6)]. However,the chemical-potential dependence of the average particlenumber per site n α enables us to evaluate the gap sizeaccurately, which may be calculated as n α = 1 L L (cid:88) i =1 (cid:104) n iα (cid:105) = 1 N s L c (cid:73) C d z πi (cid:88) K L c (cid:88) i =1 G ii,α ( K , z )(8)via the diagonal term of the Green’s function G α . Theparticle number n α calculated as a function of the chem-ical potential µ α is fixed to n α = 0 . µ α, − < µ α < µ α, + , where µ α, + corresponds to the loweredge of the upper band and µ α, − corresponds to the up-per edge of the lower band. The width of the plateau isthen given by ∆ α = | µ α, + − µ α, − | , which corresponds tothe single-particle gap.The thus calculated single-particle gaps ∆ α are shownin Fig. 2 as a function of U/t at U (cid:48) /t = 4. We find thatat U/t = 0, a gap opens in all components of the spectra( ∆ α >
18, 19)
Thesizes of the gaps are U (cid:48) in the a and b components and2 U (cid:48) in the c component, as we have discussed above, sothat we obtain the relation ∆ c (cid:39) ∆ a = 2 ∆ b at U/t = 0.With increasing
U/t , the gaps ∆ α decrease linearly, andin the region corresponding to Figs. 1(d)-1(f), we have ∆ a = ∆ b = 0 and ∆ c >
0, the region of which we callCSM(II), where the color a and b fermions have metallicbehavior and the color c fermions have insulating behav-ior. Upon increasing U/t further, the gap in the c compo-nent also closes, and around U = U (cid:48) , the gaps in all thecomponents close ( ∆ α = 0), as shown in Figs. 1(g)-1(i).In the large- U/t ( (cid:29) U (cid:48) /t ) region, we have ∆ a = ∆ b > ∆ c = 0, the state of which we call CSM(I), wherethe gaps ∆ α ( α = a, b ) increase linearly with U , indicat-ing that the CSM(I) phase is caused by U . The criticalphase boundaries determined at U (cid:48) /t = 4 are U/t ≤ . . ≤ U/t ≤ . . ≤ U/t forCSM(I).Figure 2 shows the phase diagram determined by thethus calculated gaps ∆ α , where we find three Mottphases, PM, CSM(I), and CSM(II), as well as the para-magnetic metallic phase. The CSM(I) phase appears at U (cid:29) U (cid:48) and the PM phase appears at U (cid:28) U (cid:48) , whereasaround the SU(3) symmetric point U = U (cid:48) , we find the
3. Phys. Soc. Jpn.
FULL PAPERS / t ω / t ω / t ω / t ω U / t = ( ) ω ρ α = c α = c α = c α = c α = a , b α = a , b α = a , b α = a , b α = a , b α = c -10 -5 5 100 -10 -5 5 100 -10 -5 5 100 -10 -5 5 100 -10 -5 5 100 k ΓΓ XM k ΓΓ XM (a) PM(b)(c) (d) CSM(II)(e)(f ) (g) Metal(h)(i) (j) CSM(I)(k)(l) (m) CSM(I)(n)(o) U / t = U / t = U / t = U / t = / t ω Fig. 1. (Color online) Calculated DOSs and single-particle spectra in the normal state of our model [Eq. (1)]. The
U/t dependenceat U (cid:48) /t = 4 is shown, where the solid (dotted) lines indicate the results for the color a and b (color c ) components. The dispersions ofthe spectra are shown along the line connecting the Γ(0 , π, π, π ) points of the Brillouin zone. The artificial Lorentzianbroadening of the spectra η/t = 0 . paramagnetic metallic phase. The newly found CSM(II)phase appears between the PM and paramagnetic metal-lic phases, where only the color c fermions are gapful.This phase appears because the single-particle excitationrequires twice the energy for the c component as for the a and b components, as shown above. This phase is absentin the DMFT calculations,
18, 19) suggesting its absencein the case of infinite dimensions; thus, we consider thatthe intersite spatial correlations between fermions in two-dimension, which the VCA takes into account properly,may induce this CSM(II) phase. U / t Δ α α = a , b α = c MetalPM CSM(I)CSM(II) U' / t = 4.0 Fig. 2. (Color online) Left panel: Calculated single-particle gap ∆ α for color α fermions as a function of U/t at U (cid:48) /t = 4. Rightpanel: Calculated phase diagram of the paramagnetic state in theparameter space ( U/t, U (cid:48) /t ), which includes the paired Mott (PM)and two types of color-selective Mott (CSM) phases. Illustratedbelow are their schematic representations. U / t a )CSAF ( c )CSAF ( b )CDW ( a )CDW ( b )CDW ( c ) Fig. 3. (Color online) Calculated ground-state energies E /t (left panel) and staggered magnetizations M α (right panel) of thecolor-density-wave (CDW) and color-selective antiferromagnetic(CSAF) states as a function of U/t at U (cid:48) /t = 4. Illustrated be-low are schematic representations of the CDW and CSAF states. Next, let us discuss the symmetry-broken staggeredordered states, which are obtained by adding the Weissfields defined in Eqs. (2) and (3) to the Hamiltonian inEq. (1). We thereby calculate the ground-state energyper site, E = Ω + (cid:80) α µ α , and the staggered magnetiza-tion of color α fermions defined as M α = 1 L (cid:88) i (cid:104) n iα (cid:105) e i Q · r i = 1 N s L c (cid:73) C d z πi (cid:88) K L c (cid:88) i =1 G ii,α ( K , z ) e i Q · r i (9)for the optimized Weiss fields.
4. Phys. Soc. Jpn.
FULL PAPERS -10 -5 0 5 10 ρ ( ω ) -10 -5 0 5 10XXMM kk α = a,b α = c α = a,b α = c U / t = 0.0 U / t = 8.0 ω / t ω / t Fig. 4. (Color online) Calculated DOSs and single-particle spec-tra in the CDW (at
U/t = 0 and U (cid:48) /t = 4) and CSAF (at U/t = 8and U (cid:48) /t = 4) states. The solid (dotted) curves indicate the spec-tra for color a and b fermions (color c fermions). The dispersions ofthe spectra are shown along the line connecting the Γ(0 , π, π, π ) points of the Brillouin zone. The artificial Lorentzianbroadening of the spectra η/t = 0 . The calculated results for E and M α are shown inFig. 3 as a function of U at U (cid:48) /t = 4. We find thatthe ground-state energies E of the CDW and CSAFstates cross each other at U = U (cid:48) , indicating that thephase transition between the two is of the first order. TheCDW (CSAF) is thus realized as the ground state when U (cid:48) ( U ) is larger than U ( U (cid:48) ), in accordance with pre-vious DMFT studies. Around the SU(3) symmetricpoint (or around U (cid:39) U (cid:48) ), the grand potential Ω has sta-tionary points both at M (cid:48) CDW (cid:54) = 0 and at M (cid:48) CSAF (cid:54) = 0,indicating that either the CDW or CSAF phase can ap-pear. The calculated staggered magnetization indicatesthat M (cid:48) CDW (cid:54) = 0 at
U < U (cid:48) , where the CDW state isstable. In particular, we find that M a = M b > M c <
0, which indicate that the color a and b fermionsare located on the same sites and the color c fermionsare located alternately on other sites, resulting in thestaggered order of the modulation vector Q = ( π, π ). At U > U (cid:48) , we find that M (cid:48) CSAF (cid:54) = 0, where the CSAF stateis stable. In particular, we find that M a > M b <
0, and M c = 0, which indicate that the color a and b fermionsshow staggered antiferromagnetic orderings but the color c fermions do not.We also calculate the single-particle spectra and DOSsfor the staggered ordered CDW and CSAF phases, wherewe use the optimized Weiss fields M (cid:48) CDW and M (cid:48) CSAF .The results are shown in Fig. 4, where we find that thespectral peak positions do not change markedly in com-parison with those of the corresponding PM and CSM(I)phases, whereas sharp coherence peaks appear at theedges of the gap in both the CDW and CSAF phases. These gaps become wider than those of the correspond-ing PM and CSM(I) phases, reflecting the stabilizationof the ordered phases.
U/t -0.10.00.1-0.10.00.1-0.10.00.1 2 4 6 8 104 6 8 10 12
U/tU/t U' / tU' / tU' / t (f) U'/t = 4.0(b)
U/t = 2.0(c)
U/t = 4.0 (d)
U'/t = 0.0(e)
U'/t = 2.0CDWCDW CSAFCSAFCSAF(a)
U/t = 0.0CDW0.00.10.2-0.1-0.20.00.10.2-0.1-0.20.00.10.2-0.1-0.2
Fig. 5. (Color online) Calculated kinetic and potential energydifferences in the CDW (left panels) and CSAF (right panels)phases compared with the paramagnetic normal phase. Plotted are ∆E U ab ( ◦ ), ∆E U bc + ∆E U ca ( (cid:3) ), ∆E K a + ∆E K b ( (cid:52) ), and ∆E K c ( × ) in units of t . U/t and U (cid:48) /t dependences are shown. In the previous subsection, we showed that the CDWand CSAF phases are stabilized for
U < U (cid:48) and
U > U (cid:48) ,respectively, and that the first-order phase transition oc-curs between the two phases at U = U (cid:48) . If we assumethe paramagnetic phase, the Mott insulating phases suchas PM and CSM are stabilized when the interactionstrengths are strongly anisotropic (see Fig. 2), but theparamagnetic metallic phase is maintained around theSU(3) symmetric point ( U = U (cid:48) ) even if the interactionsare very strong. Therefore, we may anticipate that themechanisms of the stabilization of the staggered ordersare different in two regions: the region around U = U (cid:48) where the system is metallic and the region where the in-teraction strengths are strongly anisotropic and the sys-tem is Mott insulating.
5. Phys. Soc. Jpn.
FULL PAPERS -0.4-0.2 0.2 0.0 0.4 0 2 4 6 8-0.4-0.2 0.2 0.0 0.4 0 2 4 6 8 U / t = U’ / t U / t = U’ / t Fig. 6. (Color online) Same as in Fig. 5 but for the dependenceon
U/t = U (cid:48) /t . It is known that there are two mechanisms for thestabilization of staggered orders.
One is the Slatermechanism, which occurs in itinerant systems, where theFermi surface instability causes band folding and gapopening due to the interactions between fermions, lead-ing to staggered orderings in the system, resulting ingains in the potential energy but losses in the kineticenergy. The other is the Mott mechanism, which oc-curs in insulating systems, where the virtual hopping offermions aligns their colors antiferromagnetically due tothe Pauli principle, leading to staggered orderings in thesystem, resulting in gains in the kinetic energy but lossesin the potential energy. Below, we calculate the kineticand potential energies in the paramagnetic and staggeredordered phases of the system, and consider the mecha-nisms of the staggered orderings from the energetic pointof view.Let us define the potential E U αβ and kinetic E K α en-ergies per site as E U αβ = 1 L (cid:32) U αβ (cid:88) i (cid:104) n iα n iβ (cid:105) (cid:33) = U αβ ∂E ∂U αβ (10) E K α = 1 L (cid:32) − t α (cid:88) (cid:104) i,j (cid:105) (cid:104) c † iα c jα (cid:105) (cid:33) = − t α ∂E ∂t α , (11)and their energy gains caused by the staggered orderingsas ∆E DW U αβ = E DW U αβ − E N U αβ (12) ∆E DW K α = E DW K α − E N K α (13)for the CDW phase at U ≤ U (cid:48) , and as ∆E AF U αβ = E AF U αβ − E N U αβ (14) ∆E AF K α = E AF K α − E N K α (15)for the CSAF phase at U ≥ U (cid:48) , where the superscriptsDW, AF, and N stand for the CDW, CSAF, and param-agnetic normal phases, respectively. Thus, comparing thesigns and magnitudes of ∆E U αβ and ∆E K α , we can eval-uate the energy gains in the formation of the staggeredlong-range orders.Figure 5 displays the calculated results for the quan-tities defined above as a function of U/t ( U (cid:48) /t ) at afixed value of U (cid:48) /t ( U/t ). The same results along the line
U/t = U (cid:48) /t are also shown in Fig. 6. First, in Figs. 5(a)-5(c), where a comparison is made between the CDW andnormal phases, we find that ∆E DW U bc + ∆E DW U ca < ∆E DW K a + ∆E DW K b > ∆E DW K c > . ≤ U (cid:48) /t < . U/t = 0, indicating that the gainin potential energy leads to the staggered CDW order.For 4 . ≤ U (cid:48) /t ≤ .
0, however, the signs are inverted andwe find that ∆E DW U bc + ∆E DW U ca > ∆E DW K a + ∆E DW K b < ∆E DW K c < , (21)which indicates that the gain in kinetic energy leads tothe staggered order. Thus, the stabilization mechanismof the CDW phase shows a crossover from the Slatermechanism to the Mott mechanism. With increasing U/t ,the interaction between the a and b components increasesand ∆E DW U ab varies considerably as shown in Figs. 5(b)and 5(c). We then find that ∆E DW U ab > U (cid:39) U (cid:48) and ∆E DW U ab < U (cid:48) (cid:29) U . We also find that around U = U (cid:48) , although the loss in the potential energy is large, ∆E DW U ab >
0, the effect of ∆E DW U bc + ∆E DW U ca < U = U (cid:48) values [see Fig. 6(a)], suggesting that the effects of Fermisurface nesting are important here.Next, in Figs. 5(d)-5(f) where a comparison is madebetween the CSAF and normal phases, we find that ∆E AF U ab < ∆E AF K a + ∆E AF K b > . ≤ U/t < . U (cid:48) /t = 0, indicating that the gainin potential energy leads to the staggered CSAF order.For 3 . ≤ U/t ≤ .
0, however, the signs are inverted andwe find that ∆E AF U ab > ∆E AF K a + ∆E AF K b < , (25)which indicates that the gain in kinetic energy leadsto the staggered CSAF order. Thus, the stabilizationmechanism of the CSAF phase also shows a crossoverfrom the Slater mechanism to the Mott mechanism.With increasing U (cid:48) /t , the c -component-related quantities ∆E AF U bc + ∆E AF U ca and ∆E AF K c vary considerably as shown inFigs. 5(e) and 5(f). We then find that ∆E AF U bc + ∆E AF U ca > ∆E AF K c < U (cid:48) (cid:39) U , indicating that the c -component-related quantity loses its potential energy.However, we find that the a - and b -component-relatedquantity ∆E AF U ab < U (cid:29) U (cid:48) , on the other hand, we find that the gain inthe kinetic energy of the a - and b -components is domi-nant, ∆E AF K a + ∆E AF K b <
0, leading to the Mott mecha-
6. Phys. Soc. Jpn.
FULL PAPERS nism of the CSAF ordering. Note that the point wherethe signs of ∆E AF U αβ and ∆E AF K αβ are inverted approaches U = U (cid:48) with increasing U (cid:48) /t , but the Slater mechanismof the CSAF stabilization even occurs at large U = U (cid:48) values [see Fig. 6(b)], suggesting that the effects of Fermisurface nesting are important around the SU(3) symmet-ric point, as in the case of the CDW stabilization.We thus find that the Slater mechanism (Mott mech-anism) of the staggered ordering predominantly occursin the metallic (Mott insulating) region of the paramag-netic phase diagram given in Fig. 2. In particular, theSlater mechanism even occurs in the strong coupling re-gion when U (cid:48) (cid:39) U , which is in contrast to the SU(2) sym-metric Hubbard model. We also find that the fermioniccomponents that show staggered orderings depend on theanisotropy of the interaction strengths. t' / t = 0.0 t' / t = 0.1 t' / t = 0.2 α = a , b α = c k x k y t' / t = 0.2 Q = ( π , π ) π - π π - π Fig. 7. (Color online) Left panel: Calculated order parameter ofthe CDW phase M α in the presence of the next-nearest-neighborhopping t (cid:48) . The U/t dependence is shown at U (cid:48) /t = 4. Right panel:Noninteracting Fermi surfaces at t (cid:48) /t = 0 and 0 .
2. The nestingvector at t (cid:48) /t = 0 is indicated by arrows. Finally, let us discuss the effects of Fermi surface nest-ing on the staggered orders. In the previous subsection,we showed that the Slater mechanism for the stabiliza-tion of the staggered orderings occurs in the region of theisotropic interaction strengths around the SU(3) sym-metric point, where we may expect that Fermi surfacenesting plays an essential role in the formation of thestaggered orders. Here, we confirm this expectation byintroducing the next-nearest-neighbor hopping term tothe Hamiltonian and destroying the Fermi surface nest-ing of Q = ( π, π ). The Hamiltonian then reads H = − t (cid:88) (cid:104) i,j (cid:105) (cid:88) α c † iα c jα − t (cid:48) (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) (cid:88) α c † iα c jα − (cid:88) i (cid:88) α µ α n iα + 12 (cid:88) i (cid:88) α (cid:54) = β U αβ n iα n iβ , (26)where t (cid:48) is the next-nearest-neighbor hopping parame-ter and (cid:104)(cid:104) i, j (cid:105)(cid:105) indicates the summation over the next-nearest-neighbor pairs of sites. The Fermi surface of thenoninteracting system at t (cid:48) /t = 0 . H (cid:48) = H + H (cid:48) on + H (cid:48) CDW with the on-site poten-tial H (cid:48) on = (cid:80) i,α (cid:15) (cid:48) α n iα . We optimize the grand potentialΩ with respect to both M (cid:48) CDW and (cid:15) (cid:48) α ; the latter is nec-essary to keep the average particle number at n α = 0 . t (cid:48) term. The calculated results for the order parameter M α areshown in Fig. 7, where we find that the region with a non-vanishing order parameter shrinks with increasing t (cid:48) /t and that the paramagnetic phase without the CDW orderactually appears around U = U (cid:48) . We thus demonstratethat, in the region of U (cid:39) U (cid:48) , the Fermi surface nestingof Q = ( π, π ) plays an essential role in the formation ofthe staggered CDW order, which is not important when U (cid:28) U (cid:48) . The Slater mechanism thus has a contrastingeffect to the Mott mechanism for the stabilization of thestaggered orders.
4. Summary
We have investigated the ground-state properties andexcitation spectra of the three-component fermionicHubbard model defined on the two-dimensional squarelattice at half filling. We used the VCA, which enablesus to study the effects of the lattice geometry and Fermisurface topology in the low-dimensional systems in thethermodynamic limit, precisely taking into account spa-tial fermion correlations.First, we presented the ground-state phase diagram ofthe paramagnetic state of the model, whose phases in-clude the paired Mott (PM) phase at U (cid:28) U (cid:48) , the color-selective Mott (CSM) phase at U (cid:29) U (cid:48) , and the para-magnetic metallic phase between them. We also showedthat the Mott transition does not occur in the SU(3) sym-metric point U = U (cid:48) and that a different CSM phase ap-pears between the PM and paramagnetic metallic phases,where the color a and b fermions are metallic and color c fermions are localized.Next, we introduced the Weiss fields to find the sponta-neous symmetry-broken phases and found that the color-density-wave (CDW) and color-selective antiferromag-netic (CSAF) phases appear at U < U (cid:48) and
U > U (cid:48) ,respectively, and that the energies of the two phases crossat U = U (cid:48) . We also examined the kinetic and potentialenergy gains in the staggered orderings and showed thatthe Slater mechanism with a predominant potential en-ergy gain occurs in the region around U = U (cid:48) , where themetallic state is realized in the paramagnetic phase, andthat the Mott mechanism with a predominant kineticenergy gain occurs in the region where the interactionsare highly anisotropic and the Mott insulating state isrealized in the paramagnetic phase.By introducing the next-nearest-neighbor hopping pa-rameters, we demonstrated that the Fermi surface nest-ing is essential in the region around U = U (cid:48) , wherethe Slater mechanism occurs for the staggered orderings.This result indicates that the staggered orders near theSU(3) symmetric point are fragile, protected only by theFermi surface nesting. A recent DMFT calculation hassuggested that the s -wave superfluid state occurs as ametastable state at U < U (cid:48) in the three-component Hub-
7. Phys. Soc. Jpn.
FULL PAPERS bard model at half filling.
We may therefore suggestthat this superfluid state can be most stable if the Fermisurface nesting is destroyed to suppress the CDW orderbecause the pairing of two fermions at k and − k for thesuperfluidity is not affected strongly by the Fermi surfacenesting. The exotic pairing mechanism for superfluidityin multicomponent fermionic systems of N >
Acknowledgments
We thank A. Koga for enlightening discussions. Thiswork was supported in part by KAKENHI GrantNo. 26400349 from JSPS of Japan. T. K. and S. M. ac-knowledge support from the JSPS Research Fellowshipfor Young Scientists.
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