Coherent state path integral approach to correlated electron systems with deformed Hubbard operators: from Fermi liquid to Mott insulator
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Coherent state path integral approach to correlated electron systems with deformedHubbard operators: from Fermi liquid to Mott insulator
Xiao-Yong Feng ∗ and Jianhui Dai † Department of Physics, Hangzhou Normal University, Hangzhou 310036, China (Dated: April 27, 2019)In strongly correlated electron systems the constraint which prohibits the double electron occu-pation at local sites can be realized by either the infinite Coulomb interaction or the correlatedhopping interaction described by the Hubbard operators, but they both render the conventionalfield theory inapplicable. Relaxing such the constraint leads to a class of correlated hopping modelsbased on the deformed Hubbard operators which smoothly interpolate the locally free and strongcoupling limits by a tunable interaction parameter 0 ≤ λ ≤
1. Here we propose a coherent statepath integral approach appropriate to the deformed Hubbard operators for arbitrary λ . It is shownthat this model system exhibits the correlated Fermi liquid behavior characterized by the enhancedWilson ratio for all λ . It is further found that in the presence of on-site Coulomb interaction afinite Mott gap appears between the upper and lower Hubbard bands, with the upper band spectralweight being heavily reduced by λ . Our approach stands in general spatial dimensions and revealsan unexpected interplay between the correlated hopping and the Coulomb repulsion. A major challenge in interacting electron systemscomes from the lack of proper theoretical approaches forvarious kinds of many body correlations which are ubiq-uitous and variable in solids. The electron correlationsare of fundamental importance in the formation of novelquantum phases in the condensed matter physics[1, 2].They can be captured by the short-ranged Coulomb in-teraction and the correlated hopping (CH) interaction asdescribed in the Hubbard model[3, 4]or its variants[5–8]as for the electron systems with narrow bandwidths[9].For the free hopping system with a weak Coulomb in-teraction the Landau’s Fermi liquid[10] develops contin-uously from the free electron gas. An opposite extremecase is the strong coupling limit when the Coulomb in-teraction tends to infinity so that the perturbation treat-ment around the non-interacting limit is invalid. In thiscase the Fermi liquid scenario might either persist butwith strong correlation effect or breakdown, driving thehalf-filled system to an insulator [4, 11]. The interaction-driven many-body features are relevant to a number ofunconventional quantum phenomena ranging from Motttransition[8, 11], Peierls dimerization[12], high temper-ature superconductivity[13, 14], heavy fermion physics[15, 16], to quantum magnetism[17, 18].Theoretically, the main difficulty in tackling the strongcoupling limit comes from the constraint that the possi-bility of the double occupation at the same lattice sitefor two electrons with opposite spin polarizations shouldbe completely excluded. This no-double-occupation con-straint can be imposed by applying the Gutzwiller pro-jection operator P G = Q j (1 − n j ↑ n j ↓ ) [19] on a givenHamiltonian ˆ H so that the resulting model system ˜ˆ H = P G ˆ H P G is defined on the truncated Hilbert space with-out any double occupation states. The truncation of sys- ∗ [email protected] † [email protected] tem’s Hilbert space such as in the t-J model[20]severelyprohibits applications of the conventional many-bodytechniques including the path integral and diagram ex-pansion. The slave-particle method is then devised inorder to implement the constraint, but at the cost of in-troducing additional slaved gauge degrees of freedom[21].Another approach is to apply the local Hubbard X opera-tors in a concise way [22, 23], among which the operators X σi = c iσ (1 − n i ¯ σ ) obviously exclude the double occu-pation at local sites. However, the Hubbard X operatorspossess no conventional anti-commutation rules, leadingto a cumbersome Lagrangian [24] and complicated X-operator-based diagram technique [25].Given the fact that the Coulomb interaction is actu-ally not infinite, the many-body correlation effect in re-alistic materials is frequently investigated by the vari-ational Gutzwiller projection method using the partialprojection operator P G ( λ ) = Q j (1 − λn j ↑ n j ↓ ) with0 ≤ λ ≤ λ as a variational param-eter can optimize the effect of the Coulomb U whichdrives the Mott transition at half filling[27, 28]. TheGutzwiller variational approach combined with the den-sity functional theory has been shown powerful in un-derstanding the many-body correlation effect in 3d or 4felectronic materials[29–32].These developments naturally stimulate the researchinterest in a class of correlated electron systems whoseHamiltonians are constructed in terms of the deformedHubbard X operators ˜ X σi = c iσ (1 − λn i ¯ σ ), the operatorswhich smoothly interpolate the conventional electron op-erators ( λ = 0) and the Hubbard operators ( λ = 1)[33].The deformation induces a complicated CH motion in-volving four- and six-fermions interacting terms in thekinetic part of the original Hamiltonian when λ = 0.This is what the CH interaction means in the presentstudy. More generally, CH interaction emerges rathernaturally in the construction of tight-binding Hamil-tonians involving variable electron-electron or electron-phonon interactions[3–5, 7, 12, 13]. In one-dimensionalsystems, such CH interaction has been examined by usingthe bosonization technique and renormalization group forsmall λ [33–36]. For λ close to unit these analytical meth-ods are inapplicable again. In spite of this fact, the limit λ = 1 has been long expected equivalent to the case ofinfinite Coulomb interaction imposed by the Gutzwillerprojection although the effect caused by CH interactionhas not been fully examined so far.In order to clarify the basic physics of CH interactionand its interplay with Coulomb interaction in general di-mensions, it is highly desirable to study the correlatedhopping models (CHMs) for arbitrary λ . In the presentpaper, we shall develop a new path integral approachfor these models using the coherent state representationof the deformed Hubbard ˜ X σ operators. We will showhow the path integral in this representation can provideinteresting results for the thermodynamics property inarbitrary spatial dimensions.To begin with, let us reiterate the definition of thedeformed Hubbord X σ operators [22, 33]˜ X σi = c iσ (1 − λn i ¯ σ ) , (1)where c iσ is the annihilation operator for the electronwith spin σ (= ↑ or ↓ ) at site i and n i ¯ σ = c † i ¯ σ c i ¯ σ is theparticle number operator at the same site with oppositespin, 0 ≤ λ ≤
1. Parallel to the path integral represen-tation for the conventional electron operators[37, 38], weseek for the coherent states | ξ i i defined as the eigenstatesof the deformed Hubbard operators˜ X σi | ξ i i = ξ iσ | ξ i i , (2)with the eigenvalues ξ iσ being Grassmann numbers. Notethat ˜ X , ↑ i and ˜ X , ↓ i have the common eigenstates becausethey are anti-commutative to each other. We find thatthese eigenstates can be constructed in the form | ξ i i = exp " − X σ ξ iσ c † iσ + λ − λ ξ i ↑ ξ i ↓ c † i ↑ c † i ↓ ! | i (3)where | i is the null state of the original annihilationoperators c iσ . The coherent states are well-defined forall λ except for the singular point λ = 1, nevertheless,the limit of λ → h ξ i | ξ ′ i i = exp "X σ ξ ∗ iσ ξ ′ iσ + ( u − ξ ∗ i ↓ ξ ∗ i ↑ ξ ′ i ↑ ξ ′ i ↓ (4)with u = (1 − λ ) − . The set of all these coherent statesis overcomplete, satisfying the following relationship1 u Z Y σ dξ ∗ iσ dξ iσ e − u P σ ξ ∗ iσ ξ iσ − ( u − u ) Q σ ξ ∗ iσ ξ iσ | ξ i ih ξ i | = 1 . (5)Now we consider a Hamiltonian ˆ H ( { ˜ X † } , { ˜ X } ), ex-pressed in terms of the deformed X operators ( { ˜ X } indi-cates the set of all ˜ X σi ). In the thermal equilibrium state at temperature T , the partition function is Z = tre − β ˆ H , β = 1 /k B T . Dividing the (imaginary) time interval ( β )by M number of slices and evaluating the trace in therepresentation of the coherent states[37, 38], the partitionfunction is expressed as the following path-integral[39] Z = Z D ( ξ ∗ , ξ ) e − S ( ξ ∗ ,ξ ) . (6)Where, the classical action takes the form S = X i,m,σ ( u ˜ n iσ,m − ξ ∗ iσ,m ξ iσ,m − )+ X i,m ( u − u ) Y σ ˜ n iσ,m − X i,m ( u − Y σ ξ ∗ iσ,m ξ iσ,m − + X m ǫH [ { ξ ∗ m } , { ξ m − } ] , (7) D ( ξ ∗ , ξ ) = lim M →∞ Q i,m ( u ) dξ ∗ i ↑ ,m dξ i ↑ ,m dξ ∗ i ↓ ,m dξ i ↓ ,m is themeasure, ξ iσ,m the Grassmann eigenvalues of the coher-ent states | ξ i i in the m -th slice, ˜ n iσ,m = ξ ∗ iσ,m ξ iσ,m , H [ { ξ ∗ m } , { ξ m − } ] the corresponding classical Hamilto-nian, and ǫ = β/M the width of the time slices.The obtained classical action looks absurd: there areadditional four-fermions interaction terms (on the sec-ond line of Eq.(7)) which are seemingly divergent at thelimit M → ∞ , hindering the direct application of thecontinuous field theory. To overcome this difficulty, weshall calculate the partition function by the Grassmannintegration over the discrete time slices. Here the keyobservation is that the contribution from the classicalaction is actually measured by the pre-factor 1 /u whicheventually guarrantees the finiteness of the problem.In order to show the validity of our approach, we firstconsider a simple toy model which contains the homoge-neous on-site quadratic term:ˆ H w = − X iσ w σ ( ˜ X σi ) † ˜ X σi . (8)The corresponding action contributed from ˆ H w is S w = − P iσ,m ǫw σ ξ ∗ iσ,m ξ iσ,m − . Owing to the site-independence of the present case, we just focus on a givensite i and expand the exponential in the partition func-tion Eq.(6) to Y m − X σ f σ ˜ n iσ,m + f Y σ ˜ n iσ,m ! X σ f σ ξ ∗ iσ,m ξ iσ,m − + f Y σ ξ ∗ iσ,m ξ iσ,m − ! (9)with f σ = f = u , f σ = 1 + ǫw σ , and f = u + ǫ P σ w σ .All possible contributions to the partition function canbe schematically summarized in Fig.1 where the dots rep-resent the positions of time slices; the lines with arrowscoming from or to the slice m represent ξ iσ,m and ξ ∗ iσ,m , (a) (b) m m -1 (i)(ii)(iii)(iv) Figure 1: (a)The building blocks. (b)The contributing dia-grams for the toy model. respectively (the lines above/below the dots are for thespin up/down components).Fig.1 (a) represents the various terms in (9) which con-stitute the building blocks in the path integration. Whenall the dots are connected by a pair of out- and in-linesfor each spin components, the corresponding Grassmannintegration is non-vanishing. We call such a configura-tion the contributing diagram. There are four distinctcontributing diagrams as shown in the Fig.1 (b), withthe corresponding contributions being given by( i ) : lim M →∞ u M f M = 1 , ( ii ) : lim M →∞ u M f M = e β P σ wσu , ( iii ) : lim M →∞ u M ( f ↓ f ↑ ) M = e βw ↑ , ( iv ) : lim M →∞ u M ( f ↑ f ↓ ) M = e βw ↓ . Therefore, we reproduce the desired exact result Z w = 1 + e β P σ wσu + X σ e βw σ . (10)Now, we consider a non-trivial CHM: ˆ H = ˆ H t + ˆ H w ,where ˆ H w is given by (8), andˆ H t = − X h ij i σ t h ( ˜ X σi ) † ˜ X σj + h.c. i , (11) h ij i indicates the nearest neighbor sites. The corre-sponding action is S t = − P h ij i σ,m ǫt ( ξ ∗ iσ,m ξ jσ,m − + ξ ∗ jσ,m ξ iσ,m − ). Performing the Fourier transformation ξ iσ,m = √ Nβ P k ξ kσ e − iω n τ m + i~k · ~r i , k = ( ω n , ~k ), ω n =(2 n − π/β ( n = 1 , , · · · , M ), and τ m = mǫ , we have S t = P kσ ε k ξ ∗ kσ ξ kσ , where ε k = ε ~k e iω n ǫ and ε ~k is theenergy band of free fermions. This contribution can becombined with that of ˆ H w if we introduce a pair of aux-iliary Grassmann fields ( η kσ , η ∗ kσ ) by using the identity:exp ( − S t ) = Y kσ ( − ε k ) Z Y kσ ( dη ∗ kσ dη kσ ) (12)exp X kσ ( ε − k η ∗ kσ η kσ + η ∗ kσ ξ kσ + ξ ∗ kσ η kσ ) . Fourier transformed back to the real lattice space, P kσ ( η ∗ kσ ξ kσ + ξ ∗ kσ η kσ ) becomes ǫ P iσ ( η ∗ iσ ξ iσ + ξ ∗ iσ η iσ ).Then, expanding the exponential we find that the con-tributions to the partition function come only from theeven power terms of ǫ . The ǫ terms generate two newtypes of contributing diagrams as illustrated in Fig. 2,with the following contributions( v ) : lim M →∞ ǫ u M M − X σ,l =0 M X m =1 f l +11¯ σ f M − l − f l σ η ∗ iσ,m + l η iσ,m , ( vi ) : lim M →∞ ǫ u M M − X σ,l =0 M X m =1 f M − l − σ f M − l σ f l η ∗ iσ,m + l η iσ,m . In the frequency space, the summation of the above termsgives rise to − P ω n ,σ g σ ( ω n ) η ∗ iσ ( ω n ) η iσ ( ω n ), with g σ ( ω n ) = 1 + e βw σ iω n + w σ + u (cid:16) e βw ¯ σ + e β P σ ′ wσ ′ u (cid:17) iω n + P σ ′ w σ ′ u − w ¯ σ . (13) m+l m (a)(b) Figure 2: Two types of contributing diagrams generated bythe ǫ terms. Integrating out the ξ fields, the partition function is Z = Y kσ ( − ε k ) Z Y kσ ( dη ∗ kσ dη kσ ) exp X kσ ( ε − k η ∗ kσ η kσ ) Y i Z w − X ω n ,σ g σ ( ω n ) η ∗ iσ ( ω n ) η iσ ( ω n ) + · · · ! , (14)where Z w is given by Eq. (10) and the ” · · · ” representsthe contribution from the high order terms of ǫ .So far our approach is rigorous if the high or-der terms of ǫ are included. For the present pur-pose, we are interested in a closed analytical expres-sion of Z , which can be obtained by using the ap-proximation: Z w − P ω n ,σ g σ ( ω n ) η ∗ iσ ( ω n ) η iσ ( ω n ) + · · · ≈ Z w Q ω n ,σ (cid:2) − Z − w g σ ( ω n ) η ∗ iσ ( ω n ) η iσ ( ω n ) (cid:3) . This is un-derstood as a re-summation in the Eq.(14) under theladder approximation. Integrating out the η fields, wefinally obtain the following expression Z = Z Nw Y kσ (cid:2) − Z − w g σ ( ω n ) ε ~k (cid:3) . (15) (d)(c) (b) : = 0.9 : = 0.75 : = 0.5 : = 0 T (a) R W / R w ( ) : = : = 80 : = 50 : = : = 80 : = 50 C / T Figure 3: The susceptibility (a,b), specific heat coefficient(c),and Wilson ratio(d) for the CHM.
Interestingly, when u = 1, the partition function re-turns to the exact result for the free hopping model(the case with λ = 0), demonstrating the validity ofthe above approximation. Using Eq.(15), the low tem-perature specific heat for w σ = 0 is obtained as C = u N (0) k B T , where N (0) is the density of states forthe corresponding free hopping electrons around ε ~k = 0.While, the susceptibility at zero temperature is obtainedas χ = h u + π − u ) (1+ u ) i N (0). Figs. 3(a-c) show the λ -dependence of the susceptibility and specific heat co-efficient at several temperatures using the square latticedispersion ε ~k = − t (cos k x + cos k y ) and t = 1 (see moredetails in Appendices E and F). These results show thatthe CHM displays the Fermi liquid behavior with therenormalized specific heat coefficient and Pauli suscep-tibility, both enhanced by λ . In particular, the Wilsonratio R W = χ : C/T is enhanced to R W = 1 .
82 for λ = 1as shown in Fig. 3(d) (where the Wilson ratio for λ = 0 isscaled to unit). The nearness of this value to the one forthe magnetic instability ( R W = 2) indicates the strongcorrelation effect in this Fermi liquid phase.Finally, we consider the Hubbard model in the presenceof CH interaction, described by the Hamiltonianˆ H = ˆ H t − X iσ µ σ n iσ + X i U n i ↑ n i ↓ , (16)where µ σ is the spin-dependent chemical potential and U the on-site Coulomb repulsion. It is remarkable thatthese two terms can be reexpressed in terms of the de-formed Hubbard operators as − X σ µ σ ( ˜ X σi ) † ˜ X σi + " uU − ( u − X σ µ σ ( ˜ X ↓ i ) † ( ˜ X ↑ i ) † ˜ X ↑ i ˜ X ↓ i . (17)Therefore, the previous approach can be applied directlyby setting f σ = f = u , f σ = 1 and f = u (1 − U ǫ ) − ( u − P σ µ σ ǫ in Eq. (9). It immediately leads to thefollowing partition function Z = Z NU Y kσ (cid:2) − Z − U g ′ σ ( ω n ) ε ~k (cid:3) , (18)where Z U = 3 + e − β ( U − P σ u − u µ σ ) ,g ′ σ ( ω n ) = 2 iω n + u (cid:18) e β ( P σ ′ u − u µ σ ′ − U ) (cid:19) iω n − U . -505 -4048 (d)(c) (b)
U=5 : = 0 : = 0.5 : = 0.75 U=1 : = 0 : = 0.5 : = 0.75 E k (a) A ( + ) A (-) k E k A (-) k A ( + ) Figure 4: The Hubbard bands and corresponding spectralweights for U = 1 (a,c) and U = 5 (b,d), respectively. We here are mainly interested in the formation of theHubbard bands at the zero temperature in the paramag-netic phase. By setting µ σ = µ and when u − u µ < U ,we obtain the Green’s function of the ξ fields at the lowesttemperature limit in the standard form[39]: h ξ ∗ kσ ξ kσ i = X ν = ± A ( ν ) ~kσ iω n − E ( ν ) ~kσ . (19)The dispersions of the quasi-particles in the upperand lower Hubbard bands are E ( ± ) ~kσ = [ U + (2 + u )( ε ~k − µ ) ± q ( U + (2 + u ) ε ~k − µ )) − ( ε ~k − µ ) U ],with A ( ± ) ~kσ = ± (2+ u ) E ( ± ) ~kσ − U √ ( U + (2+ u )( ε ~k − µ )) − ( ε ~k − µ ) U being thespectral weights. The dispersions and spectral weightsare plotted Fig. 4 for the square lattice with various U and λ , fixing t = 1 and µ = 0. We find that theband gap appears between the upper and lower Hubbardbands almost independent of λ , meaning that the Mottgap opens due to U but not λ . However, λ influencesthe Hubbard bands asymmetrically because it suppressesthe local double occupation states. Specifically, it signif-icantly flattens the upper Hubbard band and reduces thecorresponding spectral weight. This λ -driven correlationeffect is in agreement with that exhibited by the enhancedWilson ratio discussed previously.Summarizing, we have developed a new coherent statepath integral approach for a class of CHMs constructedin terms the deformed Hubbard operators. It allows afaithful description of the complicated hopping processin the whole region of the deformation-induced interac-tion parameter λ and overcomes the divergence problemin the conventional continuous field theory approach. In-terestingly, the chemical potential and the short-rangedCoulomb interaction U can be also described in this ap-proach. Our results show that the CH interaction alonealways leads to the renormalization effect of the Fermiliquid even though the local double occupation states aresuppressed at the Hubbard limit ( λ = 1) where the sys- tem locates on the verge of the correlated Fermi liquidphase. On the other hand, in the presence of finite U theMott gap opens at half-filling as usual for any λ . Increas-ing λ significantly reduces the bandwidth of the upperHubbard band and the corresponding spectral weight.These results reveal the distinct roles played by CH andCoulomb interactions, the two prototype driving forcesbehind the rich many-body physics. We hope that ourapproach can pave a way to understand the delicate in-terplay among various interactions in a wider family ofcorrelated electron systems.The authors thank C. Cao for useful discussions. Thiswork was supported in part by the National ScienceFoundation of China under the grant Nos. 11874136 and11474082. [1] P. Fulde, Electron Correlations in Molecules and Solids (Springer-Verlag, Berlin-Heidelberg, 1991).[2] P. Coleman,
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In this appendix, we briefly derive the path-integralrepresentation of the partition function in terms of thecoherent states of the deformed Hubbard operators. Thetrace of an operator ˆ A , when initially carried out in agiven representation such as the energy representation inthe basis of eigenstates | n i satisfying P n | n ih n | = 1, canbe reformulated in the coherent state representation asfollowing tr ˆ A = X n h n | ˆ A | n i = X n h n | Y i u Z Y σ dξ ∗ iσ dξ iσ C ( ξ ∗ iσ , ξ iσ ) | ξ i ih ξ i | ˆ A | n i = Y i u Z Y σ dξ ∗ iσ dξ iσ C ( ξ ∗ iσ , ξ iσ ) h ξ i | ˆ A X n | n ih n | − ξ i i = Y i u Z Y σ dξ ∗ iσ dξ iσ C ( ξ ∗ iσ , ξ iσ ) h ξ i | ˆ A | − ξ i i where the completeness relation of the coherent statesof the deformed Hubbard operators (5) is used, and C ( ξ ∗ iσ , ξ iσ ) = exp [ − u P σ ξ ∗ iσ ξ iσ − ( u − u ) ξ ∗ i ↑ ξ i ↑ ξ ∗ i ↓ ξ i ↓ ].The minus sign in | − ξ i i comes from the interchangeof two Grassmann numbers. Applying this formula tothe partition function of a thermodynamic equilibriumsystem at temperature TZ = tre − β ˆ H , (A1)where β = 1 /k B T is the imaginary time interval and k B the Boltzmann constant, we have Z = Y i u Z Y σ dξ ∗ iσ dξ iσ C ( ξ ∗ iσ , ξ iσ ) h ξ i | e − β ˆ H | − ξ i i . (A2) … … 01m-1mM-1M Figure 5: The imaginary time slices with width ǫ = β/M . Dividing β by M number of slices with equal width ǫ = β/M using e − β ˆ H = ( e − ǫ ˆ H ) M and inserting the com-pleteness relation between each operator e − ǫ ˆ H , we have Z = Z Y i M Y m =1 u Z Y σ dξ ∗ iσ dξ iσ C ( ξ ∗ iσ,m ξ iσ,m ) h ξ i,M | e − ǫ ˆ H | ξ i,M − ih ξ i,M − | e − ǫ ˆ H | ξ i,M − i· · · h ξ i,m | e − ǫ ˆ H | ξ i,m − i · · ·h ξ i, | e − ǫ ˆ H | ξ i, ih ξ i, | e − ǫ ˆ H | − ξ i,M i , where m denotes the location of each inserts, playingthe role of the discrete imaginary time. The ξ fields inEq.(A2) is denoted by ξ iσ,M .In the limit ǫ →
0, the matrix element within neigh-boring time slices is given by h ξ i,m | e − ǫ ˆ H | ξ i,m − i = h ξ i,m | − ǫ ˆ H | ξ i,m − i = e − ǫH ( ξ ∗ iσ,m ,ξ iσ,m − ) h ξ i,m | ξ i,m − i = e − ǫH ( ξ ∗ iσ,m ,ξ iσ,m − )+ P σ ξ ∗ iσ,m ξ iσ,m − +( u − Q σ ξ ∗ iσ,m ξ iσ,m − . Here, the Hamiltonian is normal ordered with the el-ement H ( ξ ∗ iσ,m , ξ iσ,m − ). Therefore, taking the anti-periodic boundary condition ξ iσ, = − ξ iσ,M , the par-tition function can be expressed in the following path-integral Z = Z D ( ξ ∗ , ξ ) e − S ( ξ ∗ ,ξ ) , (A3)where D ( ξ ∗ , ξ ) = lim M →∞ Q i,m ( u ) dξ ∗ i ↑ ,m dξ i ↑ ,m dξ ∗ i ↓ ,m dξ i ↓ ,m and S = X i,m,σ ( u ˜ n iσ,m − ξ ∗ iσ,m ξ iσ,m − )+ X i,m ( u − u ) Y σ ˜ n iσ,m − X i,m ( u − Y σ ξ ∗ iσ,m ξ iσ,m − + X m ǫH [ { ξ ∗ m } , { ξ m − } ] , (A4)with ˜ n iσ,m = ξ ∗ iσ,m ξ iσ,m . Appendix B: The Properties of Grassmannintegrations
The most useful Grassmann integrations used in thepresent path-integral approach are listed below: Z dξ ∗ dξe − hξ ∗ ξ = h, (B1) Z Y σ dξ ∗ σ dξ σ e − Uξ ∗↑ ξ ↑ ξ ∗↓ ξ ↓ = − U, (B2) Z M Y m =1 dξ ∗ m dξ m M Y m =1 ξ ∗ m ξ m − = Z M Y m =1 dξ ∗ m dξ m ( ξ ∗ M ξ M − · · · ξ ∗ ξ ξ ∗ ( − ξ M ))= Z M Y m =1 dξ ∗ m dξ m ( ξ M ξ ∗ M ξ M − · · · ξ ∗ ξ ξ ∗ ) = 1 , (B3)where the anti-periodic boundary condition ξ = − ξ M isimposed. Appendix C: the partition function for the singlesite toy model
Here we exactly solve the single site problem using theconventional particle number representation. The Hamil-tonian of this toy model isˆ H = − X σ w σ ( ˜ X σ ) † ˜ X σ = − X σ w σ (1 − λ ˆ n ¯ σ )ˆ n σ (1 − λ ˆ n ¯ σ )= − X σ w σ (cid:2) ˆ n σ + ( λ − λ )ˆ n ↑ ˆ n ↓ (cid:3) (C1)It is diagonal in the particle number representation witheigenvalues E = 0, E = − w ↑ , E = − w ↓ and E = − P σ (1 − λ ) w σ respectively. Therefore, we have Z = tre − β ˆ H = X i =1 e − βE i = 1 + e β P σ wσu + X σ e βw σ (C2)where u = (1 − λ ) − . Appendix D: Derivation of g σ ( ω n ) in Eq. (13) The contribution from diagrams illustrated in Fig. 2(a) is given by ǫ M − X l =0 M X σ,m =1 (1 + ǫw σ ) l η ∗ iσ,m + l η iσ,m = ǫ M − X l =0 X σ,ω n (1 + ǫw σ ) l e iω n ǫl η ∗ iσ ( ω n ) η iσ ( ω n )= ǫ X σ,ω n − (1 + ǫw σ ) M e iω n ǫM − (1 + ǫw σ ) e iω n ǫ η ∗ iσ ( ω n ) η iσ ( ω n ) , (D1)where the Fourier transformation for the η fields is ap-plied η iσ,m = 1 √ β X ω n η iσ ( ω n ) e − iω n τ m . (D2)Since e iω n ǫM = − M →∞ (1 + ǫw σ ) M = e βw σ ,Eq.(D1) becomes − X σ,ω n e βw σ iω n + w σ η ∗ iσ ( ω n ) η iσ ( ω n ) . (D3)The contribution from diagrams illustrated in Fig. 2(b) is given by ǫ M − X σ,l =0 M X m =1 (1 + ǫw ¯ σ ) M u ǫ P σ ′ w σ ′ u ǫw ¯ σ l η ∗ iσ,m + l η iσ,m . Taken summation over m , it becomes ǫ M − X σ,l =0 X ω n e βw ¯ σ u ǫ P σ ′ w σ ′ u ǫw ¯ σ l e iω n ǫl η ∗ iσ ( ω n ) η iσ ( ω n ) . Again, taken summation over l , it becomes − X σ,ω n u e βw ¯ σ + e β P σ ′ wσ ′ u ! iω n + P σ ′ w σ ′ u − w ¯ σ η ∗ iσ ( ω n ) η iσ ( ω n ) (D4) The sum of Eq. (D3) and Eq. (D4) leads to thecontributions from the quadratic term of the ξ fields inexpanding the exponential of expression (12), namely, − P ω n ,σ g σ ( ω n ) η ∗ iσ ( ω n ) η iσ ( ω n ) , with g σ ( ω n ) = 1 + e βw σ iω n + w σ + u (cid:16) e βw ¯ σ + e β P σ ′ wσ ′ u (cid:17) iω n + P σ ′ w σ ′ u − w ¯ σ . (D5) Appendix E: Magnetization and Susceptibility forCHM
According to Eq. (C1),ˆ n i ↑ − ˆ n i ↓ = ( ˜ X ↑ ) † ˜ X ↑ − ( ˜ X ↓ ) † ˜ X ↓ . (E1)This operator is commutative with the CH term definedin the Hamiltonian (11). By choosing w σ = σh with h being the external magnetic field and σ = 1 for up spinand σ = − m = 1 N β ∂ ln Z∂h . (E2)The partition function Z is given by Eq. (15), where Z w = (1 + e βσh )(1 + e − βσh ) and Z − w g σ ( ω n ) ε ~k = u e − βσh e − βσh ε ~k iω n + σh . Therefore, we have m = σ ( e βσh − e − βσh ) Z w − N β X kσ σiω n + σh + 1 N β X kσ σ − σ (1 − u ) Z w βε ~k iω n + σh − u e − βσh e − βσh ε ~k = 1 N X ~kσ σ (1 − − u Z βε ~k ) F ( − σh + 1 + u e − βσh e − βσh ε ~k ) , where F ( x ) = e βx is the Fermi-Dirac distribution func-tion. The susceptibility χ = ∂m∂h | h =0 is then given by βN X ~kσ (1 − βε ~k − u F ( ε ~k u − F ( ε ~k u . This summation can be represented by the integralover the energy by introducing the density of state N ( ε )for the free fermions χ = 2 β Z Λ − Λ dεN ( ε ) (cid:18) − βε − u βε − u (cid:19) F ( ε u − F ( ε u u N (0) + π − u ) (1 + u ) N (0) , (E3)with the 2Λ being the bandwidth of the free electrons.In above, we have set µ B = 1 and used the δ -functionrepresentation β ( e βx + 1)( e − βx + 1) = δ ( x ) (E4)which is valid in the low temperature limit β → ∞ . Theintegration range of βε tends to ∞ in this limit, too,leading to Z ∞−∞ x dx ( e x + 1)( e − x + 1) = π . (E5) Appendix F: Internal energy and specific heat forCHM
In order to calculate the internal energy, E = − ∂ ln Z∂β , (F1)we can set µ σ = 0 so that Z w = 4 and Z − w g σ ( ω n ) = (1 + u ) iω n . Then we haveln Z = N ln Z w + X kσ ln [1 − Z − w g σ ( ω n ) ε ~k ]= N ln Z w + X kσ ln [ iω n −
12 (1 + 1 u ) ε ~k ] − ln ( iω n ) . Introducing a temporary variable α such thatln Z α = N ln Z w ++ X kσ ln [ iω n − α −
12 (1 + 1 u ) ε ~k ] − ln ( iω n − α ) . The internal energy can be reexpressed as E = − (cid:18) ∂∂β Z dα ∂∂α ln Z α (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α =0 . Because − ∂∂β Z dα ∂∂α ln Z α = ∂∂β Z dα X kσ (cid:18) iω n − α − (1 + u ) ε ~k − iω n − α (cid:19) = ∂∂β Z d ( βα ) X ~kσ (cid:20) F (cid:18) α + 12 (1 + 1 u ) ε ~k (cid:19) − F ( α ) (cid:21) , we have E = X ~k (1 + 1 u ) ε ~k F (cid:18)
12 (1 + 1 u ) ε ~k (cid:19) . (F2) Then the specific heat can be derived as in the following: C = ∂E∂T = − k B β ∂E∂β = 2 k B β X ~k (cid:18)
12 (1 + 1 u ) ε ~k (cid:19) F (cid:18)
12 (1 + 1 u ) ε k (cid:19) (cid:20) − F (cid:18)
12 (1 + 1 u ) ε ~k (cid:19)(cid:21) = 2 k B β Z Λ − Λ dεN ( ε ) (cid:18)
12 (1 + 1 u ) ε (cid:19) F (cid:18)
12 (1 + 1 u ) ε (cid:19) (cid:20) − F (cid:18)
12 (1 + 1 u ) ε (cid:19)(cid:21) = 41 + u N (0) k B T. (F3) Appendix G: The Green’s function of ξ fields To calculate the Green’s function of the ξ fields, weintroduce the corresponding external sources by adding P kσ ( J ∗ kσ ξ kσ + ξ ∗ kσ J kσ ) to S t in Eq. (12). Then, thepartition function becomes Z = Z NU Y kσ [1 − Z − U g ′ σ ( ω n ) ε ~k − Z − U g ′ σ ( ω n ) J ∗ kσ J kσ ] . (G1)It leads to h ξ ∗ kσ ξ kσ i = − ∂ ln Z∂J kσ ∂J ∗ kσ (cid:12)(cid:12)(cid:12)(cid:12) J =0 = Z − U g ′ σ ( ω n )1 − Z − U g ′ σ ( ω n ) ε ~k . (G2)This can be expressed in the following standard form h ξ ∗ kσ ξ kσ i = X ν = ± A ( ν ) ~kσ iω n − E ( ν ) ~kσ . (G3)By taking the zero temperature limit and setting µ σ = µ , we have E ( ± ) ~kσ = 12 (cid:18) U + 2 + u ε ~k − µ ) ± s ( U + 2 + u ε ~k − µ )) −
83 ( ε ~k − µ ) U . (G4)These are the dispersions of the quasi-particles in theupper and lower Hubbard bands respectively, with thespectral weights A ( ± ) ~kσ = ± u E ( ± ) ~kσ − U q ( U + u ( ε ~k − µ )) − ( ε ~k − µ ) U (G5)satisfying A ( ± ) ~kσ > P ν = ± A ( ± ) ~kσ = (2 + u ). Appendix H: Chemical potential and Hubbardinteraction expressed in terms of deformed Hubbardoperators
The on-site Coulomb interaction can be expressed asˆ n ↑ ˆ n ↓ = u ( ˜ X ↓ ) † ( ˜ X ↑ ) † ˜ X ↑ ˜ X ↓ . (H1)where u = (1 − λ ) − . In order to investigate the genericcase with tunable electron filling, we express the particlenumber operators ˆ n σ in terms of the deformed Hubbard operators. According to Eq. (C1)( ˜ X σ ) † ˜ X σ = ˆ n σ + ( λ − λ )ˆ n ↑ ˆ n ↓ , (H2)and combining it with Eq.(H1), we obtain the followingexpression for the particle number operatorsˆ n σ = ( ˜ X σ ) † ˜ X σ + ( u − X ↓ ) † ( ˜ X ↑ ) † ˜ X ↑ ˜ X ↓ ..