Fermi surface topology of the two-dimensional Kondo lattice model: a dynamical cluster approximation approach
FFermi surface topology of the two-dimensional Kondo lattice model: a dynamicalcluster approximation approach
L. C. Martin, M. Bercx and F. F. Assaad
Institut f¨ur Theoretische Physik und Astrophysik,Universit¨at W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany
We report the results of extensive dynamical cluster approximation calculations, based on aquantum Monte Carlo solver, for the two-dimensional Kondo lattice model. Our particular clusterimplementation renders possible the simulation of spontaneous antiferromagnetic symmetry break-ing. By explicitly computing the single-particle spectral function both in the paramagnetic andantiferromagnetic phases, we follow the evolution of the Fermi surface across this magnetic tran-sition. The results, computed for clusters up to 16 orbitals, show clear evidence for the existenceof three distinct Fermi surface topologies. The transition from the paramagnetic metallic phase tothe antiferromagnetic metal is continuous; Kondo screening does not break down and we observea back-folding of the paramagnetic heavy fermion band. Within the antiferromagnetic phase andwhen the ordered moment becomes large the Fermi surface evolves to one which is adiabaticallyconnected to a Fermi surface where the local moments are frozen in an antiferromagnetic order.
PACS numbers: 71.10.Fd, 71.10.Hf, 71.27.+a, 73.43.Nq, 75.30.Kz, 87.15.ak
I. INTRODUCTION
Heavy-fermion systems are characterized by a hier-archy of distinctive energy scales . The Kondo scale, T K ∝ e − W/J with bandwidth W and superexchange J ,marks the screening of local magnetic moments. Thisscreening is a many-body effect which entangles the spinsof the conduction electrons and local moments . Belowthe coherence temperature, which is believed to track theKondo scale , the paramagnetic (PM) heavy-fermionliquid emerges and corresponds to a coherent, Blochlike, superposition of the screening clouds of the individ-ual magnetic moments. Even in the Kondo limit, wherecharge fluctuations of the impurity spins are completelysuppressed, this paramagnetic state is characterized by alarge Fermi surface with Luttinger volume including boththe magnetic moments and conduction electrons .The coherence temperature of this metallic state is smallor, equivalently, the effective mass large.Kondo screening competes with the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which indirectlycouples the local moments via the magnetic polarizationof the conduction electrons. The RKKY energy scale isset by J χ c ( q , ω = 0) where χ c corresponds to the spinsusceptibility of the conduction electrons .The competition between Kondo screening - favoringparamagnetic ground states - and the RKKY interaction- favoring magnetically ordered states - is at the heart ofquantum phase transitions , the detailed understand-ing of which is still under debate (for recent reviews seeRef. 16–19).Here, two radically different scenarios have been put for-ward to describe this quantum phase transition. In the standard Hertz-Millis picture , the quasi-particles ofthe heavy-fermion liquid remain intact across the tran-sition and undergo a spin-density wave transition. Inparticular, neutron scattering experiments of the heavy- fermion system Ce − x La x Ru Si show that fluctuationsof the antiferromagnetic order parameter are responsi-ble for the magnetic phase transition and that the tran-sition is well understood in terms of the Hertz-Millisapproach .On the other hand, since many experimental observa-tions such as the almost wave vector independent spinsusceptibility in CeCu − x Au x , or the jump in the low-temperature Hall coefficient in YbRh Si are not ac-counted for by this theory alternative scenarios have beenput forward . In those scenarios, the quantum crit-ical point is linked to the very breakdown of the quasi-particle of the heavy-fermion state , and a topologicalreorganization of the Fermi surface across the transitionis expected .Recent experiments on CeIn or CeRh − x Co x In show that a change in Fermi surface (FS) topology mustnot necessarily occur only at the magnetic order-disorderquantum critical point (QCP). In fact, even in YbRh Si it has since been shown that the Fermi surface recon-struction can be shifted to either side of the QCP viaapplication of positive or negative chemical pressure .In this paper, we address the above questions throughan explicit calculation of the Fermi surface topology inthe framework of the Kondo lattice model (KLM). In itssimplest form the KLM describes an array of localizedmagnetic moments of spin 1 /
2, arising from atomic f -orbitals, that are coupled antiferromagnetically (AF) viathe exchange interaction J to a metallic host of mobileconduction electrons.We present detailed dynamical cluster approximation(DCA) calculations aimed at the investigation of theKLM ground state. For the simulations within the mag-netically ordered phase, we have extended the DCA to al-low for symmetry breaking antiferromagnetic order. Wemap out the magnetic phase diagram as a function of J/t and conduction electron density n c , with particularinterest in the single-particle spectral function and the a r X i v : . [ c ond - m a t . s t r- e l ] O c t evolution of the Fermi surface. The outline is as follows.The model and the DCA implementation is discussed inSec. II. Results for the case of half-band filling and hole-doping are discussed in Sec. III and IV. Section V isdevoted to a summary. This paper is an extension to ourprevious work, where part of the results have alreadybeen published . II. MODEL HAMILTONIAN AND DYNAMICALCLUSTER APPROXIMATION
The Kondo lattice model (KLM) we consider reads H = (cid:88) k ,σ ( (cid:15) ( k ) − µ ) c † k ,σ c k ,σ + J (cid:88) i S c i · S f i . (1)The operator c † k ,σ denotes creation of an electron ina Bloch state with wave vector k and a z-componentof spin σ = ↑ , ↓ . The spin 1 / J >
0, are represented with the aid ofthe Pauli spin matrices σσσ by S c i = (cid:80) s,s (cid:48) c † i ,s σσσ s,s (cid:48) c i ,s (cid:48) and the equivalent definition for S f i using the localizedorbital creation operators f † i ,σ . The chemical potentialis denoted by µ . The definition of the KLM excludescharge fluctuations on the f -orbitals and as such a strictconstraint of one electron per localized f -orbital has tobe included. For an extensive review of this model werefer the reader to Ref. 36.Particle-hole symmetry at half-filling is given if hoppingis restricted to nearest neighbors on the square lattice andthe chemical potential is set to zero. We introduce a next-nearest neighbor hopping with matrix element t (cid:48) to givea modified dispersion (cid:15) ( k ) = − t [cos( k x ) + cos( k y )] − t (cid:48) [cos( k x + k y ) + cos( k x − k y )]. As we will see, thesuppression of particle-hole symmetry due to a finitevalue of t (cid:48) leads to dramatic changes in the scaling of thequasi-particle gap at low values of J/t and at half-bandfilling. We have considered the value t (cid:48) /t = − .
3. Thischoice guarantees that the spin susceptibility of the hostmetallic state at small dopings away from half-filling ispeaked at wave vector Q = ( π, π ). Thus, antiferromag-netic order as opposed to an incommensurate spin stateis favored.To solve the model we have used the DCA ap-proach which retains temporal fluctuations and henceaccounts for the Kondo effect but neglects spatial fluc-tuations on a length scale larger than the cluster size.The approach relies on the coarse-graining of momentumspace, and momentum conservation holds only betweenthe k -space patches. By gradually defining smaller sizedpatches the DCA allows to restore the k -dependency ofthe self-energy.The standard formulation of DCA is naturally transla-tionally invariant. To measure orders beyond translationinvariance the DCA can be generalized to lattices with asupercell containing N u -unit cells of the original lattice . A general unit cell is addressed by ¯ x = x + r µ , x denotingthe supercell and r µ the relative points with µ = 1 ...N u .In this work we opted for N u = 2 which is the minimumrequirement for the formation of antiferromagnetic or-der. The reduced Brillouin zone (RBZ) is then spannedby b = π (1 ,
1) and b = π (1 , −
1) and we have set thelattice constant to unity. This supercell as well as theRBZ (also coined magnetic Brillouin zone (MBZ)) areplotted in Fig. 1.The self-energy and Green function are to be understoodas spin dependent matrix functions, with an index for theunit cell within the supercell as well as an orbital indexfor the c - and f -orbitals in each unit cell. The k -spacediscretization into N p patches with momentum conserva-tion only between patches yields the coarse-grained lat-tice Green function¯ G σ ( K , i ω ) = N p N (cid:88) ˜ k G σ ( K + ˜ k , i ω ) (2)and ensures that the self-energy is only dependent on thecoarse-grained momentum K : ¯Σ ≡ ¯Σ σ ( K , i ω ). Here, thereciprocal vector K denotes the center of a patch andthe original k vectors are given by k = K + ˜ k . Theself-energy is extracted from a real-space cluster calcu-lation with periodic boundaries yielding the quantized K values. Let G ,σ ( K , i ω ) be the bath (non-interacting)Green function of the cluster problem and G σ ( K , i ω ) thefull cluster Green function. Hence, Σ = G − − G − andself-consistency requires that: G σ ( K , i ω ) = ( G − ,σ ( K , i ω ) − ¯Σ σ ( K , i ω )) − (3)= N p N (cid:88) ˜ k ( G − ,σ ( K + ˜ k , i ω ) − ¯Σ σ ( K , i ω )) − . The non-interacting lattice Green function is denoted by G ,σ ( K + ˜ k , i ω ).To summarize, our implementation of DCA approximatesthe Fourier space of a lattice with two-point basis bypatching. Regarding the KLM this leads to real-spaceclusters of size N p , each encompassing 2 × N p c - and f -orbitals. In the present work cluster sizes N p = 1 and N p = 4 are considered (Fig. 1).The lattice green function G σ ( k , i ω n ) with k ∈ M BZ can equally be expressed as g σ (¯ k , i ω n ) in an extendedBrillouin zone scheme, with ¯ k ∈ BZ and ¯ k = k + m b + n b : g σ (¯ k , i ω n ) = 12 N u =2 (cid:88) µ,ν e i ¯ k ( r ν − r µ ) [ G σ ( k , i ω n )] µν . (4)In order to implement the DCA one has to be able tosolve, for a given bath, the Kondo model on the effec-tive cluster. The method we have chosen is the auxil-iary field Quantum Monte Carlo (QMC) version of theHirsch-Fye algorithm, following precisely the same real-ization as in Ref. 39. The implementation details of theself-consistency cycle can be found in Ref. 40. The per-formance of the QMC cluster solver has been enhanced FIG. 1. (Color online) Definition of the antiferromagnetic unitcell (a) and DCA patching of the Brillouin zone (b). The lat-tice vectors a = (1 ,
1) and a = (1 , −
1) connect the AF unitcells. The two inequivalent c -orbitals are denoted by filled(empty) circles and the localized f -orbitals with arrows. Thereciprocal-space lattice vectors b and b span the magneticBrillouin zone. The MBZ is uniformly discretized in N p = 4patches, corresponding to real-space clusters each containing N p magnetic unit cells. The color coding refers to constantmomentum dependency of the self-energy and cluster Greenfunction. The dashed square denotes the extended Brillouinzone. by almost an order of magnitude by implementing themethod of delayed updates . During the QMC Markovprocess the full updates of the Green function are delayeduntil a sufficiently large number of local changes in theauxiliary field have been accumulated. This enhances theperformance of the whole QMC algorithm since consid-erable CPU time is spent in the updating section. To ef-ficiently extract spectral information from the imaginarytime discrete QMC data a stochastic analytic continua-tion scheme is employed .Ground state properties of the model are accessed by ex-trapolation of the inverse temperatures β to infinity. Thisis a demanding task since all relevant energy scales, theKondo scale, the coherence scale and the RKKY scale,become dramatically smaller with decreasing J/t . In theparamagnetic phase and below the coherence scale a clearhybridization gap is apparent in the single particle spec-tral function and we use this criterion to estimate thecoherence temperature. Since the computational timerequired by the QMC cluster solver increases propor-tional to ( βN u N p ) , this limits us in the resolution ofenergy scales to values of J/t ≥ .
8. It is importantto note that for small dopings away from half-band fill-ing and for the considered cluster sizes, the negative signproblem is not severe and hence is not the limiting factor.
III. THE HALF-FILLED KLM
We consider the KLM at half-filling in two cases: eitherwith or without particle-hole symmetry. That is t (cid:48) /t = 0and t (cid:48) /t = − . t (cid:48) = 0 lattice QMC sim- ulations do not suffer from the negative sign problem,and it is well established that the Kondo screened phasegives way to an AF ordered phase at J c /t = 1 . .This magnetic order-disorder transition occurs betweeninsulating states and it is reasonable to assume that it be-longs to the three-dimensional O (3) universality class .Here, the dynamical exponent takes the value of unitysuch that the correlation lengths in imaginary time andin real space are locked in together and diverge at thecritical point. Due to the very small cluster sizes consid-ered in the DCA it is clear that we will not capture thephysics of this transition. In fact as soon as the correla-tion length exceeds the size of the DCA cluster, symme-try breaking signaled by a finite value of the staggeredmoment m fs = 12 N p (cid:88) i (cid:104) n f i , ↑ − n f i , ↓ (cid:105) e − i Qi (5)sets in and mean-field exponents are expected. The nor-malization is chosen such that the staggered magnetiza-tion of the fully polarized state takes the value of unity.The above quantity is plotted in Fig. 2 at t (cid:48) /t = 0 and N p = 1 As apparent, the critical value of J/t overes-timates ( J c /t ≈ . exact lattice QMC result.Switching off nesting by including a finite value of t (cid:48) /t shifts the magnetic order-disorder transition to lower val-ues of J/t .To improve on this result, we can systematically enhancethe cluster size. However, our major interest lies inthe single-particle spectral function. As we will see be-low, and well within the magnetically ordered phase, thisquantity compares very well with the exact lattice QMCresults.
A. Single-particle spectrum - t (cid:48) /t = 0 We have calculated the single-particle spectral func-tion A cc ( k , ω ) = − π (cid:80) σ Im [ g ccσ ( k , ω )] of the conductionelectrons along a path of high symmetry in the extendedBrillouin zone. In all our spectral plots, on the energyaxis, ω values are given relative to the chemical potential µ .In Fig. 3 we plot the spectrum for J/t = 2 . k = ( π, π ), with rel-atively low spectral weight (note the logarithmic scaleof the color chart) in comparison to the other parts ofthe band which are mostly unchanged from the non-interacting case. This feature is associated with Kondoscreening of the impurity spins and the resultant largeeffective mass of the composite quasi-particles. Since noband crosses the Fermi energy ( ω/t = 0) we classify thisregion of parameter space as a Kondo insulator. The ob-served dispersion relation is well described already in theframework of the large- N mean-field theory of the Kondolattice model . At the particle-hole symmetric point this FIG. 2. The staggered magnetization m fs of the local mo-ment spins in the ground state as a function of coupling J/t with next-nearest neighbor hopping t (cid:48) /t = − . t (cid:48) /t = 0,respectively. The low temperature limit is reached by per-forming simulations at various inverse temperatures β . Con-vergence to the ground state has been achieved for the fol-lowing inverse temperatures: βt = 20 (2 . ≥ J/t ≥ . βt = 40 (1 . ≥ J/t ≥ . βt = 80 ( J/t = 1) and βt = 100( J/t = 0 . approximation, which in contrast to the DCA+QMC re-sults of Fig. 3 neglects the constraint of no double occu-pancy of the f -orbitals, yields the dispersion relation ofhybridized bands: E ± ( k ) = 12 (cid:104) (cid:15) ( k ) ± (cid:112) (cid:15) ( k ) + ∆ (cid:105) . (6)Within the mean-field theory the quasi-particle gap (seeEq. 7), ∆ qp ∝ ∆ , tracks the Kondo scale.At J/t = 2 . m fs = 0 . ± . k = ( π, π ) and in the lower bandaround k = (0 , shadow bands arise due to thescattering of the heavy quasi-particle off the magneticfluctuations centered at wave vector Q = ( π, π ).At J/t = 1 . on the left and our DCA result on the right,both only for the photoemission spectrum ( ω/t < k = ( π, π ) ( k = (0 , J/t .It is noteworthy that already with the smallest possible
FIG. 3. (Color online) Single-particle spectral function A cc ( k , ω ) at half-filling ( n c = 1) and with particle-hole sym-metry ( t (cid:48) /t = 0) close to the magnetic phase transition on theparamagnetic side, J/t = 2 . A cc ( k , ω ) at half-filling ( n c = 1) and with particle-hole sym-metry ( t (cid:48) /t = 0) close to the magnetic phase transition on theAF side, J/t = 2 . m fs = 0 . ± . A cc ( k , ω ) with the T = 0 spectral function.Data (a) taken from Ref. 39 which used a projective auxiliaryfield Monte Carlo (BSS) method for the KLM on a 12 × βt = 40, both for the sameparameter set - t (cid:48) /t = 0, n c = 1, J/t = 1 . cluster capable of capturing AF order, N p = 1, we areable to produce a single-particle spectrum which is essen-tially the same as the lattice QMC result. This confirmsthat the DCA is indeed a well suited approximation foruse with the KLM: the essence of the competition be-tween RKKY-mediated spacial magnetic order and thetime-displaced correlations responsible for Kondo screen-ing is successfully distilled to a small cluster dynamicallyembedded in the mean-field of the remaining bath elec-trons. B. Single-particle spectrum - t (cid:48) /t = − . We now set t (cid:48) /t = − .
3, but remain at half-filling viacareful adjustment of the chemical potential µ . With de-creasing J/t we can divide the phase diagram into fourregions on the basis of the characteristic spectrum in eachregion. For
J/t = 2 . t (cid:48) /t = 0 case suchthat the model is a Kondo insulator with an indirect gap.The minimum of the valence band lies at k = (0 ,
0) andthe maximum of the conduction band is at k = ( π, π ).For J/t = 1 . J/t = 1 . k = (0 , π ). The final plot in the series (Fig. 6(d)), with J/t = 1 . k = ( π/ , π/ k = (0 ,
0) and k = ( π, π ) with an enlargementof the energy axis around the Fermi energy. The localminimum energy dip at k = ( π/ , π/
2) becomes less pro-nounced as the heavy-fermion band flattens until by thetime we reach
J/t = 1 . k = ( π, π ) to k = ( π/ , π/ J/t = 1 .
1, 1 .
0, and 0 . k = ( π/ , π/
2) and becomes more pro-nounced. Assuming a rigid band picture, this evolutionof the band structure maps onto a topology change ofthe Fermi surface at small dopings away from half filling.We will see by explicit calculations at finite dopings thatthis topology change indeed occurs.The position of the minima and maxima in the valenceand conduction bands of the spectra for t (cid:48) /t = − . C. The quasi-particle gap
The quasi-particle gap corresponds to the energy dif-ference between the top of the conduction band and the bottom of the valence band. To be more precise,2∆ qp = min k (cid:0) E N +10 ( k ) − E N (cid:1) − max k (cid:0) E N − E N − ( k ) (cid:1) . (7)Here, E N ± ( k ) corresponds to the ground state energyof the N ± k and E N = min k E N ( k ). Technically, the energy differencesare at best extracted from the low temperature single-particle Green functions:lim β →∞ (cid:104) c ( τ ) k c † k (cid:105) ∝ e − τ ( E N +10 ( k ) − E N − µ ) , τ >> β →∞ (cid:104) c † k ( τ ) c k (cid:105) ∝ e τ ( E N − E N − ( k ) − µ ) , τ << − . In the large
J/t limit, each impurity spin traps a con-duction electron in a Kondo singlet. The wave functioncorresponds to a direct product of such Kondo singletsand the quasi-particle gap is set by the energy scale 4 J/ J/t limit the quasi-particle gap does not depend on the de-tails of the band structure. The situation is more subtlein the weak-coupling limit, where the underlying nest-ing properties of the Fermi surface play a crucial role.Assuming static impurity spins locked into an antiferro-magnetic order, (cid:104) S f i (cid:105) = e z e i Q · i one obtains the singleparticle dispersion relation E ± ( k ) = (cid:15) + ( k ) ± (cid:113) (cid:15) − ( k ) + ( J/ , (8)with (cid:15) ± ( k ) = (cid:15) ( k ) ± (cid:15) ( k + Q )2 . In one dimension and at t (cid:48) /t = 0 nesting leads to (cid:15) + ( k ) ≡ qp = J/
4. Both theDCA results presented in Fig. 9 as well as the BSS resultsof Ref. 39 support this point of view. In one-dimension,this scaling of the quasi-particle gap is also observed .At t (cid:48) /t = − . t (cid:48) /t = − . N mean-field calculations . IV. THE HOLE-DOPED KLM
In this section we concentrate on the hole-doped KLMat t (cid:48) /t = − .
3. We will first map out the magnetic phasediagram in the doping versus coupling plane and thenstudy the evolution of single-particle spectral functionfrom the previously discussed half-filled case to the heav-ily doped paramagnetic heavy-fermion metallic state.
FIG. 6. (Color online) Single-particle spectral functions A cc ( k , ω ) at half-filling ( n c = 1) with t (cid:48) /t = − .
3. As the Kondo cou-pling
J/t is decreased, the spectrum changes from a paramagnetic Kondo insulator with indirect gap (a) to an antiferromagneticordered state (d).
A. Magnetic phase diagram
Fig. 10 shows the staggered magnetization as a func-tion of conduction electron density for
J/t = 0 .
8, 1 .
0, 1 . .
4. The magnetically ordered state found at half-filling initially survives when doping with holes. At allcoupling values a continuous magnetic phase transition isobserved with the AF order decreasing gradually as thesystem is doped and vanishing smoothly at a quantumcritical point. The results are summarized in the mag-netic phase diagram shown in Fig. 11. With decreasingvalues of
J/t and increasing dopings the RKKY interac-tion progressively dominates over the Kondo scale andthe magnetic metallic state is stabilized. Checks weremade by varying the temperature, 1 /β , of the simula- tions to ensure that the results can be considered to beground state. Below J/t = 0 . algorithm, ( βN p ) andnot the negative sign problem. B. Single-particle spectrum and topology of theFermi surface
We have calculated and followed the evolution of thesingle-particle spectrum, plotted in an extended Bril-louin zone scheme. To begin with we set
J/t = 1 andshow results in Fig. 12 for the spectral function, as it
FIG. 7. (Color online) Closeup for low energies around the Fermi energy (given by ω/t = 0). Development of the single-particlespectral function A cc ( k , ω ) at half-filling with t (cid:48) /t = − . k = ( π, π )to k = ( π/ , π/
2) with decreasing
J/t .FIG. 8. (Color online) Schematic representation of the posi-tion of the upper band minima (top row) and lower band max-ima (bottom row) of the conduction electron single-particlespectral function with next-nearest neighbor hopping t (cid:48) /t = − . k = ( − π, − π ) and k = ( π, π ),respectively. is here that a change in the Fermi surface topology be-comes evident. Starting from the previously discussedhalf-filled case (Fig. 12(a)) a rigid band approximationproduces hole pockets around the k = ( ± π/ , ± π/ n c = 0 .
926 (Fig. 12(d)). Doping further
FIG. 9. The quasi-particle gap ∆ qp /t as a function of Kondocoupling J/t . At t (cid:48) = − . t , the magnetic order-disordertransition takes place at J (cid:48) c /t (cid:39) .
85. In the coupling range0 . < J/t < J (cid:48) c /t Kondo screening coexists with magnetismand is at the origin of the quasiparticle gap. gives rise to a Fermi surface with holes centered around k = ( ± π, ± π ). Since at n c = 0 .
908 (Fig. 12(e)) and n c = 0 .
898 (Fig. 12(f)) we still have non-zero magne-tizations of m fs = 0 .
340 and m fs = 0 . k = (0 ,
0) are expected.The weight of those shadow features progressively di-minishes as the staggered magnetization vanishes. Inparticular, in Fig. 12(g) the magnetization is very small
FIG. 10. The staggered magnetization m fs of the f -electronsas a function of n c at different constant couplings J/t .FIG. 11. (Color online) Ground state magnetic phase dia-gram of the hole-doped KLM showing simulation results forthe staggered magnetization m fs (color-coded) as a functionof coupling J/t and conduction electron occupancy n c . Tri-angles: PM region, large FS. Squares: AF, large FS. Circles:AF, small FS. Here t (cid:48) /t = − . N p = 1 cluster. Below, the FS topologiescorresponding to the numbered regions are shown schemati-cally. ( m fs = 0 . n c = 1 to n c =0 .
856 the quasiparticle weight Z ( k = ( π, π )) = |(cid:104) Ψ N +10 | c † k =( π,π ) ,σ | Ψ N (cid:105)| as obtained from the behaviourof g cc ( k = ( π, π ) , τ ) at large imaginary times decreasesalbeit it shows no sign of singularity. This observation stands in agreement with results obtained at half-bandfilling and excludes the occurrence of a Kondo break-down.This evolution of the single-particle spectral function at J/t = 1 points to three distinct Fermi surface topolo-gies, sketched in Fig. 11. In the paramagnetic phase(Fermi surface (1) in Fig. 11) the Fermi surface consistsof hole pockets around the k = ( ± π, ± π ) points in theBrillouin zone. Even though in our simulations chargefluctuations of the f -sites are completely prohibited, theFermi surface volume accounts for both the conductionelectrons and impurity spins. This Fermi surface topol-ogy maps onto that of the corresponding non-interactingperiodic Anderson model with total particle density givenby 1 + n c and is coined large Fermi surface. In theantiferromagnetic metallic phase close to the magneticorder-disorder transition, (Fermi surface (2) in Fig. 11)the Fermi surface merely corresponds to a backfolding ofthe paramagnetic Fermi surface as expected in a genericspin-density wave transition. Here, a heavy quasi-particlewith crystal momentum k can scatter off a magnon withmomentum Q = ( π, π ) to produce a shadow feature at k + Q . It is only within the magnetically ordered phasethat we observe the topology change of the Fermi sur-face to hole pockets centered around k = ( ± π/ , ± π/ f -spin mean-field calculation as presented in Eq.(9). We have equally plotted the single-particle spectralfunction at higher values of J/t = 1 . k = ( π, π ) with accompanying shadow bandsat k = (0 , k = (0 , N p = 4, seeFig. 14. In particular, we observe the small Fermi sur-face topology in the strong AF region with small coupling J/t and evidence for a large Fermi surface in the weaklyordered region (
J/t (cid:38) . C. Mean-field modeling of the single-particlespectral function
We resort to a mean-field modelling in order to proposea scenario for the detailed nature of the topological tran-sition of the FS within the ordered phase. Within thismodel the topology change will correspond to two Lif-shitz transitions. Aspects of the single-particle spectralfunction can be well accounted for within the following
FIG. 12. (Color online) Single-particle spectra A cc ( k , ω ) of the conduction electrons for J/t = 1 and t (cid:48) /t = − .
3, at βt = 80.The conduction electron density is reduced progressively from half-filling n c = 1 (a) to n c = 0 .
880 (h). The respective FStopologies are shown in Fig. 11.FIG. 13. (Color online) Single-particle spectra A cc ( k , ω ) ofthe conduction electrons for J/t = 1 . t (cid:48) /t = − .
3, at βt = 40. The shadow features present in (a) vanish as theoccupation number n c is reduced. FIG. 14. (Color online) Single-particle spectrum A cc ( k , ω ) ofthe conduction electrons obtained using cluster size N p = 4.confirming that the small Fermi surface topology in the lightlydoped AF phase with coupling J/t = 1 is not an artifact ofthe smaller cluster results. The staggered magnetisation ofthe f -electrons is m fs = 0 . ± . :˜ H = (cid:88) k σ c k σ c k + Q σ f k σ f k + Q σ † × (cid:15) k − µ Jm fs σ JV Jm fs σ (cid:15) k + Q − µ JV JV λ − Jm cs σ JV − Jm cs σ λ × c k σ c k + Q σ f k σ f k + Q σ . (9)Here λ , ( µ ) are Lagrange multipliers fixing the f - ( c -)particle number to unity ( n c ) and m cs , m fs correspondsto the staggered magnetization of the conduction and f -electrons. The order parameter for Kondo screening is V ∝ (cid:104) f † i,σ c i,σ + c † i, − σ f i, − σ (cid:105) . The large- N mean-field sad-dle point corresponds to the choice m cs = m fs = 0 and V (cid:54) = 0. As already mentioned this saddle point givesa good account of the hybridized bands we observe nu-merically in the paramagnetic phase. In the magneticphase, one could speculate that Kondo screening breaksdown such that the f -spins are frozen and do not par-ticipate in the Luttinger volume. This corresponds tothe parameter set V = 0, m cs (cid:54) = 0 and m fs (cid:54) = 0 andleads to the dispersion relation of Eq. (8). Throughoutthe phase diagram of Fig. 11 the single-particle spectralfunction never shows features following this scenario. Infact, many aspects of the spectral function in the mag-netically ordered phase can be understood by choosing m cs (cid:54) = 0, m fs (cid:54) = 0 and V (cid:54) = 0. This explicitly accounts fora heavy-fermion band in the ordered phase. It is in thissense that we claim the absence of Kondo breakdownwithin our model calculations. Fig. 15 plots the four-band energy dispersion relation E n ( k ) obtained from themean-field calculation in the magnetically ordered phaseat t (cid:48) /t = − . J/t = 1 and at band fillings n c = 0 . n c = 0 .
975 (Fig. 15(c,d)) correspond-ing to the Fermi surface topologies labeled (2) and (3),respectively, in Fig. 11. In each case we have used theDCA value for the magnetization and varied V to obtainthe best qualitative fit to the respective DCA spectrum.The change in the topology of the Fermi surface stemsfrom a delicate interplay of the relative magnitudes ofthe magnetization m c,fs and Kondo screening parameter V .In the DCA calculations, we are unable to study theprecise nature of the topology change of the Fermi sur-face within the magnetically ordered phase since the en-ergy scales are too small. However we can draw on themean-field model to gain some insight. Fig. 16 plots thedispersion relation along the k = (0 ,
0) to k = ( π, π )line in the Brillouin zone as a function of V . We wouldlike to emphasize the following points. i) The transition FIG. 15. Mean-field band structures E n ( k ) with J/t = 1( t (cid:48) /t = − . V = 0 . , m cs = 0 . , m fs = 0 . n c =0 . A n ( k , ω ) . The Fermi surface topology (seecloseup (b)) results from backfolding of the large- N mean-fieldbands. This corresponds to the Fermi surface (2) in Fig.11.(d) V = 0 . , m cs = 0 . , m fs = 0 . n c = 0 . k = ( ± π/ , ± π/
2) as depictedby the Fermi surface (3) in Fig. 11. from hole pockets around the wave vector k = ( π, π ) to k = ( ± π/ , ± π/
2) corresponds to two Lifshitz transitionsin which the hole pockets around k = ( π, π ) disappearsand those around k = ( ± π/ , ± π/
2) emerge. The Lut-tinger sum rule requires an intermediate phase with thepresence of pockets both at k = ( ± π/ , ± π/
2) and at k = ( π, π ). This situation is explicitly seen in Fig. 16(b)at V = 0 . k = ( ± π/ , ± π/
2) there isan overall flatting of the band prior to the change inthe Fermi surface topology. Hence on both sides of the1
FIG. 16. (Color online) Topology change (a) of the Fermisurface as obtained from the mean-field modeling E n ( k ) of theDCA results ( J/t = 1, t (cid:48) /t = − . m cs = 0 . m fs = 0 . n c = 0 . k = ( ± π/ , ± π/
2) as well as around k = ( π, π ) are present at V = 0 . transition, an enhanced effective mass is expected. Ow-ing to point i) the effective mass does not diverge. iii)The two Lifshitz transition scenario suggested by thismean-field modeling, leads to a continuous change of theHall coefficient. However, since a very small variation ofthe hybridization V suffices to change the Fermi surfacetopology one can expect, as a function of this control pa-rameter, a rapid variation of the Hall coefficient as it isobserved in experiment . D. Discussion
The evolution of the Fermi surface across the magneticorder-disorder transition has been investigated by othergroups and especially in the framework of a Gutzwillerprojected mean-field wave function corresponding to the ground state of the single particle Hamiltonian of Eq. 9.Both a variational quantum Monte Carlo calculation as well as a Gutzwiller approximation show a rich phasediagram which bears some similarities but also impor-tant differences with the present DCA calculation. Atsmall doping away from half-filling, the magnetic transi-tion as a function of the coupling J/t is continuous andof SDW type as marked by the back-folding of the Fermisurface. Within the magnetically ordered phase a firstorder transition occurs between hole (AF h ) and electron(AF e ) like Fermi surfaces. At larger hole dopings, themagnetic transition is of first order and the Fermi sur-face abruptly changes from large to small (AF e ). Thecalculations in Refs. 27–29 are carried out at t (cid:48) = 0. Atfinite values of t (cid:48) = − . t one expects the AF e Fermi-surface topology to correspond to hole pockets centeredaround (cid:126)k = ( ± π/ , ± π/
2) as shown in Fig. 12. On theother hand, the AF h Fermi surface arises from a back-folding of the large Fermi surface (Fig. 13). With thisidentification, our DCA results bear some similarity withthe variational calculation in the sense that the sameFermi surface topologies are realized. However, in ourcalculations we see no sign of first order transitions andno direct transitions from the large paramagnetic Fermi-surface topology to the AF e phase. It is equally impor-tant to note that in both the variational and DCA ap-proaches no Kondo breakdown is apparent. In particularthe variational parameter ˜ V which encodes hybridizationbetween the f - and c -electrons never vanishes . Onthe other hand, T = 0 CDMFT calculations of the three-dimensional periodic Anderson model on a two site clus-ter and finite sized baths have put forward the ideathat the magnetic order-disorder transition is driven byan orbital selective Mott transition or in other words aKondo breakdown . In particular even in the presenceof spin symmetry broken baths allowing for antiferromag-netic ordering a big enhancement of the f -effective massis apparent in the vicinity of the magnetic transition. Inthe paramagnetic phase, the low energy decoupling ofthe f - and c -electrons stem from the vanishing of the hy-bridization function at low energies. Our results, whichmakes no approximation of the number of bath degreesof freedom and which allow to treat larger cluster sizesdo not support this point of view. In particular as shownin Ref. 8 no singularity in the quasiparticle weight is ap-parent across the transition. Equally, Fig. 12 which plotsthe single particle spectral function at finite doping showsno singularity in Z ( k = ( π, π )) . V. CONCLUSION
We have presented detailed DCA+QMC results forthe Kondo Lattice model on a square lattice. Our DCAapproach allows for antiferromagnetic order such thatspectral functions can be computed across magnetictransitions and in magnetically ordered phases. Atthe particle-hole symmetric point (half-band filling and2 t (cid:48) = 0) we can compare this approach to previous exact lattice QMC auxiliary field simulations . Unsur-prisingly, with the small cluster sizes considered in theDCA approach the critical value of J/t at which themagnetic order-disorder transition occurs is substantiallyoverestimated. The important point however is that theDCA+QMC approximation gives an extremely goodaccount of the single-particle spectral function both inthe paramagnetic and antiferromagnetic phases. Hence,and as far as we can test against benchmark results, thecombination of static magnetic ordering and dynamicalKondo screening, as realized in the DCA, provides avery good approximation of the underlying physics.As opposed to exact lattice QMC auxiliary field simula-tions, which fail away from the particle-hole symmetricpoint due to the so-called negative sign problem, theDCA+QMC approach allows us to take steps away fromthis symmetry either by doping or by introducing a finitevalue of t (cid:48) . It is worthwhile noting that for cluster sizesup to 16 orbitals the limiting factor is the cubic scalingof the Hirsch-Fye algorithm rather than the negativesign problem which turns out to be very mild in theconsidered parameter range.Our major findings are summarized by the followingpoints.i) We observe no Kondo breakdown throughout thephase diagram even deep in the antiferromagneticordered state where the staggered magnetization takeslarge values. This has the most dramatic effect athalf-band filling away from the particle-hole symmetricpoint and at small values of J/t where magnetic order isrobust. Here, magnetism alone will not account for theinsulating state and the observed quasi-particle gap canonly be interpreted in terms of Kondo screening. Theabsence of Kondo breakdown throughout the phase dia-gram is equally confirmed by the single-particle spectralfunction which always shows a feature reminiscent of theheavy-fermion band. ii) The transition from the paramagnetic to the antifer-romagnetic state is continuous and is associated with thebackfolding of the large heavy fermion Fermi surface.iii) Within the antiferromagnetic metallic phase thereis a Fermi surface topology change from hole pocketscentered around k = ( π, π ) at small values of the mag-netization m fs to one centered around k = ( π/ , π/ larger values of m fs . The latter Fermi surface isadiabatically connected to one where the f -spins arefrozen and in which Kondo screening is completelyabsent. This transition comes about by continuouslydeforming a heavy-fermion band such that the effectivemass grows substantially on both sides of the transition.Within a mean-field modelling of the quantum MonteCarlo results the transition in the Fermi surface topologycorresponds to two continuous Lifshitz transitions. Asa function of decreasing mean-field Kondo screeningparameter V and at constant hole doping, the volumeof the hole pocket at k = ( π, π ) decreases at theexpense of the increase of volume of the hole pocket at k = ( π/ , π/ ACKNOWLEDGMENTS
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