Infinity scatter infinity: Infinite critical boson non-Fermi liquid
IInfinity scatter infinity:Infinite critical boson non-Fermi liquid
Xiao-Tian Zhang and Gang Chen
Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong,The University of Hong Kong, Hong Kong, China
We study a novel type of non-Fermi liquid where there exist an infinite number of critical bosonicmodes instead of finite bosonic modes for the conventional ones. We consider itinerant magnetswith both conduction electrons and fluctuating magnetic moments in three dimensions. WithDzyaloshinskii-Moriya interaction, the moments fluctuate near a boson surface in the reciprocalspace at low energies when the system approaches an ordering transition. The infinite number ofcritical modes on the boson surface strongly scatter the infinite gapless electrons on the Fermi surfaceand convert the metallic sector into a non-Fermi liquid. We explain the novel physical propertiesof this non-Fermi liquid. On the ordered side, a conventional non-Fermi liquid emerges due to thescattering by the gapless Goldstone mode from the spontaneous breaking of the global rotationalsymmetry. We discuss the general structure of the phase diagram in the vicinity of the quantumphase transition and clarify various crossover behaviors.
Introduction. —Landau Fermi liquid theory is the ma-jor milestone of modern condensed matter physics, andillustrates the triumph of physical intuition. The short-ranged repulsive interaction between the fermions wasargued to be irrelevant as one approaches the low energytowards the Fermi surface. The singular long-range in-teractions, however, are not well coped in the frameworkof Fermi liquid theory and signifies the possibility of non-Fermi liquid (NFL) metals.
These singular interactionscan come from (partially screened) long-range Coulombinteraction, the fluctuations of the gapless bosonic modesat the criticality, Goldstone boson from the continuoussymmetry breaking, and U(1) gauge boson. Most ex-isting theories for NFL metals, particularly the exper-imentally relevant ones, involve the coupling be-tween gapless fermions near the Fermi surface and criti-cal bosons. If the number of the gapless fermions is finitesuch as Dirac fermion, Weyl fermion and quadratic bandtouching, a controlled calculation with the perturbativerenormalization group theory can be performed. In con-trast, when the fermion sector is a Fermi surface, thephysics become rather complex and this topic is underan active investigation in recent years.So far there are two types of Fermi surface critical-ity associated NFLs. The first involves an ordering at afinite wavevector, e.g. an antiferromagnetic (AFM) or-der or spin density wave. The ordering wavevectorconnects a few “hotspots” on the Fermi surface, and thetheoretical analysis will focus on the coupling betweenthe hotspot fermions and the critical bosons. The sec-ond involves the critical bosons at the zero wavevector.This captures, for instance, the Ising-nematic critical-ity, spinon-gauge coupling in the spinon Fermi sur-face U(1) spin liquid, ferromagnetic(FM) criticality. It was observed that, the bosons that are tangential tothe Fermi surface scatter the fermions strongly at lowenergies. Thus, the theoretical analysis of this Ising-nematic criticality is further reduced to the so-called“patch theory” where the tangential critical boson scat- ters the fermions from one patch or two patches. Vari-ous analytic techniques were developed. The early ran-dom phase approximation (RPA) type of large- N expan-sion was questioned as this scheme of taming quantumfluctuations and organizing Feynman diagrams missesthe contribution from the processes involving fermionson the Fermi surface. The remedy was madeby double expansion that combines the large- N expan-sion and the (cid:15) -expansion, and a controlled regime wasfound. Another remedy introduces the dimensionaland co-dimensional regularization to the Fermi surface,and develops a systematic framework to regulate thequantum fluctuations.
It is hoped that, thephysical cases are located in the regimes where the newdevelopments are applicable. Inspired by these new de-velopments in this exciting field, we turn our attention toa novel type of Fermi surface criticality and NFL. Com-pared to the great efforts in the literature, we are moreinclined to thinking about the new mechanism of NFLsin this work. q k F
FM-NFL metal Paramagnet (FL metal) Helical order (Goldstone-NFL)
QCP
T r ~q FIG. 2. (Color online.) The global phase diagram. The leftcorner is a helimagnet and a Goldstone-NFL where the NFLis induced by the gapless Goldstone boson. The right corneris a paramagnet and Fermi liquid metal. The central regionis the quantum critical regime where a novel NFL with infi-nite critical bosons on the boson surface is realized. “QCP”refers to quantum critical point. As the temperature rises tothe point where the thermal fluctuation submerges the bosonsurface, the system experiences a crossover to a FM criticality-like NFL behavior. The solid (dashed) line refers to the phasetransition (thermal crossover). ity. We supplement the expansion with a Dzyaloshinskii-Moriya (DM) interaction with a strength D , due to a lackof inversion symmetry in the system. We show that thisDM interaction fundamentally modifies all the physicalproperties in the boson and fermion sectors and leads toa rich phase diagram in Fig. 2.To tackle with the magnetic fluctuation in the bosonicsector, a saddle point solution of (cid:126)φ is required, on topof which the lowest order expansion counts for the fluc-tuations. As shown in Fig. 3, we first integrate out thefermions in the coupled system that generates the dy-namics for the bosons. We obtain an effective action forthe bosonic sector S B = 12 (cid:88) q ,iω l Π µν ( q , iω l ) φ µ ( q , iω l ) φ ν ( − q , − iω l )+ u (cid:90) dτ (cid:90) d r (cid:2) (cid:126)φ ( r , τ ) (cid:3) (3)where µ, ν, λ label the vector components of (cid:126)φ , and wehave converted the 3D lattice index to a continuous, realspace coordinate r . The polarization bubble in Fig. 3takes the formΠ µν ( q , iω l ) = f ( q, iω l ) δ µν − iD(cid:15) µνλ q λ ,f ( q, iω l ) = r + Jq + | ω l | Γ q , (4)where ω l = 2 πl/β ( l ∈ Z ) is the bosonic Matsubara fre-quency, and the | ω l | / Γ q term comes from the Landaudamping. In general, the function Γ q takes a form (cid:1) a (cid:2) (cid:1) b (cid:2) qk kk+qq qq+kk FIG. 3. (a) The fermion bubble induced boson dynamics. (b)The renormalized fermion propagator from the renormalizedboson correlator. The light and bold curly line represent thebare and renormalized boson correlator, respectively. Γ q = Γ q with q = | q | , that is identical to the one for aFM criticality at this stage. Critical boson surface. —The DM interaction compli-cates the low-energy theories by introducing the vectorindex into the bosonic sector. The dispersion of thebosonic modes are modified compared to the D = 0 case.Diagonalizing the bare quadratic boson part at ω l = 0, weobtain three branches of bosonic modes with dispersionsgiven by E n ( q ) = r + Jq + nDq, n = − , , +1. Thelowest branch E − ( q ) is of particular interest, whichreaches its minima on a spherical surface in the momen-tum space q = q ≡ D/ (2 J ). Approaching the critical-ity at r c = D / (4 J ), the lowest mode E − ( q ) becomesasymptotically gapless. The boson modes on the wholesurface at q = q become critical simultaneously. Thissurface at q = q is dubbed “critical boson surface”, andthe finite radius of the boson sphere, namely q (cid:54) = 0, isguaranteed by the presence of the DM interaction. We fo-cus on the low-energy regime where the surface of bosonmodes are condensed, and then Γ q ≈ Γ q with Γ being aconstant due to the finite density of state at the Fermilevel. In the limit T →
0, the critical fluctuations will begoverned by the boson surface, i.e. will be around theboson surface.
Self-consistent renormalization. —To analyze the novelcritical fluctuations at the critcality, we first implement a self-consistent renormalization to incorporate the effectof the interaction on the criticality. Around the criticalpoint, it is well-known that a naive perturbation withrespect to the interaction in Eq. (2) breaks down andhigher order terms exhibit divergence. A phenomenolog-ical technique, dubbed the self-consistent renormaliza-tion (SCR) theory, is used below. The spirit of SCRis to find the most appropriate quadratic action that en-codes the effective renormalized non-linear interactions.To search for the best action, one relies on Feynman’svariational approach to optimize the free energy. We con-sider a trial quadratic action with the following form˜ S B [˜ r ] = 12 (cid:88) q ,iω l (cid:104)(cid:0) ˜ r + Jq + | ω l | Γ q (cid:1) δ µν − iD(cid:15) µνλ q λ (cid:105) × φ µ ( q , iω l ) φ ν ( − q , − iω l ) , (5)where we replace r with a variational parameter ˜ r thatis to be determined. The boson correlator is given as ahermitian matrix M µν , (cid:104) φ µ ( q , iω l ) φ ν ( q (cid:48) , iω l (cid:48) ) (cid:105) ˜ S B (cid:39) M µν ( q , iω l ) δ q , − q (cid:48) δ iω l , − iω l (cid:48) (6)where (cid:104)· · · (cid:105) ˜ S B refers to the statistical average against ˜ S B .This correlation conceives the information of the criti-cal boson surface, and serves as a key ingredient for themechanism of the proposed NFL that can be detected bythe neutron scattering experiment. At the criticality, thepowder-averaged neutron scattering spectrum is given asTr M ( q , ω ) ∼ ( ω/ Γ q ) / (cid:2) J ( q − q ) + ( ω/ Γ q ) (cid:3) . Aroundthe critical boson surface | q | = q , the spectrum displaysa divergent behavior. In the real space, the boson sur-face momentum q provides a characteristic length scale1 /q , which endows the correlation function with a spa-tial modulation in all directions , and the correlator fromthe elastic neutron scattering is given by an envelop func-tion on top of the usual power law decaying in the long-distance limit as (cid:80) µ (cid:104) φ µ ( r ) φ µ (0) (cid:105) ˜ S B ∼ sin( q | r | ) / | r | .The variational free energy for the bosonic sector is F (˜ r ) ≡ ˜ F (˜ r ) + 1 β (cid:104)S B − ˜ S B (cid:105) ˜ S B = ˜ F (˜ r ) + 12 β ( r − ˜ r )Tr (cid:88) q ,iω l M ( q , iω l ) + u β V (cid:110)(cid:2) Tr (cid:88) q ,iω l M ( q , iω l ) (cid:3) + 2Tr (cid:2) (cid:88) q ,iω l M ( q , iω l ) (cid:3) (cid:111) , (7)where ˜ F (˜ r ) is the free energy corresponding to the trial quadratic action ˜ S B , and V is the system volume.The variational parameter ˜ r is determined from thesaddle point equation ∂ ˜ r F (˜ r ) = 0. After some cumber-some calculation that is detailed in the supplementarymaterial, the optimization procedure results in the fol-lowing self-consistent equation for the parameter ˜ r ,˜ r − r = ucβV (cid:88) n = ± , (cid:88) q ,iω l | ω l | / Γ q + E n ( q ) , (8) where E n ( q ) = E n ( q ) | r → ˜ r and c is a numerical constant. We further set δ ≡ ˜ r − r c and δ ≡ r − r c . Here δ mea-sures the distance from the quantum critical point δ c = 0and is related to the correlation length that is expectedto diverge at the criticality. The quantum fluctuationat finite temperatures is encoded in the function δ ( T ),which is shown to behave as δ ( T ) ∼ T α , α = 4 / α = 4 / α = 3 /
2, the current exponentis much smaller, indicating much stronger fluctuationsdue to the large density of the low-energy bosonic modesfrom the boson surface.
NFL from critical boson surface. —Unlike the cases forFM or AFM criticalities where the low-energy fluctua-tions are at discrete momentum points, the low-energyfluctuations near a finite boson surface strongly scattersthe itinerant electrons, reducing the lifetime of the elec-tron quasiparticles. We use the renormalized boson cor-relator and the Feymann diagram in Fig. 3 to calculatethe electron self-energy,Σ( k , i(cid:15) n ) (cid:39) g βV (cid:88) q ,iω l G ( k + q , i(cid:15) n + iω l )Tr[ M ( q , iω l )] , (10)where (cid:15) n ≡ (2 n + 1) π/β ( n ∈ Z ) is the fermion Matsub-ara frequency, and G ( k , i(cid:15) n ) = ( i(cid:15) n − ξ k ) − is a bareGreen’s function of the electrons with ξ k ≡ k / (2 m ) − (cid:15) F being the eigen-energy of electrons subtracted by theFermi energy (cid:15) F . At the criticality, it is the lowestbranch, E − ( q ), that is responsible for the critical fluctu-ation and divergent behaviors. One could then single outthis branch in the boson propagator and perform the cal-culation in a simpler manner. By performing an analyticcontinuation i(cid:15) n → ω + iη , we obtain the T -dependenceof the retarded self-energy Σ R ( k ,
0) in the static limit ω → k at the Fermi surface and | k | = k F ,Im[Σ R ( k , ∼ T δ − ∼ T − α . (11)The function in Eq. (12) is known as the inverse life-time function, which determines the electron lifetime onthe Fermi surface | k | = k F . The electron resistivity, pro-portional to the total lifetime of electrons according tothe Drude formula, shows a power-law temperature de-pendence ρ ( T ) ∼ τ − ∼ T / . (12)This peculiar power-law temperature dependence can becompared with the FM and AFM criticalities, wherethe SCR theory yields ρ ( T ) ∼ T / and ∼ T / , respec-tively. The much slower power-law decay of the resistiv-ity is due to the large scattering density of states from thecritical boson surface and indicates the NFL behaviors.
Crossover to FM-NFL at high- T —The NFL propertiesthat were discussed above arise from the strong fluctua-tions near the boson surface at low temperatures and arequoted as an infinite critical boson NFL. When the tem-perature is further increased from the criticality and islarge compared with the characteristic energy scale asso-ciated with the boson sphere radius q , i.e. T (cid:29) Γ q , thefluctuation is no longer dominated by the boson modesnear the critical surface and does not feel the curvatureof the boson surface. In fact, the structure of the criticalboson surface is no longer discernible at the high tem-peratures. The boson sphere can be regarded as a point object in the reciprocal space, resembling the case of theFM criticality. In this high temperature regime above thecriticality, the temperature dependence of the variationalparameter crossovers to scale as δ ( T ) ∼ T / , whichcoincides with the case of the NFL from the FM fluc-tuations in 3D. Thus, the system undergoes a crossoverbetween two distinct types of NFLs, and the crossovertemperature can be approximately set by the differenceof boson energies at the center and at the surface of theboson sphere, T c ∼ E − ( q = 0) − E − ( q ) ∼ O ( D J ) . (13) NFL from Goldstone mode. —When r < r c , the systemwould develop the magnetic order by spontaneously se-lecting the ordering wavevector from the degenerate bo-son surface. In the left corner of phase diagram in Fig. 2,a helical order with an ordering wavevector at q = q ˆ n is picked up where ˆ n is the propagating direction of thehelimagnet. The original model in Eq. (1) is invariantunder a combined rotation with respect to the real spaceand the internal space of the magnetic orders. The heli-magnet spontaneously breaks this continuous symmetryand thus generates gapless Goldstone modes. In the he-limagnetic phase, a small fluctuation above the helicalorder parameter couples to the itinerant electrons with ∼ g (cid:90) d r f † α ( r ) (cid:126)σ αβ f β ( r ) · δ(cid:126)φ ( r ) , (14)where δ(cid:126)φ ( r ) = φ [ − ϕ ( r ) sin q z, ϕ ( r ) cos q z, θ ( r )] with( θ, ϕ ) describing the polar and azimuthal phase fluctu-ations against the helical order (cid:104) (cid:126)φ ( r ) (cid:105) = φ [cos( q z )ˆ x +sin( q z )ˆ y ] with ˆ n = ˆ z . These phase fluctuations give riseto gapless Goldstone modes as promised. The Yukawacoupling, g , remains finite in the low-energy limit. Thisis due to the fact that the generator for the continuoussymmetry involves the orbital angular momentum andthus does not commute with the translation operator, i.e. the total momentum. Applying the general criteriain Ref. 4, we conclude this Goldstone mode converts thefermion sector into a NFL and dub it the “Goldstone-NFL” in Fig. 2. The electronic resistivity is shownto acquire a non-analytic correction ρ ∼ T / on top ofthe Fermi liquid contribution in the helimagnet. Discussion and outlook. —We have uncovered a noveltype of NFL in the 3D itinerant quantum magnets thatare not captured by the conventional patch and hotspottheories. The infinite many critical boson modes on a bo-son sphere connects all momentum points on the Fermisurface for the itinerant electrons, leading to striking con-sequences for both the local moments and the conductionelectrons. This work initiates the study of infinite criticalboson induced NFLs. Down this road, many further in-vestigation and generalization to other physical systemsare expected. The effect of external fields, such as pres-sure and magnetic field, is keenly anticipated due to itsrelevance in the experimental observation of MnSi.
An anomalous NFL transport behavior is observed in awidespread region of the phase diagram spanned by pres-sure and magnetic field. The role of a finite uniform mag-netic field is two-folded. Firstly, it leads to the processionof the spin order parameter, which competes with theLandau damping caused by the fermion. Secondly, thisexternal field introduces anisotropy for the quantum crit-ical points, which may effectively reduce the fluctuatingdimension of the boson sector. Intuitively, the 2D coun-terpart of the present problem can be readily consideredwhere a Fermi circle is coupled to a 1D critical boson con-tour. This critical boson contour can appear for instancein 2D Rashba-type of DM interaction coupled magnets orin 2D frustrated magnets.
Particular interest lies inthe situation where the radius of the boson ring is com-mensurate/incommensurate (C/IC) to the Fermi circle. For the commensurate case, only finite Fermi points areconnected by the boson contour. The IC-C relation likely bridges the infinite critical boson NFL with the conven-tional one described by the “hotspot” theory. More ex-otically, one can consider the fermion-boson-coupled sys-tem in “mixed dimensions”, namely, the dimension of theFermi surface is incompatible with critical boson modes.One intuitive example can be found in the 3D fractionalquantum Hall system. Soft gauge bosons in 3D bulk cancouple with the chiral Fermi level on the 2D surface, which triggers a possible NFL instability on the surface. Acknowledgments. —We thank Yonghao Gao, SungsikLee, Michael Hermele, Arun Paramekanti, and Leo Radz-ihovsky, for discussion. This work is supported by theMinistry of Science and Technology of China with GrantNo.2016YFA0301001,2016YFA0300500,2018YFE0103200,by Shanghai Municipal Science and Technology MajorProject with Grant No.2019SHZDZX04, and by theResearch Grants Council of Hong Kong with GeneralResearch Fund Grant No.17303819. L. D. Landau, E. M. Lifˇsic, E. M. Lifshitz, andL. Pitaevskii,
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SUPPLEMENTARY MATERIALSA. Variational free energy
Let us start from the following free energy, F [˜ r ] = ˜ F [˜ r ] + 1 β (cid:104)S B − ˜ S B (cid:105) ˜ S B . (15)where the complete boson action S B and the variational quadratic boson action ˜ S B are given in Eq.(3) and Eq.(5)respectively. The variational free energy ˜ F [˜ r ] is defined as,˜ F [˜ r ] ≡ − β ln Tr e − ˜ S B = − β ln (cid:90) D φ µ e − (cid:80) q ,iωl A µν ( q ,iω l ) φ µ ( − q , − iω l ) φ ν ( q ,iω l ) = 12 β (cid:88) q ,iω l ln (cid:2) Det A ( q , iω l ) (cid:3) , (16)where the Greek characters µ, ν, λ label the 3D vector space. The trace Tr over the vector space is converted to afunctional path integral. A ( q , iω l ) denotes a 3 × A ( q , iω l ) = f ( q , iω l ) δ µν − iD(cid:15) µνλ q λ f ( q , iω l ) =˜ r + Jq + | ω l | Γ q (17)with the determinant taking a form Det A = f ( f − Dq )( f + Dq ) . (18)The second part in Eq. (15) is given by (cid:104)S B − ˜ S B (cid:105) ˜ S B = 12 ( r − ˜ r ) (cid:88) q ,iω l (cid:104) φ µ ( − q , − iω l ) φ µ ( q , iω l ) (cid:105) ˜ S B + (cid:104)S intB (cid:105) ˜ S B = 12 ( r − ˜ r ) (cid:88) q ,iω l Tr (cid:2) M ( q , iω l ) (cid:3) + (cid:104)S intB (cid:105) ˜ S B , (19)where S intB stands for the interacting term in the action S B . The matrix M ( q , iω l ) is the inverse of quadratic bosonHamiltonian A , which defines a boson propagator M ( q , iω l ) = 1Det A f − D q x iDq z f − D q x q y − iDq y f − D q x q z − iDq z f − D q x q y f − D q y iDq x f − D q y q z iDq y f − D q x q z − iDq x f − D q y q z f − D q z . (20)The interacting part in Eq. (19) is calculated by contracting pairs of real boson fields according to the Wick expansion, (cid:104)S intB (cid:105) ˜ S B = u (cid:90) d r dτ (cid:104) φ µ ( r , τ ) φ ν ( r , τ ) (cid:105) ˜ S B = u (cid:90) d r dτ (cid:110)(cid:2) (cid:104) φ µ ( r , τ ) φ µ ( r , τ ) (cid:105) ˜ S B (cid:3) + 2 (cid:104) φ µ ( r , τ ) φ ν ( r , τ ) (cid:105) ˜ S B (cid:104) φ µ ( r , τ ) φ ν ( r , τ ) (cid:105) ˜ S B (cid:111) = u βV (cid:110)(cid:2) (cid:88) q ,iω l Tr M ( q , iω l ) (cid:3) + 2 (cid:88) µν (cid:88) q ,iω l M µν ( q , iω l ) (cid:88) q (cid:48) ,iω (cid:48) l M µν ( q (cid:48) , iω (cid:48) l ) (cid:111) = 5 u
12 1 βV (cid:2) (cid:88) q ,iω l Tr M ( q , iω l ) (cid:3) . (21)If we collect the third line of Eq. (21) with Eq. (19), we will arrive with the expression of the variational free energy inthe main text. In the second last equality, we note that the off-diagonal terms M µν , µ (cid:54) = ν depend on the componentsof the momentum q x , q y , q z and frequency ω l linearly. Upon summation over the momentum variables q and q (cid:48) , theseterm vanish leaving us with the diagonal terms (cid:88) µ (cid:88) q ,iω l M µµ ( q , iω l ) (cid:88) q (cid:48) ,iω (cid:48) l M µµ ( q (cid:48) , iω (cid:48) l )= (cid:88) q ,iω l (cid:88) q (cid:48) ,iω (cid:48) l f ( f − D q ) 1 f (cid:48) ( f (cid:48) − D q ) × (cid:110) ( f − D q x )( f (cid:48) − D q (cid:48) x ) + ( y ) + ( z ) (cid:111) = (cid:88) q,iω l (cid:88) q (cid:48) ,iω (cid:48) l f ( f − D q ) 1 f (cid:48) ( f (cid:48) − D q ) × (cid:110) ( f − D q )( f (cid:48) − D q (cid:48) ) + ( y ) + ( z ) (cid:111) = 13 (cid:104) (cid:88) q ,iω l f − D q f ( f − D q ) (cid:105) = 13 (cid:2) (cid:88) q ,iω l Tr M ( q , iω l ) (cid:3) , (22)In the second line, the 2nd and 3rd terms in the bracket are denoted by ( y ) and ( z ) which stands for the sameexpression as the 1st term with q x replaced by q y and q z respectively. Here, we use the abbreviation f (cid:48) ≡ f ( q (cid:48) , iω (cid:48) l ).Thus, we have demonstrated that the second term in Eq. (21) can be written in the same form as the first term.Collecting terms in Eqs. (16)-(21), we arrive at an simplified expression for the variational free energy, F [˜ r ] = 12 β (cid:88) q ,iω l ln (cid:2) Det A ( q , iω l ) (cid:3) + 12 β ( r − ˜ r ) (cid:88) q ,iω l Tr (cid:2) M ( q , iω l ) (cid:3) + 1 β V u (cid:2) (cid:88) q ,iω l Tr M ( q , iω l ) (cid:3) . (23)Next, we derive the saddle point equation ∂ ˜ r F = 0 fromEq. (23) which yields˜ r = r + cuβV (cid:88) q ,iω l Tr M ( q , iω l ) (24)where c = 5 /
3. Explicitly, the self-consistent equationhas an expression,˜ r = r + cuβV (cid:88) q ,ω l f − D q f ( f + Dq )( f − Dq ) (25)= r + cuβV (cid:88) q,iω l f + 1 f + Dq + 1 f − Dq , (26)After decomposing the integrand into three terms, it isreadily seen that they come from three eigenvalues of theboson Hamiltonian.The third term ( f − Dq ) − has a divergence due tothe presence of the critical boson surface (CBS) | q | = q , which is considered to be important in the followingcalculation. The momentum q in a shell around the CBS(Fig. 1), can be uniquely represented by, q = q + δ q ⊥ , (27)where the momentum q on CBS fixes a longitudinal di-rection ˆ q , and δ q ⊥ is a small momentum along this di-rection. The momentum shell width q c imposes a bound | δ q ⊥ | = δq ⊥ ≤ q c . The lowest band of the boson mode E − ( q ) that conceiving the critical surface is abbreviatedas E q and approximated due to | δ q ⊥ | (cid:28) q , E q = ˜ r − Dq + Jq (cid:39) δ + J ( δq ⊥ ) . (28) Without loss of generality, we take J = 1 henceforthwhen evaluating the temperature dependence of variousquantities. Here, we set δ ≡ ˜ r − r c and δ ≡ r − r c with r c = ( D/ . Thus, δ measures the distance from thequantum critical point δ c = 0.Accordingly, the boson propagator reduces to a formaround the CBS, M − ( q , iω l ) (cid:39) f − Dq (cid:39) δ + ( δq ⊥ ) + | ω l | Γ q . (29)Finally, by rewriting Eq. (26) we arrive at the expressionin Eq. (8), δ = δ + cuβV (cid:88) q,iω l f − Dq . (30)
B. Calculation of δ ( T ) at low temperatures We are in a position to calculate the temperature de-pendence of δ ( T ) based on the saddle point equation inEq. (30). The contribution from the CBS modes is sin-gled out, δ (cid:39) δ + cuβV (cid:88) q,iω l f − Dq = δ + 2 cuβV (cid:88) q,iω l > E q + ω l / Γ q . (31) T. Moriya ’s variational method has been presented ina simpler language by N. Nagaosa , which we shall usein the following derivation. Instead of performing thesummation over Matsubara frequency directly, we invokea dispersive representation by using the Kramer-Kronigrelation, 1 f ( q , iω l ) − Dq = 1 π (cid:90) + ∞−∞ d(cid:15) ˜ f ( (cid:15), q ) (cid:15) − iω l , (32)where the Matsubara frequency is always positive. Thefunction ˜ f ( (cid:15), q ) is defined as an imaginary part of a re-tarded function˜ f ( (cid:15), q ) ≡ Im 1 f ( q , iω l → (cid:15) + iη ) − Dq = (cid:15)/ Γ q E q + ( (cid:15)/ Γ q ) . (33)Owing to the fact that ˜ f ( (cid:15), q ) = − ˜ f ( − (cid:15), q ), we rewritethe integral as the following and carry out the ω l -summation1 β (cid:88) ω l (cid:90) + ∞−∞ d(cid:15) ˜ f ( (cid:15), q ) 1 (cid:15) − iω l = 12 (cid:90) + ∞−∞ d(cid:15) ˜ f ( (cid:15), q ) 1 β (cid:88) ω l (cid:110) (cid:15) − iω l − − (cid:15) − iω l (cid:111) = 12 (cid:90) + ∞−∞ d(cid:15) ˜ f ( (cid:15), q ) coth( (cid:15)/ T ) . (34)With the results from Eqs. (33) and (34), we arrive at anexpression δ = δ + cu (cid:90) d q (2 π ) (cid:90) Γ q d(cid:15) π coth (cid:15) T (cid:15)/ Γ q E q + ( (cid:15)/ Γ q ) . (35)
1, thatsets up an upper limit for the frequency integral. Underthe low temperature approximation, the function in thefirst term admits an expansioncoth( (cid:15)/ T ) − f B ( (cid:15) ) (cid:39) T /(cid:15), (38)where f B ( x ) is the boson distribution function. The first term is denoted as δ ( T ) and is calculated as, δ (cid:39) cu (cid:90) d q (2 π ) T Γ q (cid:90) T d(cid:15)π δ + r c − Dq + q ) + ( (cid:15)/ Γ q ) = cu (cid:90) d q (2 π ) T Γ q | δ + ( q − q ) | tan − T / Γ q | δ + ( q − q ) | = cu (cid:90) Σ d q (2 π ) (cid:90) q c − q c d ( δ q ⊥ )(2 π ) 2 T Γ q × δ + ( δq ⊥ ) tan − T / Γ q δ + ( δq ⊥ ) ∼ cuT q (cid:90) q c d ( δq ⊥ ) 1 δ + ( δq ⊥ ) tan − T / Γ q δ + ( δq ⊥ ) , (39)where we have invoked the relation r c = q = ( D/ .As illustrated in Fig. 4(a), the q -integral in the momen-tum shell is decomposed into a q -integral on the bosonsphere surface and a δ q ⊥ -integral along the radicaldirection which is cutoff at at a bound ± q c /
2. We notethat the integrand in Eq. (41) only depends on δq ⊥ ,therefore, the integration over q are carried out trivially.The T -dependence in δ ( T ) can be extracted by per-forming a transformation x = δq ⊥ / ( T / Γ q ) , δ ∼ uT (cid:90) x c dx x + x tan − x + x ∼ uT (cid:90) x c x dx x tan − x , (40)where we have approximated the integral by introduc- ing a lower bound x = (cid:112) δ Γ q /T , and the upper boundis rescaled to be x c = q c / √ T . In low temperature limit T →
0, the upper bound x c ∼ q c /T can be taken to in-finity although q c is relatively small. Let us assume asa prior that δ ( T ) ∼ T α with α <
1. Similarly, the lowerbound diverges in the low- T limit with a weaker power.At large x , the integrand in Eq. (40) is approximatedas x − , the integrations can be performed to x c → + ∞ yielding a result δ ( T ) ∼ c T + c T [ δ ( T )] / ∼ T [ δ ( T )] / , (41)where we have used the fact that α < δ in Eq. (37) has an implicit tem-perature dependence through δ ( T ). This fact can be seenfrom, δ = cu π (cid:90) d q (2 π ) Γ q (cid:90) ds (cid:110) s ( δ + r c − Dq + q ) + s − s ( r c − Dq + q ) + s (cid:111) ∼ u (cid:90) d q ln (cid:12)(cid:12)(cid:12) q − Dq + q δ + q − Dq + q (cid:12)(cid:12)(cid:12) = u (cid:90) q c / d ( δq ⊥ ) ln ( δq ⊥ ) δ + ( δq ⊥ ) . (42)To extract the δ dependence, we perform a transforma-tion x = δq/δ . The integrals are truncated at a upperbound x c ∼ δ − / , and the integrals are evaluated ap-proximately at large x , ∼ uδ (cid:90) x c dx ln x x ∼ uδ x − c ∼ uδ. (43)Conclusively, the second term takes a form δ ∼ uδ. (44)Combining the results from Eq. (41) and Eq. (44), wededuce that δ ( T ) = T α , α = 45 . (45) C. δ ( T ) at high temperatures above the criticality For high temperatures T (cid:29) Γ q , the expansion inEq. (38) is no longer legitimate. The dominant contri-bution comes from the frequency regime (cid:15) ∼ Γ q (cid:29) Γ q .Namely, we focus on the large momentum limit q (cid:29) q , and carry out the routine calculation starting fromEq. (37).The first term δ ( T ) is calculated as, δ ∼ u (cid:90) + ∞ dqq Γ q (cid:90) + ∞ d(cid:15)f B ( (cid:15) Γ q ) (cid:15) ( δ + r c − Dq + q ) + (cid:15) = u Γ (cid:90) + ∞ dqq F (cid:2) | q ( δ + r c − Dq + q ) | Γ / (2 πT ) (cid:3) (cid:39) u Γ (cid:90) + ∞ dqq F (cid:2) q Γ / (2 πT ) (cid:3) ∼ u Γ − / T / (cid:90) x c dxx F ( x ) (46)0where we have made a change of variable x = q/T / , and used the following integral with the parameters given by β = δ + r c − Dq + q and µ = Γ q /T . We have (cid:90) + ∞ dx e µx − xx + β = F ( | βµ | / π ) F ( z ) ≡ (cid:104) ln z − z − ψ ( z ) (cid:105) , (47) ψ is the digamma function that has an asymptotic expansion ψ ( z ) (cid:39) ln z − z + + ∞ (cid:88) n =1 ζ (1 − n ) z n (48)The function F ( x ) ∼ x − in the limit x → + ∞ . The integration can be performed to infinity, which leads to thetemperature dependence δ ∼ T / . (49)The second part δ is evaluated in a way, δ = cuπ (cid:90) d q (2 π ) Γ q (cid:90) + ∞ ds (cid:110) s ( δ + r c − Dq + q ) + s − s ( r c − Dq + q ) + s (cid:111) ∼ u (cid:90) d q (2 π ) q ln r c − Dq + q δ + r c − Dq + q ∼ u (cid:90) + ∞ dqq ln q δ (cid:48) + q = uδ (cid:90) x c dxx ln x x ∼ uδ x c ∼ uδ, (50)where we have made a transformation of variable x = q/δ . The integration can not be performed to infinity, therefore,a cutoff x c ∼ δ − / is imposed.With results from Eq. (49) and Eq. (50), we concludethat in high temperature regime, the temperature depen-dence δ ( T ) takes a form r ( T ) − r c ≡ δ ( T ) ∼ T / , (51)which leads to a temperature dependence of the resistiv-ity, ρ ( T ) ∼ T / . This result is the same as the case ofitinerant FM in 3D. D. Temperature dependence of the imaginary partof self-energy
The self-energy of the fermion is defined as,Σ( k , i(cid:15) n ) = g βV (cid:88) q ,iω l G ( k + q , i(cid:15) n + iω l ) D ( q , iω l ) , (52)where (cid:15) n and ω l are the fermion and boson Matsubarafrequency, respectively. The bare Green function and theboson propagator are given by G ( k, i(cid:15) n ) = 1 i(cid:15) n − ξ k , (53) D ( q, iω l ) (cid:39) E q + | ω l | / Γ q , (54) E q ≡ δ + ( q − q ) , (55)where the boson propagator is approximated with theeigen-mode that comprises the critical boson surface. Byinvoking the Kramer-Kronig relation for both functionsas in Eq. (32), we derive a dispersive representation ofthe self-energyΣ( k , i(cid:15) n ) = g V (cid:88) q β (cid:88) ω l > iπ (cid:90) + ∞−∞ dω π (cid:90) + ∞−∞ dω (cid:48) ω − iω l ω (cid:48) − iω l (cid:110) i(cid:15) n + ω − ξ k + q + 1 i(cid:15) n − ω − ξ k + q (cid:111) ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) . (56)1The summation over the frequency iω l is carried out,1 β (cid:88) ω l > ω − iω l ω (cid:48) − iω l = − β (cid:88) ω l < ω − iω l ω (cid:48) − iω l − f B ( ω ) 1 ω − ω (cid:48) + f B ( ω (cid:48) ) 1 ω − ω (cid:48) . (57)Then, the integrals over ω are evaluated for the last two terms in Eq. (57). The first term in Eq. (57) is calculated as, f B ( ω (cid:48) ) (cid:90) + ∞−∞ dωiπ ω − ω (cid:48) (cid:110) i(cid:15) n + ω − ξ k + q + 1 i(cid:15) n − ω − ξ k + q (cid:111) = f B ( ω (cid:48) ) (cid:110) i(cid:15) n − ω (cid:48) − ξ k + q − i(cid:15) n + ω (cid:48) − ξ k + q (cid:111) . (58)The second term in Eq. (57) is calculated as, (cid:90) + ∞−∞ dω − iπ f B ( ω ) 1 ω − ω (cid:48) (cid:110) i(cid:15) n + ω − ξ k + q + 1 i(cid:15) n − ω − ξ k + q (cid:111) = − f B ( ω (cid:48) ) (cid:110) i(cid:15) n + ω (cid:48) − ξ k + q + 1 i(cid:15) n − ω (cid:48) − ξ k + q (cid:111) − f F ( ξ k + q ) 1 i(cid:15) n + ω (cid:48) − ξ k + q − β (cid:88) ω l > iω l − ω (cid:48) (cid:110) iω l + i(cid:15) n − ξ k + q + 1 − iω l + i(cid:15) n − ξ k + q (cid:111) , (59)where f F ( x ) is the fermion distribution function, and we have used the relation f B ( ξ k + q − i(cid:15) n ) = − f F ( ξ k + q ) . (60)Collecting the terms from Eqs. (58) and (59), and restoring the ω (cid:48) integral, we have ∼ − π (cid:90) + ∞−∞ dω (cid:48) (cid:2) f B ( ω (cid:48) ) + f F ( ξ k + q ) (cid:3) i(cid:15) n + ω (cid:48) − ξ k + q ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) − β (cid:88) ω l > π (cid:90) + ∞−∞ dω (cid:48) iω l − ω (cid:48) (cid:110) ω (cid:48) + i(cid:15) n − ξ k + q + 1 − ω (cid:48) + i(cid:15) n − ξ k + q (cid:111) ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) . (61)Combining results from Eqs. (57) and (61), we arrive at1 β (cid:88) ω l > (cid:110) iω l + i(cid:15) n − ξ k + q + 1 − iω l + i(cid:15) n − ξ k + q (cid:111) E q + ω l / Γ q = 2 π (cid:90) + ∞−∞ dω (cid:48) (cid:2) f B ( ω (cid:48) ) + f F ( ξ k + q ) (cid:3) i(cid:15) n + ω (cid:48) − ξ k + q ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) + 1 β (cid:88) ω l < iπ (cid:90) + ∞−∞ dω π (cid:90) + ∞−∞ dω (cid:48) ω − iω l ω (cid:48) − iω l (cid:110) i(cid:15) n + ω − ξ k + q + 1 i(cid:15) n − ω − ξ k + q (cid:111) ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) , (62)where the second term on the r.h.s turns out to to beexact same as the l.h.s. by applying a series of transfor-mations iω l → − iω l ,ω → − ω, ω (cid:48) → − ω (cid:48) . (63)Finally, we derive the expression of the self-energy function given in Eq. (56)Σ( k , i(cid:15) n ) = g (cid:90) d q (2 π ) (cid:90) + ∞−∞ dω (cid:48) π (cid:2) f B ( ω (cid:48) ) + f F ( ξ k + q ) (cid:3) × i(cid:15) n + ω (cid:48) − ξ k + q ω (cid:48) / Γ q E q + ( ω (cid:48) / Γ q ) . (64)Next, we perform an analytic continuation and calculate2the imaginary part of the retarded self-energyImΣ R ( k , ω ) = g V (cid:88) q (cid:90) +Γ q − Γ q d(cid:15)π (cid:2) f B ( (cid:15) ) + f F ( (cid:15) + ω ) (cid:3) × δ ( ω + (cid:15) − ξ k + q ) (cid:15)/ Γ q E q + ( (cid:15)/ Γ q ) . (65)For the present, we are interested in the static limit ω = 0at low temperatures T (cid:28) − ImΣ R ( k ,
0) = g V (cid:88) q (cid:90) + T − T d(cid:15)π δ ( (cid:15) − ξ k + q ) T / Γ q E q + ( (cid:15)/ Γ q ) , (66)where the temperature sets a range for the (cid:15) -integral, i.e. | (cid:15) | (cid:28) T , that is transferred to the fermionic dispersionfunction by imposing the delta function. The function isapproximated as, ξ k + q = ( k + q + δ q ⊥ ) m − (cid:15) F (cid:39) ( k + q ) · δ q ⊥ m ∼ v F δq cos θ (cid:28) T, (67)where we have taken that the scattered Fermi quasi-particle momentum k + q is still close to the Fermisurface, i.e. | k + q | = k F . The polar variable θ hereis a relative angel suspended by k + q and δ q ⊥ . Theschematic illustration of the scattering process can befound in Fig. 4(b). The relation in Eq. (67) gives rise toan upper bound for the δq ⊥ -integral, which plays an im-portant role in evaluating the temperature dependenceof the resistivity q c = T /v F .Adopting the upper bound for the δq ⊥ -integral, wehave − ImΣ R ( k ,
0) = g πV (cid:88) q T / Γ q E q + ( ξ k + q / Γ q ) , (68)where the dominant contribution in the integral comesfrom the region δ + 12 ( δq ⊥ ) ∼ ξ k + q Γ q ∼ v F Γ q δq ⊥ , (69)therefore, we can neglect the term with O ( δq ⊥ ) . And,we first take an average over the angle θ , − ImΣ R ( k F , ≡ (cid:10) − ImΣ R ( k , (cid:11) θ ∼ T (cid:90) q c − q c d ( δq ⊥ ) (cid:90) +1 − d cos θ δ + ( δq ⊥ cos θ ) ∼ Tδ (cid:90) q c − q c d ( δq ⊥ ) 1 δq ⊥ tan − δq ⊥ δ , (70)where we have neglected the unimportant factor v F / Γ q .To further carry out the δ q ⊥ -integral, we make a changeof variable x = δq ⊥ /δ and find that the upper bound for the new variable x c = q c /δ ∼ T − α is small in the low- T regime. − ImΣ R ( k F , ∼ Tδ (cid:90) x c − x c dx x tan − ( x ) ∼ Tδ x c ∼ (cid:16) Tδ (cid:17) ∼ T − α . (71)We have substituted the expression of δ in Eq. (45), andend up with a sub-linear T -dependence for the imaginarypart of self-energy in the low temperature limit.The imaginary part of self-energy in Eq. (71) corre-sponds to the inverse lifetime function of the electron onthe Fermi surface | k | = k F . According to the Drudeformula, the electronic resistivity is proportional to theinverse mean free time, which exhibits a temperature de-pendence ρ ( T ) ∼ τ ∼ T − α = T / . (72) E. Boson correlation function
In this section, we carry out the calculation of the bo-son correlation function. We start from the variationalaction ˜ S B [˜ r ] and evaluate the equal-time spin correlationfunction, C ( r ) ≡ (cid:104) φ z ( r , φ z (0 , (cid:105) ˜ S B = 1( βV ) (cid:88) q , q (cid:48) (cid:88) iω l ,iω n e i q · r (cid:104) φ z ( q , iω l ) φ z ( q (cid:48) , iω n ) (cid:105) ˜ S B = 1 βV (cid:88) q ,iω l e i q · r M − zz ( q , iω l ) (cid:39) βV (cid:88) q ,iω l e i q · r f − Dq , (73)where we have used the expression in Eq.(20) and ap-plied the powder average q z → q . We single out theterm that conceives the critical boson surface in the lastequality. The above expression admits a dispersive rep-resentation as dictated in Eq. (32) and thereafter, C ( r ) = (cid:90) d q (2 π ) e i q · r (cid:90) Γ q d(cid:15)π coth( (cid:15)/ T ) (cid:15)/ Γ q E q + ( (cid:15)/ Γ q ) . (74)From this expression, we extract the spin correlationfunction in the momentum space F ( q , ω ) ∼ ω/ Γ q E q + ( ω/ Γ q ) . (75)At zero temperature, the correlation function shows adivergent behavior around q = q in the static limit ω =0 F ( q , ω → ∼ δ [( q − q ) ] , (76)3which is nothing but a Dirac delta function. At zerotemperature, we evaluate the correlation function in real-space analytically C ( r ) (cid:39) (cid:90) Σ d q (cid:90) q c − q c d ( δ q ⊥ ) e i ( q + δ q ⊥ ) · r (cid:90) d(cid:15)π (cid:15) ( δq ⊥ ) + (cid:15) (cid:39) (cid:90) q c − q c d ( δq ⊥ ) (cid:90) +1 − d (cos θ ) e i ( q + δq ⊥ ) r cos θ ln ( δq ⊥ ) δq ⊥ ) = (cid:90) q c − q c d ( δq ⊥ ) sin[( q + δq ⊥ ) r ]2( q + δq ⊥ ) r ln ( δq ⊥ ) δq ⊥ ) = sin( q r ) rq (cid:90) q c d ( δq ⊥ ) cos( δq ⊥ r ) ln ( δq ⊥ ) δq ⊥ ) . (77) We are interested in the long-distance limit r → + ∞ .By making a change of variable x = δq ⊥ r , and takingthe cutoff x c = q c r to infinity, we arrive at C ( r ) ∼ sin( q r ) r (cid:90) x c dx cos x ln x x + r ∼ sin( q r ) r , (78)where the x -integral is (cid:82) + ∞ dx cos x ln[ x / ( x + r )] = (cid:2) − e − ( − / r + + e ( − / r (cid:3) π and we have retainedthe leading order contribution in the long-distance limit r → + ∞∞