Universal renormalization group flow toward perfect Fermi-surface nesting driven by enhanced electron-electron correlations in monolayer vanadium diselenide
Iksu Jang, Ganbat Duvjir, Byoung Ki Choi, Jungdae Kim, Young Jun Chang, Ki-Seok Kim
UUniversal renormalization group flow toward perfect Fermi-surface nesting driven byenhanced electron-electron correlations in monolayer vanadium diselenide
Iksu Jang , Ganbat Duvjir , Byoung Ki Choi , Jungdae Kim , Young Jun Chang , and Ki-Seok Kim Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea Department of Physics, BRL, and EHSRC, University of Ulsan, Ulsan 44610, Korea Department of Physics, University of Seoul, Seoul 02504, Korea (Dated: October 23, 2018)Reducing thickness of three dimensional samples on appropriate substrates is a promising way tocontrol electron-electron interactions, responsible for so called electronic reconstruction phenomena.Although the electronic reconstruction has been investigated both extensively and intensively inoxide heterostructure interfaces, this paradigm is not well established in the van der Waals hetero-interface system, regarded to be important for device applications. In the present study we examinenature of a charge ordering transition in monolayer vanadium diselenide (
V Se ), which would bedistinguished from that of V Se bulk samples, driven by more enhanced electron-electron corre-lations. We recall that V Se bulk samples show a charge density wave (CDW) transition around T CDW ∼ K , expected to result from Fermi surface nesting properties, where the low tempera-ture CDW state coexists with itinerant electrons of residual Fermi surfaces. Recently, angle resolvedphotoemission spectroscopy measurements [Nano Lett. , 5432 (2018)] uncovered that the Fermisurface nesting becomes perfect, where the dynamics of hot electrons is dispersionless along theorthogonal direction of the nesting wave-vector. In addition, scanning tunneling microscopy mea-surements [Nano Lett. , 5432 (2018)] confirmed that the resulting CDW state shows essentiallythe same modulation pattern as the three dimensional system of V Se . Here, we perform the renor-malization group analysis based on an effective field theory in terms of critical CDW fluctuationsand hot electrons of imperfect Fermi-surface nesting. As a result, we reveal that the imperfectnesting universally flows into perfect nesting in two dimensions, where the Fermi velocity along theorthogonal direction of the nesting vector vanishes generically. We argue that this electronic recon-struction is responsible for the observation [Nano Lett. , 5432 (2018)] that the CDW transitiontemperature is much more enhanced to be around T CDW ∼ K than that of the bulk sample. I. INTRODUCTION
Strongly correlated electrons had been expected to re-alize in transition metal dichalcogenides (TMDCs), in-volved with the quasi two dimensional lattice structure[1]. Actually, several compounds of the TMDC fam-ily have shown physics of strong correlations, for exam-ple, local-moment signatures in Ir-dichalcogenides [2] andMott insulating physics in transition metal sulfides [3], re-garded to be emergent phenomena at low temperatures.However, it turns out to be rather challenging to realizestrong correlations of electrons in the TMDC family. Na-ture of phase transitions seems to be determined by theband structure essentially, i.e., within the weak-couplingapproach. The quasi two dimensional nature of the lat-tice structure does not cause sufficient anisotropy in theelectronic structure except for several cases mentionedabove.Recent measurements based on angle resolved pho-toemission spectroscopy (ARPES) and scanning tunnel-ing microscopy (STM) [4] have claimed that reducingthickness of three dimensional or quasi two dimensionalTMDCs on appropriate substrates can give rise to dras-tic enhancement of correlation effects, responsible fornovel nature of phase transitions. Such experimentsfound two types of charge ordering transitions drivenby enhanced electron-electron and electron-phonon in-teractions in monolayer vanadium diselenide (
V Se ) ongraphene substrates. It is well established that V Se three dimensional bulk samples show a charge densitywave (CDW) transition around T c ∼ K , originatingfrom Fermi surface nesting properties of “hot” electrons[5–7]. The low temperature CDW state coexists with“cold” electrons of residual Fermi surfaces in this bulksystem, keeping their metallicity. In comparison withthis three dimensional case, ARPES measurements un-covered that the Fermi surface nesting becomes perfectin the monolayer limit, that is, the dynamics of such hotelectrons is dispersionless along the orthogonal directionof the nesting wave-vector [4]. This perfect Fermi-surfacenesting property has been speculated to cause noticeableenhancement of the CDW transition temperature in twodimensions, even above the room temperature, implyingthe dynamics of strongly correlated electrons beyond thedynamics of three dimensional hot electrons. More inter-estingly, the ARPES experiment revealed that residualFermi surfaces around cold zones disappear at T c ∼ K , where an insulating phase is realized, never observedin the bulk system [4]. STM measurements showed thatthis metal-insulator transition is driven by lattice dis-tortions along a particular one dimensional direction [4].These experimental results are summarized in Fig. 1.Nature of phase transitions in two dimensions turnsout to differ from that in three dimensions, where theeffective theory referred to as Hertz-Moriya-Millis theory[8–11], regarded to be a mean-field theory, does not func-tion. Even if electrons are weakly correlated at high tem-peratures, they become strongly correlated in the vicin- a r X i v : . [ c ond - m a t . s t r- e l ] O c t T T CDW T MIT ⃗ Q Hot spotCold spotInsulating ( ∼ 350 K )( ∼ 135 K ) d d T CDW ( ∼ 105 K ) ⃗ Q ≈ FIG. 1: Schematic phase diagram of monolayer
V Se and comparison with that of three-dimensional bulk V Se . Reducing thickness of V Se from threedimensions to two dimensions, electron-electroncorrelations are enhanced to cause strongrenormalization of the Fermi velocity. As a result, theweakly nested Fermi surface in three dimensions evolvesinto perfect Fermi surface nesting in two dimensions,observed in recent ARPES measurements [4]. Anotherphase transition has been observed in monolayer V Se [4], identified with a metal-insulator one. In this studywe focus on the high-temperature CDW transition.ity of two dimensional phase transitions involved withFermi-surface instabilities [12]. This implies that theband structure itself can be renormalized rather drasti-cally beyond the mean-field theoretical framework in themonolayered system.In the present study we investigate how two dimen-sionality in dynamics of hot electrons affects the natureof the CDW transition in monolayer V Se . We con-struct an effective field theory in terms of hot electronsand critical CDW fluctuations. Recalling that the STMexperiment confirmed that the CDW state shows essen-tially the same modulation pattern as the three dimen-sional system of V Se [4], we assume imperfect Fermisurface nesting for hot electrons, where the nesting vec-tor is given by the “three dimensional” (quasi two dimen-sional) CDW ordering structure. Based on this effectivefield theory, we perform the renormalization group anal-ysis in the scheme of a recently developed dimensionalregularization for a Fermi-surface problem [13]. Ourrenormalization group analysis confirms that imperfectnesting universally flows into perfect nesting in two di-mensions, where the Fermi velocity along the orthogonaldirection of the nesting vector vanishes generically. Weargue that this electronic reconstruction is responsiblefor the observation that the much higher CDW transi-tion temperature T d,CDW ∼ K in two dimensionalsample compared to the CDW transition temperature ofthree dimensional sample T d,CDW ∼ K .Before going further, we would like to speculate on therole of disorder in the high-temperature CDW and the low-temperature metal-insulator transitions of monolayer V Se [4]. It is well accepted that the diffusive dynamicsof electrons gives rise to enhancement of electron-electroncorrelations, resulting from reinforcement of the interac-tion vertex and referred to as the Altshuer-Aronov cor-rection [14, 15]. One may expect that electron-phononinteraction vertices would be also amplified, responsiblefor both the drastic enhancement of the CDW criticaltemperature and the appearance of the metal-insulatortransition in monolayer V Se . However, we point outthat the CDW state shows essentially the same modula-tion pattern as the three dimensional system of V Se ,revealed by the STM measurement [4], which impliesthat the CDW ordering results from the mechanism ofFermi surface nesting. In addition, it turns out that theCDW gap follows the Bardeen-Cooper-Schrieffer (BCS)type description well. These two experimental resultssuggest that the monolayer V Se of Ref. [4] does notlie in the strong disorder regime. Based on this discus-sion, we focus on the scenario of two dimensional CDWcriticality in the present study. II. EFFECTIVE FIELD THEORY
To construct an effective field theory, we recall theFermi surface map of monolayer
V Se , recently clari-fied in ARPES measurements [4]. Compared with theelectronic structure of three-dimensional bulk V Se [7],the Fermi-surface structure of monolayer V Se shows itsqualitatively distinguished feature in the respect that thestrong k z dispersion in three dimensions disappears tobecome almost dispersionless in two dimensions and theFermi-surface nesting property is more enhanced enoughto be called “perfect” in monolayer V Se when measuredat 180 K [4]. This ARPES experiment suggests a sim-plified Fermi-surface model for monolayer V Se , whichshows a hexagonal Brillouin zone with six cigar-shapedelectron pockets centered at the M points. See Fig. 2for the two-dimensional Fermi-surface structure of mono-layer V Se .An important point in this simplified Fermi-surfacemodel is that the Fermi-surface nesting is assumed tobe weakly realized, which may sound to be contradictorywith the emergence of perfect Fermi surface nesting intwo dimensions. Moreover, this weak Fermi-surface nest-ing property has not been verified in recent ARPES mea-surements for monolayer V Se [4]. Based on the ARPESmeasurement and conventional BCS fitting for the CDWgap, the critical temperature for the CDW ordering isestimated (extrapolated) to be T d,CDW ∼ K formonolayer V Se , which lies above the measurement tem-perature [4]. In other words, the electronic structureonly below the critical temperature of the CDW metallicphase has been verified. Even if there exists a CDW gapin the Fermi surface structure, one can trace the nest-ing property experimentally, suggesting the emergenceof perfect Fermi-surface nesting in monolayer V Se . Inthis study we assume a general Fermi surface structure,expected to appear much above the critical temperature T d,CDW ∼ K for monolayer V Se . Starting fromthis high-temperature Fermi-surface structure, we showthe emergence of perfect Fermi surface nesting at lowtemperatures in two dimensions while the evolution ofthe Fermi surface structure with respect to temperaturedoes not occur in three dimensions. xy −
1+ ¯1+¯1 − −
2+ ¯2 − ¯2+3 −
3+ ¯3 − ¯3+ Q Q Q Q Q Q b b FIG. 2: Schematic diagram for the Fermi surfacestructure of monolayer 1
T V Se . Blue colored regionsshow electron pockets, and blue and red dots refer to”hot” spots, where b and b are two reciprocal latticevectors. Q , Q , and Q are three different CDWnesting vectors.Based on this information, we construct an effectivefield theory as follows S = (cid:88) n =1 (cid:88) m = ± (cid:90) d k (2 π ) (cid:104) ψ ( m ) ∗ n ( k ) (cid:16) ik + (cid:15) ( m ) n ( k ) (cid:17) ψ ( m ) n ( k ) + ψ ( m ) ∗ ¯ n ( k ) (cid:16) ik + (cid:15) ( m )¯ n ( k ) (cid:17) ψ ( m )¯ n ( k ) (cid:105) + 12 (cid:88) n =1 (cid:90) d q (2 π ) [ q + c | q | ]Φ Q n ( q )Φ Q n ( − q ) (1) S int − bf = e (cid:88) n =1 (cid:90) d k (2 π ) (cid:90) d q (2 π ) Φ Q n ( q ) (cid:104) ψ ( − ) ∗ n ( k + q ) ψ (+) n ( k ) + ψ (+) ∗ n ( k + q ) ψ ( − ) n ( k )+ ψ (+) ∗ ¯ n ( k + q ) ψ ( − )¯ n ( k ) + ψ ( − ) ∗ ¯ n ( k + q ) ψ (+)¯ n ( k ) (cid:105) (2) S int − b = u (cid:88) n =1 (cid:90) (cid:89) i =1 d q i (2 π ) Φ Q n ( q )Φ Q n ( q )Φ Q n ( q )Φ Q n ( q )(2 π ) δ ( q + q + q + q ) (3) S int − b = u (cid:90) (cid:89) i =1 d q i (2 π ) (cid:104) Φ Q ( q )Φ Q ( q )Φ Q ( q )Φ Q ( q ) + Φ Q ( q )Φ Q ( q )Φ Q ( q )Φ Q ( q )+ Φ Q ( q )Φ Q ( q )Φ Q ( q )Φ Q ( q ) (cid:105) (2 π ) δ ( q + q + q + q ) (4) S int − b = γ (cid:90) (cid:89) i =1 d q (2 π ) (cid:104) Φ Q ( q )Φ Q ( q )Φ Q ( q ) + Φ Q ( q )Φ Q ( q )Φ Q ( q ) (cid:105) . (5)Here, ψ ( m ) n ( k ) represents an electron field living in a hotspots denoted by n (¯ n ) = 1 , , , ¯2 , ¯3) and m = ± , asshown in Fig. 2. These ”hot” electrons are described bythe dispersion relation (cid:15) ( ± ) n ( k ) = (cid:15) ( ± )1 ( R − θ n k ) = ± k x,θ n + vk y,θ n and (cid:15) ( ± )¯ n ( k ) = − (cid:15) ( ± ) n ( k ), where (cid:16) k x,θ n k y,θ n (cid:17) = (cid:16) cos θ n sin θ n − sin θ n cos θ n (cid:17)(cid:16) k x k y (cid:17) ≡ R − θ n k with θ n = π ( n − C rotational symmetry of the Fermi-surface structure. It is clear that the Fermi-surface nest-ing becomes perfect when the velocity v vanishes. Φ Q n ( q )is a bosonic order parameter field to describe CDW fluc-tuations, where Q n with n = 1 , , (cid:126)Q = (cid:126)b + (cid:126)b , whichis commensurate. Generally, charge density fluctuationswith a nesting vector Q can be described as follows ρ ( r ) = e i Q · r Φ Q ( r ) + e − i Q · r Φ ∗ Q ( r ) , (6)where Φ − Q ( r ) = Φ ∗ Q ( r ) has been used. IntroducingΦ Q ( r ) = δρ ( r ) e iθ ( r ) into the above expression, we ob-tain ρ ( r ) = δρ ( r ) cos( Q · r + θ ( r )). Here, δρ ( r ) and θ ( r )represent amplitude and phase fluctuations of the CDWorder parameter. In the case of commensurate CDWordering, such phase fluctuations are irrelevant and ne-glected. As a result, we take into account Φ Q ( r ) as areal valued function, i.e., Φ Q ( r ) = Φ ∗ Q ( r ) [16]. TheseCDW fluctuations are assumed to follow the relativisticdispersion with their velocity c , regarded to be an effec-tive field theory of the Ising model. Electrons connectedby Fermi surface nesting are strongly correlated and de-scribed by S int − bf with an effective interaction parame-ter e . In addition, such CDW order parameters interactwith themselves, constructed by symmetry consideration. S int − b ( S int − b ) describes the self-interactions betweenCDW order parameters with the same momentum (differ-ent momenta) while S int − b gives cubic self-interactions.Here, we do not take into account the S int − b interactionfor simplicity.A conventional way solving this complex Fermi-surfaceproblem is to take into account both self-energy correc- tions of electrons and order parameters self-consistentlywithout considering vertex corrections, referred to as ei-ther Eliashberg theory or self-consistent random phaseapproximation [8–11]. This Fermi-surface problem hasbeen regarded to be controlled in the so called large-Nlimit, where the spin degeneracy of electronic degrees offreedom is extended from 2 to N [17]. In other words, theEliashberg theory is supposed to be exact in the N → ∞ limit, where finite N quantum corrections can be intro-duced in a controllable way, based on the solution ofthe Eliashberg theory. However, it turns out that thisFermi surface problem remains strongly correlated evenin the N → ∞ limit [12], regarded to be a characteris-tic feature in two dimensions, which means that vertexcorrections should be introduced self-consistently. Un-fortunately, we do not know how to re-sum such quan-tum corrections consistently. Recently, the technique of“graphenization” has been proposed as a way of control-lable evaluation for Feynman diagrams, which generalizesthe dimensional regularization technique for interactingboson problems into the Fermi surface problem, wherethe density of states is reduced to allow us to controleffective interactions of electrons [13, 18].In order to prepare for the dimensional regularizationscheme in the present problem, we introduce the two-component spinorΨ ( χ ) n ( k ) = (cid:16) ψ ( χ ) n ( k ) χψ ( χ )¯ n ( k ) (cid:17) (7)and rewrite the above effective action in the followingway S = (cid:88) n =1 (cid:88) m = ± (cid:90) d k (2 π ) Ψ ( m ) † n ( k )[ ik τ + (cid:15) ( m ) n ( k ) τ ]Ψ ( m ) n ( k )+ 12 (cid:88) n =1 (cid:90) d q (2 π ) [ q + c | q | ]Φ n ( q )Φ n ( − q ) (8) S int − bf = e (cid:88) n =1 (cid:90) d k (2 π ) (cid:90) d q (2 π ) Φ n ( q ) (cid:34) Ψ ( − ) † n ( k + q ) τ Ψ (+) n ( k ) + Ψ (+) † n ( k + q ) τ Ψ ( − ) n ( k ) (cid:35) (9) S int − b = u (cid:88) n =1 (cid:90) (cid:89) i =1 d q i (2 π ) Φ n ( q )Φ n ( q )Φ n ( q )Φ n ( q )(2 π ) δ ( q + q + q + q ) (10) S int − b = u (cid:90) (cid:89) i =1 d q i (2 π ) (cid:104) Φ ( q )Φ ( q )Φ ( q )Φ ( q ) + Φ ( q )Φ ( q )Φ ( q )Φ ( q )+ Φ ( q )Φ ( q )Φ ( q )Φ ( q ) (cid:105) (2 π ) δ ( q + q + q + q ) , (11)where τ is the Pauli matrix and Φ Q n ( q ) ≡ Φ n ( q ) in theshort-hand notation. Following S.-S. Lee’s co-dimensional regularizationmethod [13], we write down the above two dimensionaleffective field theory in general d dimensions S = (cid:88) n =1 (cid:88) m = ± N f (cid:88) j =1 (cid:90) dk ¯Ψ ( m ) n,j ( k )[ i Γ · K + iγ d − (cid:15) ( m ) n ( k d − , k d )]Ψ ( m ) n,j ( k )+ 12 (cid:88) n =1 (cid:90) dk [ | K | + c ( k d − + k d )]Φ n ( k )Φ n ( − k ) (12) S int − bf = ie (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk (cid:90) dq Φ n ( q ) (cid:104) ¯Ψ ( − ) n,j ( k + q ) γ d − Ψ (+) n,j ( k ) + ¯Ψ (+) n,j ( k + q ) γ d − Ψ ( − ) n,j ( k ) (cid:105) (13) S int − b = u (cid:88) n =1 (cid:90) (cid:89) i =1 dq Φ n ( q )Φ n ( q )Φ n ( q )Φ n (2 π ) d +1 δ ( q + q + q + q ) (14) S int − b = u (cid:90) (cid:89) i =1 dq i (cid:104) Φ ( q )Φ ( q )Φ ( q )Φ ( q ) + Φ ( q )Φ ( q )Φ ( q )Φ ( q )+ Φ ( q )Φ ( q )Φ ( q )Φ ( q ) (cid:105) (2 π ) d +1 δ ( q + q + q + q ) . (15)Here, we increase the spatial dimension from 2 to d .This procedure is encoded into the extension of mo-mentum from ( k , k x , k y ) to ( K , k d − , k d ), where K =( k , k , · · · , k d − , k d − ) ≡ ( k , K ⊥ ). Accordingly, theDirac gamma matrix is changed from ( γ , γ , γ ) to( Γ , γ d − , γ d ), where Γ = ( γ , γ , · · · , γ d − , γ d − ) ≡ ( γ , Γ ⊥ ) with { γ i , γ j } = 2 δ ij . Although the number ofcomponents in the Dirac spinor should be enhanced tofollow this dimensional generalization, we keep the na-ture of the two-component spinor. As shown below, itturns out that the upper critical dimension is d = 3,which enforces us to perform the renormalization groupanalysis slightly below this upper critical dimension. Asa result, the two-component spinor is allowed. We alsoincrease the number of fermion flavors from 1 to N f . III. RENORMALIZATION GROUP ANALYSISA. Classical scaling
It is straightforward to perform the scaling analysis inthe effective action, resulting in K = K (cid:48) b , k d − = k (cid:48) d − b , k d = k (cid:48) d b Ψ( k ) = b d +22 Ψ (cid:48) ( k (cid:48) ) , Φ( k ) = b d +32 Φ( k (cid:48) ) ,e = b d − e (cid:48) , u = b d − u (cid:48) , u = b d − u (cid:48) . Here, b is the scaling parameter usually utilized in theWilsonian scheme of the renormalization group analysis.It is related with µ as b = µ − , where µ is the scaling pa-rameter conventionally used in the high-energy physicsscheme of the renormalization group analysis. See ap-pendixes A and B. As shown clearly in these equations, we observe that the upper critical dimension of all inter-action parameters is d c = 3, which allows us to performthe perturbative analysis in d = 3 − (cid:15) near the uppercritical dimension, where (cid:15) is an expansion parameter. B. Renormalization group equations in theone-loop level
We perform the renormalization group analysis basedon the scheme usually utilized in high energy physics.We introduce an effective bare action in general d − dimensions. Introducing quantum corrections intothis effective field theory, various ultraviolet (UV) di-vergences appear, but hidden in the intermediate stage,where the dimensional regularization is employed in thisstudy. Such UV divergences are canceled by so calledcounterterms, where UV divergences are absorbed intosome coefficients. Extracting all counterterms from thebare action, we have an effective renormalized action,where UV divergences disappear to be well defined.Then, it is straightforward to find relations betweenbare and renormalized quantities, where UV divergencesare taken into account. Based on these relations, onecan find renormalization group equations, referred to as β − functions, which describe how interaction parametersevolve from the high-temperature regime to the low-temperature region. Since this procedure is quite con-ventional, we would like to refer all details to appendixesA and B.First, we consider the dynamical critical exponent z and β − functions for the fermion velocity v , the boson(CDW order parameter) velocity c , and the effective cou-pling constant e between electrons of hot spots and CDW (a) (b) (c) FIG. 3: One-loop diagrams for (a) Fermion self-energy,(b) Boson self-energy, and (c) Yukawa-type interactionvertex. Here, the solid (wavy) line represents thefermion (boson) propagator. See our Feynman rules inappendix A.fluctuations, given by z = 11 + e π cN f [ h ( c, v ) − h ( c, v )] (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( a ) (16) β v = ve zh ( c, v )4 π cN f (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( a ) (17) β c = e z π N f (cid:104) h ( c, v ) − h ( c, v )) (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( a ) + πcN f v (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( b ) (cid:105) (18) β e = ze (cid:104) − (cid:15) + e πv (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( b ) + e π cN f (cid:16) h ( c, v ) (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( a ) + 12 h ( c, v ) (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( c ) (cid:17)(cid:105) . (19)We recall that the dynamical critical exponent tells usanisotropic scaling between space and time, related with the dispersion relation of critical CDW fluctuations orcritical hot electrons. Here, quantum corrections to thedynamical critical exponent result from the fermion self-energy correction given by Fig. 3 (a), which leads thedynamical critical exponent to be larger than one, con-sistent with the causality condition of any local field the-ories. We used the short-hand notation for h ( c, v ) = (cid:90) dx (cid:114) xxc + (1 − x )(1 + v ) , (20) h ( c, v ) = (cid:90) dxc (cid:114) x [ xc + (1 − x )(1 + v )] , (21)both of which are positive definite. The renormalizationgroup flow of the fermion velocity is given by the fermionself-energy correction [Fig. 3 (a)], where the fermion ve-locity decreases to vanish in the low-temperature limit.The boson velocity renormalization occurs from the bo-son self-energy correction [Fig. 3 (b)] while the fermionself-energy correction [Fig. 3 (a)] also contributes tothe boson velocity renormalization, originating from thespacetime anisotropic scaling described by the dynami-cal critical exponent. It turns out that the space-timeanisotropic scaling, if combined with the anomalous scal-ing dimension of the boson field given by the boson wave-function renormalization constant, enhances the bosonvelocity while the boson self-energy correction reduces it.The coupling constant for the Yukawa-type interactionvertex is renormalized by not only fermion [Fig. 3 (a)]and boson self-energy corrections [Fig. 3 (b)] but alsovertex corrections [Fig. 3 (c)], where both self-energycorrections appear as anomalous scaling dimensions offields, given by each wave-function renormalization con-stant. In addition, the anisotropic scaling between spaceand time also contributes. Here, the boson-fermion ver-tex function is given by h ( c, v ) = c (cid:90) dx (cid:90) − x dy g ( c, v, x, y ) g ( c, v, x, y ) − v ( x − y ) + g ( c, v, x, y ) − v g ( c, v, x, y )[ g ( c, v, x, y ) g ( c, v, x, y ) − v ( x − y ) ] / , (22) g ( c, v, x, y ) = c (1 − x − y ) + x + y, g ( c, v, x, y ) = c (1 − x − y ) + v ( x + y ) . (23)All quantum fluctuations, given by Fig. 3 and appropri-ately combined, screen out the boson-fermion effectiveinteraction, thus reduced. We refer explicit calculationsfor frequency and momentum integrals given by the Feyn-man diagram Fig. 3 to appendix B.Second, we consider renormalizations of both boson in-teractions denoted by u and u , shown in Fig. 4 for u and 5 for u . We recall that u is the self-interactionstrength of CDW fluctuations with the same nesting mo-mentum Q n while u is that between Q n and Q ¯ n bosons. One-loop beta functions for u and u are given as follows β u = zu (cid:104) − (cid:15) + Fig.3( a ) (cid:122) (cid:125)(cid:124) (cid:123) e π cN f [ h ( c, v ) − h ( c, v )]+ e πv (cid:124)(cid:123)(cid:122)(cid:125) Fig.3( b ) + 3 u π c (cid:124) (cid:123)(cid:122) (cid:125) Fig.4(a),(b),(c) + 3 u π c u (cid:124) (cid:123)(cid:122) (cid:125) Fig.4( d ) , ( e ) , ( f ) (cid:105) (24) β u = zu (cid:104) − (cid:15) + e π cN f [ h ( c, v ) − h ( c, v )] (cid:124) (cid:123)(cid:122) (cid:125) Fig.3( a ) + e πv (cid:124)(cid:123)(cid:122)(cid:125) Fig.3( b ) + u π c (cid:124) (cid:123)(cid:122) (cid:125) Fig.5( a ) , ( b ) + 5 u π c (cid:124) (cid:123)(cid:122) (cid:125) Fig.5( c ) , ( d ) , ( e ) (cid:105) . (25) nn nn (a) nn nn (b) n nn n (c) nn nn (d) nn nn (e) n nn n (f) FIG. 4: All one-loop diagrams for the u bosoninteraction. The black (white) dot represents the u ( u ) boson interaction vertex. See our Feynman rules inappendix A. nn ¯ n ¯ n (a) nn ¯ n ¯ n (b) nn ¯ n ¯ n (c) nn ¯ n ¯ n (d) n ¯ nn ¯ n (e) FIG. 5: All one-loop diagrams for the u bosoninteraction. One may be confused with diagrams (c),(d), and (e). We recall n = 1 , , n = 1and ¯ n = 3, respectively, for example, the intermediatepropagator line is indexed with ˜ n = 2.The renormalization group flow of the u boson in-teraction is governed by the anisotropic scaling of thespacetime involved with the dynamical critical exponent[Fig. 3 (a)], the anomalous scaling exponent of theboson field given by the wave-function renormalizationconstant [Fig. 3 (b)], and renormalizations of the u interaction vertex resulting from both u [Fig. 4 (a), (b),(c)] and u [Fig. 4 (d), (e), (f)] effective interactions. Itturns out that all types of quantum corrections give rise to screening effects to the u interaction except for thefact that the effect of the spacetime anisotropic scaling,if combined with the anomalous scaling dimension ofthe boson field, enhances the interaction strength. Werecall our convention that beta functions with positivevalues mean that the corresponding coupling constantdecreases as approaching to the low energy limit. Theevolution behavior of the u boson interaction is quitesimilar to that of u except for the fact that screeningeffects are reduced slightly, compared with the u case.
1. Beta function in the absence of the fermion-bosoncoupling
Although it is not difficult to solve these coupledrenormalization group equations, we start from the casein the absence of the fermion-boson interaction vertex,i.e., e = 0. Then, we focus on the renormalizationgroup equations for u and u with e = 0. As a re-sult, we find four fixed points shown in Fig. 6. Thegaussian fixed point ( u , u ) = (0 , π c (cid:15)/ , u interactions, falling into a modifiedWilson-Fisher fixed point (48 π c (cid:15)/ , π c / u modified Wilson-Fisher fixedpoint can be regarded as a critical point in the presentcontinuous CDW transition. On the other hand, we finda line of separation, in the above of which a runawayflow is observed toward a negative value of the bosoninteraction u , but in the below of which the renormal-ization group flow arrives at the modified Wilson-Fisherfixed point. The fixed point on this line of separationis (32 π c (cid:15)/ , π c (cid:15)/ u interac-tion leads us to identify this fixed point with the param-eter point for the fluctuation-induced first-order phasetransition [20–23]. This may result from interactions be-tween competing CDW fluctuations with several orderingwave vectors.
2. Beta function in the presence of the fermion-bosoncoupling
Since β c , β v , and β e do not depend on u and u , weanalyze the renormalization group flow of c , v , and e first.It is easy to figure out the renormalization group flow ofthe fermion velocity. β v > v always decreases as we approach to lower energies. This confirms the emergence of perfect Fermi surface nestingin two dimensions. In other words, the fermion dynamicsis localized in one direction, giving rise to effective onedimensional dynamics. According to our simple anal-ysis, we find that c also decreases as the energy scale FIG. 6: Renormalization group flow diagram of u and u with N f = 1 and (cid:15) = 0 .
01 when e = 0. The gaussianfixed point ( u , u ) = (0 ,
0) denoted by the black dot isunstable against the presence of weak repulsive u interactions, showing the renormalization group flowtoward the conventional Wilson-Fisher fixed point(16 π c (cid:15)/ ,
0) represented by the green dot. ThisWilson-Fisher fixed point is destabilized by weakrepulsive u interactions, falling into a modifiedWilson-Fisher fixed point (48 π c (cid:15)/ , π c /
11) givenby the red dot. There exists a line of separation, thefixed point on which is given by (32 π c (cid:15)/ , π c (cid:15)/ u above this line while the renormalization group flowarrives at the modified Wilson-Fisher fixed point below.is lowered. Not only fermions but also bosons becomeheavy and localized at low temperatures. Solving thesethree coupled renormalization group equations, we findthe renormalization group flow of c , v , and e as shownin Fig. 7. We point out that the coupling constant be-tween fermions and bosons is also converging to zero asthe energy scale is lowered.Now, we consider the renormalization group flow of twokinds of boson self-interactions, u and u , in the pres-ence of the fermion-boson coupling, i.e., e (cid:54) = 0. We recallthat there appear four fixed points when e = 0, as shownin Fig. 6. An essential point is that the Wilson-Fisherfixed point (green dot), the modified Wilson-Fisher fixedpoint (red dot), and the first order transition point (bluedot) are all proportional to c . As a result, they con-verge into the gaussian fixed point (black dot) as theboson velocity renormalizes to vanish in the low energylimit. Figure 8 shows the renormalization group flow ofeffective self-interactions, u and u , for four differentinitial values in the presence of the fermion-boson inter-action. The blue, red, and green dashed lines show thatthese effective interaction coefficients vanish to fall intothe gaussian fixed point. On the other hand, when therenormalization group flow line is placed above the bluedashed line, u and u show the runaway renormaliza-tion group flow toward a negative value for u , which im- plies the fluctuation induced first-order phase transition[20–23] as the case in the absence of the fermion-bosoncoupling.The origin of this potential existence of the first orderphase transition can be traced back to the nature ofthe fermion-boson interacting vertex. To clarify thephysical mechanism of the first order phase transition,we compare the present Fermi surface problem of theCDW transition with that of the spin density wave(SDW) transition [24]. The SDW transition may beregarded to be the O(3) symmetry version while ourproblem belongs to the Ising symmetry class. Com-paring renormalization group equations of the presentproblem with those of the SDW transition [24], one findsthat both share quite a similar structure, where mostterms have their correspondences in renormalizationgroup flow equations. However, there exists an essentialdifferent aspect between these two problems. The vertexcorrection for the effective interaction between electronsand order parameters gives rise to screening, reducingsuch interactions in the case of the Z symmetry. Onthe other hand, it results in anti-screening for the caseof the SDW transition. More concretely, the sign ofthe fermion-boson vertex function h ( c, v ) [Eq. (19)]differs from each other, where it is positive in the CDWtransition while it is negative in the SDW transition.The anti-screening nature of the SDW case results in theenhancement of the fermion-boson coupling constant,which gives rise to more effective screening of bosonself-interaction constants, u and u . On the otherhand, the screening nature of the CDW case reducesthe screening effect for u and u interactions. Asa result, both self-interaction parameters vanish toallow the second order phase transition in the SDWtransition while the first order and the second orderphase transition both seem to be able to appear in theCDW transition.
3. Beta functions for relative (dimensionless) parameters
Even if the interaction parameter between electronsand order parameters flows to zero, this does not meanthat the nature of the fixed point is Gaussian, i.e., non-interacting for itinerant electrons. In order to resolvethis question, we introduce ratios of coupling parametersin the following way of w = vc , λ = e v , κ = u c , and κ = u c , respectively. The dynamical exponent z and (a) RG flow of c (b) RG flow of v (c) RG flow of e FIG. 7: Renormalization group flow of c , v , and e with N f = 1 and (cid:15) = 0 .
01 for different initial conditions, whichcorrespond to ( c , v , e ) = (0 . , . , .
2) for blue, (0.4,0.01,0.15) for orange, and (0.05,0.15,0.05) for green,respectively. l is − ln µ , which increases as energy is lowered. FIG. 8: Renormalization group flow of u and u when e (cid:54) = 0 with four different initial values of ( c, v, e, u , u );(0 . , . , . , . , . . , . , . , . , . . , . , . , . , .
02) and (0 . , . , . , . , . N f = 1 and (cid:15) = 0 .
01. Theblue, red and green lines denote the flow of fixed pointsshown in Fig. 6; ( u ∗ , u ∗ ).beta functions for w , λ , κ ,and κ are given by z = 8 π N f π N f + λw [ h ( c, cw ) − h ( c, cw )] (26) β w = λwz π N f (cid:104) w [ h ( c, cw ) + h ( c, cw )] − πN f (cid:105) (27) β λ = zλ (cid:104) − (cid:15) + λ π + wλh ( c, cw )8 π N f (cid:105) (28) β κ = zκ (cid:104) − (cid:15) + λ π + 3 κ π + 3 κ π κ (cid:105) (29) β κ = zκ (cid:104) − (cid:15) + λ π + κ π + 5 κ π (cid:105) (30)Since c flows to zero in the low energy limit, weconsider the case of c →
0. Resorting to the factthat lim c → h ( c, cw ) = π , lim c → h ( c, cw ) = 0, and lim c → h ( c, cw ) = π w , we obtain z = 8 π N f π N f − πλw (31) β w = λwz π N f (cid:104) πw − πN f (cid:105) (32) β λ = zλ (cid:104) − (cid:15) + λ π + λ πN f w w (cid:105) (33) β κ = zκ (cid:104) − (cid:15) + λ π + 3 κ π + 3 κ π κ (cid:105) (34) β κ = zκ (cid:104) − (cid:15) + λ π + κ π + 5 κ π (cid:105) . (35)Since z , β w , and β λ do not depend on κ and κ , weanalyze β w and β λ first. It is straightforward to finda fixed point given by ( w ∗ , λ ∗ ) = (cid:16) N f , π (2+ N f ) (cid:15) N f (cid:17) asshown in Fig. 9. Putting this value to z , β κ , and β κ ,we obtain z ∗ = 22 − N f N f (cid:15) (36) β κ = z ∗ κ (cid:104) −
46 + N f (cid:15) + 3 κ π + 3 κ π κ (cid:105) (37) β κ = z ∗ κ (cid:104) −
46 + N f (cid:15) + κ π + 5 κ π (cid:105) . (38)The renormalization group flow diagram for β κ and β κ is shown in Fig. 10. There are four fixed points; onestable fixed point and three unstable fixed points. Thestable fixed point (red point) is given by ( κ ∗ , κ ∗ ) =( π (cid:15) N f ) , π (cid:15) N f ) ), identified with the critical point ofour CDW transition. IV. PHYSICAL PROPERTIESA. Scaling theory of the Green’s function
Correlation functions in terms of renormalized fermionand boson fields are described by the Callan-Symanzik0
FIG. 9: Renormalization group flow diagram of w and λ with N f = 1 and (cid:15) = 0 .
01. Fixed points are given bythe black dot (0 , , π(cid:15) ), the greendot ( N f / , N f / , π (2+ N f ) (cid:15) N f ). FIG. 10: Renormalization group flow diagram of κ and κ with N f = 1 and (cid:15) = 0 .
01. Fixed points are given bythe black dot (0 , π (cid:15) N f ) , π (cid:15) N f ) , π (cid:15) N f ) ), and the blue dot( π (cid:15) N f ) , π (cid:15) N f ) ).equation [19], the derivation of which is shown in ap-pendix A. Solving Eq. (A31) in appendix A, we ob-tain the scaling theory for the correlation function in thevicinity of the critical point as follows˜ G (2 n f ,n b ) r (˜ k ( µ ) , ˜ K ( µ ) , ˜ k d − ( µ ) , ˜ k d ( µ ))= ˜ G n f ,n b r (˜ k ( µ ) , ˜ K ( µ ) , ˜ k d − ( µ ) , ˜ k d ( µ )) × (cid:16) µµ (cid:17) n f (cid:16) d +22 − η ψ + n b (cid:16) d +32 − η Φ + z ( d − (cid:17)(cid:17) . (39)The subscript r means “renormalized”. Here, renormal-ized correlation functions of 2 n f fermion fields and n b boson fields have been taken into account. ˜ k ( µ ), ˜ K ⊥ ( µ ),˜ k d − ( µ ), and ˜ k d ( µ ) are solutions of equations A32 ∼ A35in the scaling limit, where µ is the scaling parameter. They are given by˜ k ( µ ) µ z τ = ˜ k ( µ ) µ z τ , (40)˜ K ⊥ ( µ ) µ z ⊥ = ˜ K ⊥ ( µ ) µ z ⊥ , (41)˜ k d − ( µ ) µ = ˜ k d − ( µ ) µ, (42)˜ k d ( µ ) µ = ˜ k d ( µ ) µ (43)near the critical point. η ψ ( η Φ ) is the anomalous scal-ing dimension of the fermion (boson) field, and z is thedynamical critical exponent.Based on this general equation for the correlation func-tion, it is straightforward to find the scaling expressionof the one-particle Green’s function, given by G f (˜ k , ˜ K ⊥ , ˜ k d − , ˜ k d ) = 1 | ˜ k d − | − η Ψ ˜ G (cid:16) ˜ k | ˜ k d − | z , ˜ K ⊥ | ˜ k d − | z (cid:17) (44)for the fermion propagator of the hot spot +1 and G b (˜ k , ˜ K ⊥ , ˜ k d − , ˜ k d ) = C (˜ k + | ˜ K ⊥ | ) − η Φ4 (45)for the boson Green’s function. Here, C is a positiveconstant. We have anomalous scaling for both fermionand boson Green’s functions, characterized by ˜ η Ψ = η Ψ + (2 − (cid:15) )( z − and ˜ η Φ = η Φ + (2 − (cid:15) )( z − , where both fermionand boson anomalous scaling dimensions are given by η Ψ = η Φ = −
12 2 + N f N f (cid:15) (46)up to the O ( (cid:15) ) order. The dynamical critical exponent is z = 1 + 12 2 + N f N f (cid:15). (47)We would like to emphasize that the fermion Green’sfunction does not depend on k d and the boson propagatordoes not rely on k d − and k d in the low energy limit. Thisscaling theory originates from the fact that the fermionvelocity v and the boson velocity c go to vanish in thelow energy limit. B. Enhancement of Fermi surface nesting in twodimensions
Our beta function analysis showed that the fermionvelocity perpendicular to the nesting vector Q i decreasesas we approach to lower energies. It means that there ap-pear effective Fermi lines which can be more connectedby the nesting vector Q i at low energies, as shown in Fig.11. However, we emphasize that this occurs away fromthree dimensions. More precisely, we find the fermion1 Q Q v ? = 0 v ? = 0 v ? FIG. 11: Schematic description on how Fermi surfacenesting is enhanced by electron-electron correlations.Here, the blue regime refers to the phase spaceconnected by the nesting vector Q (orange arrow). v ⊥ is one component of the Fermi velocity, orthogonal tothe nesting vector. As v ⊥ approaches to the zero value,the Fermi surface nesting becomes much stronger.velocity as a function of an energy scale µ with the di-mensional regularization parameter (cid:15)(cid:15) (cid:54) = 0 : v ( µ ) ∼ v (cid:114) − ln (cid:16) µµ (cid:17) (cid:15) = 0 : v ( µ ) ∼ v (cid:114) (cid:104) ln (cid:16) µ µ (cid:17)(cid:105) . This different form of v ( µ ) originates from whether λ = e v flows to zero or non-zero, respectively.
10 20 30 40 50 l0.20.40.60.81.0v d = = FIG. 12: Evolution of the fermion velocity v ( l ) asapproaching to lower energies ( l ↑ ) for two differentcases; (cid:15) = 0 ( d = 3) and (cid:15) = 1 ( d = 2).Fig. 12 shows v ( l ) as l (= ln( µ /µ )) increases, i.e., en-ergy decreased, for two cases; d = 3 ( (cid:15) = 0) and d = 2( (cid:15) = 1). v decreases faster in d = 2, compared to thecase of d = 3. It means that the two dimensional systemis more favorable for the nesting effect than three dimen-sional systems. However, this argument is not correctcompletely. Since we used the co-dimensional regulariza-tion, the Fermi surface remains to be one dimensional lineeven in the three dimensional case. It is different fromthe real Fermi surface of this system. The actual Fermisurface for the V Se bulk system has been well known[5–7]. Although we cannot give quantitative analysis forthis real case, we can deduce some rigorous statementsbased on the scaling analysis. Since three dimensional V Se is basically given by stacking of two dimensional V Se layers, it is weakly dispersive along the stackingdirection. This can be confirmed in the actual Fermi surface which turns out to be rather “flat” along thestacking direction. Based on this discussion, we assumethat the dispersion relation for hot electrons is given by (cid:15) ( k , k , k ) ∼ k + vk + (cid:80) n> k n , where k is the co-ordinate along the stacking direction. As a result, wededuce the scaling dimension of the coupling constant e :[ e ] = (cid:16) n − (cid:17) . This scaling relation gives rise to [ e ] < n >
1, which implies that e is always irrele-vant in the low energy regime. Since e is essential in therenormalization of v as shown in β v , we conclude that theperfect Fermi-surface nesting would not occur in three di-mensional V Se . This argument is consistent with whythe Hertz-Moriya-Millis theory [8–11] works well in threedimensions although it breaks down in two dimensions.One may criticize that the emergence of perfect nestingof the Fermi surface results from just reducing the dimen-sionality and involved with the lattice structure. Actu-ally, this is related with the main point of the presentstudy, reflected in the title. Although the initial Fermisurface nesting structure is not “good” at high temper-atures (UV), there exists a temperature evolution forthe Fermi velocity near CDW criticality in two dimen-sions: The Fermi surface nesting becomes perfect at lowtemperatures (IR), entitled with “Universal renormaliza-tion group flow toward perfect Fermi-surface nesting nearCDW criticality in two dimensions.” On the other hand,this renormalization group flow, i.e., the temperatureevolution of the Fermi surface structure does not occurnear criticality in three dimensions as discussed above.Furthermore, if the “approximately” perfect Fermi sur-face nesting is regarded to be just an effect of the bandstructure involved with the dimensional reduction, theredo not exist temperature evolutions toward perfect Fermisurface nesting.Suppose that the Fermi surface nesting property isquite nice in either three or two dimensions, describedby a band structure calculation. This can happen ei-ther accidentally or inevitably, which can originate frominterference effects due to the lattice structure. An im-portant point is that there is no such strong renormaliza-tion group flow, i.e., temperature evolution for the Fermisurface structure both near CDW criticality in three di-mensions and in the band structure effect of two dimen-sions, where the band structure does not change muchfrom high temperatures to low temperatures. However,the Fermi velocity acquires strong renormalization effectsfrom enhanced interactions between electrons, responsi-ble for the emergence of the perfect Fermi surface nest-ing in two dimensions. This temperature evolution maybe resolved in the ARPES experiment, which requireshigh-precision energy and momentum resolution. Unfor-tunately, recent experiments did not verify this issue [4].2 V. BEYOND THE ONE-LOOP ORDER:DISCUSSION ON CONTROLLABILITY
To check the validity of our renormalization groupanalysis up to the one-loop order, we investigate the con-trollability of the present Fermi-surface problem, follow-ing the S.-S. Lee’s paper [24]. For a generic Feynmandiagram, the amplitude of the diagram I is given by I ∼ e V e u V u i (cid:90) (cid:104) L (cid:89) i =1 dp i (cid:105) I f (cid:89) j =1 (cid:110) Γ · K j + γ d − [ ± k d − ,j + vk d,j ] (cid:111) × I b (cid:89) l =1 (cid:104) | Q l | + c ( q d − ,l + q d,l ) (cid:105) . Here, V e and V u are the number of interaction verticesfor e and u , respectively. I f ( I b ) is the number offermion (boson) propagators, and L is the total num-ber of loops. k and q are momentum of fermions andbosons, respectively, which consist of loop momentum p and external momentum. If we denote the external mo-mentum as P ext , we obtain { k } = { α k } p + { β k } P ext and { q } = { α q } p + { β q } P ext .First, we transform p d into v p d . Under this transfor-mation, we obtain k d,j → v ( α k d ,j p d + vβ k d ,j P ext,d ) ≡ v k (cid:48) d,j (48) q d,j → v ( α q d ,j p d + vβ q d ,j P ext,d ) ≡ v q (cid:48) d,j . (49)Then, we rewrite the above expression as follows I ∼ e V e u V u i v L (cid:90) (cid:104) L (cid:89) i =1 dp i (cid:105) I f (cid:89) j =1 (cid:110) Γ · K j + γ d − [ ± k d − ,j + k (cid:48) d,j ] (cid:111) × I b (cid:89) l =1 (cid:104) | Q l | + c q d − ,l + w q (cid:48) d,l (cid:105) The above integral converges when there are no loopswhich consist of only bosonic propagators. If there areloops which consist of only bosonic propagators, thisgives rise to the divergence in the zero c limit. In otherwords, this loop integral is proportional to c Lb , where L b is the number of loops consisting of only boson propaga-tors. As a result, we reach the following expression I ∼ e V e u V u i v L c L b = w − V u λ Ve +2 − E κ V u i e E − c − L b + V u , (50)where E is the number of external lines. In the above, L + V − I = 1 with V = V e + V u and I = I f + I b , and 3 V e +4 V u = 2 I + E have been utilized. Based on this result,we obtain the magnitude of renormalization constantsfor the propagator ( E = 2), the Yukawa coupling vertex( E = 3), and the quartic vertex ( E = 4) I E =2 ∼ w − V u λ Ve κ V u i c δ ,I E =3 ∼ w − V u λ Ve − κ V u i c δ e,I E =4 ∼ w − V u λ Ve − κ V u i e c δ , where δ = V u − L b ≥ A i coefficients in counterterms, we shouldalso consider some coefficients in front of A i . Then, weobtain A , A ∼ ∂I E =2 ∂P ext, ∼ ∂I E =2 ∂P ext, ⊥ ∼ w − V u λ Ve κ V u i c δ ,A = ∂I E =2 ∂P ext,d − ∼ w − V u λ Ve κ V u i c δ (1 + c ) ,A = 1 v ∂I E =2 ∂P ext,d ∼ w − V u λ Ve κ V u i c δ (1 + w − ) ,A , A ∼ ∂ I E =2 ∂P ext, ∼ ∂ I E =2 ∂P ext, ⊥ ∼ w − V u λ Ve κ V u i c δ ,A ∼ c (cid:16) ∂ I E =2 ∂P ext,d − + ∂ I E =2 ∂P ext,d (cid:17) , ∼ w − V u λ Ve κ V u i c δ ( c − + 1 + c + w + w − ) ,A ∼ e I E =3 ∼ w − V u λ Ve − κ V u i c δ ,A , A ∼ u i I E =4 ∼ w − V u +1 λ Ve κ V u − i c δ − . Up to the one-loop order, we find w ∼ O (1) and λ, κ i ∼O ( (cid:15) ), where c goes to the zero limit. Therefore, higher-order loop contributions to A , · · · , A and A can beignored in the small (cid:15) limit. On the other hand, suchhigher-order loop contributions to A and A , A cannotbe neglected when δ is smaller than two for A and one for A , A , respectively, in the limit of non-zero (cid:15) since c goesto zero in the low energy limit. Fortunately, even theseterms can be ignored in three dimensions since c con-verges to zero slower than λ and κ . We reall c ∼ l ) / , λ, κ i ∼ l , where l ∼ ln µ . However, such higher-order di-agrams should be taken into account for two dimensionalsystems. VI. SUMMARY
Dimensionality and hetero-interface structure of quan-tum material are essential factors to control bothelectron-electron and electron-phonon interactions, re-sponsible for electronic reconstruction phenomena, whichserves as the basic principle for device applications. Com-pared with the electronic reconstruction paradigm in ox-ide hetero-structured quantum materials, such phenom-ena appear as rather a simple fashion in the van derWaals hetero-interface system, thus expected to be anideal flat form testing the basic principle in the stronglycorrelated regime, for example, metallic quantum criti-cality in two dimensions. Actually, recent ARPES andSTM measurements demonstrated that physics of strongcorrelations arises in monolayer
V Se [4]. In particular,the ARPES experiment has shown perfect Fermi-surfacenesting, implying further dimensional reduction that onedimensional motion of electrons is realized instead of twodimensional dynamics.3In order to understand this strongly correlated dynam-ics of electrons, we constructed an effective field theoryin terms of itinerant electrons and CDW critical fluctua-tions. Resorting to a novel dimensional regularizationtechnique for this Fermi surface problem [13, 18, 24],we performed the renormalization group analysis to re-veal the mechanism for perfect Fermi surface nesting inthe monolayer V Se system. The renormalization groupflow for the curvature parameter gives rise to the emer-gence of the perfect Fermi surface nesting universally onlyin two dimensions beyond the Hertz-Moriya-Millis de-scription in three dimensions [8–11]. We claim that thisfurther dimensional reduction from the two dimensionalFermi surface with imperfect Fermi surface nesting to theone dimensional Fermi surface with perfect Fermi sur-face nesting is responsible for the drastic enhancement of the CDW ordering transition temperature althoughthe CDW ordering itself follows that of the bulk parent.We point out that this further dimensional reduction inthe dynamics of electrons has been also reported in twodimensional SDW transitions [24]. ACKNOWLEDGEMENT
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We start from an effective bare action given by S b, = (cid:88) n =1 (cid:88) m = ± N f (cid:88) j =1 (cid:90) dk b ¯Ψ ( m ) b,n,j ( k b )[ iγ k b, + i Γ ⊥ · K b, ⊥ + iγ d − (cid:15) ( m ) n ( k b,d − , k b,d , v b )]Ψ ( m ) b,n,j ( k )+ 12 (cid:88) n =1 (cid:90) dk b [ | k | + | K b, ⊥ | + c b ( k b,d − + k b,d )]Φ b,n ( k b )Φ b,n ( − k b ) (A1) S b,int − bf = ie b (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk b (cid:90) dq b Φ b,n ( q b ) (cid:104) ¯Ψ ( − ) b,n,j ( k b + q b ) γ d − Ψ (+) b,n,j ( k b ) + ¯Ψ (+) b,n,j ( k b + q b ) γ d − Ψ ( − ) b,n,j ( k b ) (cid:17)(cid:105) (A2) S b,int − b = u b (cid:88) n =1 (cid:90) (cid:89) i =1 dq b Φ b,n ( q b, )Φ b,n ( q b, )Φ b,n ( q b, )Φ b,n ( q b, )(2 π ) d +1 δ ( q b, + q b, + q b, + q b, ) (A3) S b,int − b = u b (cid:90) (cid:89) i =1 dq b,i (cid:104) Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, ) + Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, )+ Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, )Φ b, ( q b, ) (cid:105) (2 π ) d +1 δ ( q b, + q b, + q b, + q b, ) . (A4)Introducing quantum corrections into this effective field theory, various ultraviolet (UV) divergences appear. SuchUV divergences are canceled by so called counterterms S ct, = (cid:88) n =1 (cid:88) m = ± N f (cid:88) j =1 (cid:90) dk b ¯Ψ ( m ) r,n,j ( k r )[ iA γ k r, + iA Γ ⊥ · K r, ⊥ + iA γ d − (cid:15) ( m ) n ( k r,d − , k r,d , A A v r )] × Ψ ( m ) r,n,j ( k ) + 12 (cid:88) n =1 (cid:90) dk r [ A | k | + A | K r, ⊥ | + A c r ( k r,d − + k r,d )]Φ r,n ( k r )Φ r,n ( − k r ) (A5) S ct,int − bf = iA ˜ e r µ (cid:15)/ (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r (cid:90) dq r Φ r,n ( q r ) (cid:104) ¯Ψ ( − ) r,n,j ( k r + q r ) γ d − Ψ (+) r,n,j ( k r ) + ¯Ψ (+) r,n,j ( k r + q r ) γ d − Ψ ( − ) r,n,j ( k r ) (cid:17)(cid:105) (A6) S ct,int − b = A ˜ u r µ (cid:15) (cid:88) n =1 (cid:90) (cid:89) i =1 dq r Φ r,n ( q r, )Φ r,n ( q r, )Φ r,n ( q r, )Φ r,n ( q r, )(2 π ) d +1 δ ( q r, + q r, + q r, + q r, ) (A7) S ct,int − b = A ˜ u r µ (cid:15) (cid:90) (cid:89) i =1 dq r,i (cid:104) Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, ) + Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )+ Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, ) (cid:105) (2 π ) d +1 δ ( q r, + q r, + q r, + q r, ) , (A8)where UV divergences are absorbed into A n coefficients with n = 0 , ..., S r = S b − S ct , we have an effective renormalized action, givenby5 S r, = (cid:88) n =1 (cid:88) m = ± N f (cid:88) j =1 (cid:90) dk b ¯Ψ ( m ) r,n,j ( k r )[ iγ k r, + i Γ ⊥ · K r, ⊥ + iγ d − (cid:15) ( m ) n ( k r,d − , k r,d , v r )]Ψ ( m ) r,n,j ( k )+ 12 (cid:88) n =1 (cid:90) dk r [ | k | + | K r, ⊥ | + c r ( k r,d − + k r,d )]Φ r,n ( k r )Φ r,n ( − k r ) (A9) S r,int − bf = i ˜ e r µ (cid:15)/ (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r (cid:90) dq r Φ r,n ( q r ) (cid:104) ¯Ψ ( − ) r,n,j ( k r + q r ) γ d − Ψ (+) r,n,j ( k r ) + ¯Ψ (+) r,n,j ( k r + q r ) γ d − Ψ ( − ) r,n,j ( k r ) (cid:17)(cid:105) (A10) S r,int − b = ˜ u r µ (cid:15) (cid:88) n =1 (cid:90) (cid:89) i =1 dq r Φ r,n ( q r, )Φ r,n ( q r, )Φ r,n ( q r, )Φ r,n ( q r, )(2 π ) d +1 δ ( q r, + q r, + q r, + q r, ) (A11) S r,int − b = ˜ u r µ (cid:15) (cid:90) (cid:89) i =1 dq r,i (cid:104) Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, ) + Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )+ Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, )Φ r, ( q r, ) (cid:105) (2 π ) d +1 δ ( q r, + q r, + q r, + q r, ) , (A12)where UV divergences disappear and well defined. Here, we introduce an energy scale µ to make e r , u r , and u r bedimensionless quantities, redefined by ˜ e r , ˜ u r , and ˜ u r . The upper critical dimension for all interaction parametersof ˜ e r , ˜ u r , and ˜ u r turns out to be d c = 3, where the expansion parameter is given by (cid:15) = 3 − d in the dimensionalregularization scheme.It is straightforward to find equations between bare and renormalized quantities. First, we consider the scalingtransformation, given by k b, = Z τ k r, , K b, ⊥ = Z ⊥ K r, ⊥ ,k b,d − = k r,d − , k b,d = k r,d . (A13)Here, Z τ and Z ⊥ are rescaling parameters for frequency and “transverse” momentum in fictitious extra dimensions.Second, we introduce field renormalization constants of Z Ψ and Z Φ , which relate bare fields with renormalized onesin the following way Ψ b = Z / Ψ r , Φ b = Z / Φ r . (A14)Then, resorting to S b = S r + S ct , we define all renormalized constants Z n with n = 0 , ..., Z Ψ Z d − ⊥ Z τ = Z , Z Ψ Z d − ⊥ Z τ = Z ,Z Ψ Z d − ⊥ Z τ = Z , Z Ψ Z d − ⊥ Z τ v b = Z v r (A15) Z Φ Z d − ⊥ Z τ = Z , Z Φ Z d ⊥ Z τ = Z ,Z Φ Z d − ⊥ Z τ c b = Z c r , (A16) Z / Z Ψ Z d − ⊥ Z τ e b = Z ˜ e r µ (cid:15)/ (A17) Z Z d − ⊥ Z τ u b = Z ˜ u r µ (cid:15) ,Z Z d − ⊥ Z τ u b = Z ˜ u r µ (cid:15) , (A18)where such renormalization constants are given by counterterm coefficients as Z n = 1 + A n . (A19)
2. Feynman rules
In order to perform the perturbative renormalization group analysis systematically, we introduce Feynman rulesbased on the renormalized effective action and counterterms. Here, we express the fermion-involved sector in a morecompact way as follows:6 S r, = (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r ¯Ψ r,n,j ( k r ) (cid:32) i Γ · K r + iγ d − (cid:15) ( − ) n i Γ · K r + iγ d − (cid:15) (+) n (cid:33) Ψ r,n,j ( k r )+ 12 (cid:88) n =1 (cid:90) dk r [ | K r | + c r ( k r,d − + k r,d )]Φ r,n ( k r )Φ r,n ( − k r ) (A20) S r,int − bf = i ˜ e r µ (cid:15)/ (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r (cid:90) dq r ¯Ψ r,n,j ( k r + q r )Φ r,n ( q r ) γ d − ⊗ σ Ψ r,n,j ( k r ) (A21) S ct, = (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r ¯Ψ r,n,j (cid:32) iA k r, + iA Γ ⊥ · K r, ⊥ + iA γ d − (cid:15) ( − ) n ( A A v r ) 00 iA k r, + iA Γ ⊥ · K r, ⊥ + iA γ d − (cid:15) (+) n ( A A v r ) (cid:33) × Ψ r,n,j + 12 (cid:88) n =1 (cid:90) dk r [ A | k r, | + A | K r, ⊥ | + A c r ( k r,d − + k r,d )]Φ r,n ( k r )Φ r,n ( − k r ) (A22) S ct,int − bf = i A ˜ e r µ (cid:15)/ (cid:112) N f (cid:88) n =1 N f (cid:88) j =1 (cid:90) dk r (cid:90) dq r ¯Ψ r,n,j ( k r + q r )Φ r,n ( q r ) γ d − ⊗ σ Ψ r,n,j ( k r ) (A23)where we introduced Ψ r,n,j ≡ (cid:32) Ψ ( − ) r,n,j Ψ (+) r,n,j (cid:33) and ¯Ψ r,n,j ≡ (cid:32) ¯Ψ ( − ) r,n,j ¯Ψ (+) r,n,j (cid:33) T .Based on this effective field theory, we construct Feynman rules= (cid:104) Ψ r,n,j ( k ) ¯Ψ r,n (cid:48) ,j (cid:48) ( k (cid:48) ) (cid:105) = (2 π ) d +1 δ ( k − k (cid:48) ) δ n,n (cid:48) δ j,j (cid:48) − i Γ · K r + γ d − (cid:15) ( − ) n | K r | +( (cid:15) ( − ) n ) − i Γ · K r + γ d − (cid:15) (+) n | K r | +( (cid:15) (+) n ) = (cid:104) Φ r,n ( k )Φ r,n (cid:48) ( k (cid:48) ) (cid:105) = (2 π ) d +1 δ ( k + k (cid:48) ) δ n,n (cid:48) | K r | + c r ( k r,d − + k r,d ) e = − i ˜ e r µ (cid:15)/ (cid:112) N f γ d − ⊗ σ , n nn n = − ˜ u r µ (cid:15) , n n ¯ n ¯ n = − ˜ u r µ (cid:15) , for fermion and boson propagators and their interaction vertices and= − (cid:32) iA k r, + iA Γ ⊥ · K r, ⊥ + iA γ d − (cid:15) ( − ) n ( A A v r ) 00 iA k r, + iA Γ ⊥ · K r, ⊥ + iA γ d − (cid:15) (+) n ( A A v r ) (cid:33) = − [ A | q r, | + A | Q r, ⊥ | + A c r ( q r,d − + q r,d )] e = − i A ˜ e r µ (cid:15)/ (cid:112) N f γ d − ⊗ σ , n nn n = − A ˜ u r µ (cid:15) , n n ¯ n ¯ n = − A ˜ u r µ (cid:15) ( n (cid:54) = ¯ n )for counterterms, respectively. Resorting to these Feynman rules, one can take into account quantum fluctuationsperturbatively, where the co-dimensional regularization scheme is utilized.
3. Renormalization group equations
Correlation functions in terms of bare & renormalized fermion and boson fields are defined by7 (cid:104) Ψ b ( k b, ) · · · Ψ b ( k b,n f ) ¯Ψ b ( k b,n f +1 ) · · · ¯Ψ b ( k b, n f )Φ b ( q b, ) · · · Φ b ( q b,n b ) (cid:105) = G (2 n f ,n b ) b ( k b,i , q b,i ; v b , c b , e b , u b , u b ) δ ( d +1) (cid:16) n f (cid:88) i =1 ( k b,i − k b,i + n f ) + n b (cid:88) j =1 q b,j (cid:17) (A24) (cid:104) Ψ r ( k r, ) · · · Ψ r ( k r,n f ) ¯Ψ r ( k r,n f +1 ) · · · ¯Ψ r ( k r, n f )Φ r ( q r, ) · · · Φ r ( q r,n b ) (cid:105) = G (2 n f ,n b ) r ( k r,i , q r,i ; v r , c r , e r , u r , u r ) δ ( d +1) (cid:16) n f (cid:88) i =1 ( k r,i − k r,i + f ) + n b (cid:88) j =1 q r,j (cid:17) , (A25)respectively.In order to make the scaling dimension be apparent, we take into account classical scaling (engineering dimension)explicitly as follows K r = µ ˜ K , k r,d − = µ ˜ k d − , k r,d = µ ˜ k d (A26)Ψ r = µ − d +22 ˜Ψ r , Φ r = µ − d +32 ˜Φ r , (A27)where µ is an energy scale, introduced before. Then, we obtain the renormalization group equation for correlationfunctions G (2 n f ,n b ) b ( k b,i , q b,i ; v b , c b , e b , u b , u b ) = Z τ Z d − ⊥ Z n f Ψ Z nb Φ µ − n f ( d +2) − n b d +32 + d +1 × ˜ G (2 n f ,n b ) r (˜ k r,i , ˜ q r,i ; v r , c r , ˜ e r , ˜ u r , ˜ u r ; µ ) , (A28)where G (2 n f ,n b ) r ( k r,i , q r,i ; v r , c r , e r , u r , u r ) = µ − n f ( d +2) − n b d +32 + d +1 × ˜ G (2 n f ,n b ) r (˜ k r,i , ˜ q r,i ; v r , c r , ˜ e r , ˜ u r , ˜ u r ; µ ) . (A29)Resorting to dG (2 n f ,n b ) b ( k b,i , q b,i ; v b , c b , e b , u b , u b ) d ln µ = 0 , (A30)we reformulate the integral form of the renormalization group equation for the correlation function into the differentialequation in the following way (cid:104) n f (cid:88) i =1 (cid:16) z τ ˜ k ∂ + z ⊥ ˜ K ⊥ ,i · ∇ i + ˜ k d − ∂ d − + ˜ k d ∂ ˜ k d (cid:17) + n b (cid:88) i =1 (cid:16) z τ ˜ q ∂ + z ⊥ ˜ Q ⊥ ,i · ∇ i + ˜ q d − ∂ ˜ q d − + ˜ q d ∂ ˜ q d (cid:17) − β v ∂ v − β c ∂ c − β e ∂ e − β u ∂ u − β u ∂ u + 2 n f (cid:16) d + 22 − η Ψ (cid:17) + n b (cid:16) d + 32 − η Φ (cid:17) − ( z ( d −
1) + 2) (cid:105) ˜ G (2 n f ,n b ) r = 0 , (A31)referred to as the Callan-Symanzik equation for the correlation function [19]. Here, we used dk b, d ln µ = 0 → d ˜ k d ln µ = − (cid:16) d ln Z τ d ln µ (cid:17) ≡ − z τ ˜ k (A32) d K b, ⊥ d ln µ = 0 → d ˜ K ⊥ d ln µ = − (cid:16) d ln Z ⊥ d ln µ (cid:17) ≡ − z ⊥ ˜ K ⊥ (A33) dk b,d − d ln µ = 0 → d ˜ k d − d ln µ = − ˜ k d − (A34) dk b,d d ln µ = 0 → d ˜ k d d ln µ = − ˜ k d . (A35)8Anomalous scaling dimensions for fermion and boson fields are given by η Ψ = 12 ∂ ln Z Ψ ∂ ln µ , η Φ = 12 ∂ ln Z Φ ∂ ln µ , (A36)respectively. Beta functions are β v ≡ dv r d ln µ , β c ≡ dc r d ln µ , β e ≡ d ˜ e r d ln µ , β u ≡ d ˜ u r d ln µ , β u ≡ d ˜ u r d ln µ . (A37)We obtain beta functions based on d ln O b d ln µ = 0. Suppose O b = µ y Z y Z y · · · Z y N N O r . Then, the corresponding betafunction is given by β O ≡ d O r d ln µ = − (cid:16) y + y d ln Z d ln µ + y d ln Z d ln µ + · · · + y N d ln Z N d ln µ (cid:17) O r . Following this renormalizationgroup equation, we obtain − (ln Z ) (cid:48) + 2 η Ψ + ( d − z ⊥ −
1) + 2( z τ −
1) = 0 (A38) − (ln Z ) (cid:48) + 2 η Ψ + ( d − z ⊥ −
1) + ( z τ −
1) = 0 (A39) − (ln Z ) (cid:48) + 2 η Ψ + ( d − z ⊥ −
1) + ( z τ −
1) = 0 (A40) − (ln Z ) (cid:48) + 2 η Φ + ( d − z ⊥ −
1) + 3( z τ −
1) = 0 (A41) − (ln Z ) (cid:48) + 2 η Φ + d ( z ⊥ −
1) + ( z τ −
1) = 0 (A42)and β v = [(ln Z ) (cid:48) − (ln Z ) (cid:48) ] v r (A43) β c = 12 [2 η Φ + ( d − z ⊥ −
1) + ( z τ − − (ln Z ) (cid:48) ] c r (A44) β e = [ − (cid:15) η Φ + 2 η Ψ + 2( d − z ⊥ −
1) + 2( z τ − − (ln Z ) (cid:48) ]˜ e r (A45) β u = [ − (cid:15) + 4 η Φ + 3( d − z ⊥ −
1) + 3( z τ − − (ln Z ) (cid:48) ]˜ u r (A46) β u = [ − (cid:15) + 4 η Φ + 3( d − z ⊥ −
1) + 3( z τ − − (ln Z ) (cid:48) ]˜ u r . (A47)Here, we used the short-hand notation of (ln Z i ) (cid:48) ≡ d ln Z i d ln µ . Since ( A − ( A
39) and ( A − ( A
42) give two redundantequations, there are actually 9 equations with 9 variables; z τ , z ⊥ , η Ψ , η Φ , β v , β c , β e , β u , and β u .Solving these coupled equations, we find renormalization group equations for z τ , z ⊥ , η Ψ , η Φ , β v , β c , β e , β u , and β u as follows z ⊥ = (cid:104) F (1) e, − F (1) e, )˜ e r + ( F (1) u , − F (1) u , )˜ u r + ( F (1) u , − F (1) u , )˜ u r (cid:105) − (A48) z τ = z ⊥ (cid:104) F (1) e, − F (1) e, )˜ e r + ( F (1) u , − F (1) u , )˜ u r + ( F (1) u , − F (1) u , )˜ u r (cid:105) (A49) η Ψ = − (cid:104) z ⊥ (cid:16) e r F (1) e, + ˜ u r F (1) u , + ˜ u r F (1) u , (cid:17) + 2 z τ − (cid:105) + z ⊥ − (cid:15) (A50) η Φ = − (cid:104) z ⊥ (cid:16) e r F (1) e, + ˜ u r F (1) u , + ˜ u r F (1) u , (cid:17) + 3 z τ − (cid:105) + z ⊥ − (cid:15) (A51)and β v = vz ⊥ (cid:104)
12 ˜ e ( F (1) e, − F (1) e, ) + ˜ u ( F (1) u , − F (1) u , ) + ˜ u ( F (1) u , − F (1) u , ) (cid:105) (A52) β c = c r (cid:104) − z τ ) + z ⊥ (cid:110)
12 ˜ e r ( F (1) e, − F (1) e, ) + ˜ u r ( F (1) u , − F (1) u , ) + ˜ u r ( F (1) u , − F (1) u , ) (cid:111)(cid:105) (A53) β e = ˜ e r (cid:104) z ⊥ − z τ + z ⊥ (cid:110)
12 ˜ e r (cid:16) F (1) e, − F (1) e, − F (1) e, (cid:17) + ˜ u r (cid:16) F (1) u , − F (1) u , − F (1) u , (cid:17) + ˜ u r (cid:16) F (1) u , − F (1) u , − F (1) u , (cid:17)(cid:111)(cid:105) − (cid:15) z ⊥ ˜ e r (A54) β u = ˜ u r (cid:104) z ⊥ − z τ + z ⊥ (cid:110)
12 ˜ e r ( F (1) e, − F (1) e, ) + ˜ u r ( F (1) u , − F (1) u , ) + ˜ u r ( F (1) u , − F (1) u , ) (cid:111)(cid:105) − (cid:15)z ⊥ ˜ u r (A55) β u = ˜ u r (cid:104) z ⊥ − z τ + z ⊥ (cid:110)
12 ˜ e r ( F (1) e, − F (1) e, ) + ˜ u r ( F (1) u , − F (1) u , ) + ˜ u r ( F (1) u , − F (1) u , ) (cid:111)(cid:105) − (cid:15)z ⊥ ˜ u r . (A56)9Here, we used the short-hand notation, given by F O ,i ≡ ∂ O ln Z i = ∂ O ln (cid:16) ∞ (cid:88) n =1 Z ( n ) i (cid:15) n (cid:17) = ∂ O ∞ (cid:88) m =1 ( − m +1 m (cid:16) ∞ (cid:88) n =1 Z ( n ) i (cid:15) n (cid:17) m ≡ ∞ (cid:88) n =1 F ( n ) O ,i (cid:15) n . (A57) Appendix B: Calculation of Feynman diagrams a. One-loop fermion self-energy correction
The fermion self-energy correction is given byΣ f (1) n,j ( p ) = − (˜ e r ) µ (cid:15) N f (cid:90) dkγ d − ⊗ σ G fn ( k + p ) γ d − ⊗ σ G bn ( k )= i ˜ e r π cN f (cid:15) (cid:32) − h ( c, v ) Γ · P + h ( c, v ) γ d − (cid:15) (+) n ( p ) 0 − h ( c, v ) Γ · P + h ( c, v ) γ d − (cid:15) ( − ) n ( p ) (cid:33) (B1)in the one-loop level. From now on, we omit the subscript r in both fermion and boson velocities of v r and c r ,respectively, for simplicity. As a result, we obtain A = A = − ˜ e r h ( c, v )8 π cN f (cid:15) ,A = − ˜ e r h ( c, v )8 π cN f (cid:15) , A = − A . (B2)Here, we used the short-hand notation for h ( c, v ) = (cid:90) dx (cid:114) xxc + (1 − x )(1 + v ) , h ( c, v ) = (cid:90) dxc (cid:114) x [ xc + (1 − x )(1 + v )] . (B3) b. One-loop boson self-energy correction The boson self-energy correction is given byΠ (1) n ( q ) = N f (cid:16) ˜ e r µ (cid:15)/ N / f (cid:17) (cid:90) dp tr (cid:16) G fn ( p ) γ d − ⊗ σ G fn ( p + q ) γ d − ⊗ σ (cid:17) = − ˜ e r | Q | πv (cid:15) + · · · . (B4)We recall that both self-interaction vertices of u and u do not cause any self-energy corrections in the one-loop level[19]. They result from two loops. Then, we obtain A = A = − ˜ e r πv(cid:15) , A = 0 . (B5) c. One-loop boson-fermion vertex correction The boson-fermion vertex correction is given by e (1) = i (˜ e r ) µ (cid:15)/ N / f (cid:90) dkγ d − ⊗ σ G fn ( k ) γ d − ⊗ σ G fn ( k + q ) γ d − ⊗ σ G bn ( k − p )= i ˜ e r π cN / f (cid:15) h ( c, v ) γ d − ⊗ σ (B6)0in the one-loop level, where h ( c, v ) = c (cid:90) dx (cid:90) − x dy g ( c, v, x, y ) g ( c, v, x, y ) − v ( x − y ) + g ( c, v, x, y ) − v g ( c, v, x, y )[ g ( c, v, x, y ) g ( c, v, x, y ) − v ( x − y ) ] / (B7) g ( c, v, x, y ) = c (1 − x − y ) + x + y, g ( c, v, x, y ) = c (1 − x − y ) + v ( x + y ) . (B8)As a result, we find A = ˜ e r h ( c, v )16 π cN f (cid:15) . (B9) d. One-loop u and u vertex corrections One-loop u and u vertex corrections are essentially the same as those of the Φ theory [19]. The u vertexrenormalization is given by u (2 , = ˜ u r µ (cid:15) (cid:90) dk (cid:104) G bn ( k ) G bn ( Q + k ) + G bn ( k ) G bn ( P + k ) + G bn ( k ) G bn ( K + k ) (cid:105) = 3˜ u r π c (cid:15) (B10) u (0 , = ˜ u r µ (cid:15) (cid:90) dk (cid:88) ¯ n (¯ n (cid:54) = n ) (cid:104) G b ¯ n ( k ) G b ¯ n ( Q + k ) + G b ¯ n ( k ) G b ¯ n ( P + k ) + G b ¯ n ( k ) G b ¯ n ( K + k ) (cid:105) = 3˜ u r π c (cid:15) (B11)in the one-loop level, where the external momenta are assigned to be P = p + p , Q = p + p , and K = p + p .The superscript ( i, j ) means ˜ u i r ˜ u j r in the perturbative analysis. As a result, we obtain A = 38 π c (cid:15) (cid:16) ˜ u r u r ˜ u r (cid:17) . (B12)Similarly, the u vertex renormalization is given by u (1 , = ˜ u r ˜ u r µ (cid:15) (cid:90) dk (cid:104) G bn ( k ) G bn ( Q + k ) + G b ¯ n ( k ) G b ¯ n ( Q + k ) (cid:105) = ˜ u r ˜ u r π c (cid:15) (B13) u (0 , = ˜ u r µ (cid:15) (cid:90) dk [ 12 G b ¯¯ n ( k ) G b ¯¯ n ( k + Q ) + G n ( k ) G ¯ n ( k + P ) + G n ( k ) G ¯ n ( k + K )] = 5˜ u r π c (cid:15) (B14)in the one-loop level, where the external momentum is assigned to be Q = p − p . As a result, we find A = 5˜ u r + 2˜ u r π c (cid:15) . (B15)Additionally, there can appear quantum corrections in the one-loop order from the Yukawa vertex as shown in Fig.13. However, it turns out to vanish.FIG. 13: Possible self-interaction boson vertex from the Yukawa coupling in the one-loop order. It turns out tovanish.1Inserting the A n coefficient of the counterterm into the renormalization factor of Z n = 1 + A n , we obtain Z (1)0 = Z (1)1 = − ˜ e r h ( c, v )8 π cN f (B16) Z (1)2 = − ˜ e r h ( c, v )8 π cN f (B17) Z (1)3 = − Z (1)2 = ˜ e r h ( c, v )8 π cN f (B18) Z (1)4 = Z (1)5 = − ˜ e r πv (B19) Z (1)6 = 0 (B20) Z (1)7 = ˜ e r h ( c, v )16 π cN f (B21) Z (1)8 = 38 π c (cid:16) ˜ u r u r ˜ u r (cid:17) (B22) Z (1)9 = 5˜ u r + 2˜ u r π c . (B23)where Z n = (cid:80) ∞ i =1 Z ( i ) n (cid:15) i . Introducing these results into the equations of (A48) ∼ (A56), we obtain one-loop betafunctions, Eqs. (16) ∼∼