Non-Landau Fermi Liquid induced by Bose Metal
NNon-Landau Fermi Liquid induced by Bose Metal
SangEun Han and Yong Baek Kim
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada (Dated: February 22, 2021)Understanding non-Landau Fermi liquids in dimensions higher than one, has been a subject ofgreat interest. Such phases may serve as parent states for other unconventional phases of quan-tum matter, in a similar manner that conventional broken symmetry states can be understood asinstabilities of the Landau Fermi liquid. In this work, we investigate the emergence of a novel non-Landau Fermi liquid in two dimensions, where the fermions with quadratic band-touching dispersioninteract with a Bose metal. The bosonic excitations in the Bose metal possess an extended nodal-line spectrum in momentum space, which arises due to the subsystem symmetry or the restrictedmotion of bosons. Using renormalization group analysis and direct computations, we show that theextended infrared (IR) singularity of the Bose metal leads to a line of interacting fixed points ofnovel non-Landau Fermi liquids, where the anomalous dimension of the fermions varies continuously,akin to the Luttinger liquid in one dimension. Further, the multi-patch generalization of the modelis used to explore other unusual features of the resulting ground state.
Introduction–
Classification of gapless quantum groundstates of interacting fermions is an outstanding questionin modern theory of quantum matter. Deciphering theorigin and instability patterns of such phases holds thekey for understanding quantum critical phases and novelbroken symmetry or topological phases that may arisethereof [1–3]. In one dimension, the Luttinger liquid [4] isa well-known example going beyond the paradigm of theLandau Fermi liquid, the standard model of conventionalmetals. It does not have well-defined quasiparticles andis characterized by a continuously-varying exponent thatdescribes the algebraic correlations in space-time [4]. Inhigher dimensions, there have been numerous studies ofnon-Landau Fermi liquids that may arise from the long-range interaction between gapless fermions and criticalbosonic excitations [5–22]. Nonetheless, these systemsare often in the strong coupling limit, which makes ithard to find a controlled theoretical framework [17–22].Considering the fermion-boson interactions in two andthree dimensions, the bosons would condense at zero tem-perature if they are gapless at a specific momentum (un-less they are Goldstone modes or gauge fields). This isthe reason why such interactions are mostly studied neara quantum critical point for the bosons, where they re-main gapless down to zero temperature. It would beinteresting to consider an interacting boson system thatremains gapless even without the quantum critical pointand consider their interaction with fermions. The so-called Bose metal is precisely such a system. Here the el-ementary excitations are gapless in an extended region inthe momentum space, which prohibits the condensation[23–26]. It is a consequence of the subsystem symmetry,where the boson number for each row and column of theunderlying lattice is separately conserved [23, 27]. Thisphase was shown to arise in a model of bosons with aring-exchange interaction in two-dimensional square lat-tice [23], where the correlated two-particle hoping pre-serves the subsystem symmetry mentioned above. Be-cause of the restricted motion of the bosons and the sub-system symmetry, the Bose metal can be considered as an example of fracton phases [27–40].In this work, we investigate non-Landau Fermi liquidsthat may arise from the interaction between the fermionswith a quadratic dispersion and Bose metal in two dimen-sions. In the ring-exchange model of Bose metal on thesquare lattice [23], the dispersion of the low energy exci-tations are given by ω q ∼ | sin( q x / q y / | (Fig. 1(a)).The IR singularity along the nodal lines at q x = 0 and q y = 0 may provide a seed for non-Landau Fermi liquidswhen the Bose metal is coupled to fermions.We first consider the interaction between the fermionswith a quadratic dispersion at the Brillouin zone centerand the Bose metal. Focusing on the low-energy con-tinuum limit of the bosonic excitations, we show thatthe one-loop fermion self-energy acquires the logarith-mic correction, ReΣ ∼ ω ln ω , while the bosonic disper-sion is not renormalized. Motivated by this discovery,we then consider the continuum limit of the two bandmodel of fermions with p x , p y orbitals and the quadraticband-touching (Fig. 1(b)), which interact with the Bosemetal. Using the renormalization group analysis, we findthat a line of interacting fixed points arises (see Fig. 2(a)),where the fermions acquire an anomalous dimension that (a) (b) FIG. 1. Plots of the dispersions in the Bose metal and two-band model of fermions with p x and p y orbitals. (a) TheBose metal has nodal lines at k x = 0 and k y = 0 (red lines).(b) The dispersion of the fermions with A x /A = 0 . A z /A = 0 .
4. The blue and red bands are the top and bottombands touching at ( k x , k y ) = (0 , a r X i v : . [ c ond - m a t . s t r- e l ] F e b varies continuously with the fixed point values of the di-mensionless interaction parameter. In addition, the Bosemetal is not renormalized or the subsystem symmetry ispreserved at the fixed point. The continuously-varyinganomalous dimension is akin to the Luttinger liquid [4].In order to further explore the influence of the IRsingularity from the nodal lines, we also consider thequadratic-band-touching fermions located at four differ-ent symmetric positions in the Brillouin zone, namely k = ( ± k , , (0 , ± k ) (see Fig. 2(b)). The lattice bosondispersion, ω q ∼ | sin( q x / q y / | , allows two kindsof IR singularities. The first kind acts mainly withinthe same patch, defined as a small area in momentumspace around each quadratic band-touching point. Thisinvolves the small momentum transfer q = ( δq x , δq y ),where | δq x | , | δq y | (cid:28) k , with the relevant bosonic exci-tations described by ω q ∼ | δq x δq y | . The second kind in-volves the large momentum transfer (in one of the two di-rections) between the reflection-related patches, namely q = ( δq x , k ) or q = (2 k , δq y ), with the correspondingbosonic soft mode ω q ∼ | δq x sin( k ) | or ω q ∼ | sin( k ) δq y | .Considering the one-loop self-energy correction, it isfound that the small momentum process in each patchdominates the low energy scaling. Hence the same lowenergy fixed point found earlier should apply in the scal-ing limit. On the other hand, the large momentum trans-fer in one of the momentum directions leads to a non-analytic correction to the fermion self-energy. For exam-ple, at the one-loop level, we find ReΣ ∼ ω ln ω + ω / and ImΣ ∼ ω + ω / . Here the subleading ω / isfrom the large momentum transfer between the patches.Hence the second derivative ∂ ImΣ /∂ω ∼ ω − / di-verges, which is in principle visible in the fermion scat-tering rate measurement in ARPES [41]. In this sense,the large momentum inter-patch process is dangerouslyirrelevant. and may play an important role in the char-acterization of the underlying fermionic ground state. Bose metal–
We first consider the ring-exchange modelof bosons on the square lattice [23], H = (cid:88) r (cid:20) U n r − ¯ n ) − K cos(∆ xy φ r ) (cid:21) , (1)where ∆ xy φ r = φ r − φ r +ˆ x − φ r +ˆ y + φ r +ˆ x +ˆ y is defined oneach plaquette. Here n r and φ r correspond to the bosonnumber and the phase of the boson wave function at eachlattice site r = ( x, y ), which satisfy the canonical com-mutation relation, [ φ r , n r (cid:48) ] = iδ rr (cid:48) . ¯ n is the mean bosondensity. The phase φ r is 2 π periodic, φ r = φ r + 2 π , sothat n r takes the integer values. The ring-exchange inter-action correspond to the two-particle correlated hoppingin each plaquette such that the boson number in eachrow and column of the lattice is preserved.It was shown that the Bose metal phase is a stableground state of this model for a range of parameters [23,27], where the effective low energy action can be obtained via the expansion, cos(∆ xy φ ) ∼ (∆ xy φ ) , S = 12 (cid:90) d q (2 π ) (cid:90) ∞−∞ dω n π ( ω n + ω q ) | φ ( ω n , q ) | , (2)where ω q ≡ u | sin( q x /
2) sin( q y / | for q x , q y ∈ ( − π, π ).Here we set U = 1 and u = √ U K . The nodal linesat q x = 0 , q y = 0 reflect the presence of infinite num-ber of conserved quantities (Fig. 1(a)). The correspond-ing symmetries involve the invariance of the action under φ ( x, y ) → φ ( x, y )+Φ x ( x )+Φ y ( y ), where Φ x ( x ) , Φ y ( y ) arearbitrary functions of x and y [23, 27]. The lattice actionfor the Bose metal is symmetries under C rotation, in-version, time reversal, and reflection about x, y axes. Thecontinuum limit is rather subtle and ∆ xy φ → ∂ x ∂ y φ iswell defined while ∂ x φ and ∂ y φ are not [27]. The resultingcontinuum action, S ∼ (cid:82) dτ (cid:82) d x [( ∂ τ φ ) + u ( ∂ x ∂ y φ ) ]is also consistent with ω q = u | q x q y | for small momen-tum q x , q y . The symmetries mentioned above imply thatthe coupling to fermion bilinears, ψ † M ψ , where M isa symmety-allowed matrix representation, involves theform factor, F ( q ) = 4 sin( q x /
2) sin( q y / (cid:80) k , q ψ † k + q M ψ k F ( q ) φ q , which leads to ψ † M ψ ( ∂ x ∂ y φ ) inthe continuum limit. p-wave orbital model and preliminary analysis– Letus consider the fermions in p x , p y orbitals on the two-dimensional square lattice. In the continuum limit, themodel can be written as H = (cid:88) k Ψ † k H ( k )Ψ k , H ( k ) = A k σ + A x (2 k x k y ) σ x + A z ( k x − k y ) σ z , (3)where Ψ (cid:124) = ( c x , c y ) is a two-component spinor, c x,y is the fermion annihilation operator for p x,y orbital, σ i is the Pauli matrix, and σ is the 2 × E ( k ) = A k ± (cid:113) A x k x k y + A z ( k x − k y ) . We assume A ,x,z arepositive. We focus on the case of A > max( A x , A z ), inwhich two bands are quadratically touching at k = (0 , C ro-tation, U C Ψ = iσ y Ψ, and the reflection about x and y axes, U R x,y = ± σ z Ψ.When these fermions interact with the bosonic exci-tations in the Bose metal, the action for the interactingmodel can be written as S = S + S int , S = (cid:90) x,τ Ψ † ( ∂ τ + H )Ψ + 12 [( ∂ τ φ ) + u ( ∂ x ∂ y φ ) ] , S int = g (cid:90) x,τ ( ∂ x ∂ y φ )(Ψ † σ x Ψ) , (4)where φ is the phase field of the bosons, Ψ and H are thefermion fields and their Hamiltonian introduced earlier.The tree-level scaling dimensions of fields and parametersare [Ψ] = d/
2, [ φ ] = ( d − z ) /
2, [ A i ] = [ u ] = z −
2. Inparticular, the scaling dimension of g is [ g ] = (3 z − d − /
2. Hence, for d = 2 and z = 2, this interaction ismarginal, [ g ] = 0. Before going further, we introduce thedimensionless parameters, α g ≡ g /π u , and a i = A i /u where i = 0 , x, z . Below, we use the fermion and bosonpropagators given by G ( iω n , k ) = ( − iω n σ + H ) − and D ( iω n , q ) = ( ω n + u q x q y ) − , respectively.We first consider the model with A (cid:54) = 0, but all other A x , A z = 0. The one-loop boson self-energy isΠ( iω n , q ) = − g q x q y (cid:90) Λ d p (2 π ) (cid:90) ∞−∞ d Ω m π × Tr[ σ x G ( i Ω m + iω n , p + q ) σ x G ( i Ω m , p )] , (5)where Λ is the UV cutoff. The integration over Ω m givesΠ = 0. Therefore, the bosonic action is not renormalized.The one-loop fermion self-energy is given byΣ( iω n ) = g (cid:90) Λ σ x G ( iω n + i Ω m , p ) σ x D ( i Ω m , p )[ F ( p )] = α g u Λ (cid:90) dx (cid:90) − d Ω (1 − | Ω | / (cid:112) Ω + x / − i (Ω + ( ω n /u Λ )) + a x σ , (6)where (cid:82) Λ = (cid:82) Λ d p (2 π ) (cid:82) u Λ − u Λ d Ω m π . When | ω n | /u Λ (cid:28) iω n → ω + iη , Eq. 6 leads tothe following fermion self-energy in real frequency.Σ( ω ) ≈ α g (cid:20) C I , iω + C R , log ω ln (cid:18) | ω | u Λ (cid:19)(cid:21) σ , (7)where C i ’s are constants depending on a . The loga-rithmic correction encourages the renormalization groupanalysis, which we present below. Renormalization group analysis–
Based on the aboveobservation, we perform the momentum-shell renormal-ization group analysis for the full model Hamiltonian.The renormalization of the boson self-energy at the one-loop level is given by δ Π = − g q x q y (cid:90) ∂ Λ Tr[ σ x G ( i Ω m + iω n , p + q ) σ x G ( i Ω m , p )] . (8)where (cid:82) ∂ Λ = (cid:82) ΛΛ e − (cid:96) pdp (2 π ) (cid:82) π dθ (cid:82) ∞−∞ d Ω m π , (cid:96) ≡ ln(Λ /µ ).Here Λ and µ are UV and IR cutoffs, respectively. When A > max( A , A ), one can show that δ Π = 0. Thus thebosonic action is not renormalized, just like the previousexample with A (cid:54) = 0 and A x , A z = 0.The renormalization correction to the fermion self-energy is given by δ Σ( iω n , k ) = g (cid:90) ∂ Λ ( p x − k x ) ( p y − k y ) σ x G ( i Ω m , p ) σ x × D ( i Ω m − iω n , p − k ) ≈ − α g (cid:96) [ F ω ( − iω n ) σ + F A k σ a z α g a a ∗ (a) k y k x k − k k − k (b) FIG. 2. (a) RG flow diagram in terms of a , a z , α g for a = a ∗ and a z = 0 slices. Here, the initial value a x, init = 0 isused. The red solid line represents the line of fixed points,( a ∗ , a ∗ x , a ∗ z , α ∗ g ) = (1 . , , , α ∗ g ), where α ∗ g depends on theinitial values of other parameters. (b) The location of thefermion band-touching points, in momentum space, in themulti-patch model. The locations of band-touching pointsare k = ( ± k ,
0) and (0 , ± k ). + F x (2 A x k x k y ) σ x + F z A z ( k x − k y )] , (9)where F i ’s are functions of a , a x , and a z , and their def-initions are provided in Supplementary Materials. Notethat F ω, are positive, but F z is negative for a > a x,z .The vertex correction is given by δ Γ g = g (cid:90) ∂ Λ p x p y D ( i Ω m , p )Tr[ σ x G ( i Ω m , p ) σ x G ( i Ω m , p )]= α g F g (cid:96). (10)where F g is a function of a , a x , a z , and positive for a > a x,z .In terms of dimensionless parameters a i and α g , theRG equations up to one-loop order are given by1 a i da i d(cid:96) = − α g ( F ω − F i ) , α g dα g d(cid:96) = − α g ( F ω − F g ) . (11)We now analyze the above RG equations. In the firstplace, da z /d(cid:96) is negative because F ω is positive and F z is negative. Thus, a z decreases under the RG flow andmay approach a z →
0. With a z →
0, we find fixed pointvalues of a and a x , ( a ∗ , a ∗ x ) = (1 . , α g decreases from an initial value as a z diminishesin the RG flow. As a z → F ω and F g approach thesame value. It is a consequence of the Ward identity,lim iω n → Tr[ ∂δ Σ /∂ ( iω n )] / δ Γ g , because the interactionvertex (Eq. 4) commutes with the fermion Hamiltonian(Eq. 3) when a z →
0. This, combined with the factthat the bosonic action is not renormalized, makes α g toapproach a finite value under the RG flow as a z →
0. Thefixed point value of α g , however, depends on the initialvalues of other dimensionless parameters, as shown inFig. 2(a). This means there exists a line of stable fixedpoints, ( a ∗ , a ∗ x , a ∗ z , α ∗ g ) = (1 . , , , α ∗ g ), characterizedby continuously varying fixed point values of α ∗ g . Somenumerical solutions of the RG equations are shown inFig. S1 in Supplementary Material.Along the line of stable fixed points, the anomalous di-mension of the fermions is finite as α g (cid:54) = 0 and is given by η f = α g C f , where C f = F ω ( a = 1 . , a x = a z = 0) =0 . G ( iω n , k ) ∼ / ( − iω + E k ) − η f / in the low-energy limit,and there is no quasiparticle pole. The absence of thequasiparticle pole is the signature of the non-Fermi liq-uid behavior [22]. Moreover, the anomalous dimension ofthe fermions changes continuously as α g is varied alongthe line of stable fixed points. This behavior is remi-niscent of the Luttinger liquids in one dimension [4]. Inthis sense, we may consider the line of fixed points as atwo-dimensional version of the Luttinger liquids. Multi-patch model–
In order to examine further theinfluence of the nodal lines in the Bose metal, we con-sider multiple fermions at symmetry-related locations inthe momentum space. Let us assume that the fermionswith the quadratic dispersion, A k , are located at k =( ± k , , (0 , ± k ) (Fig. 2(b)). We use the term “patch”to call a small area in momentum space near each band-touching point. For small momentum transfer in both k x and k y directions, we only need to consider the scat-tering processes within the same patch. However, whenthe momentum transfer is large in one of the directionsand small in the other, two patch areas related by thereflection about k x or k y axis, can be connected by sucha momentum transfer in the low energy limit.To illustrate this point, let us focus on the patch near k = ( k ,
0) and compute the fermion self-energy. In thecase of the small momentum transfer, q = ( δq x , δq y ),the dispersion and form factor for the bosonic excita-tions are given by ω q = u | δq x δq y | and F ( q ) = δq x δq y .The one-loop fermion self-energy due to the small mo-mentum transfer within the same patch has the sameform as Eq. 6, and after analytic continuation,Σ small ( ω ) ≈ α g [ C small R , ω + C small R , log ω ln (cid:18) | ω | u Λ (cid:19) + C small I , iω ] . (12)Now let us consider the large momentum transfer be-tween the patches near k = ( k ,
0) and k = ( − k , q = ( − k , δq y ). The dispersion and form factor aregiven by ω q = 2 u | ( δq y ) sin k | and F ( q ) = 2( δq y ) sin k .Note that we do not consider the large momentum trans-fer between the patches located near k = (0 , ± k ) and( k ,
0) as it is not a low energy process. Now the one-loopfermion self-energy at k = ( k ,
0) due to the momentumtransfer q = ( − k , δq y ) is given byΣ large ( iω n ) = g (cid:90) Λ G ( i Ω n + iω m , p ) D ( i Ω m , p − k ˆ x ) F ( p − k ˆ x ) = α g u Λ (cid:90) dx (cid:90) − d Ω (1 − | Ω | / (cid:112) Ω + ξx ) − i (Ω + ( ω n /u Λ )) + a x (13) ≈ α g (cid:32) C large R , / ω / n ( u Λ ) / + C large I , iω n + C large I , / i sgn( ω n ) | ω n | / ( u Λ ) / (cid:19) , (14)where ξ ≡ − sin ( k ), D ( iω n , p − k ˆ x ) = ( ω n +4 p y sin k ) − .After analytic continuation, the total fermion self-energy in real frequency, Σ tot ( ω ) ≡ Σ small ( ω ) + Σ large ( ω ),is obtained. The real part of this fermion self-energy isReΣ tot ( ω ) = α g (cid:20) C tot R , ω + C tot R , log ω ln (cid:16) | ω | u Λ (cid:17) + C tot R , / ω / ( u Λ ) / (cid:21) . (15)The leading contribution, ω ln ω , comes from the smallmomentum transfer process within the same patch. Itimplies that the same renormalization group fixed pointobtained earlier for the single patch model would describethe low energy properties of this non-Fermi liquid state.On the other hand, the imaginary part of the totalfermion self-energy is given byImΣ tot ( ω ) = α g (cid:20) C tot I , ω + C tot I , / ω / ( u Λ ) / (cid:21) . (16)While the leading behavior is linear in ω , the non-analyticcorrection ω / comes from the large momentum process.Note that the second derivative, ∂ ImΣ( ω ) /∂ω ∝ ω − / ,is singular in the low energy limit. If we only kept theleading order contribution from the small momentumtransfer within the same patch, we would not be ableto find such a singular behavior. In this sense, the largemomentum transfer process is dangerously irrelevant. Itshould also be noted that both the small and large mo-mentum processes do not renormalize the bosonic actionof the Bose metal (see Supplementary Material). Discussion–
In the two-dimensional Bose metal phase,one can define the currents, J = ∂ τ φ , J xy = u ( ∂ x ∂ y φ ),and the continuity equation, ∂ τ J = ∂ x ∂ y J xy , can be de-rived from the equation of motion, ∂ τ φ = ∂ x ∂ y ( u ∂ x ∂ y φ )[27, 39, 40]. Noting that the conjugate momentum π = ∂ τ φ ∼ n ( x, y, τ ) is the boson number, one can ob-tain the conserved quantities, Q x ( τ ) = (cid:82) dyJ ( x, y, τ ), Q y ( τ ) = (cid:82) dxJ ( x, y, τ ), which are related to the totalnumber of bosons on each row and column of the lattice.Since ∂Q x /∂τ = (cid:82) dy∂ x ∂ y J xy , ∂Q y /∂τ = (cid:82) dx∂ x ∂ y J xy ,and J xy is well defined in the continuum limit [27], weobtain ∂Q x /∂τ = ∂Q y /∂τ = 0, with the appropriateboundary condition, and hence Q x , Q y are conserved.In the presence of the interaction with fermions, weshowed that the non-Landau Fermi (or non-Fermi forshort) liquid fixed point exists, where the bosonic actionis not renormalized as long as A > max( A x , A z ) in themicroscopic model. Therefore, at the fixed point g = g ∗ ,the continuity equation is simply modified to ∂ τ J = ∂ x ∂ y (cid:101) J xy , where (cid:101) J xy = J xy + g ∗ ∂ x ∂ y (Ψ † σ x Ψ). This leadsto ∂Q x /∂τ = (cid:82) dy∂ x ∂ y (cid:101) J xy , ∂Q y /∂τ = (cid:82) dx∂ x ∂ y (cid:101) J xy ,where the additional contribution from the fermion alsovanishes because it is a total derivative of the fermionbilinear. Thus, Q x and Q y remain conserved so that thesubsystem symmetry of the Bose metal is intact for thenon-Fermi liquid fixed point.In the multi-patch model, where the quadratic-band-touching fermions are located at multiple places in themomentum space, we showed that the nodal-line excita-tions can also influence the large momentum (in one ofthe two momentum directions) scattering processes be-tween different patches. While the contribution fromthe inter-patch scattering process is irrelevant at the T = 0 fixed point, the non-analytic T / correction fromsuch processes may appear in the fermion self-energy,ImΣ ∼ T + T / , at finite temperature, before one reachesthe low temperature scaling regime. 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Supplemental Material for “Non-Landau Fermi Liquid induced by Bose Metal”
SangEun Han and Yong Baek Kim
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
1. TIGHT-BINDING HAMILTONIAN FOR p -ORBITALS Here, we introduce the tight-binding Hamiltonian of the fermions with p -orbitals on the square lattice. First, thenearest neighbor hopping Hamiltonian is given by H NN = (cid:88) i t σ ( c † x,i +ˆ x c x,i + c † y,i +ˆ y c y,i + h.c.) + (cid:88) i t π ( c † x,i +ˆ y c x,i + c † y,i +ˆ x c y,i + h.c.)= (cid:88) k (cid:16) ˆ c † x,k ˆ c † y,k (cid:17) (cid:18) t σ cos( k x ) + 2 t π cos( k y ) 00 2 t σ cos( k y ) + 2 t π cos( k x ) (cid:19) (cid:18) ˆ c x,k ˆ c y,k (cid:19) , where h.c. stands for the hermitian conjugate and c ( x,y ) ,i and c ( x,y ) ,k are the fermion annihilation operators in theposition and momentum space for x ( y ) orbital, respectively. t σ and t π are the hopping parameters for the σ and π bonds.The next nearest neighbor hopping Hamiltonian is given by H NNN = (cid:88) i (cid:88) α = x,y ˜ t ( c † α,i +ˆ x +ˆ y c α,i + c † α,i − ˆ x +ˆ y c α,i + c † α,i +ˆ x − ˆ y c α,i + c † α,i − ˆ x − ˆ y c α,i + h.c)+ (cid:88) i (cid:88) α (cid:54) = β ˜ t ( c † α,i +ˆ x +ˆ y c β,i − c † α,i − ˆ x +ˆ y c β,i − c † α,i +ˆ x − ˆ y c β,i + c † α,i − ˆ x − ˆ y c β,i + h.c)= (cid:88) k (cid:16) ˆ c † x,k ˆ c † y,k (cid:17) (cid:18) t cos( k x ) cos( k y ) − t (cid:48) sin( k x ) sin( k y ) − t (cid:48) sin( k x ) sin( k y ) 4˜ t cos( k x ) cos( k y ) (cid:19) (cid:18) ˆ c x,k ˆ c y,k (cid:19) , where ˜ t and ˜ t (cid:48) are the hopping parameters for the next nearest neighbor hopping. The total tight-binding Hamiltonianis H tb = (cid:88) k (cid:16) ˆ c † x,k ˆ c † y,k (cid:17) H ( k ) (cid:18) ˆ c x,k ˆ c y,k (cid:19) , where H ( k ) = (cid:18) t σ cos( k x ) + 2 t π cos( k y ) + 4˜ t cos( k x ) cos( k y ) − t (cid:48) sin( k x ) sin( k y ) − t (cid:48) sin( k x ) sin( k y ) 2 t σ cos( k y ) + 2 t π cos( k x ) + 4˜ t cos( k x ) cos( k y ) (cid:19) . Then, near k = (0 , H =2( t σ + t π + 2˜ t ) σ + (cid:18) − ( t σ + 2˜ t ) k x − ( t π + 2˜ t ) k y − t (cid:48) k x k y − t (cid:48) k x k y − ( t π + 2˜ t ) k x − ( t σ + 2˜ t ) k y (cid:19) = E Γ σ + (cid:0) A ( k x + k y ) σ + A x (2 k x k y ) σ x + A z ( k x − k y ) σ z (cid:1) where E Γ ≡ t σ + t π + 2˜ t ), A = − ( t σ + t π + 4˜ t ) / A x = − t (cid:48) , and A z = ( t π − t σ ) a /
2. DEFINITION OF F i FUNCTIONS
The functions F ω, ,x,z in Eq. 9 and 10 in the main text are defined as F ω ( a , a x , a z ) = (cid:90) ∞−∞ dω π (cid:90) π/ dθ ω S ( θ )( ω + a + a x S ( θ ) + a z C ( θ ))( ω + S ( θ ) / ( ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) (S1) F ( a , a x , a z ) = (cid:90) ∞−∞ dω π (cid:90) π/ dθ ω ( − ω + 3 S ( θ ) / ω + a − a x S ( θ ) − a z C ( θ ))( ω + S ( θ ) / ( ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) , (S2) F x ( a , a x , a z ) = (cid:90) ∞−∞ dω π (cid:90) π/ dθ ω S ( θ )( − ω + S ( θ ) / ω − a + a x S ( θ ) + a z C ( θ ))( ω + S ( θ ) / ( ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) , (S3) F z ( a , a x , a z ) = (cid:90) ∞−∞ dω π (cid:90) π/ dθ ω C ( θ )( − ω + 3 S ( θ ) / ω − a + a x S ( θ ) + a z C ( θ ))( ω + S ( θ ) / ( ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) , (S4) F g ( a , a x , a z ) = (cid:90) ∞−∞ dω π (cid:90) π/ dθ S ( θ ) / a + a ( ω − a x S ( θ ) − a z C ( θ )) − ( ω − a x S ( θ ) + a z C ( θ ))( ω + a x S ( θ ) + a z C ( θ )) )( ω + S ( θ ) / ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) − (cid:90) ∞−∞ dω π (cid:90) π/ dθ a S ( θ ) / ω + 2 ω (5 a x S ( θ ) − a z C ( θ )) + ( a x S ( θ ) + a z C ( θ ))( a x S ( θ ) − a z C ( θ )))( ω + S ( θ ) / ω + 2( a + a x S ( θ ) + a z C ( θ )) + ( a − a x S ( θ ) − a z C ( θ )) ) , (S5)where S ( θ ) = sin (2 θ ) and C ( θ ) = cos (2 θ ). Here, F ω, ,g are positive, but F z is negative for a > a x , a z .When a z → F ω, ,x,g can be written as F ω ( a , a x , ≡ − a − a x ((1 − a ) − a x )((1 + 2 a ) − a x ) (S6)+ (1 − a x ) tan − (cid:20) − a − a x √ a − (1 − a x ) (cid:21) (4 a − (1 − a x ) ) / + (1 + 2 a x ) tan − (cid:20) − a +2 a x √ a − (1+2 a x ) (cid:21) (4 a − (1 + 2 a x ) ) / , (S7) F ( a , a x , ≡ − a + (1 − a x ) (1 − a x ) + 16 a (9 + 20 a x ) + 8 a ( − − a x + 16 a x )4 a (16 a + (1 − a x ) − a (1 + 4 a x )) (S8) − (1 + 8 a + 2 a x − a x ) tan − (cid:20) − a − a x √ a − (1 − a x ) (cid:21) (4 a − (1 − a x ) ) / − (1 + 8 a − a x − a x ) tan − (cid:20) − a +2 a x √ a − (1+2 a x ) (cid:21) (4 a − (1 − a x ) ) / , (S9) F x ( a , a x , ≡ − a − a x + 48 a x − a (1 + 12 a x ))(16 a + (1 − a x ) − a (1 + 4 a x )) (S10) − a + a x (1 + 2 a x )(1 − a x ) + 4 a (1 + a x − a x )) tan − (cid:20) − a − a x √ a − (1 − a x ) (cid:21) a x (4 a − (1 − a x ) ) / (S11)+ 2(8 a − a x (1 − a x )(1 + 2 a x ) + 4 a (1 − a x − a x )) tan − (cid:20) − a +2 a x √ a − (1+2 a x ) (cid:21) a x (4 a − (1 + 2 a x ) ) / , (S12) F g ( a , a x ,
0) = F ω ( a , a x , . (S13)
3. RG FLOW EQUATIONS OF PARAMETERS
From the one-loop corrections introduced in the main text, we obtain the following RG flow equations for theparameters of the model, 1 A i dA i d(cid:96) =( z − − α g ( F ω − F i ) , (S14)1 u dud(cid:96) =( z − , (S15)1 g dgd(cid:96) = 12 (3 z − d − − α g ( F ω − F g ) . (S16)Combining the RG flow equations above, we obtain the RG flow equations of the dimensionless parameters,1 a i da i d(cid:96) = − α g ( F ω − F i ) , (S17) (a) (b) (c) FIG. S1. (a,b) The RG flows for some initial values of the dimensionless parameters. (a) The initial value ( a , a x , a z , α g ) =(3 , , , . a ∗ , a ∗ x , a ∗ z , α ∗ g ) = (1 . , , , . a , a x , a z , α g ) = (3 , , , . a ∗ , a ∗ x , a ∗ z , α ∗ g ) = (1 . , , , . a and a x for an arbitrary fixed α g when a z = 0. The red point represents the stable fixed point values of a and a x , ( a ∗ , a ∗ x ) = (1 . , α g dα g d(cid:96) =(2 − d ) − α g ( F ω − F g ) . (S18)As mentioned in the main text, the RG flow equations Eq. S17 and S18 lead to a line of stable fixed points,( a ∗ , a ∗ x , a ∗ z , α ∗ g ) = (1 . , , , α ∗ g ), where α ∗ g depends on the initial values of the dimensionless parameters. Forexample, the RG flows for several initial values of the parameters are shown in Fig. S1(a) and S1(b). The fixed pointvalues of the parameters in Fig. S1(a) and S1(b) belong to the line of stable fixed points. As a z → a and a x approach the fixed point values, ( a ∗ , a ∗ x ) = (1 . ,
4. ANOMALOUS DIMENSION BY FIELD-THEORETICAL RG
Here, using the field-theoretical RG, we find the anomalous dimension of the fermions from the one-loop fermionself-energy. The Callan-Symanzik equation for the fermion propagator is given by (cid:32) Λ ∂∂ Λ + β g ∂∂g + (cid:88) i β A i ∂∂A i + β u ∂∂u + η f (cid:33) G = 0 , (S19)where η f is the anomalous dimension of the fermions, and Λ is the UV cutoff. β g , β A i , and β u are the beta equationsof g , A i , and u , and they are defined as β g = Λ ∂g∂ Λ , β A i = Λ ∂A i ∂ Λ , β u = Λ ∂u∂ Λ , η f = − Λ ∂ ln Z − ψ ∂ Λ , (S20)respectively, and Z ψ is wavefucntion the renormalization constant of the fermions. Assuming that the parametersare near the fixed point values, we have β A i = β u = β g = 0. Then, Eq. S19 can be written as η f = − Λ G ∂G∂ Λ . Theone-loop fermion self-energy has the form, Σ( iω n ) ≈ − iω n α g C log ln (cid:16) | ω n | u Λ (cid:17) + regular + · · · , as shown in the maintext (where C log > Λ G ∂G∂ Λ = − α g C log . Hence the anomalous dimension of the fermions is η f = 2 α g C log . For example, when we have the fixed point value ( a , a x , a z ) = (1 . , , C log = 0 . η f = 0 . α g . Note that along the line of stable fixed points, C log does not change.From the perturbative Wilsonian RG, we can also obtain the anomalous dimension of the fermions, η f = − Λ ∂ ln Z − ψ ∂ Λ = α g F ω , where Z ψ = 1 + α g F ω ln(Λ /µ ) and F ω (1 . , ,
0) = C f = 0 . C log ≈ C f .
5. ONE-LOOP FERMION SELF-ENERGIES FROM SMALL AND LARGE MOMENTUM TRANSFER
The one-loop fermion self-energy for the small momentum transfer within the same patch isΣ small ( iω m )= g (cid:90) Λ G ( i Ω m + iω n , p ) D ( i Ω m , p ) F ( p )= g (cid:90) Λ dp x dp y (2 π ) (cid:90) u Λ − u Λ d Ω m π − i (Ω m + ω n ) + A p p x p y Ω m + u p x p y = g Λ π u (cid:90) Λ0 pdp Λ (cid:90) u Λ − u Λ d Ω m u Λ (cid:90) π dθ π − i (Ω m + ω n ) /u Λ + ( A /u )( p/ Λ) ( p/ Λ) sin (2 θ ) / m /u Λ ) + ( p/ Λ) sin (2 θ ) / α g u Λ (cid:90) dx (cid:90) − d Ω (cid:90) π dθ π − i (Ω + ( ω n /u Λ )) + a x x sin (2 θ ) / + x sin (2 θ ) / α g u Λ (cid:90) dx (cid:90) − d Ω 1 − i (Ω + ( ω n /u Λ )) + a x (cid:32) − | Ω | (cid:112) Ω + x / (cid:33) (S21) ≈ α g (cid:18) C small R , | ω n | + C small R , ω n u Λ + C small I , iω n + C small I , log iω n ln (cid:16) | ω n | u Λ (cid:17)(cid:19) , (S22)where x ≡ ( p/ Λ) and Ω ≡ Ω m /u Λ . The coefficients C small i ’s depend on a . The logarithmic correction comes fromthe last term in Eq. S21.The one-loop fermion self-energy for the large momentum transfer between different patches isΣ large ( iω m )= g (cid:90) Λ G ( i Ω m + iω n , p ) D ( i Ω m , p − k ) F ( p − k )= g (cid:90) Λ dp x dp y (2 π ) (cid:90) u Λ − u Λ d Ω m π − i (Ω m + ω n ) + A p ( k ) p y Ω m + 4 u sin ( k ) p y = g π u u Λ (cid:90) Λ0 pdp Λ (cid:90) u Λ − u Λ d Ω m u Λ (cid:90) π dθ π − i (Ω m + ω n ) /u Λ + ( A /u )( p/ Λ) − sin ( k ) sin θ ( p/ Λ) (Ω m /u Λ ) + 4Λ − sin ( k ) sin θ ( p/ Λ) = α g u Λ (cid:90) dx (cid:90) − d Ω (cid:90) π dθ π − i (Ω + ( ω n /u Λ )) + a x ξx sin θ Ω + ξx sin θ = α g u Λ (cid:90) dx (cid:90) − d Ω 1 − i (Ω + ( ω n /u Λ )) + a x (cid:32) − | Ω | (cid:112) Ω + ξx (cid:33) (S23) ≈ α g (cid:18) C large R , / | ω n | / ( u Λ ) / + C large R , ω n u Λ + C large I , iω n + C large I , / i sgn( ω n ) | ω n | / ( u Λ ) / (cid:19) , (S24)where ξ ≡ − sin ( k ) and k = (2 k , C large i ’s depend on a and ξ . The non-analytic | ω n | / comes form the last term in Eq. S23. Note that in the real parts of Σ small and Σ large have constant terms whichdepend on the cutoff, but we drop those.The comparison between the numerical values and fitting functions of Σ small ( iω n ) and Σ large ( iω n ) for a = 1 . ξ = 1 is shown in Fig. S2. The coefficients are C small R , = − . C small R , = − . C small I , = 0 . C small I , log = − . C large R , / = − . C large R , = 1 . C large I , = 0 . C large I , / = − . iω n → ω + iη ,Σ small ( ω ) ≈ α g (cid:18) C small R , ω + C small R , log ω ln (cid:18) | ω | u Λ (cid:19) + C small R , ω u Λ + C small I , iω (cid:19) , (S25)Σ large ( ω ) ≈ α g (cid:18) C large R , ω + C large R , / ω / ( u Λ ) / + C large R , ω u Λ + C large I , / iω / ( u Λ ) / (cid:19) . (S26) × -4 × -4 × -4 × -4 × -5 -2 × -5 -3 × -5 -4 × -5 -5 × -5 -6 × -5 (a) × -4 × -4 × -4 × -4 × -4 × -4 (b) × -4 × -4 × -4 × -4 × -6 -10 × -6 -15 × -6 (c) × -4 × -4 × -4 × -4 × -4 × -4 × -4 (d) FIG. S2. Comparison between the numerical calculation and the functional forms for Σ small ( iω n ) and Σ large ( iω n ) when a =1 . ξ = 1. (a), (b) The real and imaginary parts of Σ small ( iω n ). The red circles and diamonds are the numerical valuesand solid lines stand for the fitting of the functional forms. (c), (d) The real and imaginary parts of Σ large ( iω n ). The bluecircles and diamonds are the numerical values and solid lines stand for the fitting of the functional forms. (a-d) are well-fittedby Eq. S22 and S24. The total one-loop fermion self-energy isΣ tot ( ω ) ≈ α g (cid:18) C tot R , ω + C tot R , log ω ln (cid:16) | ω | u Λ (cid:17) + C tot R , / ω / ( u Λ ) / + C tot R , ω u Λ + C tot I , iω + C tot I , / iω / ( u Λ ) / (cid:19) . (S27)
6. BOSON SELF-ENERGY IN MULTI-PATCH MODEL
Here, we show that both the small and large momentum transfer processes do not renormalize the bosonic excitationsin the multi-patch model. The one-loop boson self-energy for the small momentum transfer is given byΠ( iω n , q ) = − g q x q y (cid:90) Λ d p (2 π ) (cid:90) ∞−∞ d Ω m π G ( i Ω m + iω n , p + q ) G ( i Ω m , p )= − g q x q y (cid:90) Λ d p (2 π ) (cid:90) ∞−∞ d Ω m π − i (Ω m + ω n ) + A ( p + q ) − i Ω m + A p =0 . In terms of Ω m , the integrand has poles at − ω n − iA ( p + q ) and − iA p . If we choose to close the d Ω m contour inthe upper half-plane, the integral becomes zero because two poles are in the lower half-plane. Therefore, Π = 0.The one-loop boson self-energy for the large momentum transfer is given byΠ( iω n , q ) = − g q x q y (cid:90) Λ d p (2 π ) (cid:90) ∞−∞ d Ω m π G ( i Ω m + iω n , p + k + q ) G ( i Ω m , p )= − g q x q y (cid:90) Λ d p (2 π ) (cid:90) ∞−∞ d Ω m π − i (Ω m + ω n ) + A ( p + k + q ) − i Ω m + A p = 0 ..