Novel Quantum Criticality in Two Dimensional Topological Phase transitions

Topological quantum phase transitions intrinsically intertwine self-similarity and topology of many-electron wave-functions, and divining them is one of the most significant ways to advance understanding in condensed matter physics. Our focus is to investigate an unconventional class of the transitions between insulators and Dirac semimetals whose description is beyond conventional pseudo relativistic Dirac Hamiltonian. At the transition without the long-range Coulomb interaction, the electronic energy dispersion along one direction behaves like a relativistic particle, linear in momentum, but along the other direction it behaves like a non-relativistic particle, quadratic in momentum. Various physical systems ranging from TiO2-VO2 heterostructure to organic material α-(BEDT-TTF)2I3 under pressure have been proposed to have such anisotropic dispersion relation. Here, we discover a novel quantum criticality at the phase transition by incorporating the long range Coulomb interaction. Unique interplay between the Coulomb interaction and electronic critical modes enforces not only the anisotropic renormalization of the Coulomb interaction but also marginally modified electronic excitation. In connection with experiments, we investigate several striking effects in physical observables of our novel criticality.

In this supplemental method, we present the detailed calculation of the polarization, or the boson self-energy Π(k) appearing in the main text. For convenience, we assume N -copies of the electrons coupled to the gauge boson. If we are interested in the only one copy, i.e., physical limit, one can simply take N = 1 at the end of the calculation.

A. Deivation of Polarization
In the leading order in 1/N expansion, the boson selfenergy is simply the two-leg one-loop diagram. The fermion propagator appearing in the loop diagram is where H(k) is the free fermion Hamiltonian equation (6) in the main text. It is convenient to write the propagator in the following form g f (k, ω) = 1 −iω + H(k) , with the energy E s (k) = sE(k) = s v 2 k 2 x + A 2 k 4 y and the projection P s (k) Now the polarization is given by Performing the contour integral over ω and the trace Tr[·], we finally have in which P(k, p) is the projection operator where 1 (k) = vk x and 2 (k) = Ak 2 y are the appropriate functions.

B. Calculation of Polarization
To evaluate the function Π(k) ≡ Π(k, ω = 0) in equation(5), we use the change of variables as following: which gives rise to the following expression.

II. SCREENING EFFECTS IN QUANTUM CRITICALITIES
Here, we summarize bare and self-energy of Coulomb boson propagators in various quantum criticalities. The one-loop boson self-energy is obtained in the standard way. With different electron Hamiltonian for each case, we obtain the self-energy which is illustrated in Table I.

III. FERMION SELF-ENERGY AND RG EQUATIONS IN STRONG-COUPLING LIMIT
Here we compute the fermion self-energy at the large-N limit to extract the RG equations. We start with the expression for the fermion self-energy with the corrected boson propagator .
To extract out the correction to the bare parameters {v, A}, we expand the Σ f (k, ω) in k near k = 0, Here {Ĩ x ,Ĩ y } are the integrals Apparently, the integrals are divergent and needed to be regulated. We regulate the integrals by where the UV and IR cutoffs {Λ, µ} are introduced. With the expression equation (15), we find A. Strong-coupling limit We first evaluate the integrals equation(16) when α → ∞.
in which the cutoff scheme equation (17) is assumed. We demonstrate the detail of evaluations only forĨ x . where The integral is well-defined in that it has no divergence in the integral because The integral can be evaluated numerically by using the numeric value of G(ξ), and we have obtained J x ≈ 0.18. The numeric value is not important but it is important to remember J x > 0 because of G(|ξ|) > 0. Similarly, we can writeĨ y as following.
where J y is the definite integral The integeral is well-defined because Hence there is no more IR divergence than log(Λ/µ) appearing in equation(23) in the renormalization of A in the strong-coupling limit. The numeric value of J y is obtained as 0.03. Hence, under the RG step, we find that the velocity v and the inverse mass A are renormalized as follows.
from which we deduce the RG flow equations, which have been shown in the main text. Notice that the frequency dependence is trivial since only the instantaneous Coulomb propagator is used. If one incorporates frequency dependence, only the numeric numbers J x , J y are modified but not the structure of our calculation. The absence of the frequency dependence indicates no vertex correction by the Ward identity.

B. Finite Coupling Constant
Here we will discuss the fate of the finite coupling constant α which is now slightly away from the strongcoupling limit. In the prsence of the finite coupling constant α, the bare term ∼ |q| in the boson propagator cannot be ignored. Hence the above calculations of α → ∞ should be properly changed to see the scaling behaviors of the parameters {v, A}.
Performing the straightforward calculation as the above, we find The integrals {K x , K y } are whereĀ(q x ) = Aqx v is the dimensionless number depending on the momentum q x .
Before marching further into the details, we first make an assumption,Ā(q x ) <Ā(Λ) 1. The condition A(Λ) 1 should be interpreted as following. First of all, vΛ can be roughly thought as the bandwidth where the low-energy Hamiltonian equation (6) in the main text is the qualitatively correct description of the system. Note that A determines the curvature along k y and v determines the velocity along k x . ThusĀ(Λ) 1 means that the velocity along k y is always much smaller than the velocity along k x within the energy E ≤ vΛ, i.e., the dispersion along the k y direction is always much flatter than that along the k x direction. In other words, the spectrum is extremely anisotropic in that the equal energy contour is extended along k y and shrinked along k x .
At this stage, the assumptionĀ(Λ) 1 looks like an ad-hoc assumption to evaluate the integrals equation(30). However, employeeing the momentum-shell RG approach at the weak-coupling limit α → 0 (as present in the main text and the next supplemental method), we can show A 1 at the fixed point. With this condition in hand, we proceed to evaluate the integrals equation(30) approximately by takingĀ → 0, whereJ x > 0 andJ y are the constants depending on α whose detailed calculation can be found below. From these, we find Hence we find that Thus the parameters {v, A} in the bare fermion Hamiltonian equation (6) in the main text receive the logarithmic corrections, and the coupling constant α decreases along the renormalization group process.

Calculation ofJx andJy
Below we only present the detail for evaluatingJ x but it is straightforward to generalize toJ y .
in whichĀ(q x ) = A v q x and q y = vqx A ξ. We continue to evaluate this Now assumingĀ(q x ) <Ā(Λ) 1 which we will justify in the low-energy limit later, we can ignore the dependence onĀ in the integral to obtaiñ .

(37)
This integral is well-defined and finite as On the other hand, we can evaluateJ y similarly in A 1 and obtainJ in which The integral is finite and well-defined as Hence there is no more divergence in the renormalization in the inverse mass than ∝ log( Λ µ ) in equation(39).

IV. DETAILED CALCULATION OF MOMENTUM SHELL RG
In this supplemental method, we will perform the oneloop RG by the momentum-shell method. We start with the following bare action S = S ψ + S φ + S ψ,φ present in the main text, where η is introduced to facilitate the engineering dimensional analysis of the anisotropic scaling along x and y in the bare boson propagator. The physical value of η, as mentioned in the main text, is 1 because the (bare) Coulomb interaction is isotropic. On the other hand, we have introduced κ into the bare action to allow the possible anomalous screening term ∼ |q x |, via the anomalous scaling dimension appearing in ∼ q 2 x term, under the RG steps.
We start with the dimensional analysis. First of all, we set the engineering dimensions [x] = −z 1 , [y] = −1 and [τ ] = −z 2 . Then it is straightforward to show where Λ is the momentum cutoff in the momentum q y .
Here the fields have the following dimensions.
On the other hand, in terms of physical unit, we have The field operators have the following scaling dimensions.
From the above dimensional analysis we have the following dimensionless constants, which will be used to parametrize the renormalization flow Before proceeding to the details of the RG calculation, we first note that η and e are protected under the RG process due to the non-analytic structure of the propagator of the boson and the gauge invariance. Hence, the only non-trivial corrections are the renormalizations of {v, A} in the fermion propagator and κ in the boson propagator. The renormalization can be deduced from the self-energies of the fermion and the boson, which we calculate below. We start with the fermion self-energy.
such that where the integrals should be properly regularized as following.
as the integrands of equation (49) are even under q i → −q i , i = x, y. Here µ = Λe −l ≈ Λ(1 − l) with l 1 is assumed by following the standard strategy of the momentum-shell calculations.
With the assumptionĀ 1 which will be justified selfconsistently in the end of the calculation, we can evalute the integrals straightforwardly as following. where We remark that in γ → 0 limit (κ → 0), the inverse mass correction F 2 diverges logarithmically in ∼ log γ. This indicates instability of the non-interacting fermion ground state with the bare Coulomb interaction without operator insertions. We now compute the polarization by the momentumshell integral method in which Now we have the renormalized action S = S ψ + S φ + S φ,ψ obtained by integrating out the high-energy modes as the following. We start with the fermion part S ψ where On the other hand, the boson part S φ is where On the other hand, the vertex correction vanishes and thus S φ,ψ is the same as S φ,ψ from equation (42).
From these, we can obtain the flow equations.
We now derive the following renormalization flow equations The fixed point of these equations is (α, γ, β) → (0, 0, 3). We plot the numerical solutions also in Fig.2. This also implies that (αF where we have used η = 1 for physical limit and (1 + is the scaling factor of q x . Now we justifyĀ 1 which has been used in the derivations of the fermion self-energy. We calculate and we can approximate J 2 = J 1 ≈ 0 near the fixed point. Then we clearly have which dictates thatĀ → 0 1 at the fixed point.

V. STABILITY OF NEW FIXED POINT
In this supplemental method, we diagnose the stability and the scaling properties near the new intermediate in which where the momentum integral q is regularized as in equation (50).
After the straightforward algebra withĀ 1 near the fixed point, we find where From these, we can derive the flow equations The attractive fixed point of these flow equations is The numerical solution can be found in Fig.3. Hence we find that the velocity and inverse mass at the energy scale µ are where 8. This logarithmic divergence as µ → 0 is reminiscent of graphene case.

VI. DETAILED CALCULATION OF SCREENING CHARGE
In this supplemental method, we show the detailed calculation of the screening charge dictated by equation (18) in the main text and equation (20) in the main text when a single impurity charge Z is introduced at r = 0. We calculate the free and interacting cases at the level of the linear response theory. The details of the calculation here will closely follow Biswas, Son, and Sachdev 8 .

Free Theory
We first compute the screening charge of the noninteracting theory from equation (7) in the main text. The screening charge is given by in which D 0 (q) = 1 |q| and Π 0 (q) is the one-loop polarization. The real space configuration of the screening charge, we perform the Fourier transformation We will be mainly interested in the directional behavior of the scaling in the integrated screening charges Q x (x) and Q y (y) alongx andŷ, which are obtained from We now evaluate these screening charges. We start with Now from we can take the derivative with respect to x on the both sides. After the straightforward calculation, we have Similarly, we have the following expression for the other direction By plugging we find that Hence we see that the free theory has completely different scaling behaviors of the integrated screening charge alonĝ x andŷ directions, i.e., whose sign is the opposite of the impurity charge Z.

Interacting Theory
Now we use the non-trivial anisotropic fixed point equation (18) in the main text obtained from the interactions between the fermion and the boson. We will use the renormalized boson propagator dictated from equation (18) in the main text and the scaling behavior equation (20) in the main text to deduce the asymptotic behaviors of the integrated screening charges.
From this, we now calculate the integrated screening charge Q x (x). To calculate it, we need in which {v(q), A(q)} are the running parameters at the scale q. From equation(71), we have where {v 0 , A 0 } are the bare parameters. Hence we find where we have v(q x ) = v 0 (1 + αC f log( 1 r0qx )) with the r −1 0 ∼ E 0 , the bandwidth of the bare action equation (6) in the main text. From the relationship we find whose sign is the same as the impurity charge Z. From we find ρ ind (0, q y ) = − e 2 Zc y 4π 2 v(q y ) .
Thus we see that whose sign is the same as the impurity charge Z.

VII. DETAILED CALCULATION OF OPTICAL CONDUCTIVITY
In this supplemental method, we compute the optical conductivity σ jj (Ω), j = x, y at the level of the linear response theory. The expression for the optical conductivity is given by in which Π R ii (Ω + iδ) ≡ Π ii (iΩ → Ω + iδ) where the latter is obtained from the imaginary-time formalism. Hence we need to compute in which M j is the appropriate vertex along the jdirection and the factor P k,α = 1 comes from the Green's function g f ( k, ω). By performing summation over {α, β}, it is straightforward to find Here we perform the Wick rotation iΩ → Ω + iδ.
Now it is straightforward to take the imaginary part from this expression. With the assumption Ω > 0, we have which can be evaluated by performing elementary integrals over k. By combining with equation(91), we finally obtain Now we evaluate this for M x = veσ x and M y = Aek y σ y to calculate σ xx (Ω) and σ yy (Ω) for this anisotropic semimetal.
Plugging M x = veσ x into equation(96), we have By the change of variables ζ x = vk x and ζ y = Ak 2 y and then performing the trace, we have Performing the change of the variables ζ x = Ωx and ζ y = √ Ωy, we find with the constant I xx defined as following We can plug M y = Aek y σ y to calculate σ yy (Ω). After the straightforward calculation, we find where I yy is a constant In summary, we found anisotropic scalings in the optical conductivities alongx andŷ directions

VIII. INDEPENDENCE OF CUTOFF SCHEME AT QCP
In this supplementary material, we show clearly that the RG flow equations near the QCP in V, i.e., Eq (8) in main text and Eq(69) of V, are cutoff scheme independent. To show this, we first note that the only nontrivial renormalization to the bare action (60) is from the fermion self-energy, and thus we only need to show the independence of the cutoff scheme in the renormalization of the velocity v and the mass A, which can be calculated from We introduce a soft cutoff function J( qy Λ ) = J(− qy Λ ) which is a smooth function (which is monotonically decreasing for q y > 0) satisfying Then the fermion self-energy can be written as We will explicitly demonstrate that the renormalizations to the velocity v and mass A are independent of the detailed form of the function J(·), i.e., the flow equations near the QCP (see V), are independent of the precise form of the cutoff function J(·). Note that the flow equations in V are originally obtained by using the hard cutoff function.
A. Correction to the velocity v: we first calculate the correction to the velocity. We only need to calculate where We now perform the derivative D Λ on the self-energy to find Hence we calculate To evaluate the integral, we first perform the change of the variable y = qy Λ to find Next we scale out q x → q x y to find where the integral over y can be performed analyticall to find which is nothing but the momentum-shell integral for the renormalization of the velocity v obtained in (66) of V, i.e., I 1 in Eq.(66). Notice that the expression is manifestly independent of the precise form of J(·). After successive change of variables as in V, we can show that the same integral f 1 [κ] in Eq.(107) appears in the renormalization of the velocity v. B. Correction to the mass A: we next calculate the correction to the mass A. After the straightforward series expansion of (109) for k = (0, k y ) to O(k 2 y ) and then taking D Λ , we find where {K 1 , K 2 , K 3 } are the following integrals to be evaluated.
in which f (q) = Aq 2 y √ v 2 q 2 x +A 2 q 4 y and we have used the nota- and similarly for f (x), f (x). For K 2 , we have and, for K 3 , we have Performing the change of the variable q y = Λy and then scaling q x → q x y, we can show that On the other hand, we find By performing the integral over y explicitly, we find which is nothing but the momentum-shell integral for the renormalization of the mass A obtained in (66) of V, i.e., I 2 in Eq.(66). Notice that the expression is manifestly independent of the precise form of J(·). After successive change of variables as in V, we can show that the same integral f 2 [κ] in Eq.(107) appears in the renormalization of the mass A.

IX. WARD IDENTITY AND NON-RENORMALIZATION OF ELECTRIC CHARGE
Here, we discuss the Ward identity and its consequences in two dimensional systems with the Coulomb interaction, and we also discuss difference between our work and Isobe et. al. 9 . The effective action after redefinition eφ → φ is where S ins represents an additional inserted operator. For the discussion of the Ward identity, the inserted operator part is not important but it becomes crucial to investigate the stability of the non-interacting fixed point. Note that the action has gauge-invariance under After incorporating quantum fluctuation, the effective action becomes Notice that non-analytic dependence of the 1 e 2 |q| term in q prohibits its renormalization under integrating out the high-energy modes. Here we did not explicitly write out the renormalized HamiltonianH(−i∇) as it is not important in the discussion of the Ward identity.
It is clear that the gauge invariance gives the Ward identity, Moreover the non-renormalizable e 2 gives that the correction terms are absorbed by redefining ψ, ψ = Z ψ ψ r with Z −1 ψ = 1 + δ ω , and parameters in the Hamiltoniañ H(−i∇) such as velocity and effective mass.
Therefore, in our calculation, the electric charge is always non-renormalizable but eZ ψ receives correction even though it appears in higher loop calculation with the bare Coulomb interaction. This is one main difference between work by Isobe et. al. 9 and ours. In their work, they claim electric charge (g in their notation) is renormalized but gZ (eZ ψ in our notation) is not.
Another related difference between our work here and theirs 9 is their emphasis on the wavefunction renormal- However, in the strongcoupling limit αN → ∞, it is not difficult to see that contains the log-squared divergence (see below for the detail), which originates from the poor screening of the Coulomb interaction at finite frequency with zero momentum. In this limit, the only effect of the Coulomb interaction is to rotate the phase of the fermions as well explained in Son 1 and thus should not be taken as physical. Such log-squared dependence is also present in Isobe et.al. 9 in the limit αN → ∞ and taken to calculate the fermion wavefunction renormalization Z. They interpret the wavefunction renormalization as single-particle residue. But it cannot be adiabatically connected to one of the strong coupling fixed point due to the additional divergence.
One can see another difference from insertion operators S ins . As shown above, the perturbative RG with the bare Coulomb potential without the inserted operator gives log-squared divergence in the inverse effective mass correction. This log-squared correction indicates necessity of additional operators. In our calculation, we insert the operator which is natural in the Wilsonian RG, while they 9 insert an approximated polarization function even for weak coupling analysis, We believe that the insertion of an approximated polarization function is questionable in the weak coupling limit since it is not controlled by the 1 N f factor, which disappeared in their weak coupling analysis.
Note that the insertion of the approximated polarization which is contributed by the fermion modes from all energy scales is valid in strong coupling limit since it is well controlled by the 1 N f factor. In the current problem, the strong coupling limit is indeed controlled, and all physical quantities receives 1 N f corrections which are well captured in Isobe et. al. and ours with qualitative difference from different approximation. However, as shown in the main text, the strong coupling fixed point becomes immediately unstable due to the anomalous dimension of the velocity, thus the validity of the large-N f expansion is not guaranteed in the infrared limit.
We also believe that unusual cutoff (scale) dependence of physical quantities in their analysis (for example, log α in RG equations) indicates their RG scheme is not conventional and it cannot be perturbatively obtained from the non-interacting theory. On contrary, in our analysis, all physical quantities receive logarithmic cutoff (scale) dependence as ones of quantum criticality described by renormalizable field theories.
Apart from the differences, we find the ground state and its excitation in the IR limit are marginally stable, so-called marginal Fermi liquid, which is also obtained by Isobe et. al. 9 in spite of different renormalization of the inverse mass.

A. Log-squared Divergence in Fermion Self-Energy
Here we present the log-squared divergence in the strong-coupling limit, αN f → ∞, where g −1 b (q, Ω) → −Π(q, Ω) by keeping the dependence in Ω.
Such log-squared divergence has been observed in the graphene case as discussed by Son 1 . The fermion selfenergy in Son is given by where with q = |(q 0 , q)|, the size of the three-component vector (q 0 , q), which is not to be confused with the size of the spatial two-component vector |q| = |(q x , q y )|. Naively, this expression is single-log divergent, but it is in fact log-squared divergent. The log-squared divergence is apparent in the limit |q| q0 → 0 because Eq.(134) can be approximated which has the two sources of the logarithmic divergence, one from the integral over q 0 and the other from the integral over q.
We expect the similar log-squared divergence to be present if we keep the dependence of the polarization Π(q, Ω) in Ω. To exhibit this explicitly, we reconsider the fermion self-energy and the boson polarization in the strong-coupling limit α → ∞. Here we show the logsqaured divergence in Σ 0 where Here we work in the unit v = A = 1 for convenience and the clarity of the discussion. We start from the expression where E q = (q 2 x + q 4 y ) 1/2 . In the limit Ω E q , we have , where a x ≈ 0.66 and a y ≈ 0.75 and we have used the following form of the polarization which is asymptotically correct as far as Ω E q . Now in Eq.(138), we can perform the change of the variables x = √ ax |Ω| 1/4 q x and y = √ a y |Ω| 1/4 q y to find which clearly exhibits the log-sqaured divergence. Now we show how we obtained the asymptotic expression of the boson self-energy Eq.(139). We start with the expression for the boson self-energy Eq.(5). By performing the following change of the variables k x = |Ω|x, k y = |Ω| 1/2 y, X = qx |Ω| and Y = qy √ |Ω| (remember that we took v = A = 1 for this subsection) and expanding Π to the lowest orders in X and Y , we find

X. WARD IDENTITY AND NON-RENORMALIZATION OF ELECTRIC CHARGE
Here, we discuss the Ward identity and its consequences in two dimensional systems with the Coulomb interaction, and we also discuss difference between our work and Isobe et. al. 9 . The effective action after redefinition eφ → φ is where S ins represents an additional inserted operator. For the discussion of the Ward identity, the inserted operator part is not important but it becomes crucial to investigate the stability of the non-interacting fixed point. Note that the action has gauge-invariance under After incorporating quantum fluctuation, the effective action becomes Notice that non-analytic dependence of the 1 e 2 |q| term in q prohibits its renormalization under integrating out the high-energy modes. Here we did not explicitly write out the renormalized HamiltonianH(−i∇) as it is not important in the discussion of the Ward identity.
It is clear that the gauge invariance gives the Ward identity, Moreover the non-renormalizable e 2 gives that the correction terms are absorbed by redefining ψ, ψ = Z ψ ψ r with Z −1 ψ = 1 + δ ω , and parameters in the Hamiltoniañ H(−i∇) such as velocity and effective mass.
Therefore, in our calculation, the electric charge is always non-renormalizable but eZ ψ receives correction even though it appears in higher loop calculation with the bare Coulomb interaction. This is one main difference between work by Isobe et. al. 9 and ours. In their work, they claim electric charge (g in their notation) is renormalized but gZ (eZ ψ in our notation) is not.
Another related difference between our work here and theirs 9 is their emphasis on the wavefunction renormalization Z = (1 + ∂Σ f (ω,k) ∂(iω) ) −1 . However, in the strongcoupling limit αN → ∞, it is not difficult to see that contains the log-squared divergence (see below for the detail), which originates from the poor screening of the Coulomb interaction at finite frequency with zero momentum. In this limit, the only effect of the Coulomb interaction is to rotate the phase of the fermions as well explained in Son 1 and thus should not be taken as physical. Such log-squared dependence is also present in Isobe et.al. 9 in the limit αN → ∞ and taken to calculate the fermion wavefunction renormalization Z. They interpret the wavefunction renormalization as single-particle residue. But it cannot be adiabatically connected to one of the strong coupling fixed point due to the additional divergence.
One can see another difference from insertion operators S ins . As shown above, the perturbative RG with the bare Coulomb potential without the inserted operator gives log-squared divergence in the inverse effective mass correction. This log-squared correction indicates necessity of additional operators. In our calculation, we insert the operator which is natural in the Wilsonian RG, while they 9 insert an approximated polarization function even for weak coupling analysis, S ins = −1 e 2 Π(q, ω)|φ(q, ω)| 2 .
We believe that the insertion of an approximated polarization function is questionable in the weak coupling limit since it is not controlled by the 1 N f factor, which disappeared in their weak coupling analysis.
Note that the insertion of the approximated polarization which is contributed by the fermion modes from all energy scales is valid in strong coupling limit since it is well controlled by the 1 N f factor. In the current problem, the strong coupling limit is indeed controlled, and all physical quantities receives 1 N f corrections which are well captured in Isobe et. al. and ours with qualitative difference from different approximation. However, as shown in the main text, the strong coupling fixed point becomes immediately unstable due to the anomalous dimension of the velocity, thus the validity of the large-N f expansion is not guaranteed in the infrared limit.
We also believe that unusual cutoff (scale) dependence of physical quantities in their analysis (for example, log α in RG equations) indicates their RG scheme is not conventional and it cannot be perturbatively obtained from the non-interacting theory. On contrary, in our analysis, all physical quantities receive logarithmic cutoff (scale) dependence as ones of quantum criticality described by renormalizable field theories.
Apart from the differences, we find the ground state and its excitation in the IR limit are marginally stable, so-called marginal Fermi liquid, which is also obtained by Isobe et. al. 9 in spite of different renormalization of the inverse mass.
A. Log-squared Divergence in Fermion Self-Energy Here we present the log-squared divergence in the strong-coupling limit, αN f → ∞, where g −1 b (q, Ω) → −Π(q, Ω) by keeping the dependence in Ω.