Multi-critical topological transition at quantum criticality

The investigation and characterization of topological quantum phase transition between gapless phases is one of the recent interest of research in topological states of matter. We consider transverse field Ising model with three spin interaction in one dimension and observe a topological transition between gapless phases on one of the critical lines of this model. We study the distinct nature of these gapless phases and show that they belong to different universality classes. The topological invariant number (winding number) characterize different topological phases for the different regime of parameter space. We observe the evidence of two multi-critical points, one is topologically trivial and the other one is topologically active. Topological quantum phase transition between the gapless phases on the critical line occurs through the non-trivial multi-critical point in the Lifshitz universality class. We calculate and analyze the behavior of Wannier state correlation function close to the multi-critical point and confirm the topological transition between gapless phases. We show the breakdown of Lorentz invariance at this multi-critical point through the energy dispersion analysis. We also show that the scaling theories and curvature function renormalization group can also be effectively used to understand the topological quantum phase transitions between gapless phases. The model Hamiltonian which we study is more applicable for the system with gapless excitations, where the conventional concept of topological quantum phase transition fails.


Motivation.
In this work, we are motivated to study the TQPT occurring between two gapless phases through a Lorentz symmetry breaking point. We consider transfer field Ising model (TFIM) with three spin interaction 42 , where the study of edge modes at criticality has revealed the appearance and disappearance of localized edge modes at one of the quantum critical lines with corresponding change in the parameter values 43 . In other words, both topological and non-topological characters appear on the same critical line for different parameter regimes. This provides an interesting platform to study TQPT between gapless phases as well as to understand the validity of characterizing tools in identifying this transition.
Motivation of this work is twofold. First is to prove that, indeed the critical line possess distinct gapless phases and there is a TQPT between these phases occurring through a multi-critical point which breaks the Lorentz invariance in our model Hamiltonian. Second one is to perform this using characterizing techniques that have been used to distinguish between gapped phases, thereby validating the reliability of these techniques to distinguish between gapless phases. We also show the relation between the breaking of Lorentz invariance and topological quantum phase transition at the multi-critical point. This phenomenon can be analogously understood from the topological semimetals, where the Dirac points confluence to form quadratic dispersion at a critical point which breaks the Lorentz symmetry 44,45 .
There are several studies on multi-critical behavior and topological transition using conventional RG techniques in the literature [46][47][48][49][50] . The conventional RG captures the physics of correlated topological systems with local Coulomb interaction in one, two and three dimensions. However, here we adopt CRG based on the diverging behavior of curvature function as we approach the topological quantum critical point. Since the curvature function encapsulates the topological signatures of the band structure, its prominent behavior near the transition point is promising and sufficient to address the unconventional topological transition between gapless phases in our model.

Model hamiltonian and topological quantum phase diagram
We consider transverse field Ising model with three spin interaction 42,51 where σ x,z are Pauli matrices. Performing Jordan-Wigner transformation σ x i = 1 − 2c † i c i and σ z i = − j<i (1 − 2c † j c j )(c i + c † i ) , the model Hamiltonian can be written in spinless fermionic form as where nearest neighbor superconducting gap is equal to nearest neighbor hopping amplitude 1 and next nearest neighbor superconducting gap is equal to next nearest neighbor hopping amplitude 2 . In this equation, c † i (c i ) is creation (annihilation) fermionic operator and h.c represents the Hermitian conjugate. It is a one-dimensional mean-field model for a triplet superconductor. The three spin interaction added to the transverse field Ising model can be physically realized in realistic Hamiltonians since the term is generated through real-space renormalization group treatments 42 .
The model has been studied previously in different contexts 42,43,51,52 . The model was first introduced by the authors of Ref. 42 to study the persistence of quantum criticality at high temperature in correlated systems. The authors of Ref. 52 has studied the physics of Majorana zero modes in the gapped phases of this model with both broken and unbroken time-reversal symmetry. One of the authors (S.S) has studied the quantization of geometric phase with integer and fractional topological characterization for this model in Ref. 51 . Very recently authors of Ref. 43 have solved the problem of bulk-boundary correspondence at the quantum critical lines and discussed the principle of least topological invariant number at the criticality.
In this work we intent to show explicitly that there exist a TQPT between two gapless phases (CP-1 and CP-2 in Fig. 1) on the critical line 2 = µ − 1 through a multi-critical point 1 = 2µ (point 'b' in Fig. 1). We also explore the nature of transition and critical behavior implementing the scaling law of critical theories and show that these characterizing tools, which are used to characterize the transition between gapped phases, are also efficient tools to characterize the TQPT between gapless phases.
There are two multi-critical points at the intersections of the critical lines. For the parameter value µ = 1 a multi-critical point with an emergent U(1) symmetry exist at ( 1 , 2 ) = (0, 1) 52 . This multi-critical point 'a' in the phase diagram ( Fig. 1) occurs at the intersection of the critical lines 2 = µ + 1 and 2 = µ − 1 . It posses linear spectra at the gap closing momenta k = 0 and k = ±π and does not break the Lorentz invariant. Since it does not involve any topological transition between gapless phases on a critical line, we consider it a trivial multicritical point. Another multi-critical point exist at ( 1 , 2 ) = (2, −1) . This multi-critical point 'b' in the phase diagram occurs at the intersection of critical lines 2 = µ − 1 and 2 = −µ . Since it posses quadratic spectra at k = 0 and breaks Lorentz invariance, we consider it to be a non-trivial multi-critical point. This is exactly the point 1 = 2µ , through which TQPT between gapless phases occur.
The transition can be verified by investigating behavior of pseudo spin-vector in the parameter space 51,53 . The model Hamiltonian can be expressed in terms of pseudo spin-vector as where χ z (k) = −2 1 cos k − 2 2 cos 2k + 2µ, and χ y (k) = 2 1 sin k + 2 2 sin 2k . The pseudo spin-vector takes a closed curve in the parameter space around the origin for a set of parameter values representing a gapped phase. For gapless phase the curve passes through the origin and this behavior is characteristic of criticality. In Fig. 2 we have shown the behavior of pseudo spin-vector in the parameter space on the critical line 2 = µ − 1 . The curve is always closed and passes through the origin indicating the criticality. As one goes from Fig. 2a-c, system is passing from topologically trivial gapless phase to non-trivial gapless phase through a multi-critical point (Fig. 2b). Trivial gapless phase is the phase boundary between w = 0 and w = 1 gapped phases, as well as, non-trivial gapless phase is the phase boundary between w = 1 and w = 2 gapped phases. The non-trivial gapless phase is characterized by the emergence of secondary loop which passes through the origin. Therefore this behavior of pseudo spin-vector suggest that there exist a TQPT between two gapless phases on the critical line 2 = µ − 1 . Figure 1. Topological phase diagram of model Hamiltonian for µ = 1 . Line 'ac' represents the critical line 2 = µ − 1 (blue line), line 'be' represents the critical line 2 = −µ (magenta line) and line 'ad' represents the critical line 2 = µ + 1 (red line). Points 'a' and 'b' are multi-critical points (green and black dots respectively) which differentiate between three distinct gapped phases with w = 0, 1, 2 (represented in different colors). Here CP-1 is critical/gapless phase for the transition between w = 0 and w = 1 . CP-2 is critical/gapless phase for the transition between w = 1 and w = 2.

Results and discussion
Energy dispersion and critical exponents. One can distinguish between the universality classes of the gapless phases by calculating the values of critical exponents. In this section we calculate the correlation length critical exponent ( ν ) and dynamical critical exponent (z) for the two gapless phases on the critical line The spectra of this model on the critical line 2 = µ − 1 is gapless and linear for 1 < 2µ , and quadratic for 1 ≥ 2µ . On the critical line 2 = −µ spectra has two gapless points at the two incommensurate momenta, ±k 0 , symmetric about the point k = 0 as shown in Fig. 3a-c. As we approach multi-critical point on this critical line, the two incommensurate points confluence at k 0 = 0 (i.e, ( 1 , 2 ) = (2, −1) ), as shown in Fig. 3d. Therefore the spectra is non-relativistic (breaks Lorentz invariance) and become quadratic in nature instead of linear. Energy dispersion for one dimensional system close quantum critical point can be written as E k = |g| 2νz + k 2z , where ν is correlation length critical exponent and z is dynamical critical exponent 23 . At the critical point the gap function � = |g| 2νz should go to zero, therefore E ∝ k z .
The energy dispersion expanded around the gap closing momenta k 0 = 0 can be written as where C 2 = (16 2 µ + 4 1 µ − 4 1 2 ) and C 4 = 1 3 ( 1 2 − 1 µ − 16 2 µ) . Gap function g 2νz = (2µ − 2 1 − 2 2 ) 2 implies νz = 1 . At QCP the gap function goes to zero and the shape of the spectra can be obtained as E k ∝ k z , by identifying the dominant coefficient among C 2 and C 4 . Above the multi-critical point (trivial gapless phase, i.e., 1 < 2µ ) one can observe that the coefficient of quadratic term C 2 is much larger than C 4 . Therefore quadratic term dominate implying E k ∝ k , hence z = 1 . Similarly below the multi-critical point (non-trivial gapless phase, i.e., 1 > 2µ ) one can find that C 4 dominates over C 2 and the spectra E k ∝ k 2 implying the value of z = 2 . At the multi-critical point (i.e, 1 = 2µ and 2 = −µ ) the coefficient  There are two gapless points around which the spectra is linear (i.e., E k ∝ k ) which implies z = 1 . (d) Spectra at multi-critical point with 1 = 2µ and 2 = −µ . Two gapless points confluence at k = 0 where the spectra is quadratic (i.e., E k ∝ k 2 ) and z = 2. www.nature.com/scientificreports/ Therefore the dynamical critical exponent is found to have z = 1 with linear spectra at the trivial gapless phase and z = 2 with quadratic spectra at transition point (multi-critical point) as well as non-trivial gapless phase.
Once the dynamical critical exponent z is obtained one can also obtain the value of correlation length critical exponent ν from the condition νz = 1 in our model. Thus in the trivial gapless phase the critical exponents are z = 1 and ν = 1 and in the non-trivial gapless phase z = 2 and ν = 1 2 . Note that the situation C 2 = C 4 is not possible on the critical line since it requires 1 to be complex. Equating C 2 and C 4 results in 1 ∝ 4µ − i √ 177µ , which is not possible in our model, implying This observation suggest that these two gapless phases belong to different universality classes since their critical exponents has different set of values. This entails the fact that there is a TQPT in the Lifshitz universality class with z = 2 and ν = 1 2 23,54,55 , between two distinct gapless phases through multi-critical point. Thus in this study the breaking of Lorenz invariance occurs at the Lifshitz universality class.
For completeness we also calculate the critical exponents for the critical theory at 2 = µ + 1 . The spectra on this line is found to be linear in k as shown in Fig. 4, which implies the value of dynamical critical exponent to be z = 1 . Spectra close to k 0 = ±π can be written as . At the QCP gap function goes to zero and coefficient C 2 dominates over C 4 , implying E k ∝ k . Therefore the spectra at the gap closing point is linear and dynamical critical exponent z = 1 . The gap function g 2νz = (2µ + 2 1 − 2 2 ) 2 implies ν = 1.
We have shown the breakdown of Lorentz invariant symmetry at the multi-critical point. The authors of Ref. [56][57][58] have shown explicitly that the break down of Lorentz invariance also occur for graphene and 3D Weyl semimetal. The authors of Ref. 59 have shown explicitly the transformation from the Dirac semimetal to band insulator QCP at = 0 , ( is the energy scale), where the quasiparticle spectra is two momentum space dimension. In x-direction, it is linear in k and in the y-direction it is quadratic ( k 2 ). But the model Hamiltonian which we have studied is one dimension, therefore only one component has appeared.
We confirm the results for our model by calculating the critical exponents from the Berry connection approach and also show the presence of TQPT between gapless phases using CRG analysis in the next section.
Curvature function renormalization group. At first, we briefly review the curvature function renormalization group (CRG) method which encapsulates the critical behavior of a system during topological phase transition. Let us consider a system with a set of parameters M = (M 1 , M 2 , M 3 , . . .) , which upon tuning appropriately changes the underlying topology of the system and induces topological phase transition. The curvature function F(k, M) at momentum k dictate the topological properties of the system. Integral of this curvature function over a Brillouin zone defines topological invariant number which characterizes a gapped phase. For 1D systems it reads Change in this topological invariant number involves the phase transition between the distinct gapped phases. For 1D systems Berry connection is the curvature function. Since Berry connection is gauge dependent, one can choose the gauge for which F(k, M) can be written in Ornstein-Zernike form around the high symmetry point (HSP) k 0 , where δk is small deviation from HSP, and ξ is characteristic length scale. As the system approaches critical point to undergo topological phase transition i.e, M → M c , curvature function diverges and changes sign as system moves across critical point F(k, M).  The critical point can be defined by the condition | dM dl | = ∞ , and fixed point can be defined by the condition | dM dl | = 0 . As we approach critical point, along with the divergence of the curvature function [Eq. (10)], characteristic length ξ in Eq. (9) also diverges These divergences in F(k 0 , M) and ξ give rise to divergent behavior characterized by the critical exponents In conventional Landau theory of phase transition with order parameter, correlation function plays prime role. The same can not be defined for topological phase transitions since there is no local order parameter. However, a correlation function in terms of a matrix element between Wannier states of distant home cells is proposed to characterize the topological phase transition 20 . This Wannier state correlation function R , can be obtained from Fourier transform of the curvature function for 1D systems as Substituting the Ornstein-Zernike form of curvature function yields R ∝ e − R ξ . This suggest that ξ can be treated as correlation length of topological phase transition with critical exponent ν . Similarly curvature function at HSP, F(k 0 , M) has the notion of susceptibility in the Landau paradigm with the critical exponent γ . These critical exponents define the universality class of a model undergoing topological phase transition. A generic scaling law-imposed by the conservation of topological invariant-can be deduced for the critical exponents as where D is the dimensionality of the system. Thus for 1D systems we have γ = ν 20 . The CRG method has been used to understand topological transition between gapped phases. Here we use this method to understand the topological transition between previously discussed gapless phases in our model. We calculate the RG equations and critical exponents for the critical theories between both gapped and gapless transitions and ensure the reliability of this method.
CRG for the transition between gapped phases. In this section we perform CRG for the topological transition across the critical line 2 = µ − 1 , i.e, between the gapped phases with w = 0, 1 and 2. The objective of this discussion is to distinguish between the distinct critical phases CP-1 and CP-2 . We derive RG equations to confirm the topological transition between the gapped phases (between w = 0, 2 and w = 1 ). We derive critical exponents for the CP-1 and CP-2 through Berry connection approach 23 to characterize their universality classes. Transition between the CP-1 and CP-2 through the multi-critical point 'b' is studied in the next section.
The curvature function can be calculated as e ikR F(k, M).  Direction of the RG flow, in the 1 -2 plane, is shown by the arrows, which signals the presence of critical and fixed lines. The critical line is denoted by solid line in the flow diagram which traces a line 2 = µ − 1 as predicted analytically. This line distinguish between, w = 0 and w = 1 gapped phases for 1 < 2µ and w = 2 and w = 1 gapped phases for 1 > 2µ for µ = 0 . The RG flow of coupling parameters 1 and 2 flows away from the critical line and towards the stable fixed line as shown in Fig. 6a,b. One can dubiously distinguish between w = 0 and w = 2 gapped phases based on the flow lines, which flows towards 1 = 2µ in w = 2 phase and towards the fixed line in w = 0 phase.
Multi-critical point appear exactly at the intersection of critical and fixed lines, i.e at the point ( 1 , 2 ) = (2µ, −µ) . This intersection point can be obtained analytically by equating critical and fixed line equations, which yield a quadratic equation 2 1 − 4µ 1 + 4µ 2 = 0 . The solution of this quadratic equation is 1 = 2µ which is the multi-critical point for the HSP k 0 = 0 . The curvature function is found to be diverging at this point. This multi-critical point distinguish the critical phases 1 < 2µ and 1 > 2µ on the critical line, whose physics can also be captured by the CRG method which is discussed in the next section.
In order to show the distinct nature of CP-1 and CP-2, we calculate the critical exponents, explained in Eq. (13), and characterize their universality classes. Set of critical exponents (z, ν, γ ) characterize the critical phases which governs the transition between w = 0 and w = 1 as well as w = 2 and w = 1 gapped phases. To calculate these critical exponents we first expand the Hamiltonian terms χ z and χ y from Eq. (3), around the HSP k 0 = 0.
Summarizing above results suggest that the set of critical exponents for CP-1 between w = 0 and w = 1 are (ν, z, γ ) = (1, 1, 1) and for CP-2 between w = 2 and w = 1 are (ν, z, γ ) = ( 1 2 , 2, 2) . This clearly indicate that the two gapless phases belong to different universality classes. There is a TQPT between these two gapless phases through multi-critical point which we discuss in the next section. This result coincide with the results that we obtained from energy dispersion analysis.
Note that for CP-1 the scaling law in Eq. (15) is obeyed, while for CP-2 it is violated. The dynamical critical exponent is found to take the value z = 1 for CP-1 since the spectra is linear in k around the gap closing point. In the case of CP-2, the spectra is found to be quadratic in k around the gap closing point which yields z = 2 . For this case one can write an effective form of Eq. (23) around the HSP as Integrating this over its width ξ −1 i for the conservation of topological invariant, yields the scaling law γ = 2 D i=1 ν i . Thus when z = 2 the scaling law will get modified into γ = 2ν for 1D systems (refer to "Method" section for details).
In order to verity this modification in scaling law, we perform the CRG for the HSP k = π which address the topological transition between gapped phases w = 2 and w = 1 for 2 > 0 . This transition happens through the critical line 2 = µ + 1 . As we approach this QCP the curvature function in Eq. (16), diverges at the HSP k 0 = π as shown in Fig. 7 and takes the Ornstein-Zernike form around this HSP. RG flow equations for the coupling parameters 1 , 2 and µ can be derived as (refer to "Method" section for a detailed derivation) .   where A = (4 2 − 2 1 ) and δg = (2µ + 2 1 − 2 2 ) . This clearly indicate ξ ∝ |δg| −1 , which implies the correlation length critical exponent ν = 1 . The curvature function at the HSP k 0 = π can be written as F(k 0 , δg) = 2(2 2 − 1 ) δg . As we approach the critical line 2 = µ + 1 , curvature function is F(k 0 , δg) ∝ |δg| −1 which implies the value of γ = 1 . Thus we obtain a set of critical exponents i.e, (z, ν, γ ) = (1, 1, 1) for the transition between gapped phases at k 0 = π . Note that the critical exponents obey the scaling law in Eq. (15). Since the spectra on the critical line is linear around the gap closing point with z = 1 , the scaling law ν = γ is obeyed. Even though there is a transition between w = 1 and w = 2 gapped phases for both k 0 = 0 and k 0 = π HSPs, the nature of energy spectra, critical theory and the scaling of curvature function are different. This results in the modified scaling law observed previously for CP-2 at k 0 = 0.
CRG for the transition between gapless phases. In this section we discuss the topological transition between the gapless phases through multi-critical point on the critical line 2 = µ − 1 . The gapless phases CP-1 and CP-2 are found to have different set of critical exponents. The nature of transition between these two distinct gapless phases is indeed topological and occurs through the multi-critical point 'b' (see Fig. 1). We perform CRG again and derive RG equations and critical exponents to prove the existence of topological transition between gapless phases and also to characterize the critical behavior at the multi-critical point.
Curvature function on the critical line 2 = µ − 1 can be obtained as where M = {µ, 1 } . Fig. 9a shows F(k, M) for the transition between gapless phases through multi-critical point. Surprisingly the curvature function tend to diverge as we approach the multi-critical point. For the parameter value µ = 1 multi-critical point is obtained at the critical value 1 = 2 . Curvature function shows diverging peak as we approach critical value and flips sign across it. This behavior of the curvature function allow one to perform CRG to understand the topological transition between gapless phases. The behavior of curvature function at the multi-critical point 'a' is shown in Fig. 9b. It is a trivial multi-critical point at which two critical line, 2 = µ + 1 and 2 = µ − 1 meet. Hence, as we approach this multi-critical point from either directions the curvature function diverges at both HSPs k 0 = 0 and k 0 = π . This multi-critical point preserve Lorentz invariance and no topological transition occurs between gapless phases as in the case of the multi-critical point 'b' .
The RG flow equations, which signals the topological transition between the gapless phases through multicritical point, for the coupling parameters 1 and µ , can be derived as (refer to "Method" section for a detailed derivation) . Now the Berry connection can be written as (refer to "Method" section for details) For different parameter values on the critical line, we observe the coefficient of δk 4 is dominant over δk 2 . This implies the correlation length ξ ∝ |δg| − 1 2 , suggesting the correlation length exponent and dynamical critical exponents to be ν = 1 2 and z = 2 respectively. To calculate the critical exponent γ we obtain the curvature function at HSP, which has a form F(k 0 , δg) = 4µ−3 1 2 |δg| −1 . Therefore as we approach multi-critical point the curvature function critical exponent takes the value γ = 1 . Note that the scaling law is violated here also as in the case of the transition between the gapped phases w = 2 and w = 1 for 2 < 0 . As proposed earlier the scaling law get modified as γ = 2ν since the dynamical critical exponent z = 2 . Thus the critical phase at the multi-critical point, which governs the topological transition between two gapless phases on the critical line 2 = µ − 1 , has critical exponents (ν, z, γ ) = ( 1 2 , 2, 1).  Fig. 6 we observe the fixed line at 2 = 1 ( 1 +µ) 1 −8µ . This fixed line is stable for finite range of parameter values and flow lines flows towards it. However, it is not the same case in Fig. 8. The fixed line occurs at 2 = 1 (µ− 1 ) 8µ+ 1 , which has both stable and unstable parts. This causes a major distortion in the RG flow on 1 -2 plane. Thus the nature of RG flow are different for different critical lines.
An interesting point can be observed when one set the parameter µ = 0 . RG flow in this case is shown in Fig. 11 for both HSPs. Setting µ = 0 , removes non-topological phase ( w = 0 ) completely and only topological gapped phases remain. It also eliminate the non-trivial multi-critical point along with distinct gapless phases. Hence, the RG flow at µ = 0 for both HSPs are similar in nature. The fixed lines for both HSPs are unstable with RG flow lines flowing away. It is interesting to note that for k 0 = 0 (Fig. 11a), the fixed line coincide with critical line for k 0 = π . Similarly for k 0 = π (Fig. 11b) the fixed line coincide with the critical line for k 0 = 0.
RG flow lines in Fig. 6 shows asymptotic nature around the the line 1 = 2µ . The flow direction is reversed on the opposite sides of the multi-critical point, which occurs at the intersection of fixed and critical lines. This nature of RG flow lines are due to the term 1 − 2µ in the denominator of RG equation for 2 in Eq. (18). This RG equation blows up for 1 = 2µ which accounts for the asymptotic nature of RG flow lines in Fig. 6. For 2 value above the multi-critical point, RG flow asymptotically increase for 1 < 2µ and asymptotically decrease for 1 > 2µ . This flow directions reverses for 2 value below the multi-critical point. Similar nature can be expected for HSP k 0 = π around the line 1 = −2µ.
Correlation function for gapped and gapless phases. Now we discuss the physical significance of correlation length as a length scale to determine the correlation between Wannier states. In the case of one dimensional systems, the curvature function is given by the Berry connection F(k, M) = n �u kn |iδ k |u kn � , where n is the index of all occupied bands. The Fourier transform of which gives the charge polarization correlation function ( R ), between Wannier states at a distance R apart 20,27 .
We have two bands in our model and only the lower band (n = 1) is occupied. Therefore we have R = �R|r|0� , which is a measure of overlap between Wannier states at 0 and R. The zeroth component 0 is the charge polarization, which is the topological invariant. Since Wannier state �r|R� = W(r − R) is a localized function with center at R, the quantity �R|r|0� is expected to decay with R to zero.
The correlation function R can be analytically calculated in the continuous approximation for the appropriate gauge choice of Berry connection, which takes Ornstein-Zernike form. We study the behavior of R near the critical line 2 = µ − 1 which occurs at the HSP k 0 = 0 . Since the critical line has distinct gapless phases (CP-1 and CP-2), we study the nature of R separately near these gapless phases. As we approach the CP-1 i.e, for the transition between gapped w = 0 to w = 1 phase, the correlation function R can be obtained as (refer to "Method" section for details) . www.nature.com/scientificreports/ where ξ = 2( 1 +2 2 ) 2µ−2 1 −2 2 . Similarly as we approach the CP-2 i.e, for the transition between gapped w = 1 to w = 2 phase, R can be obtained as (refer to "Method" section for details) where ξ = 2 1 +8 2 2(2µ−2 1 −2 2 ) . Behavior of correlation function near the critical lines between distinct gapped phases is depicted in Fig. 12. Figure 12a shows the decay in the correlation function in Eq. 37 as we approach a critical point at 1 = 1.5 on CP-1. We observe the decay length of the R is shorter for the parameter value away from the critical value and it gets longer as we approach the critical point. In other words the correlation function decays slower near the critical line as the decay is sharp deep inside the gapped phase. Similar behavior can be observed for the transition across CP-2 as shown in Fig. 12b. In this case the critical point is at 1 = 2.5 . R shows sharp decay for the parameter value away from the critical value and the decay length is longer as we approach the critical point. This indicate the TQPT between the gapped phases as this behavior of correlation function is universal around a QCP. Note that for w = 0 gapped phase, 2 range from −1 to 1 (see Fig. 1). In this range of 2 , we consider one attractive (− ve) and the other one repulsive (+ ve) coupling. Figure 12a is plotted for attractive coupling of 2 and Fig. 12c is plotted for repulsive coupling of 2 . The critical value of 1 = 0.7 near to which R decay slowly and sharp decay can be observed for the value away from critical value. We observe the decay in R is much slower in the repulsive case than in the attractive case at the same distance from the critical line. The topological transition across the critical line 2 = µ + 1 can also be observed in terms R . This critical line corresponds to the transition between gapped phases with w = 2 and w = 1 . Behavior of R for the HSP k 0 = π can be obtained as (refer to "Method" section for details) where ξ = Figure 12d shows oscillatory behavior of R close to the critical point at 1 = 0.5 on the critical line 2 = µ + 1 . We observe that the amplitude of the oscillation decreases, which indicate the decay in R . This decay gets slower as we approach the critical point as in Fig. 12d. This clearly confirms the presence of TQPT across the critical point between the gapped phases w = 1 and w = 2.
Behavior of correlation function R near a critical point signals the TQPT successfully. Therefore we analyze the same universal property of R for the transition between gapless phases CP-1 and CP-2. The analytical  Figure 13 shows the behavior of R as we approach the multi-critical point at 1 = 2 . R decays sharply deep within the gapless phase and the decay length increases as the 1 value approaches critical point. The decay tends to slow down with longer decay length for the value close to critical point. This behavior of R near the multi-critical point is similar to the cases of gapped phases. One can conclude from the behavior of R in Fig. 13 that it clearly indicate the presence of TQPT across the multi-critical point between the gapless phases CP-1 and CP-2.

Discussion
The theory of critical phenomena and curvature function renormalization scheme, developed for the topological phase transitions, provides an alternative platform to understand the transition between gapped phases against the conventional theory on topological invariant. We have shown explicitly that these tools can also be extended for the characterization of topological quantum phase transition occurring between gapless phases. The two distinct gapless phases of our model Hamiltonian has been analyzed and they were found to belong to different universality classes based on the values of critical exponents. Among the three quantum critical lines of the model Hamiltonian, two are topological in nature and also capture the essential TQPT across the gapless topological quantum critical line. This interesting feature is absent in the original Kitaev chain. CRG analysis confirmed the presence of topological quantum phase transition between the gapless phases through the nontrivial multi-critical point. We have shown explicitly the break down of Lorentz invariance at the topological multi-critical point. The values of critical exponents revealed that the transition is in the Lifshitz universality class. We have performed the calculation of Wannier state correlation function for the TQPT between gapped and gapless phases. Decrease in the decay rate of correlation function as we approach multi-critical point revealed the presence of TQPT between gapless phases.   χ z (k) = −2 1 cos k − 2 2 cos 2k + 2µ, and χ y (k) = 2 1 sin k + 2 2 sin 2k , are expanded around HSP k 0 = 0 as
T h e c o r r e l a t i o n l e n g t h ξ = (4 2 −2 1 ) (2µ+2 1 −2 2 ) ∝ |δg| −1 =⇒ ν = 1. Derivation of modified scaling law. In order to preserve the constant value of topological invariant, the divergence of the curvature function near HSP, as we approach the transition point ( M → M c ), has to be conserved 28 . The contribution to the topological invariant from the divergence C div of curvature function near the HSP k 0 = 0 , as we approach CP-2, can be obtained by integrating over the width ξ −1 here Thus we have (81) C div = F(k 0 , δg) ξ 2 × O(1) = constant.