Signatures of topological phase transition on a quantum critical line

In the quest of classifying novel phases of quantum matter in the absence of local order parameters, the topology of electronic band structure plays a prime role haldane1988model ; kane2005quantum ; narang2021topology ; wang2017topological . It enables the distinction between gapped phases in terms of a quantized invariant number, which counts the number of localized edge modes present thouless1982quantized . The transition between the distinct topological phases involves a bulk band gap closing at the point of topological phase transition. Across the transition the number of edge modes, protected by the bulk gap, changes hasan2010colloquium . In the gapped phases, the localization length of the edge modes diverges as the system drives towards the transition point or criticality continentino2020finite .

Interestingly, this conventional knowledge is revised recently, realizing even criticalities can host the stable localized edge modes despite the vanishing bulk gap verresen2018topology ; verresen2019gapless ; jones2019asymptotic ; verresen2020topology ; rahul2021majorana ; niu2021emergent ; PhysRevB.104.075132 ; PhysRevResearch.3.043048 ; fraxanet2021topological ; keselman2015gapless ; scaffidi2017gapless ; duque2021topological . This results in the emergence of non-trivial criticalities with unique topological properties even in the presence of gapless bulk excitations. The non-trivial criticalities can be effectively characterized in terms of the zeros and poles of complex function associated with the Hamiltonian verresen2018topology . The localized edge modes at criticality are protected by novel phenomena such as kinetic inversion verresen2020topology (in fermionic models) and finite high energy charge gap fraxanet2021topological (in bosonic models). It has been shown that they also remain robust against interactions and disorders verresen2018topology ; verresen2019gapless . This intriguing interplay between topology and criticality causes an unconventional topological transition between critical phases verresen2020topology ; kumar2021multi .

In this work, we report the possibility of a new kind of topological phase transition between critical phases, happening at a multicritical point where two critical lines intersect. Considering a two-band model as a prototype representing gapless (critical) topological insulators and superconductors, in Sec. II, we find the critical lines in the model and existence of two species of multicritical points with quadratic and linear dispersions, both corresponding to the topological transition between distinct critical phases. In support of our finding, in Sec. III, we solve the Dirac equation for criticality to obtain the localized edge mode solutions in the non-trivial critical phases. This is further supported by our proposal of obtaining the non-zero integer winding number for non-trivial critical phases which we discuss in Sec. IV.1. This proposal is supported in Sec. IV.2 by calculating the winding number using zeros and poles of a complex function. The curvature function diverges on approaching the multicritical points from critical phases indicating the existence of a transtion between critical phases. The critical exponents $(\gamma,\nu,z)$, discussed in Sec. V.1, unravel the different universality classes of two multicritical points. Using the divergence of curvature function we develop a renormalization group (RG) scheme to distinguish the different critical phases in Sec. V.2. In Sec. V.3 we show that a correlation length, extracted from the Fourier-transformed curvature function, diverges at the multicritical points indicating phase transitions between critical phases. Further, in Sec. VI we study the spatial scaling of entanglemet entropy, especially to characterize and distinguish the multicritical points. The scaling reveals an interplay between the fixed points and multicritical points. The entanglement entropy is minimum where the fixed point and multicritical point overlap reflecting the dominance of the former over the latter. Finally, we conclude in Sec. VII .

We consider a one-dimensional lattice chain of spinless fermions in momentum space PhysRevLett.42.1698 ; kitaev2001unpaired represented by a generic two-band Bloch Hamiltonian of the form

where $\boldsymbol{\Gamma}=\left\{\Gamma_{0},\Gamma_{1},\Gamma_{2}\right\}$, $\chi_{x}=\Gamma_{0}+\Gamma_{1}\cos k+\Gamma_{2}\cos 2k,$ and $\chi_{y}=\Gamma_{1}\sin k+\Gamma_{2}\sin 2k$ and $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y})$ are the Pauli matrices. The model represents extended Su-Schrieffer-Heeger (SSH) hsu2020topological and extended Kitaev models niu2012majorana by uniquely defining the parameters (see Appendix.A for detailed discussion on the physical relevance of the model). The parameters $\Gamma_{0}$, $\Gamma_{1}$ and $\Gamma_{2}$ describe the onsite potential, the nearest neighbor (NN) couplings and the next nearest neighbor (NNN) couplings respectively.

In general, the model can support three distinct gapped phases distinguished by the number of edge modes they possess. These phases can be identified with the winding numbers $w=0,1$ and $2$, which quantify the edge modes. The model undergoes transition between these phases with necessarily involving the gap closing, $E_{k}=\pm\sqrt{\chi_{x}^{2}+\chi_{y}^{2}}=0$, at the phase boundaries. The criticalities, where the bulk gap closes, occurs for the momentum $k_{0}=0,\pm\pi,\cos^{-1}(-\Gamma_{1}/2\Gamma_{2})$, which respectively gives the critical surfaces $\Gamma_{1}=-(\Gamma_{0}+\Gamma_{2})$, $\Gamma_{1}=(\Gamma_{0}+\Gamma_{2})$ and $\Gamma_{0}=\Gamma_{2}$. The model possesses three multicritical lines at which two critical surfaces intersect. Two of them ($MC_{1}$) share identical properties and shows quadratic dispersion around the gap closing point and the other one ($MC_{2}$) is identified with the linear dispersion. Uniquely, the model supports the edge modes and topological transition at critical surfaces $\Gamma_{1}=\pm(\Gamma_{0}+\Gamma_{2})$ corresponding to $k_{0}=\pi,0$ respectively (see Appendix.B for the numerical results).

To further explore these unique phenomena we propose a framework that works out for criticality without referring to any of the gapped phases of the model. Ideally driving the system to criticality involves $k\rightarrow k_{0}$ and $\boldsymbol{\Gamma}\rightarrow\boldsymbol{\Gamma}_{c}$, where $\boldsymbol{\Gamma}_{c}$ is the critical point in the parameter space. To avoid the singularities involving the exact critical point, one can define the Hamiltonian critical only in the parameter space as $\mathcal{H}(k,\boldsymbol{\Gamma}_{c})$ with $k=k_{0}+\Delta k$, where $\Delta k<<2\pi$. This situation is hereafter referred as criticality in this work.

The model at criticality can be obtained by using the critical surface relation which modifies the components into $\chi_{x}=\Gamma_{0}(1+\cos k)+\Gamma_{2}(\cos 2k+\cos k),$ and $\chi_{y}=\Gamma_{2}(\sin 2k+\sin k)+\Gamma_{0}\sin k$ for $\Gamma_{1}=(\Gamma_{0}+\Gamma_{2})$ and $\chi_{x}=\Gamma_{0}(1-\cos k)+\Gamma_{2}(\cos 2k-\cos k),$ and $\chi_{y}=\Gamma_{2}(\sin 2k-\sin k)-\Gamma_{0}\sin k$ for $\Gamma_{1}=-(\Gamma_{0}+\Gamma_{2})$. The possible topological trivial and non-trivial critical phases are separated by the phase boundaries at the multicritical lines $\Gamma_{2}=\Gamma_{0}$ ($MC_{1}$) and $\Gamma_{2}=-\Gamma_{0}$ ($MC_{2}$). Without loss of any generality, we assume $\Gamma_{0}=1$. Hence hereafter the critical surfaces and the multicritical lines will be called as the critical lines and the multicritical points respectively, as shown in Fig. 1(a). The multicriticalities, $MC_{1,2}$, are identified with quadratic and linear dispersion respectively, as shown in Fig. 1(b). They can be obtained for the following $k_{0}^{mc}$. For $\Gamma_{2}=\Gamma_{0}$ ($MC_{1}$):

For $\Gamma_{2}=-\Gamma_{0}$ ($MC_{2}$):

Interestingly, $MC_{2}$ exhibits swapping of the values of $k_{0}^{mc}$. At $MC_{2}$, one can observe that $k_{0}^{mc}=0$ for $\Gamma_{1}=(\Gamma_{0}+\Gamma_{2})$ which was obtained for $k_{0}=\pi$ and $k_{0}^{mc}=\pi$ for $\Gamma_{1}=-(\Gamma_{0}+\Gamma_{2})$ which was obtained for $k_{0}=0$. This property emerge as a result of intersection of both the critical lines at $MC_{2}$. We will show in the following that our proposed framework based on the near-critical Hamiltonian $\mathcal{H}(k,\boldsymbol{\Gamma}_{c})$ can capture the essential physics of topological transition at criticality.

The presence of edge modes in topological insulators and superconductors is lucid from the bound state solution of Dirac equation jackiw1976solitons ; lu2011non ; shun2018topological . We solve the model in Eq.1 for the bound state solution at criticality (see Appendix.C for the bound state solution at gapped phases). Interestingly, as a consequence of the near-critical approach adopted here, the Dirac Hamiltonian at criticality naturally fixes the interface at a multicritical point. Dirac Hamiltonian at criticality up to third-order expansion around $k_{0}^{mc}$ for $MC_{1}$ is

where $\epsilon_{1}=(\Gamma_{0}-3\Gamma_{2})/2$, $\epsilon_{2}=(\Gamma_{2}-\Gamma_{0})$ and $\epsilon_{3}=(7\Gamma_{2}-\Gamma_{0})/6$. We look for zero energy solution in real space (we set $\hbar=1$ throughout the discussion), $\mathcal{H}\psi(x)=0$. Identifying the spinor of the wave-function $\psi(x)=\rho_{\eta}\phi(x)$ is an eigenstate of $\sigma_{z}$ and using $\phi(x)\propto e^{-x/\xi}$, inverse of the non-zero decay length can be obtained as $\xi^{-1}_{\pm}=(-\eta\epsilon_{1}\pm\sqrt{\epsilon_{1}^{2}-4\epsilon_{2}\epsilon_{3}})/(-2\epsilon_{3})$. For both roots to be positive, it requires $\xi^{-1}_{+}+\xi^{-1}_{-}>0$, which implies $\eta=\text{sign}(\epsilon_{1}/\epsilon_{3})$. The edge mode decay length (longer one of two) is $\xi_{+}\approx|\epsilon_{1}|/\epsilon_{2}$ remains finite and positive for $\epsilon_{2}>0$ i.e., $\Gamma_{2}>\Gamma_{0}$, which means the criticality in this region possesses edge modes and is the topological non-trivial phase. Note that, the term $\epsilon_{2}$ is the gap term at criticality, which mimics the role of mass. As $\epsilon_{2}\rightarrow 0$ the decay length $\xi_{+}\rightarrow\infty$ indicating the edge mode delocalize into the bulk and topological transition takes place at $MC_{1}$ i.e., at $\Gamma_{2}=\Gamma_{0}$. To visualize this phenomenon we write the bound state solution $\psi(x)\propto\begin{pmatrix}\eta&0\end{pmatrix}^{T}(e^{-x/\xi_{+}}-e^{-x/\xi_{-}}),$ which distributes dominantly near the boundary and decay exponentially as $x\rightarrow\infty$, as shown in Fig.2(a).

To identify the topological transition at $MC_{2}$ and the corresponding topological non-trivial phase one has to consider the swapping property of $k_{0}^{mc}$, which emerge as a result of intersection of critical lines at $MC_{2}$. In this case, after expanding around $k^{mc}_{0}$ and using the swapping property, the Dirac Hamiltonian can be obtained up to second order as

where $\epsilon_{1}=2(\Gamma_{0}+\Gamma_{2})$, $\epsilon_{2}=(\Gamma_{0}+3\Gamma_{2})$ and $\epsilon_{3}=(5\Gamma_{2}+\Gamma_{0})/2$. With $\eta=\text{sign}(\epsilon_{2}/\epsilon_{3})$, the edge mode decay length $\xi_{+}\approx-(|\epsilon_{2}|/\epsilon_{1})$ is obtained using $\phi(x)\propto e^{x/\xi}$ and is positive if $\epsilon_{1}<0$. Therefore, in this case, the gap term is $\epsilon_{1}$ which vanish at the multicritical point $MC_{2}$, i.e. at $\Gamma_{2}=-\Gamma_{0}$. This implies that the criticality $\Gamma_{2}<-\Gamma_{0}$ is topological non-trivial phase and the topological transition occur at $MC_{2}$, i.e. $\Gamma_{2}=-\Gamma_{0}$ as a consequence of the delocalization of edge mode into the bulk as $\epsilon_{1}\rightarrow 0$. In this case the bound state solution $\psi(x)\propto\begin{pmatrix}0&\eta\end{pmatrix}^{T}(e^{x/\xi_{+}}-e^{x/\xi_{-}})$, distribute near the boundary and decay as $x\rightarrow-\infty$ as shown in Fig.2(b).

The topological character of a gapped phase is quantified using topological invariant numbers thouless1982quantized . The quantized values of these invariant numbers represents the number of localized stable edge modes at each end of the open chain. For one dimensional systems winding number is a good invariant number which represents the winding of psuedospin vector in the Brillion zone PhysRevLett.115.177204 .

Therefore, the edge excitations of the gapped phases can be quantified in terms of winding number hasan2010colloquium

where $F(k,\mathbf{\Gamma})=i\left\langle u_{k}\left|\partial_{k}\right|u_{k}\right\rangle$ is the Berry connection or curvature function of Bloch wavefunction $\psi_{k}(r)=u_{k}(r)e^{ikr}$.

In order to quantify the edge modes at criticality one has to define the winding number at criticality verresen2020topology . The conventional definition of winding number fails at criticalities. This is due to the non-analyticity of the curvature function (integrand in Eq.8) at criticalities. This constraint is naturally avoided in the near-critical approach and allows one to calculate the winding number in its usual integral form even at criticality.

However, it yields fractional values, as shown in Fig.3(a), which does not account correctly for the number of edge modes present at criticalities.

Alternatively, one can refer to the auxiliary space and differentiate between NN and NNN loops and consider only one among them which gives integer contribution and accounts for the edge modes at criticality rahul2021majorana . However, this method is not efficient as the auxiliary space loops gets complicated with the increasesing NN couplings PhysRevLett.115.177204 .

Topological phase transition can be induced by changing the underlying topology of the system upon tuning the parameters $\mathbf{\Gamma}$ appropriately. The information of the topological property of the system is embedded in the curvature function $F(k,\mathbf{\Gamma})$ defined at momentum $k$ chen2016scaling ; chen2016scalinginvariant ; chen2018weakly ; PhysRevB.95.075116 ; chen2019universality ; molignini2018universal ; molignini2020generating ; abdulla2020curvature ; malard2020scaling ; molignini2020unifying ; rufo2019multicritical ; kartik2021topological . The topological quantum phase transition can be identified from the quantized jump of topological invariant number as the parameter tuned across the critical point $\mathbf{\Gamma}_{c}$. As the system approaches critical point to undergo topological phase transition i.e, $\mathbf{\Gamma}\rightarrow\mathbf{\Gamma}_{c}$, curvature function diverges at $k_{0}$, with the diverging curve satisfying $F(k_{0}+\delta k,\mathbf{\Gamma})=F(k_{0}-\delta k,\mathbf{\Gamma})$. The sign of the diverging peak flips across the critical point as

Interestingly, even at criticality, the qualitative behavior of the curvature function remains the same with the fact that, now the critical point is a multicriticality which governs the topological transition between critical phases. As one tunes the parameters at criticality $\mathbf{\Gamma}_{c}$ towards a multicritical point $\mathbf{\Gamma}_{mc}$, the curvature function diverges at $k_{0}^{mc}$ with the symmetric nature $F(k_{0}^{mc}+\delta k,\mathbf{\Gamma}_{c})=F(k_{0}^{mc}-\delta k,\mathbf{\Gamma}_{c})$, as shown in Fig.5(a).

Topological transition is signalled as the sign of the diverging peak flips if the parameters tuned across the multicritical point.

This is the characteristic feature of topological transition at criticality through both the multicritical points $MC_{1,2}$. The curvature function of the generic model at criticality can be written using the critical line relations of the parameters. The pseudo-spin vectors on the two critical lines, $\Gamma_{1}=\pm(\Gamma_{0}+\Gamma_{2})$, of the model are $\chi_{x}(k)=\Gamma_{0}(1\pm\cos k)+\Gamma_{2}(\cos 2k\pm\cos k),$ and $\chi_{y}(k)=\Gamma_{2}(\sin 2k\pm\sin k)\pm\Gamma_{0}\sin k$. This defines curvature function on the critical lines $F(k,\mathbf{\Gamma}_{c})=F(k,\mathbf{\Gamma}_{\Gamma_{1}=\pm(\Gamma_{0}+\Gamma_{2})})$,

The property in Eq.14 can be observed to be obeyed by $F(k,\mathbf{\Gamma}_{\Gamma_{1}=\pm(\Gamma_{0}+\Gamma_{2})})$ as shown in the Fig.5(b,c,d,e). They show the critical behavior of curvature function around the multicritical points $MC_{1,2}$, which distinguish between distinct critical phases. The peak of the curvature function tends to diverge as the parameters approach $MC_{1,2}$ from both sides at criticality. Both the criticalities exhibit the universal nature of curvature function around the multicritical points.

The scenario around $MC_{1}$ on the critical line $\Gamma_{1}=-(\Gamma_{0}+\Gamma_{2})$ shows the divergence in curvature function at the $k_{0}^{mc}=0$, as shown in Fig.5(b). As the parameter $\Gamma_{2}$ is tuned towards its multicritical value (i.e $MC_{1}$) on both sides, the diverging peak of curvature function increases leading to a complete divergence at $MC_{1}$ and flips sign as the critical value is crossed. This signals the topological transition across $MC_{1}$ at criticality. Similar behavior of curvature function can be observed around $MC_{1}$ on the critical line $\Gamma_{1}=(\Gamma_{0}+\Gamma_{2})$, for which the divergence occurs at $k_{0}^{mc}=\pi$, as shown in Fig.5(c).

The nature of curvature function around $MC_{2}$ at both the criticalities share the same property of divergence and flipping of sign as shown in Fig.5(d) and 5(e). Note that, the $k_{0}^{mc}$ at which the diverging peak increases on approaching the multicritical value, is $k_{0}^{mc}=\pi$ instead of $k_{0}^{mc}=0$ for $\Gamma_{1}=-(\Gamma_{0}+\Gamma_{2})$ (and $k_{0}^{mc}=0$ instead of $k_{0}^{mc}=\pi$ for $\Gamma_{1}=(\Gamma_{0}+\Gamma_{2})$). This swapping of $k_{0}^{mc}$ occur as a consequence of the intersection of critical lines. Typically the multicritical point $MC_{2}$ is the same point for both the critical lines $\Gamma_{1}=\pm(\Gamma_{0}+\Gamma_{2})$ in parameter space. These critical lines intersect each other at $MC_{2}$, which results in the swapping of respective $k_{0}^{mc}$ values.

The characteristics of a criticality can be effectively quantified from the entanglement entropy (EE) of the ground state by arbitrarily dividing a system into two subsystems PhysRevLett.90.227902 ; PhysRevLett.102.255701 ; PhysRevLett.121.076802 . Taking the advantage of Wick’s theorem, the eigenvalues of the reduced density matrix can be extracted from the two point correlation matrix, which in the thermodynamic limit can be written as PhysRevLett.121.076802

with $1\leq i,j\leq L$ (where $L$ is the subsystem size). The EE ($S$) can be computed as PhysRevLett.121.076802

where $\Lambda_{i}$ are the eigenvalues of correlation matrix. The EE signals the topological transition at the multicritical points, as shown in Fig.9. The profile of EE shows, maxima at $MC_{2}$ (Fig.9(b)) and surprisingly minima at $MC_{1}$ (Fig.9(a)).

For the generic model in Eq.1, the $MC_{1}$ is the intersection point of fixed and critical lines. Remarkably, at $MC_{1}$, we observe that the fixed point characteristic is more dominant which results in the minima of entanglement entropy, in oppose to the critical point behavior where entanglement is supposed to maximize due to the enhanced correlations (see Appendix.E for more details). Besides, at $MC_{1}$, the bulk is not a CFT. This can be seen from the multiplicity factor ($m$) i.e. the degenerate zeros on the unit circle of the complex function associated with the Hamiltonian. The multiplicity at $MC_{1}$ is $m=2$ (see Fig.4(g,h,i,j)). As shown in Ref.verresen2018topology , if the complex function has degenerate zeros with multiplicity $m$, the bulk is not CFT and implies the dynamical exponent $z=m$. This is consistent with $z$ value obtained for $MC_{1}$ (see Fig.6(c)).

At $MC_{2}$, the EE is $S=S_{0}+(c/3)\log L$ PhysRevLett.121.076802 where constant $S_{0}=0.72$ and the central charge $c=1$ as shown in the inset of Fig.9(b). The value of $c$ at $MC_{2}$ is consistent with Ref. PhysRevLett.121.076802 , where $c$ was found to be the sum of the central charges of intersecting criticalities. As $MC_{2}$ is the intersecting point of the two Ising criticalities ($c=1/2$), we get $c=1$.

In this work, we reconstruct various tools to characterize the unusual topological phase transitions between distinct critical phases of an extended model that represents topological insulators and superconductors at criticality. Bound state solutions of the Dirac equation and winding number defined for criticality show that the transitions between the critical phases occur through multicritical points $MC_{1,2}$ of different universality classes as captured through the critical exponents obtained from the divergence of the curvature function. There exists an interesting swapping behavior of the critical momenta $k_{0}^{mc}$ at $MC_{2}$ which manifests in the behavior of curvature function. A scaling theory based on the curvature function unravels that the transitions at $MC_{1,2}$ can be efficiently identified from the RG flow in the parameter space and also shows that, $MC_{2}$ manifests as unstable fixed line of RG flow for $MC_{1}$ and vice versa. A diverging correlation length obtained from the Wannier state correlation function, which essentially is the Fourier transform of the curvature function, indicates the ocurrences of topological phase transitions at $MC_{1,2}$ . Moreover, the unique transitions at $MC_{1,2}$ are characterized with the minima and maxima of entanglement entropy respectively revealing an intriguing dominance of the fixed point over the criticality at $MC_{1}$.

Our proposed framework, in general, can be applied to the driven systems and higher dimensional systems. A unique advantage of having topological non-trivial criticalities is that the quantum information remains robust upon tuning the system towards it verresen2019gapless . By identifying the multicritical points one can choose a proper criticality to tune into and avoid the decoherence due to bulk gap closing and opening. Our topological model at criticality can be simulated with a good control over the tunable parameters in the suitable experimental platforms which include the superconducting circuit with a single qubit PhysRevB.101.035109 ; niu2021simulation and the ultracold atoms mimicking the topological models  goldman2016topological ; xie2019topological ; an2018engineering ; meier2018observation ; meier2016observation , especially the Kitaev model with controlled NN and NNN couplings kraus2012preparing ; jiang2011majorana ; an2018engineering .