Multi-criticality and long-range effects in non-Hermitian topological models

For topological conditions, the winding vectors form a closed loop in the parameter space, due to the periodicity in the boundary condition. In Hermitian systems, when the parameter $k$ runs from $-\pi$ to $\pi$, which gives topological invariantWilczek and Shapere (1989),

$$W=\left(\frac{1}{2\pi}\right)\int_{-\pi}^{\pi}\frac{\chi_{x}\partial_{k}\chi_{y}-\chi_{y}\partial_{k}\chi_{x}}{\chi_{x}^{2}+\chi_{y}^{2}}dk.$$

(6)

The imbalance in the intercell hopping induces non-Hermiticity in the Hamiltonian which results in the complex topological angle ($\phi$), i.e.,

$$\phi=\phi^{re}+i\phi^{im},$$

(7)

where the imaginary parts do not contributes to the argument of the topological angleYin et al. (2018). But the imaginary part creates two exceptional points (with topological angles $\phi_{1},\phi_{2}$) instead of single Dirac (gap closing) point. The angles $\phi_{1}$ and $\phi_{2}$ are real and they produce the winding number asYin et al. (2018)

$$W=\frac{1}{2}\left(\frac{1}{2\pi}\right)(\oint\partial_{k}\phi_{1}dk+\oint\partial_{k}\phi_{2}dk),$$

(8)

where $\phi_{1}=\tan^{-1}\left(\frac{\chi_{y}^{re}(k)+\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)-\chi_{y}^{im}(k)}\right)$ and $\phi_{2}=\tan^{-1}\left(\frac{\chi_{y}^{re}(k)-\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)+\chi_{y}^{im}(k)}\right)$.(For detailed derivation, refer Appendix A)
Here, we initially consider the simplest longer-range model with two neighbor couplings. We gradually increase the number of neighbor couplings to get further longer-range models and to analyze the transitions among them. At the end, we consider infinite couplings to understand the interplay of non-Hermiticity and long-range behavior.

In the vicinity of phase transition point, the physical quantities shows the diverging nature, which can be quantified by the critical exponentsContinentino (2017); Continentino et al. (2020). The set of critical exponents constitutes the universality class, which effectively categorizes the phase transitions based on their behavior. Here we analyze a few critical exponents (relevant to zero temperature TQPTs) and analyze their behavior.

The dynamical critical exponent $(z)$ defines the nature of dispersion and can be calculated by expanding Eq. II around gap closing point $k_{0}$Rufo et al. (2019). i.e.,

$$E(k,\mathbf{M})=\sqrt{|\delta g|^{2}+A_{1}k+A_{2}k^{2}+A_{4}k^{4}+...},$$

(9)

where the dominating terms among $A_{1},A_{2},A_{4}$ are real numbers which decides the nature of dispersion and corresponding $z$.
For the simplest longer-range model with two neighbors, the winding vectors are given by

	$\displaystyle\chi_{x}$	$\displaystyle=$	$\displaystyle t+\cos(k)+\frac{\cos(2k)}{2^{\alpha}},$
	$\displaystyle\chi_{y}$	$\displaystyle=$	$\displaystyle\sin(k)+\frac{\sin(2k)}{2^{\alpha}}-i\delta,$		(10)

with $t^{\prime}=1$. The criticality occurs at three values of $k$, i.e., $k=0,\pi$ and $\cos^{-1}(-2^{\alpha-1})$ which results in six exceptional points respectively. Thus the model possesses six TQCLs

	$\displaystyle k$	$\displaystyle=$	$\displaystyle 0\implies t=-\frac{1}{2^{\alpha}}+\delta-1,\hskip 5.69046ptt=-\frac{1}{2^{\alpha}}-\delta-1,$
	$\displaystyle k$	$\displaystyle=$	$\displaystyle\pi\implies t=-\frac{1}{2^{\alpha}}+\delta+1,\hskip 5.69046ptt=-\frac{1}{2^{\alpha}}-\delta+1,$
	$\displaystyle k$	$\displaystyle=$	$\displaystyle\cos^{-1}(-2^{\alpha-1})\implies t=\frac{1}{2^{\alpha}}+\delta,\hskip 5.69046ptt=\frac{1}{2^{\alpha}}-\delta.$

with red, blue, green, black, magenta and gray respectively (Fig 2 a).
Here we observe two different spectrum of real and absolute energy dispersion with different critical exponents. Due to the non Hermiticity, the energy dispersion need not to be symmetric around the gap closing values, which may result as different exponents around a same pointArouca et al. (2020). In our study, both real and absolute spectrum yield $z=1/2$ for all TQCLs except at multi-criticality (Fig 3 a). At multi-criticality the real spectrum yield $z=2,1$ and 0.5 while (Fig 3 b)absolute spectrum yield $z=1,1$ and 0.5 for MC1, MC2 and MC3 respectively (absolute spectrum is not showed here).
The longer-range non-Hermitian SSH model contains both fractional and integer WNs, where the WN is given by Eq. 8, which physically means the encircling of the winding vectors around the origin of the pseudo-spin parameter space. Due to non-Hermiticity, there occurs two exceptional points instead of single Dirac point which act as center of pseudo-spin space. If the winding vectors encircle both the exceptional points equal times, then the WN is integer. If the winding vectors encircle either one of the exceptional point, or unequal number of time, then there occur fractional WN.
In Fig. 2 b, when winding vectors encircle both the EPs once (twice), we get $W=1(W=2)$ topological phase which is equivalent to Hermitian version of topological phases. If the winding vectors encircle either one of the exceptional point, it gives $W=1/2$ and encircling first (second) EP twice (once) while other only once (twice) give rise to $W=3/2$, which do not have any Hermitian counterpart. Under some special cases, winding vectors encircle one of the exceptional point even number of times and do not encircles the other, which also results in the integer WN. With the increasing decay parameter, the longer-range reduces to its short-range version at $\alpha=1$ through the intersection of TQCLs i.e., multi-critical points.

Localization critical exponent $(\nu)$: In a topological system, the zero energy modes are localized at the edges and protected by the bulk Bloch states.

As a simple example, we consider Eq.1 with two neighbor couplings ($r=2$). For a semi-infinite system with $j=1,2,3...$, with wave function $\psi_{n}$, the energy relation yieldsYin et al. (2018)

	$\displaystyle(t+\delta)\psi_{a,j}+t^{\prime}\psi_{a,j+1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{a,j+2}=E_{n}\psi_{b,j}$
	$\displaystyle(t-\delta)\psi_{b,j}+t^{\prime}\psi_{b,j-1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{b,j-2}=E_{n}\psi_{a,j}.$		(11)

We construct the boundary condition with zero energy state ($E=0$) as

	$\displaystyle(t+\delta)\psi_{a,j}+t^{\prime}\psi_{a,j+1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{a,j+2}=0$
	$\displaystyle(t-\delta)\psi_{b,j}+t^{\prime}\psi_{b,j-1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{b,j-2}=0.$		(12)

Here the boundary condition $\psi_{b,j}=0$ is true for every $n$. The ratio of nth localized zero eigenstate to that of first is given byContinentino (2017); Continentino et al. (2020)

$$\delta\psi_{n}=\frac{\psi_{n}(E=0)}{\psi_{1}(E=0)}=\left|\frac{\delta g}{A_{1,2,4}}\right|^{n-1}.$$

(13)

These states are ensured by the condition $e^{ik_{0}}=-\left(\frac{\delta g}{A_{1,2,4}}\right)$. Thus the $k_{0}$ is a complex number with $k_{0}=i\left(\frac{\delta g}{A_{1,2,4}}\right)$, where $A_{1,2,4}$ are the real numbers and dominating term among them decides the value of $k_{0}$. With the system size $n$, the localization of the eigenstates is given by,

$$\delta\psi_{n}=e^{\frac{-(n-1)}{\xi}},$$

(14)

where $\xi$ is the localization length with $\xi=\frac{A_{n}}{|\delta g|^{\nu}}\implies\xi=\xi_{0}|\delta g|^{-\nu}$, with $\xi_{0}$ as the natural length of the system. Here $A_{n}$ is the dominating term in Eq. 9 and $\nu$ is the localization critical exponent. In non-Hermitian systems, the edge localization is highly influenced by the skin-effect which decides the bulk-edge correspondence. In our model, we get $\nu=1$ for both TQCLs and multi-critical points (Fig.3 e,f).
To understand the number of localized edge modes at each end, we write Eq.12 as

	$\displaystyle(t+\delta)+t^{\prime}\lambda+\frac{t^{\prime}}{2^{\alpha}}\lambda^{2}=0,$
	$\displaystyle(t-\delta)+t^{\prime}\lambda^{\prime}+\frac{t^{\prime}}{2^{\alpha}}\lambda^{\prime 2}=0.$		(15)

with $\lambda=\frac{\psi_{a,j+1}}{\psi_{a,j}}$ and $\lambda^{\prime}=\frac{\psi_{b,j+1}}{\psi_{b,j}}$, corresponding to the left and right localized modes respectively. The quadratic form gives two roots

$$\lambda_{\pm}=\frac{-t^{\prime}\pm\sqrt{(t^{\prime})^{2}-4(\frac{t^{\prime}}{2^{\alpha}})(t\pm\delta)}}{2^{1-\alpha}(t^{\prime})}.$$

(16)

The topological limit forms a unit circle, where the number of solutions less than unity represent the number of localized modes. The roots of $\lambda$ and $\lambda^{\prime}$ represent the left and right localized modes respectively. The combination of these two solution constitutes the phase diagram as mentioned in Ref. Yin et al. (2018).

Crossover critical exponent $(y)$: The scaling behavior near the TQCL is given byArouca et al. (2020),

	$\displaystyle E(k\rightarrow k_{0},\mathbf{M}=\mathbf{M}_{c})$	$\displaystyle\propto$	$\displaystyle k^{z},$
	$\displaystyle E(k=k_{0},\mathbf{M}\rightarrow\mathbf{M}_{c})$	$\displaystyle\propto$	$\displaystyle\left|\partial g\right|^{y},$		(17)

where $y$ is called crossover or gap critical exponent with $y=z\nu$. This exponent gives the relation between $z$ and $\nu$ and can be calculated in the limit $E(k_{0},\mathbf{M}\rightarrow\mathbf{M}_{c})$. The energy gap at criticality behaves like $\Delta\propto|\delta g|^{z\nu}$, where $\delta g$ is the distance from the criticality. In our case, both real and absolute spectrum give $y=0.5$ for TQCLs as well as for multi-critical points (Fig. 3 c,d).

Canonical critical exponent $(\alpha^{*})$: The concept of thermodynamic grand potential ($\Omega$) helps to understand the order of phase transition and corresponding critical exponentArouca et al. (2020); Kempkes et al. (2016). The divergence or non-analyticity in the derivative of the grand potential signals the order of the corresponding transition. For a system with periodic boundary condition at zero temperature, the grand potential density is given by,

$$\omega=\frac{\Omega}{N}=-\int_{-\pi}^{\pi}E(k,\mathbf{M})dk,$$

(18)

where $N$ is the system size. The above integral is convergent for all positive $\alpha$, which makes the grand potential as independent of system size in this limit. The order of the transition can be calculated through the derivative of the grand potential. The transition can be nth order if the nth derivative of grand potential shows a cusp or spike against the parameter and the (n+1)th derivative shows a discontinuityMalard et al. (2020b); Chen et al. (2008).

In Fig. 4, we study $k=\pi$ critical line (t=1-$\frac{1}{2^{\alpha}}\pm\delta$) under two different configurations. For $\alpha=0.1$ the second derivative with respect to $t$ shows a cusp at criticality followed by a discontinuous third order derivative. At multi-criticality ($\alpha=1$), the first order derivative shows cusp at MC2 and a continuous curve at MC2. The second order derivative shows discontinuity at MC2 and a cusp at MC1 respectively. This observation signals that, the multi-criticalities MC1 and MC2 exhibit different nature of transitions. To understand this in detail we study the hyperscaling relation.
Here, the grand potential ($\omega$) contains two parts, i.e., regular and singular parts. The singular part of the grand potential density scales asArouca et al. (2020)

$$\omega_{singular}\propto|g|^{2-\alpha^{*}},$$

(19)

where $\alpha^{*}$ is the canonical critical exponent and $2-\alpha^{*}$ signals the order of transition. These critical exponents are connected by the relations called scaling laws which governs the phase transitions. One such relation is the Josephson’s hyperscaling relation given byContinentino (2017)

$$2-\alpha^{*}=\nu(z+d),$$

(20)

where $z+d$ is the effective dimension. This law connects the dimensionality, dynamical and correlation critical exponents with order of transition through hyper-scaling relation. Thus, through hyper-scaling relation (Eq.20),

	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for TQCLs},$
	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(2+1)=3\hskip 14.22636pt\text{for MC1},$
	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(1+1)=2\hskip 14.22636pt\text{for MC2},$
	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for MC3}.$		(21)

which clearly signals the difference between the order of transitions among normal criticality and multi-criticality. The fraction order of transitions are the interesting observations in non-Hermitian systems, which are different than their corresponding Hermitian counterparts. In Ref. Arouca et al. (2020), the authors have tried to analyze this issue by scaling the singular part with the factor $\xi\propto|\delta g|^{-\nu}$. As the grand potential has a contribution from system size $L$, every distance is divided by the correlation length. This has showed that, for the lower range of $\delta g$ the singular part of grand potential scales as $\delta g^{1/2}$, whereas for the higher range it scales as $\delta g^{1}$. This is the reason behind the difficulty to decide whether the transitions are first order or second order. However, in our model also we find similar observation at normal criticality, but multi-criticality remains interesting and merits more attention in this regard.

Due to the non-Hermitian effects, the TQPT occurs through the exceptional points at $(0,\chi_{y}^{im})$ and $(0,-\chi_{y}^{im})$ instead of the Dirac points. The WN is given by Eq. 8, with

	$\displaystyle F_{1}(k,\mathbf{M})$	$\displaystyle=$	$\displaystyle\partial_{k}\phi_{1}=\frac{(\chi_{x}^{re}-\chi_{y}^{im})\partial_{k}\chi_{y}^{re}-\chi_{y}^{re}\partial_{k}(\chi_{x}^{re}-\chi_{y}^{im})}{(\chi_{x}^{re}-\chi_{y}^{im})^{2}+(\chi_{y}^{re})^{2}},$
	$\displaystyle F_{2}(k,\mathbf{M})$	$\displaystyle=$	$\displaystyle\partial_{k}\phi_{2}=\frac{(\chi_{x}^{re}+\chi_{y}^{im})\partial_{k}\chi_{y}^{re}-\chi_{y}^{re}\partial_{k}(\chi_{x}^{re}+\chi_{y}^{im})}{(\chi_{x}^{re}+\chi_{y}^{im})^{2}+(\chi_{y}^{re})^{2}}.$

Here we deal with an interesting quantity called curvature function (CF). CF is the quantity whose integral over the Brillouin zone gives the topological invariant, like Berry connection, Berry curvature and Pfaffian wave functionKumar et al. (2021a); Chen and Sigrist (2019). In 1D models, we consider Berry connection as our CF which is an gauge dependent quantity. The CF exhibits non-analyticity at the criticality and yields ill-defined topological invariant at these points. In case of non-Hermitian systems, exceptional points are the points of non-analyticity. Based on the behavior in the vicinity of exceptional points, CF can be classified as following.

1. 1.

   High symmetry points (HSP): When the CF exhibits a symmetric nature around the gap closing point $k_{0}$, i.e.,$F(k_{0}+\delta k,\mathbf{M})=F(k_{0}-\delta k,\mathbf{M})$ is called a HSP. Here the Lorentzian peak is is symmetric under condition $k_{0}=-k_{0}$. The CF diverges as one approach $k_{0}$ from one direction $\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}$ and flips the direction as one moves away $\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}$ from $k_{0}$ with same diverging curve. This can be represented as fixed peak with varying height (Fig. 5 a1,b1).

2. 2.

   Non-High Symmetry points (non-HSP): In this case, as one approaches criticality, there occurs diverging peak with different $k_{0}$ values. Here one can observe the flipping of peaks but the Lorentzian peak keep moving along the BZ (Fig. 5 a2,b2).

3. 3.

   Fixed point (FP): Generally this kind of behavior is observed at the multi-criticality where HSP and non-HSP meet togetherKartik et al. (2021); Kumar et al. (2021a, b). Here the CF shows a diverging peak as $\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}$ from one side, and at $k_{0}$ we observe a peak with diminished height. As we go away $\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}$ we observe a peak with constant height with increasing width. The multi-critical points MC1 and MC2 show the FP nature (Fig. 5 a3,b3).

Susceptibility critical exponent $(\gamma)$: Here the CF exhibits non-analytic property at the exceptional points, which gives the idea of critical scaling near these pointsChen and Sigrist (2019). In present case, we consider Berry connection as our CF, which is a gauge dependent quantity. Without loss of generality, we write CF in Ornstein-Zernike form as,

$$F(k_{0}+\delta k,\mathbf{M})=\frac{F(k_{0},\mathbf{M})}{1\pm\xi^{2}\delta k^{2}+\xi^{4}\delta k^{4}},$$

(23)

where $k_{0}$ is a gap closing point, $\delta k$ is a small deviation from gap closing and $\xi$ is the characteristic scale respectively. At $k=k_{0}$ and $\mathbf{M}\rightarrow\mathbf{M}$, the CF diverges signaling a TQPT where the characteristic length vanishes.

$$lim_{\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}}F(k_{0},\mathbf{M})=-lim_{\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}}F(k_{0},\mathbf{M})=\pm\infty.$$

(24)

and CF exhibiting an interesting nature

$$F(k_{0},\mathbf{M}^{\prime})=F(k_{0}+\delta k,\mathbf{M}),$$

(25)

which gives the idea of renormalization group study near criticality.

The divergence of CF and characteristic length gives the critical exponent $F(k_{0},\mathbf{M})\propto|\mathbf{M}-\mathbf{M}_{c}|^{-\gamma}$ and $\xi(k_{0},\mathbf{M})\propto|\mathbf{M}-\mathbf{M}_{c}|^{-\nu}$, where $\gamma$ and $\nu$ are the susceptibility and characteristic critical exponents respectively, where the susceptibility exponent explains the quality of the transition to be topological. Here we get $\gamma=1$ for both TQCLs and multi-critical points.
Expanding Eq 25 up to a leading order, we get CRG equation asChen and Sigrist (2019),

$$\frac{d\mathbf{M}}{dl}=\frac{(\frac{1}{2})\frac{d^{2}F(k,\mathbf{M})}{dk^{2}}|_{k=k_{0}}}{\frac{dF(k_{0},\mathbf{M})}{d\mathbf{M}}},$$

(26)

where $\left|\frac{d\mathbf{M}}{dl}\right|\rightarrow\infty$ defines criticality and $\left|\frac{d\mathbf{M}}{dl}\right|\rightarrow 0$ defines fixed point configuration respectively. In our model, at $k=\pi$ (both $F1(k,\mathbf{M})$ and $F2(k,\mathbf{M})$), there occurs a superposition of all critical and fixed lines at $\alpha=1$. Here we find a whirlpool kind of behavior where the flow lines encircle a single point (Fig. 6). At this point, $\frac{d^{2}F}{dk^{2}},\frac{dF}{d\alpha},\frac{dF}{d\mu}\rightarrow 0$ and $\frac{d\alpha}{dl},\frac{d\mu}{dl}\rightarrow\frac{0}{0}$. Interestingly at this point, we observe a different class of UCCE through multi-critical points. On the other hand, we do not observe any such behavior for $k=0$ criticality. (For detailed derivation, refer AppendixB)

Multi-criticalities are the intersection of (at least) two TQCLs and distinguish (at least) three different topological regimes. So far, multi-critical points acted as unique points, where we witnessed the breaking of Lorentz invariance, topological transition without bulk gap and as a point with zero entanglement entropyKumar et al. (2021b). There are category of multi-critical points based on the conformal field theory where we get different central charge. However, in case of non-Hermitian systems, multi-critical points may behave differently based on the nature of non-Hermitian factor. For example, recently there has been the observation on the vanishing of multi-criticality when the non-Hermiticity is introduced into an extended Kitaev chain through chemical potentialRahul (2022). In our case, we have introduced non-Hermiticity through local imbalance in the hopping, which splits the Hermitian multi-critical points into two. We can observe that multi-critical points as the superposition of HSP and non-HSP with a fixed point behavior and belong to separate UCCE. This part is in similarity with Hermitian counterpart.

The expansion of Eq. II around exceptional points $k_{0}$ gives the better understanding towards the underlying physics of multi-criticality. At normal criticalities, the dispersion behaves as $E(k\rightarrow k_{0},\mathbf{M}_{c})\propto k^{1/2}$. At multi-criticalities, the dispersion takes first ($E(k\rightarrow k_{0},\mathbf{M}_{c})\propto k^{1}$ for MC1) and second order ($E(k\rightarrow k_{0},\mathbf{M}_{c})\propto k^{2}$ for MC2) corrections as shown in Eq.9 respectively. On the other hand, the multi-critical point MC3 is no different than the normal criticalities, as it shows similar behaviors of normal criticality. However, we observe similar cases in Hermitian counterpart, where the the instance is recognized as breaking of Lorentz invarianceKumar et al. (2021a).
Exponent $z$ gives information about the massless condition of the system. In condensed matter, the energy dispersion at Fermi surface resembles the relativeistic energy form $E^{2}=c^{2}p^{2}+m^{2}c^{4}$. For gap closing points, the equation gives masless condition with $E^{2}=c^{2}p^{2}$. Under some cases, the equation takes correction form as, $E^{2}=c^{2}p^{2}+m^{2}c^{4}+Ap^{4}$ with $A$ as constantSadhukhan et al. (2020). Similar effects we observe at exceptional points. For normal criticalities, we get $E^{2}=\sqrt{AK}$ and for multi-criticalities, we obtain $E^{2}=\sqrt{Ak+Bk^{2}}$ and $E^{2}=\sqrt{Ak+Bk^{4}}$ respectively. This creates the difference in the nature of energy dispersion and thereby dynamical critical exponent. Transition among gapless phases: Localization at criticality is an important phenomenon, which is important both from the perspective of theory and experimental aspects  Jones and Verresen (2019); Thorngren et al. (2020); Kestner et al. (2011); Cheng and Tu (2011); Fidkowski et al. (2012); Sau et al. (2011); Kraus et al. (2013); Scaffidi et al. (2017); Jiang et al. (2018), especially in longer-range modelsRahul et al. (2021); Kumar et al. (2021a); Verresen et al. (2019, 2018); Verresen (2020); Kumar et al. (2021b). In our model, the multi-critical points act as an unique point, which witnesses the topological transition among critical phases. Here MC1 and MC2 are present on the $k=\pi\rightarrow t=1\pm\delta+\frac{t^{\prime}}{2^{\alpha}}$ respectively. Through Eq.15, we find the solutions of quadratic equation as shown in Fig. 7. Throughout the critical line, one of the solution remains unity, which confirms the gapless condition. For $\alpha<1$, one of the solution is less than unity ($\lambda=0.4$), indicating a localization of edge mode at criticality. At $\alpha=1$, the both the solutions are unity, indicating a transition among gapless phases through multi-criticality. For $\alpha>1$, one of the solution is greater than unity ($\lambda=1.8$), indicating the absence of localized modes. Hence, both MC1 and MC2 act as transition points among gapless phases. A similar phenomenon has been observed in the Hermitian counterpart of the similar modelRahul et al. (2021); Kumar et al. (2021a).

Due to the feature of long-range coupling in Eq.1, it is possible to achieve higher winding number with the increase of interacting neighbors, i.e., finite $r$. For the limit $r<L/2$, we can obtain higher WNs where the uppermost WN is $W=r$. This is an effective method to obtain more localized modes through static method while similar can be achieved through periodic driving which is dynamical in natureKartik et al. (2021); Tong et al. (2013). However the higher WNs are comparatively less stable and reduce to consecutive lower WNs with the increasing of $\alpha$. Here we find an interesting patter of transitions , i.e., a staircase of TQPTs.

• •

  Even-Even transition: When the number of neighbor is an even number (for $r>2$), the uppermost WN will be an even number and the transition occurs among only even WNs.

• •

  Odd-Odd transition: When the number of neighbor is an odd number (for $r>2$), the uppermost WN will be an odd number and the transition occurs among only odd WNs.

• •

  Even-Odd transition: When the number of neighbor is an even number (for $r>2$), the uppermost WN is an even number and with the increasing $\alpha$, the longer-range model reduces to short-range version. At the interface of short-range and longer-range, there occurs a transition between $W:2\rightarrow 1$. For very small $\delta$, this can be clearly observed and with the increase in $\alpha$, the transition occurs like $W:2\rightarrow\frac{3}{2}\rightarrow 1$. This transition occurs through the multi-critical points.

Here we choose parameter space ($t=0.5,\delta=0.1$), where the increase in $\delta$ suppresses the uppermost WN and creates fractional WNs. For the even $r$, the fractional WNs are observable with small $\delta$ and large $\alpha$, whereas the similar effect is not visible for odd $r$. The interface of short-range and longer-range occurs through multi-critical points as shown in Fig. 8(d). These multi-critical points can also be recognized by the CRG method also (Not shown here).

The quasi-particle energy dispersion is given by Eq. II, where the band gap vanishes for $k=0$ and $\pi$ respectively. Due to the polylogarithmic nature, the band gap closes for all values of $\alpha$ at $k=\pi$ and only $\alpha>1$ at $k=0$. The parameter space possesses two exceptional points and four TQCLs corresponding to two values of $k_{0}$. i.e.,

1. 1.

   $t=\delta-\text{Li}_{\alpha}(1)$ for $k=0$ for $\alpha>1$,

2. 2.

   $t=-\delta-\text{Li}_{\alpha}(1)$ for $k=0$ for $\alpha>1$,

3. 3.

   $t=\delta-\text{Li}_{\alpha}(-1)$ for $k=\pi$ and $\forall\alpha$,

4. 4.

   $t=-\delta-\text{Li}_{\alpha}(-1)$ for $k=\pi$ and $\forall\alpha$.

Here the pseudo-spin parameters go as $\sin(k)$ instead of $\sin(lk)$ which result in the single time encircling of the origin. Thus the phase diagram is given by Fig. 9(a,b).

Behavior of winding vectors: Due to the long-range effect, the winding vectors exhibit a discontinuity at $k=0$ for $\alpha<1$ which give rise to fractional winding number. Thus the region $\pm\delta-\text{Li}_{\alpha}(1)<\mu<\pm\delta-\text{Li}_{\alpha}(-1)$ for $\alpha>1$ gives topological (Hermitian and non-Hermitian) phase where the winding vectors encircle the origin of the parameter space without any discontinuity. Due to the polylogarithmic behavior, we find a less population of winding vectors in the region $1<\alpha<2$, technically which do not affects the winding number. In the region $\delta-\text{Li}_{\alpha}(1)<\mu<-\delta-\text{Li}_{\alpha}(-1)$ for $\alpha>1$, we find that both set of winding vectors encircle the origin of each parameter space which yield $W=1$. The region $\mu<\pm\delta-\text{Li}_{\alpha}(-1)$ for $\alpha<1$ gives topologically ill-defined (fractional) phase where winding vectors corresponding to both the parameter space shows a discontinuity at $k=0$. If both set of vectors exhibit half encirclement around their corresponding origin, then it yield $W=1/2$. If any one exhibits this behavior, it yields $W=\pm 1/4$ where the sign indicates the direction of encircling. On the other hand, due to the imbalance in intracell hopping, there arise fractional WN for $-\delta-\text{Li}_{\alpha}(1)<\mu<\delta-\text{Li}_{\alpha}(1)$ and $-\delta-\text{Li}_{\alpha}(-1)<\mu<\delta-\text{Li}_{\alpha}(-1)$ for $\alpha>1$. These are the non-Hermitian topological phases and do not posses any Hermitian counterpart. Here any one set of the winding vector encircle the origin by yielding $W=1/2$. Extended winding vectors are given by

	$\displaystyle\chi_{x}(ext)$	$\displaystyle=$	$\displaystyle\pm\delta+\frac{\text{Li}_{\alpha}(\exp(-ik))+\text{Li}_{\alpha}(\exp(ik))}{2}+t,$
	$\displaystyle\chi_{y}(ext)$	$\displaystyle=$	$\displaystyle\frac{\text{Li}_{\alpha}(\exp(ik))-\text{Li}_{\alpha}(\exp(-ik))}{2i}.$		(30)

The behavior of extended winding vectors are presented in Fig. 9(c1-c6).

Momentum space behavior: Due to the polylogarithmic nature, in the limit $k\rightarrow 0$, the $\chi_{y}$ term shows a divergence for $\alpha<1$ and converges towards zero for $\alpha>1$ and a discontinuity at $\alpha=1$ respectively(Fig. 10c). This reflects in the behavior of CF and energy dispersion. For $\alpha<1$ the CF shows a divergence (Fig. 10d) at $k=0$ which leads to the emergence of the fractional WN. The corresponding energy spectrum shows a gapped region with diverging bands(Fig. 10a).The transition from $\alpha<1$ to $\alpha>1$ occurs at $\alpha=1$, where we observe topological transition without gap closing $(W:-1/4\rightarrow 0,1/4\rightarrow 1/2,W:-1/2\rightarrow 1)$. This is an interesting behavior which can be observed in long-range models.

Curvature function and Wannier state correlation function: Substituting the values of Eq. 27 into Eq. LABEL:cf we get the CF in Ornstein-Zernike form as

$\displaystyle F(k,\mathbf{M})\propto\frac{\frac{Ak^{\alpha-2}}{Bk^{\alpha-1}}-\frac{Ck^{2\alpha-3}}{Dk^{2\alpha-2}}}{1+(\frac{Ek^{\alpha-1}}{Fk^{\alpha-1}})^{2}}.$

(31)

There are three possible cases,

• •

  When $\alpha<1$, the term $k^{\alpha-2}$ dominates and $F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$, irrespective of $\mathbf{M}\rightarrow\mathbf{M}_{c}$.

• •

  When $1<\alpha<2$, the term $k^{\alpha-2}$ dominates and $F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$, irrespective of $\mathbf{M}\rightarrow\mathbf{M}_{c}$.

• •

  When $\alpha>2$, again the term $k^{\alpha-2}$ dominates and $F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$ with $\mathbf{M}\rightarrow\mathbf{M}_{c}$.

Thus we can define the CF in the Ornstein-Zernike form and corresponding critical exponents only for $\alpha>2$ around $k=0$ regionSadhukhan and Dziarmaga (2021). For $k=\pi$, the CF can expressed Ornstein-Zernike form for all values of $\alpha$.
Wannier state correlation function is the quantity, which discusses the phase transition among different phasesKumar et al. (2021a); Chen and Sigrist (2019); Kumar et al. (2021b). Conceptually, it is the overlap between the Wannier state centering at the origin and that of distance $R$. i.e.,

$\displaystyle\langle r|R\rangle=W_{n}(r-R)$

(32)

with

$\displaystyle|R_{n}\rangle=\frac{1}{N}\sum_{k}e^{ik.(r-R)}|u_{k}\rangle$

(the form of $|u_{k}\rangle$ is expressed in Appendix A.)
Mathematically, Wannier states can be expressed as Fourier transform of the CF, i.e.,

$$\lambda_{R}=\int\frac{dk}{2\pi}e^{ik.R}F(k,\mathbf{M}).$$

(33)

In our case, we observe the decay of Wannier state correlation function around $k=\pi$ for all values of $\alpha$, with a critical exponent $\nu=1$ (Fig. 10 e). For $k=0$, there occurs no correlation decay in the region $1<\alpha<2$, which results in the undefined critical exponent. For the region $\alpha>2$, we observe the decay of correlation function, which is very sharp near the criticality with an exponent $\nu=1$ (Fig. 10 f).

Energy dispersion and Fermi velocity: By substituting Eq. 27 into Eq. II, we get energy dispersion. The first order derivative of the energy dispersion gives the Fermi velocity as

	$\displaystyle E(k,\mathbf{M})$	$\displaystyle=$	$\displaystyle\sqrt{Ak^{2\alpha-2}+Bk^{\alpha-1}+C},$
	$\displaystyle\frac{dE_{k}}{dk}$	$\displaystyle=$	$\displaystyle\frac{Ak^{2\alpha-3}+Bk^{\alpha-1}+Ck}{\sqrt{A^{2}k^{2\alpha-2}+Bk^{\alpha}+C^{2}k^{2}}}.$		(34)

where $A,B$ and $C$ are constants. For the values $\alpha<1$, the Fermi velocity is dependent on $\alpha$ as shown in Table 2

Thus, as $k\rightarrow 0$ we obtain a diverging energy spectrum and Fermi velocity for $\alpha<1$. For $1<\alpha<1.5$ we obtain gapless energy spectrum and a diverging Fermi-velocity as $k\rightarrow 0$. For $\alpha>1.5$, we obtain vanishing Fermi velocity as the system drives towards criticality.

Here we analyze the UCCE through nonlinear curve fitting method (Fig. 11). Around $k=0$, we obtain $z<0.5$ for the region $1<\alpha<2$, whereas $z=0.5$ for $\alpha>2$. In the region $1<\alpha<2$, we obtain undefined $\nu,\gamma$ and $y$ critical exponents, and $\nu=1,\gamma=1,y=0.5$ for $\alpha>2$. On the other hand, we get $z=0.5,y=0.5,\nu=1$ and $\gamma=1$ for all values of $\alpha$ around $k=\pi$. Table 3 summaries the critical exponents for different regions of non-Hermitian long-range SSH chain.

To understand the order of phase transition, we check the derivatives of grand potential density for different orders. The phase transitions act as singularities in the density parameter space and the derivatives of density signals the order of transition with non-analytic curves. The polylogarithmic nature creates a singularity in the vicinity of $k\rightarrow 0$, while the entire BZ contributes to the geometric phase. i.e., if $\omega_{singular}$ represents the grand potential density of the bulk, then it can be expressed asCats et al. (2018)

$$\omega_{singular}=\omega_{x}+\omega_{y},$$

(35)

where $\omega_{x}$ integral around $k=0$ and $\omega_{y}$ is the integral over the rest of the BZ respectively.

Through Eq. 28, we write an expanded term around $k=0$ as

$$\omega_{x}(k=0,\mathbf{M}(\alpha>2))=\int_{0-\epsilon}^{0+\epsilon}\sqrt{\mathbf{M}^{2}+k^{2}}dk\propto\mathbf{M}^{2}\ln|\mathbf{M}|.$$

(36)

In this case we can observe a divergence in the second order derivative of the above equation, signaling a second order bulk transition. For the region $1<\alpha<2$,

$$\omega_{x}(k=0,\mathbf{M}(1<\alpha<2))\approx\int_{0-\epsilon}^{0+\epsilon}\sqrt{\mathbf{M}^{2}+k^{2(\alpha-1)}}dk.$$

(37)

Through the Eq. 28, we observe the above term is proportional to $\Gamma(-\frac{\alpha}{2(\alpha-1)})\Gamma(\frac{2\alpha-1}{2(\alpha-1)})\mathbf{M}^{\frac{\alpha}{\alpha-1}}$, where $\Gamma$ function diverges for $\alpha=\frac{n}{n-1}$ where $n\in N$. Thus as one approaches $\alpha=1$ from the side of $\alpha=2$, the integral shows divergence for higher orders signaling a higher order bulk transitions. Thus as we take the higher order derivatives, the criticalities corresponding to $k=\pi$ show the peaks while $k=0$ fails to produce the same. This signals a possible higher order phase transitions as one approaches $\alpha=1$. The region with integral WN (Fig. 9) has similar Hermitian counterpart as explained in Ref. , while the fractional WN do not have Hermitian counterpart. Interestingly, the fractional WN corresponding to $k=0$ shows a very narrow phase boundary, and as $\alpha\rightarrow 1$, the non-Hermitian phase gradually vanishes. Thus at $\alpha=1$, the Hermitian phase exhibits an infinite order of bulk transition, while non-Hermitian phases shows almost none. This is also in good agreement with the hyper-scaling relation (Eq. 20) also.

	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for $k=\pi$},$
	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for $k=0,\alpha>2$},$
	$\displaystyle 2-\alpha^{*}$	$\displaystyle=$	$\displaystyle\text{Higher as $\alpha\rightarrow 1$}\hskip 14.22636pt\text{for $k=0,1<\alpha<2$}.$