Multipartite entanglement in topological quantum phases

TL;DR

This paper demonstrates how multipartite entanglement, quantified via quantum Fisher information, characterizes topological phases and phase transitions in the Kitaev chain, revealing divergence patterns even without energy gap closing.

Contribution

It introduces a method to detect topological phases and transitions through multipartite entanglement scaling, applicable to models with variable-range pairing.

Findings

Finite winding number phases show power-law divergence in entanglement scaling.

Quantum phase transitions are marked by divergence of Fisher information derivative.

Entanglement signatures persist even when the energy gap does not close.

Abstract

We witness multipartite entanglement in the Kitaev chain -- a benchmark model of one dimensional topological insulator -- also with variable-range pairing, using the quantum Fisher information. Phases having a finite winding number, both for short- and long-range pairing, are characterized by a power-law diverging finite-size scaling of multipartite entanglement. Moreover the occurring quantum phase transitions are sharply marked by the divergence of the derivative of the quantum Fisher information, even in the absence of a closing energy gap.

Equations

$$\begin{array}{rl}\hat{H}& =-\frac{J}{2}\sum _{j=1}^L\left({\hat{a}}_j^{†}{\hat{a}}_{j+1}+h.c.\right)-\mu \sum _{j=1}^L\left({\hat{n}}_j-\frac{1}{2}\right)\\ & +\frac{\Delta }{2}\sum _{j=1}^L\sum _{\ell =1}^{L-j}{d}_{\ell }^{-\alpha }\left({\hat{a}}_j{\hat{a}}_{j+\ell }+{\hat{a}}_{j+\ell }^{†}{\hat{a}}_j^{†}\right),\end{array}$$

$$\epsilon_{k}=\sqrt{\big{(}J\cos k+\mu\big{)}^{2}+\big{(}f_{\alpha}(k)\,\Delta/2\big{)}^{2}},$$

$$F_Q[|{\psi }_{\mathrm{gs}}⟩,{\hat{O}}_{\rho }]=⟨{\psi }_{\mathrm{gs}}|{\hat{O}}_{\rho }^2|{\psi }_{\mathrm{gs}}⟩-⟨{\psi }_{\mathrm{gs}}|{\hat{O}}_{\rho }|{\psi }_{\mathrm{gs}}{⟩}^2,$$

$$\hat{o}_{\rho}^{(j)}=(-{\rm i})^{\delta_{\rho,y}}\big{(}\hat{a}_{j}^{\dagger}e^{i\pi\sum_{l=1}^{j-1}\hat{n}_{l}}+(-1)^{\delta_{\rho,y}}e^{-i\pi\sum_{l=1}^{j-1}\hat{n}_{l}}\hat{a}_{j}\big{)},$$

$$f_Q[|{\psi }_{\mathrm{gs}}⟩,{\hat{O}}_{\rho }]=1+\sum _{l=1}^{L-1}{C}_{\rho }(l),$$

$$f_Q[|{\psi }_{\mathrm{gs}}⟩,{\hat{O}}_{\rho }]\le \kappa ,$$

$$f_Q[|{\psi }_{\mathrm{gs}}⟩,{\hat{O}}_{\rho }]=1+c{L}^b,$$

$$\frac{1}{f_{Q}}\frac{\partial f_{Q}}{\partial\eta}=\frac{c\,L^{b}}{1+c\,L^{b}}\times\bigg{(}\frac{1}{c}\frac{\partial c}{\partial\eta}+\frac{\partial b}{\partial\eta}\,\log L\bigg{)}.$$

$$\hat{H}=-\frac{J}{2}\sum _{j=1}^L\left({\hat{a}}_j^{†}{\hat{a}}_{j+1}+h.c.\right)-\mu \sum _{j=1}^L\left({\hat{n}}_j-\frac{1}{2}\right)+\frac{\Delta }{2}\sum _{j=1}^L\sum _{\ell =1}^{L-j}{d}_{\ell }^{-\alpha }\left({\hat{a}}_j{\hat{a}}_{j+\ell }+{\hat{a}}_{j+\ell }^{†}{\hat{a}}_j^{†}\right).$$

$${\hat{a}}_{k_n}^{†}=\frac{1}{\sqrt{L}}\sum _{j=1}^L{e}^{-\mathrm{i}{k}_{n}j}{\hat{a}}_j^{†},{\hat{a}}_j^{†}=\frac{1}{\sqrt{L}}\sum _{n=0}^{L-1}{e}^{\mathrm{i}{k}_{n}j}{\hat{a}}_{k_n}^{†},$$

$$k_{n}=\frac{2\pi}{L}\bigg{(}n+\frac{1}{2}\bigg{)},\quad n=0,1,2,...,L-1.$$

$$\displaystyle-\frac{J}{2}\sum_{j=1}^{L}\left(\hat{a}^{\dagger}_{j}\hat{a}_{j+1}+\mathrm{h.c.}\right)-\mu\sum_{j=1}^{L}\left(\hat{n}_{j}-\frac{1}{2}\right)=-\sum_{k}\bigg{(}\frac{J}{2}\cos k+\frac{\mu}{2}\bigg{)}\big{(}\hat{a}^{\dagger}_{k}\hat{a}_{k}+\hat{a}^{\dagger}_{-k}\hat{a}_{-k}\big{)}+\frac{\mu L}{2},$$

$$\sum _{j=1}^L\sum _{\ell =1}^{L-j}\frac{{\hat{a}}_j{\hat{a}}_{j+\ell }}{d_{\ell }^{\alpha }}=\frac{1}{2}\sum _{j=1}^L\sum _{\ell =1}^{L-1}\frac{{\hat{a}}_j{\hat{a}}_{j+\ell }}{d_{\ell }^{\alpha }}=\frac{1}{4}\sum _{j=1}^L\sum _{\ell =1}^{L-1}\frac{{\hat{a}}_j{\hat{a}}_{j+\ell }-{\hat{a}}_{j+\ell }{\hat{a}}_j}{d_{\ell }^{\alpha }}=\frac{\mathrm{i}}{2}\sum _k{f}_{\alpha }(k){\hat{a}}_k{\hat{a}}_{-k}$$

$$\hat{H}=\sum_{k}\big{(}\hat{a}_{k}^{\dagger},\hat{a}_{-k}\big{)}\hat{\mathcal{H}}_{k}\begin{pmatrix}\hat{a}_{k}\\ \hat{a}_{-k}^{\dagger}\end{pmatrix},$$

$$\hat{\mathcal{H}}_{k}=\begin{bmatrix}-(\tfrac{J}{2}\cos k+\tfrac{\mu}{2})&{\rm i}\tfrac{\Delta}{4}f_{\alpha}(k)\\ -{\rm i}\tfrac{\Delta}{4}f_{\alpha}(k)&(\tfrac{J}{2}\cos k+\tfrac{\mu}{2})\\ \end{bmatrix}=-\frac{\Delta}{4}f_{\alpha}(k)\hat{\sigma}_{y}-\bigg{(}\frac{J}{2}\cos k+\frac{\mu}{2}\bigg{)}\hat{\sigma}_{z},$$

$$h_y(k)=-\frac{f_{\alpha }(k)\Delta /2}{\sqrt{[J\cos k+\mu {]}^2+[f_{\alpha }(k)\Delta /2{]}^2}}\equiv \sin \Theta (k),$$

$$h_z(k)=-\frac{J\cos k+\mu }{\sqrt{[J\cos k+\mu {]}^2+[f_{\alpha }(k)\Delta /2{]}^2}}\equiv \cos \Theta (k).$$

$$\left(\begin{array}{c}{\hat{a}}_k\\ {\hat{a}}_{-k}^{†}\end{array}\right)=\left[\begin{array}{cc}\cos \Theta (k)/2& \mathrm{i}\sin \Theta (k)/2\\ \mathrm{i}\sin \Theta (k)/2& \cos \Theta (k)/2\end{array}\right]\left(\begin{array}{c}{\hat{\eta }}_k\\ {\hat{\eta }}_{-k}^{†}\end{array}\right),$$

$$\hat{H}=\sum_{k}\epsilon_{k}\bigg{(}\eta_{k}^{\dagger}\eta_{k}-\frac{1}{2}\bigg{)},$$

$$f_Q[|{\psi }_{\mathrm{gs}}⟩,{\hat{O}}_{\rho }]=1+L^{b-1}\sum _{l=1}^{L-1}{c}_{\rho }(l/L)=1+L^b{\int }_0^{1}dx{c}_{\rho }(x),$$

$$|\psi_{\rm gs}\rangle=\prod_{n=0}^{L/2-1}\bigg{(}\cos\frac{\theta_{k_{n}}}{2}-{\rm i}\sin\frac{\theta_{k_{n}}}{2}\hat{a}_{k_{n}}^{{\dagger}}\hat{a}_{-k_{n}}^{{\dagger}}\bigg{)}|0\rangle,$$

$$\mathcal{F}=\langle\psi_{\rm gs}|\tilde{\psi}_{\rm gs}\rangle=\prod_{n=0}^{L/2-1}\cos\bigg{(}\frac{\theta_{k_{n}}-\tilde{\theta}_{k_{n}}}{2}\bigg{)},$$

$$\mathcal{F}\approx 1-\frac{1}{8}\sum _{n=0}^{L/2-1}(d{\theta }_{k_n}{)}^2.$$

$$\chi_{\eta}=-\frac{\mathrm{d}^{2}\mathcal{F}}{\mathrm{d}\eta^{2}}=\frac{1}{4}\sum_{n=0}^{L/2-1}\bigg{(}\frac{\mathrm{d}\theta_{k_{n}}}{\mathrm{d}\eta}\bigg{)}^{2},$$

$$\theta_{k_{n}}=\arccos\bigg{(}\frac{J\cos k_{n}+\mu}{\sqrt{(J\cos k_{n}+\mu)^{2}+\Delta^{2}f^{2}_{\alpha}(k_{n})/4}}\bigg{)}.$$

$${\chi }_{\mu }=\frac{1}{4}\sum _{n=0}^{L/2-1}\frac{{\Delta }^2{f}_{\alpha }^2(k_n)/4}{[(J\cos k_n+\mu {)}^2+{\Delta }^2{f}_{\alpha }^2(k_n)/4{]}^2}.$$

$$\chi_{\alpha}=\frac{1}{4}\sum_{n=0}^{L/2-1}\frac{(J\cos k_{n}+\mu)^{2}}{[(J\cos k_{n}+\mu)^{2}+\Delta^{2}f^{2}_{\alpha}(k_{n})/4]^{2}}\bigg{(}\frac{\Delta}{2}\frac{\mathrm{d}f_{\alpha}(k_{n})}{\mathrm{d}\alpha}\bigg{)}^{2}$$

$${\chi }_{\Delta }=\frac{1}{4}\sum _{n=0}^{L/2-1}\frac{(J\cos k_n+\mu {)}^2}{[(J\cos k_n+\mu {)}^2+{\Delta }^2{f}_{\alpha }^2(k_n)/4{]}^2}\frac{f_{\alpha }^2(k_n)}{4}.$$

$$\chi_{\alpha}(k)=\frac{(J\cos k+\mu)^{2}}{[(J\cos k+\mu)^{2}+\Delta^{2}f^{\infty}_{\alpha}(k)^{2}/4]^{2}}\bigg{(}\frac{\Delta}{2}\frac{\mathrm{d}f^{\infty}_{\alpha}(k)}{\mathrm{d}\alpha}\bigg{)}^{2}$$

Full text

20 mars 2024

Introduction.– The characterization of quantum phases and quantum phase transitions (QPTs) via entanglement measures and witnesses HorodeckiRMP2009 ; GuhnePR2009 is an intriguing problem at the verge of quantum information and many-body physics AmicoRMP2008 ; Zeng . The study of entanglement pushes our understanding of QPTs OsbornePRA2002 ; OsterlohNATURE2001 beyond standard approaches in statistical mechanics Mussardo ; Sachdev and sheds new light on the creation, manipulation and protection of useful resources for quantum technologies.

The literature AmicoRMP2008 has mostly focused on bipartite entanglement. A benchmark is the so-called area law EisertRMP2010 that relates the amount of bipartite entanglement among two partition of a many-body system to the surface area between the two blocks VidalPRL2003 ; LatorreJPA2009 . For models with short-range interaction in one dimension, the Von Neumann block entropy is constant in the gapped phases while it increases logarithmically with the system size at criticality CalabreseJSM2004 . Yet, violations of the area law are not always related to a closing gap: a logarithmic increase of the Von Neumann entropy occurs also in some gapped phases of one-dimensional models with long-range interaction VodolaPRL2014 ; VodolaNJP2016 ; KoffelPRL2012 ; AresPRA2015 , as well as for peculiar short-range models MovassaghPNAS2016 . An alternative approach to bipartite entanglement is the analysis of the two-body reduced density matrix OsterlohNATURE2001 ; WuPRL2004 , often also indicated as pairwise entanglement.

The complex structure of a many-body quantum state is of course much richer than that caught by bipartite/pairwise entanglement. Yet, multipartite entanglement (ME) has been much less studied GuhneNJP2005 ; HofmannPRB2014 . Recently HaukeNATPHYS2016 ; LiuJPA2013 ; MaPRA2009 , ME in models exhibiting Ginzburg-Landau-type QPTs has been witnessed using the quantum Fisher information (QFI) calculated for the local order parameter, exploiting well-known relations between the QFI and ME PezzePRL2009 ; HyllusPRA2012 ; TothPRA2012 . In this case, ME diverges at criticality in benchmark spin systems, as the Ising HaukeNATPHYS2016 , XY LiuJPA2013 and the Lipkin-Meshkov-Glick MaPRA2009 ; HaukeNATPHYS2016 model. However, it has been emphasized that this approach, based on local operators, fails to detect ME at topological QPTs HaukeNATPHYS2016 . For this task, methods based on the QFI generally require the extension of entanglement criteria for nonlocal operators, along the lines of Ref. PezzePNAS2016 .

Here we focus on a paradigmatic model showing topological quantum phases: the Kitaev chain of spinless fermions in a lattice Kitaev ; AliceaRPP2012 with variable-range pairing VodolaPRL2014 ; VodolaNJP2016 . We calculate the QFI of the ground state for a suitable choice of nonlocal observables and witness ME (the parties being the single sites of the chain) when the corresponding correlation functions have a sufficiently slow decay. We show that phases identified by a nonzero winding number are characterized by a superextensive scaling of the QFI, signaling the divergence of ME with the system size. This divergence is found, for short-range pairing, only at topologically nontrivial phases, hosting massless edge modes. Instead, for long-range pairing superextensivity of the QFI is found whenever the winding number assumes semi-integer values. Furthermore, for arbitrary pairing range, ME is shown to vary suddenly at QPTs, even when the energy gap in the quasiparticle spectrum does not close at the boundary lines. Our work addresses an open problem in the literature – the detection of ME in topological phases and QPTs – and paves the way towards a characterization of strongly-correlated systems in terms of ME.

The model.– The Kitaev chain is a tight-binding model with both tunneling and superconducting pairing. This was originally studied for nearest-neighbor pairing Kitaev and extended to variable-range pairing in VodolaPRL2014 ; VodolaNJP2016 . The corresponding Hamiltonian is

[TABLE]

where $J>0$, $\hat{a}_{j}$ is a fermionic annihilation operator on the site $j$, $\hat{n}_{j}=\hat{a}^{\dagger}_{j}\hat{a}_{j}$ and $L$, assumed even, is the total number of sites. We consider a closed chain, with $d(l)=l$ for $1\leq l\leq L/2$ and $d(l)=L-l$ for $L/2\leq l<L$, and assume antiperiodic boundary conditions $\hat{a}_{j+L}=-\hat{a}_{j}$. Following Lieb1961 , the Hamiltonian (13) can be diagonalized exactly by a Bogoliubov transformation. The resulting quasiparticle spectrum is VodolaPRL2014

[TABLE]

where $k=\frac{2\pi}{L}(n+\frac{1}{2})$ with $n=0,1,...,L-1$, and $f_{\alpha}(k)=\sum_{l=1}^{L-1}\frac{\sin(kl)}{d(l)^{\alpha}}$. The function $f_{\alpha}(k)$ displays singularities at $k=0$ and $k=2\pi$ in the thermodynamic limit if $\alpha\leq 1$.

The schematic phase diagram of the model is shown in Fig. 1. Colored regions refer to phases with different (and constant) values of the winding number $W=\int_{0}^{2\pi}\tfrac{d\Theta(k)}{dk}\frac{dk}{2\pi}$, where $\tan\Theta(k)=(\Delta/2)f_{\alpha}(k)/(J\cos k+\mu)$ is defined as follows AliceaRPP2012 ; supp . After the Fourier transform $\hat{a}_{k}^{\dagger}=\tfrac{1}{\sqrt{L}}\sum_{j}e^{-ikj}\hat{a}_{j}$, Eq. (13) becomes $\hat{H}=\sum_{k}(\hat{a}_{k}^{\dagger},\hat{a}_{-k})\hat{\mathcal{H}}_{k}\binom{\hat{a}_{k}}{\hat{a}_{-k}^{\dagger}}$ where $\hat{\mathcal{H}}(k)=\bm{h}(k)\cdot\bm{\sigma}$ and $\bm{\sigma}$ is the Pauli vector. We can then associate to the quasi-momentum $k$ the vector $\bm{h}(k)=\bm{y}\sin\Theta(k)+\bm{z}\cos\Theta(k)$. Varying $k$ from 0 to $2\pi$, $\bm{h}(k)$ winds $W$ times around a unit circle in the $y-z$ plane, see Fig. 1. The value of the topological invariant $W$ changes when the system crosses the boundary between two phases.

For short-range pairing ($\alpha>1$), $W$ assumes only integer values: $W=0,\,\pm 1$. $W\neq 0$ signals a topologically nontrivial phase AliceaRPP2012 , characterized by the presence of massless (Majorana) edge modes in the open chain Kitaev . For long-range pairing ($\alpha\leq 1$), semi-integer values $W=\pm 1/2$ appear delgado2015 ; lepori2016 ; footnote1 . We quote the corresponding phases as “long-range”. A similar phase for $\mu/J<1$ supports massive edge modes in the open chain, while for $\mu/J>1$ no edge mode occurs delgado2015 ; lepori2016 . Long-range phases are all characterized by a violation of the area law for the Von Neumann entropy VodolaPRL2014 ; AresPRA2015 .

For $\Delta\neq 0$ and in the limit $L\to\infty$, the energy gap between the superfluid ground state $|\psi_{\rm gs}\rangle$ and the first excited state closes at $\mu/J=1$ and $k=\pi$ for all the values of $\alpha$, as well as at $\mu/J=-1$ and $k=0$ when $\alpha>1$ only [see Fig. 1 (a)]. There, $W$ changes between 0 and 1. Furthermore, QPT are also present along the line $\alpha=1$: between two phases with $W=1/2$ and [math] for $\mu/J<-1$, between $W=1/2$ and $1$ for $|\mu|/J<-1$, and between $W=-1/2$ and [math] for $\mu/J>1$. Remarkably, these QPTs occur without closing the energy gap; they are associated with various discontinuities, for instance in the mutual information and in the decay exponents for the two-point correlations $\langle\hat{a}_{i}^{\dagger}\hat{a}_{j}\rangle$, $\langle\hat{a}_{i}\hat{a}_{j}\rangle$ VodolaPRL2014 ; lepori2016 . In the following we will show that, similarly to the QPT at $\mu/J=\pm 1$, the QPT at $\alpha=1$ is signaled by the divergence of the derivative of the QFI with respect to $\alpha$, as well as by the divergence of the fidelity susceptibility.

Figures 1(b) and (c) show the phase diagram in the $\Delta/J$–$\mu/J$ plane for $\alpha=\infty$ and $\alpha=0$, respectively. The energy gap closes at $\Delta=0$ for $|\mu|/J\leq 1$ and where $\cos k=-\mu/J$. For $\alpha\leq 1$ and $|\mu|/J>1$, we have a transition – without closing the energy gap – between $W=\pm 1/2$ footnote1 , crossing $W=0$ on the singular line $\Delta=0$.

Multipartite entanglement.– We witness ME in the ground state of the Kitaev model (13) using the QFI,

[TABLE]

calculated with respect to the nonlocal operator $\hat{O}_{\rho}=\sum_{j=1}^{L}\hat{o}_{\rho}^{(j)}$ ($\rho=x,y$), where

[TABLE]

$\delta_{\rho,y}=1$ for $\rho=y$ and $\delta_{\rho,y}=0$ otherwise. We also calculate the QFI with respect to the staggered operator $\hat{O}_{\rho}^{({\rm st})}=\sum_{j=1}^{L}(-1)^{j}\hat{o}_{\rho}^{(j)}$: $F_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{\rho}^{(\rm st)}]$. Noticing that $\langle\psi_{\rm gs}|\hat{O}_{\rho}|\psi_{\rm gs}\rangle=\langle\psi_{\rm gs}|\hat{O}_{\rho}^{({\rm st})}|\psi_{\rm gs}\rangle=0$, we can rewrite the Fisher density $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{\rho}]\equiv F_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{\rho}]/L$ for the closed chain as

[TABLE]

where $C_{\rho}(l)=\langle\psi_{\rm gs}|\hat{o}_{\rho}^{(1)}\hat{o}_{\rho}^{(1+l)}|\psi_{\rm gs}\rangle$, and analogous expressions for $\hat{O}_{\rho}^{(\rm st)}$, with C_{\rho}^{(\rm st)}(l)=\big{\langle}\psi_{\rm gs}|(-1)^{l}\hat{o}_{\rho}^{(1)}\hat{o}_{\rho}^{(1+l)}|\psi_{\rm gs}\big{\rangle}. Equation (5) directly links the QFI to the connected correlation function for the operator $\hat{O}_{\rho}$.

The relation between ME and Eq. (3) is obtained by exploiting the convexity properties of the QFI PezzePRL2009 ; HyllusPRA2012 ; TothPRA2012 , that holds even when the QFI is calculated with respect to a nonlocal operator PezzePNAS2016 . Specifically, the violation of the inequality

[TABLE]

signals $(\kappa+1)-$partite entanglement ($1\leq\kappa\leq L-1$) between sites of the fermionic chain. In particular, separable pure states $|\psi_{\rm sep}\rangle=\bigotimes_{j=1}^{L}|n_{j}\rangle$, where $n_{j}$ is the occupation number of the $j$-th site (that can assume only values [math] or $1$), satisfy f_{Q}\big{[}|\psi_{\rm sep}\rangle,\hat{O}_{\rho}\big{]}\leq 1. Moreover, states $|\psi\rangle$ with $L-1<f_{Q}[|\psi\rangle,\hat{O}_{\rho}]\leq L$ are genuinely $L$-partite entangled footnote2 , $f_{Q}[|\psi\rangle,\hat{O}_{\rho}]=L$ being the ultimate (Heisenberg) bound.

QFI phase diagram.– We have calculated the variance Eq. (3) for different values of the parameter of Kiteav chain, using standard exact techniques Lieb1961 . The QFI follows an asymptotic power law scaling

[TABLE]

where the coefficients $b$ and $c$ depend on $\mu/J$, $\delta/J$, $\alpha$ and $\rho$, but are independent on $L$. The scaling behavior is reported schematically in Fig. 1, while Fig. 2 shows numerical results obtained up to $L\approx 1000$.

The scaling coefficient $b$ is directly related to the behavior of the correlation function. For instance, an exponentially decaying correlation, $C_{\rho}(l)=e^{-d(l)/\xi}$ on a ring, with $\xi>0$ independent on $L$, gives $b=0$ and $c=2/(e^{1/\xi}-1)$ in the thermodynamic limit. In this case, the QFI is extensive: $c>\kappa-1$ is obtained for $\xi^{-1}<\log[(\kappa+1)/(\kappa-1)]$ and witnesses $\kappa$-partite entanglement that remains constant when increasing the system size. Instead, when $b>0$, the QFI is superextensive: the larger is $L$, the larger is the witnessed $\kappa$-partite entanglement. Values $0<b<1$ can be related to a rescaling of the correlation functions with the system size, $C_{\rho}(l)=L^{b-1}c_{\rho}(l/L)$, giving $c=\int_{0}^{1}dx\,c_{\rho}(x)$ supp . This is obtained, for instance, when $C(l)\sim 1/l^{1-b}$.

In Figs. 1 and 2 we use the scaling coefficient $b$ to characterize the different phases of the Kitaev model. In particular, in Figs. 1(a) and 2(a) we plot the phase diagram of $b$ in the $\mu/J$-$\alpha$ plane, for the operator $\hat{O}_{\rho}$ maximizing the QFI (see below). For short-range pairing ($\alpha>1$), we find $b=1$ for $|\mu|/J<1$ and $b=0$ for $|\mu|/J>1$. Therefore, a superextensive QFI is observed only in the topologically nontrivial phase. On the critical lines $|\mu|/J=1$, we have $b=3/4$, that is associated to the algebraic asymptotic decay of the correlation functions, $C_{x}(l\to L)\sim L^{1/4}$ supp , implied at $\alpha>2$ by conformal invariance Mussardo ; lepori2016ann . For $\alpha\approx 1$ the numerical calculations are affected by finite-size effects supp . Yet, when $\alpha=1$, we can argue (see supp for a finite-size scaling analysis) that $b=1/2$ for $|\mu|/J>1$, and $b=3/4$ for $|\mu|/J\leq 1$. For long-range pairing ($\alpha<1$), $b=3/4$ for $\mu/J\neq 1$ and $b=1/2$ for $\mu/J=1$. The above scalings refer for every $\alpha$ to the QFI relative to an optimal choice of operators, that is $\hat{O}_{x}$ for $\mu/J\leq 1$ and $\hat{O}_{y}^{(\rm st)}$ for $\mu/J\geq 1$.

In Figs. 1 and 2 we also plot $b$ in the $\Delta/J$–$\mu/J$ plane for nearest-neighbor [$\alpha=\infty$, panel (b)] and infinite-range [$\alpha=0$, panel (c)] pairings. Notice that the regime $\Delta<0$ is mapped on the one at $\Delta>0$ by the phase re-definition $\hat{a}_{j}\to\pm i\,\hat{a}_{j}$; this operation also interchanges $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{x}]\leftrightarrow f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{y}]$ and $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{x}^{(\rm st)}]\leftrightarrow f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{y}^{(\rm st)}]$. For $\alpha=\infty$ and $\Delta\neq 0$, we have $b=1$ for $|\mu|/J<1$, $b=3/4$ for $|\mu|/J=1$, and $b=0$ elsewhere. For $|\Delta|/J=0$ and $|\mu|/J<1$ we have $b=1/2$. Again, a superextensive scaling of the QFI is found in the correspondence of phases with nontrivial winding numbers. If $\alpha=\infty$ the Kitaev Hamiltonian maps, via the Jordan-Wigner transformation Lieb1961 , to the short-range Ising model (for $\Delta=J$) and our scaling of the QFI agrees with existing calculations HaukeNATPHYS2016 ; LiuJPA2013 . In particular, the correlation function of the $\hat{O}_{x}$ operator is exponentially decaying for $|\mu|/J>1$ and constant for $|\mu|/J<1$: genuine $L$-partite entanglement is witnessed at $\mu=0$ where $C_{x}(l)=1$ $\forall l$ supp . For $\Delta=0$ (and every $\alpha$), the Kitaev chain maps to the XX model. It is known BarouchPRA1971 that the ground state is a product state for $|\mu|/J>1$ and, accordingly, we find $c=0$ in this case supp . In the full diagram of Fig.2(b), the optimal choice of operators is $\hat{O}_{x}$ for $\Delta/J\geq 0$ and $\hat{O}_{y}$ for $\Delta/J\leq 0$.

For infinite-range pairing ($\alpha=0$) we find $b=3/4$ everywhere, except at the phase boundaries: $b=1/2$ for $\mu/J=1$ and $|\Delta|/J>1$, as well as for $|\Delta|/J=0$ and $|\mu|/J<1$, while $b=0$ for $|\Delta|/J=0$ and $|\mu|/J>1$. We can distinguish four regions in the phase diagram of Fig. 2(c), singled out by the operators optimizing the QFI. For $|\mu|/J<1$, the optimal operators are $\hat{O}_{x}$ for $\Delta>0$ and $\hat{O}_{y}$ for $\Delta<0$, while, if $|\mu|/J>1$, $\hat{O}_{x,{\rm st}}$ for $\Delta>0$ and $\hat{O}_{y,{\rm st}}$ for $\Delta<0$.

Next, we show that the QPT between phases characterized by different winding numbers are signaled by a divergence of the derivative of the QFI with respect to the control parameter. This is illustrated in Fig. 3, where we plot the weighted derivative of $f_{Q}$ with respect to $\mu/J$ [panel (a)], $\alpha$ [panel (b)], and $\Delta/J$ [panel (c) and (d)]. These results can be understood by taking the derivative of Eq. (7) with respect to a parameter $\eta$ of the model (i.e. $\eta=\mu/J$, $\Delta/J$ or $\alpha$):

[TABLE]

In the interesting case $c\neq 0$ and $b\geq 0$, Eq. (8) diverges in the thermodynamic limit $L\to\infty$ either because of a divergence of $\partial_{\eta}c/c$ or, even when $\partial_{\eta}c$ is smooth, because of $\partial_{\eta}b\neq 0$. Figure 3, obtained at $L=1000$, shows that, while $\partial_{\mu}f_{Q}/f_{Q}$ and $\partial_{\Delta}f_{Q}/f_{Q}$ vary sharply at the phase transition points of Fig. 1 (see supp for a plot of the coefficients $b$ and $c$), $\partial_{\alpha}f_{Q}/f_{Q}$ varies smoothly as a function $\alpha$. Yet, a finite-size scaling analysis supp shows that, in the limit $L\to\infty$ (up to $L=5000$ in our numerics), $\partial_{\alpha}f_{Q}/f_{Q}$ tends to peaks at $\alpha=1$. Therefore, a fast change of the QFI is able to detect the QPT at $\alpha=1$ – associated to a change of the winding number – even if it occurs without closing of the gap in the quasiparticle spectrum.

The different QPTs of Fig. 1 are also detected by the fidelity susceptibility. In general, this quantity detects the qualitative changes in the ground state when crossing a QPT and should not be confused with the QFI, Eq. (3), studied here. The fidelity susceptibility is defined as $\chi_{\eta}=-\frac{d^{2}\mathcal{F}}{d\eta^{2}}$ YouPRE2007 ; ZanardiPRL2007 , where $\mathcal{F}=\langle\psi_{\rm gs}|\tilde{\psi}_{\rm gs}\rangle=\prod_{n=0}^{L/2}\cos\frac{\theta_{k_{n}}-\tilde{\theta}_{k_{n}}}{2}$ is the fidelity between $|\psi_{\rm gs}\rangle$ and $|\tilde{\psi}_{\rm gs}\rangle$, which are ground states of the Hamiltonian in Eq. (13) for different values of the parameter(s). Analytical expressions of $\chi_{\mu}$, $\chi_{\alpha}$ and $\chi_{\Delta}$ are given in the Supplementary Material supp . Plots of the fidelity susceptibly are shown in the panels (e)-(h) of Fig. 3, they can be compared with the panels (a)-(d), showing the weighted derivative of the Fisher density. Notice that both display similar divergencies at the critical transition points, also on the massive line $\alpha=1$.

Conclusions.– The quantum Fisher information detects multipartite entanglement in the topological and long-range phases (with nonvanishing winding numbers) of the Kitaev chain with variable-range pairing. A key aspect is the calculation of the quantum Fisher information relative to nonlocal operators showing long-range correlations, whereas the quantum Fisher information relative to local operators is unable to detect entanglement in this model, as noted in HaukeNATPHYS2016 .

Furthermore, we have shown that QPTs are identified by the divergence of the derivative of the quantum Fisher information with respect to different control parameters, even when the phase transition is not associated to a closing gap in the excitation spectrum. A similar behavior is also found for the fidelity susceptibility. Our results are a step forward in the study of entanglement in topological insulators by providing a clear evidence of multipartite entanglement in these systems.

Acknowledgements – We thank S. Ciuchi, S. Paganelli, and D. Vodola for useful discussions.

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See Supplemental Materials.

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In the case $\Delta<0$ the phase diagram is obtained changing $W\to-W$.

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These fractional numbers are due to the singularity of $f_{\alpha}(k)$ for $\alpha<1$ that occurs at $k=0,2\pi$ in the thermodynamic limit and do not testify topological stability, nor the presence of edge modes, in general lepori2016 . Specifically, massive edge state are found for $\mu/J<1$ for both $\Delta>0$ (a phase characterized by $W=1/2$) and $\Delta<0$ (a phase characterized by $W=-1/2$).

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Supplemental Materials

Hamiltonian and basic equations.– We discuss here the Bogoliubov Hamiltonian and the calculation of the variance giving the QFI. We start from Eq. (1) of the manuscript,

[TABLE]

The parameters $\mu$, $J$ and $\Delta$ are, respectively, the chemical potential, the hopping rate (we take $J$ real and positive) and the pairing strength, and $L$, assumed even, is the total number of sites (we consider even values of $L$). The Fourier transform and its inverse read

[TABLE]

where

[TABLE]

We have

[TABLE]

and

[TABLE]

where $f_{\alpha}(k)=\sum_{\ell=1}^{L-1}\frac{\sin k\ell}{d_{\ell}^{\alpha}}$. The function $f_{\alpha}(k)$ diverges at $k=0$ when $\alpha\leq 1$. In particular, $f_{\infty}(k)=2\sin(k)$ and $f_{0}(k)=\cot(k/2)$. In the limit $L\to\infty$, we have $f_{\infty}(k)=2\sin k$ and $f_{0}(k)=\cot(k/2)$. The Hamiltonian (9) becomes

[TABLE]

where

[TABLE]

and $\hat{\sigma}_{y}$ and $\hat{\sigma}_{z}$ are Pauli matrices. We associate this Hamiltonian to a vector $\bm{h}(k)=h_{y}(k)\bm{y}+h_{z}(k)\bm{z}$ in the unit circle, where

[TABLE]

and

[TABLE]

Finally, the Hamiltonian (13) can be diagonalized by the Bogoliubov transformation

[TABLE]

giving

[TABLE]

where $\epsilon_{k}=\sqrt{[J\cos k+\mu]^{2}+[f_{\alpha}(k)\Delta/2]^{2}}$. When $\Delta\neq 0$, the energy gap $\Delta E$ between the ground state (of energy 0) and the first excited state can close (in the limit $L\to\infty$) only when $f_{\alpha}(k)=0$ for some quasi-momentum $k$. When $\alpha>1$, this happens for $k=0$ (in which case $\Delta E=0$ at $\mu/J=-1$) and $k=\pi$ (in which case $\Delta E=0$ at $\mu/J=1$). When $\alpha\leq 1$, $f_{\alpha}(k)$ vanishes only at $k=\pi$ and the energy gap closes only when $\mu/J=1$.

$\mu/J$–$\alpha$ phase diagram.– We point out that the optimal operator for the QFI is different in the different regions of the phase diagram. In Fig. 4 we plot the Fisher density $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{x}]$ [panel (a)] and $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}_{y}^{(\rm st)}]$ [panel (b)]. The QFI calculated for the other operators, i.e. $\hat{O}_{y}$ and $\hat{O}_{x}^{(\rm st)}$, is always smaller than $L$ and it is not shown.

We define $f_{Q}[|\psi_{\rm gs}\rangle]$ as the Fisher density $f_{Q}[|\psi_{\rm gs}\rangle,\hat{O}]$ maximized over the four operators considered in the manuscript, i.e. $\hat{O}_{\rho}$ and $\hat{O}_{\rho}^{(\rm st)}$, with $\rho=x,y$. We fit $f_{Q}[|\psi_{\rm gs}\rangle]$ as a function of $L$ as $f_{Q}[|\psi_{\rm gs}\rangle]=1+cL^{b}$ (see below), where the coefficients $b$ and $c$ depend on $\mu/J$, $\Delta/J$ and $\alpha$. In Fig. 5 we plot $b$ and $\log c$ in the phase diagram. In Fig. 6 and 7 we plot the derivative of $b$ and $c$, respectively, with respect to $\mu$ [in panel (a)] and $\alpha$ [in panel (b)].

Finite-size scaling analysis around $\alpha=1$.– From Fig. 6 and the analysis of the scaling coefficients $b$ and the prefactor $c$, it is evident that the derivative of the Fisher density with respect to $\mu$ diverges at $\mu/J=1$ for $\alpha\geq 0$ and $\mu/J=-1$ for $\alpha>1$, in correspondence of the QPTs. It is less evident, from the numerical data, that the derivative of the Fisher density with respect to $\alpha$ diverges at $\alpha=1$ and any value of $\mu$, in correspondence of the QPT. We have thus performed a finite-size scaling analysis of the normalized derivative $\tfrac{1}{f_{Q}}\tfrac{df_{Q}}{d\alpha}$ around $\alpha=1$ and different system size $L$. From the results shown in Fig. 8 we clearly see that, increasing $L$, slowly $\tfrac{1}{f_{Q}}\tfrac{df_{Q}}{d\alpha}$ tends to peak at $\alpha=1$.

$\Delta/J$–$\mu/J$ phase diagram, $\alpha=\infty$.– In Fig. 9 we show the Fisher density for the optimal operator in the $\Delta/J$–$\mu/J$ phase diagram for short-range pairing ( $\alpha=\infty$). In Fig. 10 we show the corresponding coefficients $b$ and $c$.

$\Delta/J$–$\mu/J$ phase diagram, $\alpha=0$.– In Fig. 11 we show the Fisher density for the optimal operator in the $\Delta/J$–$\mu/J$ phase diagram for long-range pairing ( $\alpha=0$). In Fig. 12 we show the corresponding coefficients $b$ and $c$.

Correlation functions and polynomial fits.– Below, we show polynomial fits of the Fisher density for different values of the parameters. These and similar fits are used to extract the coefficients $b$ and $c$ discussed above and in the manuscript. We also show examples of the rescaling of the correlation function, highlighting the collapse of curves $L^{1-b}C_{\rho}(l)$ as a function of $l/L$ and for different values of $L$. This implies that the correlation function can be written as $C_{\rho}(l)=L^{b-1}c_{\rho}(l/L)$. Computing Eq. (5) gives

[TABLE]

where the last equality is obtained in the limit $L\to\infty$.

Fidelity susceptibility.– The normalized ground state of the Kitaev chain can be written as

[TABLE]

where $|0\rangle$ is the vacuum of $\hat{a}_{k}$. The ground state fidelity is

[TABLE]

where $|\psi_{\rm gs}\rangle$ and $|\tilde{\psi}_{\rm gs}\rangle$ refer to the ground state for different values of the parameter(s). Expanding in Taylor series for $\tilde{\theta}_{k_{n}}\approx\theta_{k_{n}}$, up to second order in $d\theta_{k_{n}}=\tilde{\theta}_{k_{n}}-\theta_{k_{n}}$, we find

[TABLE]

Finally, the fidelity susceptibility with respect to the variation of a generic parameter $\eta$ is defined as

[TABLE]

where

[TABLE]

We now give the explicit expression of the fidelity susceptibility considering the variation of different parameters:

• —

varying $\mu$:

[TABLE]

• —

varying $\alpha$:

[TABLE]

• —

varying $\Delta$:

[TABLE]

We want to see whether $\chi_{\alpha}$ diverges in the thermodynamic limit $L\to\infty$ when $\alpha\to 1$. We first calculate $f_{\alpha}(k)$ in the limit $L\to\infty$, giving $f_{\alpha}^{\infty}(k)=[{\rm Li}_{\alpha}(e^{{\rm i}k})-{\rm Li}_{\alpha}(e^{-{\rm i}k})]/{\rm i}$, where ${\rm Li}_{\alpha}(x)$ are polylogarithmic functions. To see the divergence of $\chi_{\alpha}$ it is enough to see that one term in the sum diverges (this because we have a sum of positive terms). We calculate

[TABLE]

in the limit $k=\pi/L\to 0$. The results of calculations are shown in Fig. 16. We see that, very slowly in $L$, $\chi_{\alpha}(k)$ peaks at $\alpha=1$. Exploiting the expansion $f_{\alpha}^{\infty}(k)\sim k^{\alpha-1}$ for $k\to 0$, it is easy to check that the found peak in $\chi_{\alpha}(k)$, developing as $(\log L)^{2}$, is directly related with the divergence of $f_{\alpha}^{\infty}(k)$ at $\alpha=1$; the same divergence, holding also at $\alpha<1$, has been found also responsible for the appearance of the semi-integer winding numbers in the same regime.

This result for the divergence of $\chi_{\alpha}(k)$ justifies the emergence of new phases below $\alpha=1$, even if no vanishing mass gap in the Bogoliubov spectrum occurs at this threshold.