After having given the parameterized exact solutions for the cluster-XY family of Hamiltonians and calculated the overlap for the ground-state entanglement, we now examine a few examples. We did verify numerically that for all the models we consider, the closest product state to the ground state using (1) single-site product ones and (2) two-site product ones can be written as (1)
and (2)
, respectively. We also compared numerical exact diagonalization for lowest two energies, indicated by points at the below figures, with our analytic solutions.
The anisotropic XY model with three-site interaction (XzY model)
The first model analyzed with the geometric entanglement is the celebrated XY model, done in Ref. [21]. It was observed that the geometric entanglement displays a singular behavior across the critical line \(h_c=1\). This model was also investigated in terms of other entanglement measures, such as the concurrence [23] and the entanglement entropy [65, 66, 68]. The behavior of concurrence is similar. The entanglement entropy shows a logarithmic scaling in the subsystem size at criticality.
As a first example in our calculation, we present the solution of the anisotropic XY model with three-site interaction (XX and YY, each mediated by one-site Z term) in the transverse field and discuss the ground-state entanglement. Similar Hamiltonians have been examined previously [55,56,57,58], with little emphasis on the entanglement behavior, except for the localizable entanglement in Ref. [55]. This model in one dimension is exactly solvable. We find that near the critical line \(h_c=1\), the global entanglement shows divergence and quantum phase transition occurs between a nontrivial SPT phase and a trivial paramagnetic phase. The existence of the continuous transition is also consistent with the behavior of the energy gap.
The model is characterized by the following parameters, which we introduced earlier,
$$\begin{aligned} N^{(x)}&=1,\ N^{(y)}=1, \end{aligned}$$
(67a)
$$\begin{aligned} J_{l}^{(x)}&=\{(1+r)/2\},\ J_{l'}^{(y)}=\{(1-r)/2\}, \end{aligned}$$
(67b)
$$\begin{aligned} n_{l}^{(x)}&=\{1\},\ n_{l'}^{(y)}=\{1\}. \end{aligned}$$
(67c)
Substituting these terms into \(H_{PXY}\) (4), we obtain the XzY model in the transverse field:
$$\begin{aligned} H_{XzY}&= - \sum _{j=1}^{N} \left[ \frac{1+r}{2} \sigma _{j-1}^{x} \sigma _{j}^{z} \sigma _{j+1}^{x} + \frac{1-r}{2} \sigma _{j-1}^{y} \sigma _{j}^{z} \sigma _{j+1}^{y} + h \sigma _{j}^{z} \right] , \end{aligned}$$
(68)
where r is a magnetic anisotropy constant between \(\sigma _{x}\) and \(\sigma _{y}\) terms with \(0 \le r \le 1\). When \(r=1\) (the Ising limit), the model reduces a cluster model [73], and in the limit \(r=0\), it becomes an isotropic XY model with three-site interaction. Using Eq. (14), we calculate \(\alpha _{k}\) and \(\beta _{k}\):
$$\begin{aligned} \beta _{k}&= \Big (\frac{1+r}{2}\Big ) \sin \varTheta ^{(x)}_{l} - \Big (\frac{1-r}{2}\Big ) \sin \varTheta ^{(y)}_{l'}, \end{aligned}$$
(69)
$$\begin{aligned} \alpha _{k}&= {h - \Big (\frac{1+r}{2}\Big ) \cos \varTheta ^{(x)}_{l} -\Big (\frac{1-r}{2}\Big ) \cos \varTheta ^{(y)}_{l'}}, \end{aligned}$$
(70)
with \( \varTheta _{1}=\varTheta ^{(x)}_{1}=\varTheta ^{(y)}_{1}=\frac{4\pi }{N}(k+b)\). We then obtain the diagonalized Hamiltonian and the exact energy spectrum (see Eqs. 16–20):
$$\begin{aligned} H&= \sum _{k=0}^{N-1} \epsilon _{k}^{(b)} \left( \gamma _{k}^{(b) \dagger } \gamma _{k}^{(b)}-\frac{1}{2} \right) . \end{aligned}$$
(71)
The eigenvalues can be obtained by carefully analyzing odd (\(b=0\), periodic boundary conditions) and even (\(b=1/2\), antiperiodic boundary conditions) sectors separately, assuming N is even or odd, respectively:
$$\begin{aligned} \epsilon _{k}^{(b)}= \left\{ \begin{array}{lr} 2 (h-1), &{} \text {for } k=0 \wedge b=0\\ 2 (h-1), &{} \text {for } k=\frac{N}{2} \wedge b=0\\ 2 (h-1), &{} \text {for } k=\frac{N-1}{2} \wedge b=1/2 \end{array}\right\} = 2 \alpha _{k}^{(b)}, \end{aligned}$$
(72)
or otherwise (N can be either even or odd):
$$\begin{aligned} \epsilon _{k}^{(b)}&= 2 \sqrt{\left( \beta _{k} \right) ^{2}+\left( \alpha _{k} \right) ^{2}}= 2 \sqrt{\left( r \sin \frac{4\pi }{N}(k+b)\right) ^{2}+\left( h-\cos \frac{4\pi }{N}(k+b)\right) ^{2}}, \end{aligned}$$
(73)
with the corresponding Bogoliubov solution:
$$\begin{aligned} \tan 2\theta _{k}^{(b)}&= \frac{\beta _{k}}{\alpha _{k}} = \frac{r \sin \varTheta _{1} }{h - \cos \varTheta _1}. \end{aligned}$$
(74)
One notices that the solution is similar to the solution of the standard XY model [51, 59, 72]. The only difference occurs in the momentum space by a factor of two, i.e., in the XY model \(\varTheta _1\) is \(2\pi (k+b)/N\) instead of \(4\pi (k+b)/N\). But there are some differences that are related to the subtlety in getting the global lowest energy state. For instance, in the XY model with \(r\ne 1\), the state of the lowest energy can come from either the even or the odd sector, as illustrated in Fig. 1b for \(r=0.5\). As a function or h, the ground state switches between the two sectors, as the lowest energy changes between \(E_0^{(b=0)}\) and \(E_0^{(b=1/2)}\). But for the XzY model, the ground state is always in the even sector with zero fermion, as illustrated in Fig. 1a. Moreover, for the odd-number fermion case (\(b=0\)), the lowest energy level in this sector depends on the field parameter (h) and the anisotropy constant (r). For example, in the Ising limit where \(r=1\), the odd sector has three-fermion occupation as the lowest energy state in the region of \(h<0\); otherwise, it is energetically favorable to occupy one fermion for even N; see Fig. 2a and also Fig. 1a. However, the true ground state arises from the \(b=1/2\) (even) sector and has no \(\gamma \) fermion. This phenomenon differs from the standard XY model, where the lowest energy in the odd sector always has one-fermion occupation. The possibility of such peculiarity is discussed in Sect. 2.1; see discussions around Eq. (21). We note that for a finite system size N (even) and \(r=0.5\), the lowest energy level in the odd sector has three fermions from negative h values up to about \(h\approx 0.4\); see Figs. 3a and 1a. Moreover, the energy gap between the ground and the first excited states is closing with an increasing system size N at \(h=1\), implying a quantum phase transition there; see Figs. 3b and 2b. For small finite sizes, the gap as a function of h is not smooth for \(r=0.5\). In contrast, the gap vs. h is smooth for \(r=1\) even with finite sizes, and in the thermodynamic limit \(N\rightarrow \infty \), the energy gap for \(r=1\) (Ising limit of the XzY model) becomes \(2\big | 1-|h| \big |\).
To examine the quantum phase transition in the phase diagram, we also calculate geometric entanglement and we plot the entanglement per site in Fig. 4 over a wide range of r and h. It is visible that the behavior of entanglement is singular across \(h=1\), similar to that in the standard XY model [21]. We illustrate this for two different r’s (\(r=0.5\) and \(r=1\)) in Figs. 3c and 2c, as well as the entanglement derivative w.r.t. h in Figs. 3d and 2d. The derivative of the entanglement develops singularity, which indicates a quantum phase transition.
From the above, it follows that for \(r=1\) the Hamiltonian reduces to
$$\begin{aligned} H=-\sum _j \left( \sigma _{j-1}^x \sigma _{j}^z\sigma _{j+1}^x + h \sigma _j^z\right) . \end{aligned}$$
(75)
The model has a \(Z_2\times Z_2\) symmetry, generated by \(U_e=\prod _{j\, \mathrm{even}} \sigma _j^z \) and \(U_o=\prod _{j\, \mathrm{odd}} \sigma _j^z\) [49]. At \(h=0\), the ground state is known to be the cluster state, which is a nontrivial SPT state. (One expects this nontrivial SPT order to hold for general n-site mediated Ising model with \(Z_2^{\otimes n+1}\) symmetry; see Ref. [50].) At large h, the ground state is a trivial paramagnetic state. As we have seen that there is a quantum phase transition at \(h=1\), detected by the gap closing and the entanglement singularity, the SPT order appears in the region \(|h|\le 1\). In fact, XzY model Eq. (68) at any r has the \(Z_2\times Z_2\) symmetry, and we expect that for \(0<r\le 1\), the phase diagram contains a nontrivial SPT phase for \(h<1\) (as there is no phase transition inside that region) and a trivial paramagnetic phase for \(h>1\), separated at a critical line at \(h=1\). The reason \(r=0\) line is excluded is because the system is gapless for \(h\in [0,1]\) at \(r=0\). This compares to the standard XY model, where \(h=1\) separates a ferromagnetic phase from a paramagnetic phase. From the results in Ref. [50], we also expect that this is generic behavior for general but finite n (where the interaction is restricted to be short ranged).
XY model with halfway interaction
In Sect. 2.2, we introduced an illustrative example of the XY model with n-site Z-mediated XX and YY interactions. For \(n=0\) and \(n=1\), we recover the standard XY model and the XY model with three-site interaction (XzY model) investigated in the previous example. In this part, we demonstrate how a specific choice of site interaction \(n=N/2-1\) (halfway interaction) exhibits different behaviors from that of \(n=0,1\) and has no quantum phase transition at \(h=1\). This is a rather interesting result since except at this arbitrary point (\(n\ne N/2-1\)), the XY model generically exhibits a quantum phase transition for each n-site interaction, as seen by vanishing of the gap there in Fig. 5a. Moreover, we also discover a first-order phase transition in the XY model with halfway interaction in the region of \(0 \le r<1\). (The halfway interaction only occurs for even system sizes N). In this limit, the first-order transition occurs at the Barouch–McCoy circle [74], namely \(r^2+h^2=1\). For example, in the case of \(r=0.7\) the phase transition occurs at \(h_{c}=\sqrt{1 - 0.7^2}\approx 0.714\) as illustrated in Fig. 5b. We note that there is an even-odd effect in N / 2 and the behavior of the gap is different.
We note that for the standard XY model, the Barouch–McCoy circle represents only a crossover that divides the ferromagnetic phase into two regions. Here, for the halfway interaction, the circle represents a curve of first-order transition points.
First, let us define the parameters that give the XY model with n-site interaction
$$\begin{aligned} N^{(x)}&=1,\ N^{(y)}=1, \end{aligned}$$
(76a)
$$\begin{aligned} J_{l}^{(x)}&=\{(1+r)/2\},\ J_{l'}^{(y)}=\{(1-r)/2\}, \end{aligned}$$
(76b)
$$\begin{aligned} n_{l}^{(x)}&=\{n\},\ n_{l'}^{(y)}=\{n\}, \end{aligned}$$
(76c)
yielding the corresponding Hamiltonian:
$$\begin{aligned} H_{XnY}&= - \sum _{j=1}^{N} \bigg ( \frac{1+r}{2} \sigma _{j-1}^{x} \sigma ^{z}_{j} \ldots \sigma ^{z}_{j+n-1} \sigma _{j+n}^{x} \nonumber \\&\quad + \frac{1-r}{2} \sigma _{j-1}^{y} \sigma ^{z}_{j} \ldots \sigma ^{z}_{j+n-1} \sigma _{j+n}^{y} + h \sigma _{j}^{z} \bigg ). \end{aligned}$$
(77)
This Hamiltonian can be diagonalized into the form, Eq. (16), and we obtain the following Bogoliubov solution (with \(\phi ^{n}_{k} \equiv \frac{2\pi }{N}(n+1)(k+b)\)):
$$\begin{aligned} \tan 2\theta ^{(b)}_{k} = \frac{r\sin \phi ^{n}_{k}}{h-\cos \phi ^{n}_{k}}. \end{aligned}$$
(78)
In the case of halfway interaction, we substitute \(n=N/2-1\) to simplify Bogoliubov solution, respectively, for the even (\(b=1/2\)) and the odd sector (\(b=0\)):
$$\begin{aligned} \tan 2\theta ^{(1/2)}_{k}&= \frac{r\sin \Big [\pi \left( k+\frac{1}{2}\right) \Big ] }{h-\cos \Big [\pi \left( k+\frac{1}{2}\right) \Big ]}= \frac{(-1)^{k} r}{h}, \end{aligned}$$
(79a)
$$\begin{aligned} \tan 2\theta ^{(0)}_{k}&= \frac{r\sin \left( \pi k\right) }{h-\cos \left( \pi k\right) }=0, \end{aligned}$$
(79b)
with following energy spectrum for odd N and \(b=1/2\) and for even N and \(b=0\):
$$\begin{aligned} \epsilon _{k}^{(b)}= \left\{ \begin{array}{lr} 2 (h-1), &{} \text {for } k=0 \wedge b=0\\ 2 \left[ h-(-1)^{N/2}\right] , &{} \text {for } k=\frac{N}{2} \wedge b=0\\ 2 \ h, &{} \text {for } k=\frac{N-1}{2} \wedge b=1/2 \end{array}\right\} = 2 \alpha _{k}^{(b)}, \end{aligned}$$
(80)
or otherwise:
$$\begin{aligned} \epsilon _{k}^{(b)}&= 2 \sqrt{\Big [h-\cos \big (\pi (b+k)\big )\Big ]^2+\Big [r \sin \big (\pi (b+k)\big )\Big ]^2}\nonumber \\&= \left\{ \begin{array}{lr} 2 |h-(-1)^k|, &{} \text {for } b=0,\\ 2 \sqrt{h^2+r^2}, &{} \text {for } b=1/2. \end{array}\right. \end{aligned}$$
(81)
To obtain the ground state and the first excited state, one should examine even and odd sectors carefully. This model shows vacua competition [59] similar to the standard XY model, meaning that odd and even sectors switch the roles of being the true ground state depending on h. This competition is lifted in the Ising limit where \(r=1\) and the ground state is certainly constructed from the even sector (\(b=1/2\)) with no fermion; except when \(N=2(2m+1)\) and at \(h=0\), another degenerate ground state is from the odd sector with one fermion; see Fig. 7. In the case of \(r=0.5\), the switching happens around \(h\approx 0.866\). The ground state becomes dominated by the odd sector in the range \(-0.87 \lesssim h \lesssim 0.87\), but outside that range the ground state comes from the even sector (\(b=0\)) with zero-fermion occupation; see Fig. 6. In particular, for \(-0.87 \lesssim h < 0\) and with \(N=4m\), the lowest energy level in the odd sector has three-fermion occupation instead of one fermion, as it is energetically favorable to occupy three fermions in the odd sector rather than just one fermion. In fact, in this region, the ground state is degenerate (not shown explicitly in Fig. 6a, but is shown in Fig. 6b), both degenerate ground states have 3 fermions. But in \(0\le h \lesssim 0.87\), the lowest one-fermion and three-fermion states become degenerate. For \(N=2(2m+1)\) and \(-\,0.87 \lesssim h \lesssim 0.87\), the lowest energy is dominated by the one-fermion state in the odd sector. This phenomenon is anticipated earlier in Eqs. (19–22). Using these equations, we also calculate the lowest energy for the odd/even sector and the true energy gap which is shown in Fig. 6. All of these suggest that there is a first-order phase transition for the halfway XY model with \(0\le r<1\), as the transition is due to a level crossing. However, for \(r=1\), the halfway Ising model, the gap closes at \(h=0\) only for \(N=2(2m+1)\), but not for \(N=4m\).
There is an interesting picture that emerges. In the standard XY model in a transverse field, there is a crossover curve, the so-called Barouch–McCoy circle, given by \(r^2+h^2=1\) [74]. The crossover curve divides the ferromagnetic phase into two regions: (i) inside the arc, the spin–spin correlation functions display oscillatory behavior, and (ii) outside the arc, the correlation functions have no oscillatory behavior. On the arc, the ground state is essentially a product state, also detected by zero geometric entanglement previously in Ref. [21]. Here for the halfway XY model, the crossover arc, \(r^2+h^2=1\) is promoted to a first-order transition curve, due to the mediated long-range Z string of a specific length \(n=N/2-1\). Thus, the transition field h for \(r=0.5\) is \(h_c(r=0.5)=\sqrt{1-0.5^2}\approx 0.886\), agreeing with our calculations of the energy gap in Fig. 6. This works for other value of \(0\le r<1\) as well, see Fig. 5b for \(r=0.7\) case. The behavior of the \(r=1\) halfway Ising model is different, as there is a closing of the energy gap at \(h=0\) only for the total site number being \(N=2(2m+1)\), as shown in Fig. 7. But in the thermodynamic limit, the energy gap \(\varDelta E\) is always finite, except at the peculiar point \(h=0\), namely that it does not close continuously. We thus do not regard this as a phase transition.
As the transition in the halfway XY model is first order, one expects that the entanglement will have a discontinuity at the transition, as it is caused by a level crossing. In this case, the ground state in the range \(-\sqrt{1-r^2}\le h \le \sqrt{1-r^2}\) involves the odd sector with either one or three fermions. One could calculate the ground-state overlap with product states. But we will not proceed with that here. For \(r=1\) halfway Ising model, as well as other Ising models with n-site interaction, the ground-state wavefunction comes from the even sector without a fermion, and for that the overlap is calculated in Sect. 3, and hence, the geometric entanglement (per site and per block of two sites) is readily available upon simple parameter optimization. As shown in Fig. 8, the entanglement develops a cusp behavior at \(h=0\) and gives rise to a jump in the derivative. However, this ‘weak’ singularity is a result that the entanglement is symmetric w.r.t. \(h=0\), but it immediately decreases as soon as h deviates from 0 (i.e., with a nonzero slope). As shown in Ref. [50], at \(h=0\), the state is the generalized cluster state, which exhibits the same geometric entanglement as the cluster state, and is expected to display the infinite localizable entanglement length [75]. Even though there is no true phase transition in the usual statistical mechanics, but there is one peculiar transition proposed by Verstraete, Martin-Delgado and Cirac [75] in that the localizable entanglement length is infinite. This kind of transition was shown to be detectable by the geometric entanglement, displaying the weak singularity, such as the cusp [71].
GHZ-Cluster model
In this part, we calculate the ground-state energy of the GHZ-Cluster model, which was introduced by Wolf et al. [47], and examine the quantum phase transition on the phase diagram, utilizing the geometric entanglement and the energy gap. We consider a local Hamiltonian with three-site interaction constructed by the following matrix product state as its ground state:
$$\begin{aligned} A_{0}=\Bigg ( \begin{matrix} 0&{}\quad 0\\ 1&{}\quad 1 \end{matrix}\Bigg ), \quad A_{1}=\Bigg ( \begin{matrix} 1&{}\quad g\\ 0&{}\quad 0 \end{matrix}\Bigg ), \end{aligned}$$
(82)
and the corresponding Hamiltonian possessing \(\mathbb {Z}_{2}\) symmetry was constructed by Wolf et al. [47] and reads:
$$\begin{aligned} H= \sum _{j=1}^{N} \bigg ( 2(g^2 - 1) \sigma _{j-1}^{z} \sigma _{j}^{z} + (g - 1)^2 \sigma _{j-1}^{z} \sigma _{j}^{x} \sigma _{j+1}^{z} - (1 + g)^2 \sigma _{j}^{x} \bigg ). \end{aligned}$$
(83)
The QPT in the model is peculiar as the ground-state energy is analytic for all range of the parameter g, even though the correlation length diverges at the critical point.
To utilize our parameterization for the model, first we rotate the Hamiltonian around the y axis such that \(\sigma _{x} \rightarrow \sigma _{z}\). Then we choose \(N^{(x)}=2\) and a list of \(J_{l}^{(x)}\), as we need two X blocks and \(N^{(y)}=0\) to eliminate Y block. We note that one can assign the value for h in terms of g to generate the required Hamiltonian. Here we give the resulting parameters that give the equivalent cluster-GHZ model:
$$\begin{aligned} h&=(1 + g)^2, \end{aligned}$$
(84a)
$$\begin{aligned} N^{(x)}&=2,\ N^{(y)}=0, \end{aligned}$$
(84b)
$$\begin{aligned} J_{l}^{(x)}&=\{-2(g^2 - 1), -(g - 1)^2\},\ J_{l'}^{(y)}=\{0\}, \end{aligned}$$
(84c)
$$\begin{aligned} n_{l}^{(x)}&=\{0,1\},\ n_{l'}^{(y)}=\{0\}. \end{aligned}$$
(84d)
Substituting above parameters into Eq. (4) yields the following Hamiltonian:
$$\begin{aligned} H= -\sum _{j=1}^{N} \bigg ( -2(g^2 - 1) \sigma _{j-1}^{x} \sigma _{j}^{x} -(g - 1)^2 \sigma _{j-1}^{x} \sigma _{j}^{z} \sigma _{j+1}^{x} + (1 + g)^2 \sigma _{j}^{z} \bigg ). \end{aligned}$$
(85)
This Hamiltonian can be diagonalized in the form of Eq. (16) with the following Bogoliubov solution, where \(\varphi ^{(b)}_{k}\equiv \frac{2 \pi (b+k)}{N}\),
$$\begin{aligned} \tan 2\theta _{k}^{(b)}&= -\frac{2 (g-1) \sin \varphi ^{(b)}_{k} \big [(g-1) \cos \varphi ^{(b)}_{k}+g+1\big ]}{2 \left( g^2-1\right) \cos \varphi ^{(b)}_{k}+(g-1)^2 \cos 2\varphi ^{(b)}_{k}+(g+1)^2}. \end{aligned}$$
(86)
The exact energy spectrum can be obtained by utilizing Eqs. (14) and (17–20). The eigenvalues in the case of even N for the odd sector (\(b=0\), periodic boundary conditions) and odd N for the even sector (\(b=1/2\), antiperiodic boundary conditions) are as follows:
$$\begin{aligned} \epsilon _{k}^{(b)}= \left\{ \begin{array}{lr} 8 g^2, &{} \text {for } k=0 \wedge b=0\\ 8, &{} \text {for } k=\frac{N}{2} \wedge b=0\\ 8, &{} \text {for } k=\frac{N-1}{2} \wedge b=1/2 \end{array}\right\} = 2 \alpha _{k}^{(b)}, \end{aligned}$$
(87)
or otherwise:
$$\begin{aligned} \epsilon _{k}^{(b)}=4 \left| 1+g^2+\left( g^2-1\right) \cos \varphi ^{(b)}_{k} \right| . \end{aligned}$$
(88)
The model exhibits quantum phase transition at \(g_{c}=0\), and the ground state is the Greenberger–Horne–Zeilinger (GHZ) state. At \(g=1\), the Hamiltonian is proportional to \(\sum _j \sigma _j^z\) where all spins are in the z-direction; this is a paramagnetic phase. At \(g=-1\), the ground state is a cluster state (disordered phase), and the Hamiltonian has a \(Z_2\times Z_2\) symmetry. The cluster state is a representative nontrivial \(Z_2\times Z_2\) SPT state. However, the model only has \(Z_2\) symmetry at \(g\ne -1\). Here we also obtain the exact energy spectrum for this model using Eqs. (17–30) and analyze what ground and first excited states are composed of by examining odd/even sector and the number of fermions occupation. If we restrict ourselves to the region \(-2<g<2\), we find that the ground state comes from the even sector (\(b=1/2\)) with no fermions and the first excited state is constructed from the odd sector (\(b=0\)) with one-fermion occupation. The ground state in the model has no three-fermion occupation in any finite g, see Fig. 9a. We remark that for any system size N (even), the energy gap is equal to \(\varDelta E=8 g^2\) in the regime of \(-1<g<1\); otherwise, outside that range the energy gap is always \(\varDelta E=8\), regardless of the system size. As already shown by construction in Ref. [47] and confirmed here by calculation, the ground-state energy displays no singularity at the critical point \(g=0\); see Fig. 9b. It is a peculiar type of quantum phase transition, as emphasized in Ref. [47].
Figure 9c shows the global entanglement upon using the solution which we derived in the previous section. It contains the global entanglement per site (red, dashed) and per block (black, L=2). We also examine the derivative of the entanglement [46] to study the divergence near the critical point. As shown in Fig. 9d, the quantum phase transition is detected at the GHZ point (\(g=0\)) by the behavior of entanglement. However, we note that at \(g=-1\), the entanglement per block shows a cusp behavior, there is no true phase transition there. However, there is a different kind of transition there in the sense of infinite localizable entanglement length [75]. As remarked earlier, this kind of transition was shown to be detectable by the geometric entanglement in the form of weak singularity, such as the cusp [71].
SPT-Antiferromagnetic transition
As the last example, we examine a particular quantum phase transition [76, 77] between a symmetry-protected topological order and an antiferromagnetic phase by using the same method we derived. The specific model we study here was first discussed by Son et al. [48], who also computed the geometric entanglement per site. They showed that the transition was detected by the singular behavior of the entanglement. For completeness, we also study the spectrum and the geometric entanglement per block.
In order to construct the Hamiltonian, we choose one X and one Y block and set \(h=0\) to eliminate the transverse field term. Parameters of the model considered are shown as follows:
$$\begin{aligned} h&=0, \end{aligned}$$
(89a)
$$\begin{aligned} N^{(x)}&=1,\ N^{(y)}=1, \end{aligned}$$
(89b)
$$\begin{aligned} J_{l}^{(x)}&=\{1\},\ J_{l'}^{(y)}=\{-\lambda \}, \end{aligned}$$
(89c)
$$\begin{aligned} n_{l}^{(x)}&=\{1\},\ n_{l'}^{(y)}=\{0\}. \end{aligned}$$
(89d)
Substituting above parameters into Eq. (4) yields the following Hamiltonian:
$$\begin{aligned} H= -\left( \sum _{j=1}^{N} \sigma _{j-1}^{x} \sigma ^{z}_{j} \sigma _{j+1}^{x} - \lambda \sum _{j=1}^{N}\sigma _{j-1}^{y} \sigma _{j}^{y}\right) . \end{aligned}$$
(90)
This Hamiltonian can be diagonalized in the form of Eq. (16) with the following Bogoliubov solution:
$$\begin{aligned} \tan 2\theta _{k}^{(b)}&= \frac{\lambda \sin \left( \frac{2 \pi (b+k)}{N}\right) +\sin \left( \frac{4 \pi (b+k)}{N}\right) }{\lambda \cos \left( \frac{2 \pi (b+k)}{N}\right) -\cos \left( \frac{4 \pi (b+k)}{N}\right) }. \end{aligned}$$
(91)
The exact energy spectrum can be obtained by utilizing Eq. (14) and (17–20). The eigenvalues in the case of even N for the odd sector (\(b=0\), periodic boundary conditions) and odd N for the even sector (\(b=1/2\), antiperiodic boundary conditions) are as follows:
$$\begin{aligned} \epsilon _{k}^{(b)}= \left\{ \begin{array}{lr} 2 (\lambda -1), &{} \text {for } k=0 \wedge b=0\\ -2 (\lambda +1), &{} \text {for } k=\frac{N}{2} \wedge b=0\\ -2 (\lambda +1), &{} \quad \text {for } k=\frac{N-1}{2} \wedge b=1/2 \end{array}\right\} = 2 \alpha _{k}^{(b)}, \end{aligned}$$
(92)
or otherwise:
$$\begin{aligned} \epsilon _{k}^{(b)}=2 \sqrt{1+\lambda ^2-2 \lambda \cos \left( \frac{6 \pi }{N}(k+b)\right) }. \end{aligned}$$
(93)
The even sector (\(b=1/2\)) with no fermions corresponds to the ground state energy for finite system size N (even), whereas the first excited state comes from the odd sector (\(b=0\)) with one-fermion occupation as shown in Fig. 10a. The energy gap in this case can be obtained by calculating \(\varDelta E=E^{\mathrm{lowest}}_{b=0}-E^{\mathrm{lowest}}_{b=1/2}\) which is approximately \(2(1-|\lambda |)\) in the region \(-1/2<\lambda <1/2\) for small system size (N). In the thermodynamic limit (\(N\rightarrow \infty \)), the energy gap becomes \(\varDelta E=\big (1-\left| \lambda \right| \big ) \big [1+\mathrm {sgn}(1-\left| \lambda \right| )\big ]\) for all regions \(-\infty<\lambda <\infty \). The critical point, \(\lambda _c=1\), can be deduced from the energy gap in the thermodynamic limit; see Fig. 10b. We also calculated geometric entanglement per site and per block, shown in Fig. 10. As can be seen in Fig. 10d, the derivative of the entanglement per site has singularity at \(\lambda =1\), at which the quantum phase transition occurs between the cluster and the antiferromagnetic phases. We note that as the antiferromagnetic phase is involved in the model, in order to compute entanglement per site, we use the closest product state of the form
with
. The entanglement derivative w.r.t. \(\lambda \) clearly also shows the development of divergence at \(\lambda =1\) as the system size N increases. The representative state in the SPT phase is the 1D cluster state [78, 79], which we also have seen in previous subsection. We remark that there is a weak singularity in the entanglement per block around \(\lambda \approx 0.94\), but we cannot identity the state there and do not know the nature of this singularity. It might be a transition in localizable entanglement, but that requires further investigation.
Halfway antiferromagnetic-SPT model
Beyond reproducing results by Son et al. [48], we also examine a slight variation in the model, where, instead of XZX, the halfway interaction for X blocks is considered:
$$\begin{aligned} H= -\left( \sum _{j=1}^{N} \sigma _{j-1}^{x} \sigma ^{z}_{j} \ldots \sigma ^{z}_{j+(N/2)-2} \sigma _{j+(N/2)-1}^{x} - \lambda \sum _{j=1}^{N}\sigma _{j-1}^{y} \sigma _{j}^{y}\right) . \end{aligned}$$
(94)
The parameters for this model can be defined as follows:
$$\begin{aligned} h&=0, \end{aligned}$$
(95a)
$$\begin{aligned} N^{(x)}&=1,\ N^{(y)}=1, \end{aligned}$$
(95b)
$$\begin{aligned} J_{l}^{(x)}&=\{1\},\ J_{l'}^{(y)}=\{-\lambda \}, \end{aligned}$$
(95c)
$$\begin{aligned} n_{l}^{(x)}&=\{N/2-1\},\ n_{l'}^{(y)}=\{0\}. \end{aligned}$$
(95d)
The model can be exactly diagonalized with the following Bogoliubov solution:
$$\begin{aligned} \tan 2\theta _{k}^{(b)}&= \frac{\lambda \sin \left( \frac{2 \pi (b+k)}{N}\right) +\sin \left( \pi (b+k)\right) }{\lambda \cos \left( \frac{2 \pi (b+k)}{N}\right) -\cos \left( \pi (b+k)\right) }. \end{aligned}$$
(96)
The exact energy spectrum can be obtained by utilizing Eqs. (14) and (17–20). The eigenvalues in the case of even N for the odd sector (\(b=0\), periodic boundary conditions) and odd N for the even sector (\(b=1/2\), antiperiodic boundary conditions) are as follows:
$$\begin{aligned} \epsilon _{k}^{(b)}= \left\{ \begin{array}{lr} 2 (\lambda -1), &{} \text {for } k=0 \wedge b=0\\ -2 \left[ \lambda +(-1)^{N/2}\right] , &{} \text {for } k=\frac{N}{2} \wedge b=0\\ -2 \lambda , &{} \text {for } k=\frac{N-1}{2} \wedge b=1/2 \end{array}\right\} = 2 \alpha _{k}^{(b)}, \end{aligned}$$
(97)
or otherwise:
$$\begin{aligned} \epsilon _{k}^{(b)}=2 \sqrt{1+\lambda ^2-2 \lambda \cos \left( \frac{(2+N) \pi }{N}(k+b)\right) }. \end{aligned}$$
(98)
Similar to SPT-AFM model, the ground state is constructed from the even sector (\(b=1/2\)) with no fermions as the first excited state comes from the odd sector (\(b=0\)) with one-fermion occupation for \(N=8\), see Fig. 11a. On the other hand, in the case of \(N=10\), the lowest zero-fermion and one-fermion states become degenerate except in the vicinity of \(\lambda =1\). Interestingly, at the point \(\lambda =-1\) ground-state energy is constructed by the odd sector with one-fermion occupation, whereas at \(\lambda =1\), the ground-state energy comes from the even sector with zero-fermion occupation, see Fig. 11c. This model does not exhibit the peculiarity discussed in Eq. (21), where the odd sector has three-fermion occupation as the lowest energy state. We note that the energy gap has different characteristics depending on the system sizes (even): \(N=4m\), \(N=2(4m+1)\), and \(N=2(4m-1)\) (with \(m=1,2\ldots \)). The latter is gapless for all range of \(\lambda \), whereas the case of \(N=2(4m+1)\) displays a peak at the \(\lambda =1\), as shown in Fig. 11d. With increasing system sizes, the peak approaches to zero, and in the thermodynamic limit, both cases become gapless. However, the case of \(N=4m\) exhibits similar behavior to the SPT-AFM model with critical points \(\lambda _c=\pm 1\), see Fig. 11b. The ground state is degenerate for \(|\lambda | \ge 1\) in the thermodynamic limit and the energy gap becomes \(\varDelta E=2\big (1-\left| \lambda \right| \big ) \theta (1-\left| \lambda \right| )\), where \(\theta (x)=1\) if \(x>0\) and zero otherwise. Thus, the singularity at \(\lambda =1\) signals a quantum phase transition. This is in contrast to the halfway XY model, discussed in Sec. 4.2, that the halfway interaction prevents the model from undergoing a quantum phase transition but rather helps to exhibit a first-order transition across the Barouch–McCoy circle. The quantum phase transition (for \(N=4m\) case) in the halfway SPT-AFM model can be confirmed by the behavior of entanglement as well. With an increasing system size, the derivative of the entanglement per site develops a singularity at \(\lambda _c=1\), at which the quantum phase transition takes place; see Fig. 12.