Charging a Dimerized Quantum XY Chain

Abstract

Quantum batteries are quantum systems designed to store energy and release it on demand. The optimization of their performance is an intensively studied topic within the realm of quantum technologies. Such optimization forces the question: how do quantum many-body systems work as quantum batteries? To address this issue, we rely on symmetry and symmetry breaking via quantum phase transitions. Specifically, we analyze a dimerized quantum XY chain in a transverse field as a prototype of an energy storage device. This model, which is characterized by ground states with different symmetries depending on the Hamiltonian parameters, can be mapped onto a spinless fermionic chain with superconducting correlations, displaying a rich quantum phase diagram. We show that the stored energy strongly depends on the quantum phase diagram of the model when large charging times are considered.

1. Introduction

Models of interacting spins have great importance in various contexts of physics [1]. Firstly, they enable quantitative predictions within the theory of magnetism [2,3,4,5]. Materials well described by spin models in various dimensions are indeed not rare [6,7,8,9,10,11]. At the same time, they are at the core of topological phases, both for thermal and the ground-state properties [12,13,14]. In this context, their relation to the fascinating field of frustrated systems is worth mentioning [15,16,17,18,19]. Moreover, they have recently played a crucial role in the determination of novel paradigms in the theory of thermalization and information spread through the general procedure called quantum quench [20,21,22,23]. A quantum quench is a sudden variation of a parameter of the Hamiltonian of a quantum system. Quantum quenches can be achieved experimentally within very diverse platforms [24,25,26].

Alongside the large range of physical applications, many effective techniques have been proposed for analysis of spin systems. These range from the celebrated spin wave theory based on the Holstein–Primakoff transformation [27] to numerical ones, based on even the Fedotov–Popov transformation [28,29]. Additionally, one-dimensional models can be analyzed by means of bosonization [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47] as long as the low-energy properties are considered. Even more remarkably, they sometimes might be exactly solved by means of the Bethe Ansatz technique [48] or more simply by the Wigner–Jordan transformation [49]. Such exact solutions, which are at the heart of the so-called theory of quantum integrable systems [50], can indeed help in understanding some general principles of quantum many-body interacting systems and are hence extremely valuable.

Indeed, the existence of such techniques has strongly favored the adoption of quantum spin chains in the context of quantum technologies [51]. Indeed, a two-level system can mathematically be described as a spin 1/2, so that one-dimensional collections of qubits are immediately described by interacting spin chains. This fact has been exploited in the fields of quantum information, quantum communication, quantum computation, and, more relevantly for the present article, quantum energy storage.

In this context, quantum spin chains have been proposed as quantum batteries (QBs) [52,53,54,55,56,57]. Conceived for the first time a decade ago [58], QBs are miniaturized device able to store and efficiently deliver energy, exploiting purely non-classical effects such as quantum superposition and entanglement [59,60,61]. The interest in such devices is not only motivated by fundamental issues in quantum thermodynamics [62,63,64,65,66], but it is also driven by the need to integrate quantum energy providers to improve the performance of quantum circuits [67,68,69,70,71] or quantum computing architectures [72,73,74] with the hope that QBs could outperform their classical counterparts [58,75].

Among all the mechanisms that can be envisioned to charge a QB, one is tightly connected to the vast literature on quantum quenches. Such protocol works in the following way [76,77]: a system is prepared in the ground state of a Hamiltonian $H_0$, called the battery Hamiltonian. Then, the Hamiltonian suddenly changes, say at $t=0$, to a different operator $H_1$—the charging Hamiltonian—that implements the time evolution. Finally, after a time interval $\tau$, the Hamiltonian is brought back to the original operator $H_0$. In such a way, for fixed $H_0$ and $H_1$ values, one can evaluate the energy transferred to the quantum system as a function of the charging time $\tau$. A QB working according to the mechanism just described and composed of a collection of interacting two-level systems is usually called a spin QB.

Recently, it has been shown that a dimerized XY spin chain used as a spin QB shows a rich phenomenology [56]. The energy stored as a function of $\tau$ is characterized by the presence of three regimes: (i) A short charging time regime, where prominent oscillations as a function of $\tau$ are present; (ii) A plateau regime—resembling the prethermalization plateau of the quantum quench theory—where no oscillations as a function of $\tau$ are present; (iii) a recurrence time where oscillations are restored for times large enough to probe the level spacing. This last regime is pushed to $\tau \to \infty$ in the thermodynamic limit. Among the three regimes, the second one shows an interesting behavior: the energy stored strongly depends on the quantum phase diagram of the charging Hamiltonian, and in particular, it is sensitive to a quantum phase transition (QPT) induced in the model by the dimerization strength [56]. In other words, in a specific case, it has been shown that the dependence of the stored energy on the parameters of the Hamiltonian when the battery Hamiltonian $H_0$ and the charging Hamiltonian $H_1$ are kept within the same quantum phase are strikingly different with respect to the case in which $H_0$ and $H_1$ are in different quantum phases. In a sense, the QPTs of the charging Hamiltonian appear to be manifest in the behavior of the stored energy.

In this article, we extend the analysis to a richer model, that is the dimerized XY chain in an external transverse field [78], to prove that, even within this larger model, the energy stored in the plateau regime strongly depends on the quantum phase diagram of the charging Hamiltonian.

The rest of the article is structured as follows. In Section 2, we introduce the model, its solution, and its phase diagram. In Section 3, we introduce the charging energy and discuss its behavior crossing various QPTs. In Section 4, finally, we draw our conclusions.

2. Model

The model under investigation is the so-called dimerized XY quantum chain in a transverse field [78]. The Hamiltonian $H_B$ is given by where the subscript B indicates the fact that this system will be considered as a QB.

In the above expression, N is the number of unit cells, considered here to be even for simplicity, and ${\sigma }_j^{\alpha }$ (with $\alpha =x,y$) are Pauli matrices, in the usual representation, corresponding to the j-th site spin. The parameter J is the energy scale of the system, $\gamma$, $\delta$, and h characterize the strength of the anisotropy, the dimerization, and the external field, respectively.

By focusing for simplicity on the even-parity sector of the model [23], and considering the periodic boundary conditions ${\sigma }_{j+N}^{\alpha }\equiv {\sigma }_j^{\alpha }$, it is possible to diagonalize the above Hamiltonian by means of a standard Wigner–Jordan transformation mapping spins into free spinless fermions [49]. Notice that the obtained fermionic Hamiltonian corresponds to the dimerized Kitaev chain and is characterized by interesting topological features [79], due to the competition between fractional solitons [80,81,82,83,84,85] and Majorana fermions [86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104]. In terms of two species of auxiliary fermionic annihilation operators $a_q$ and $b_q$, and assuming from now on $J=1$ as reference scale for the energies, we obtain (from now one we will consider $\hslash =1$).

$$H_B=\sum _{q\in \mathsf{\Gamma }}\left[{\omega }_{1,q}\left(a_q^{†}{a}_q-{\displaystyle \frac{1}{2}}\right)+{\omega }_{2,q}\left(b_q^{†}{b}_q-{\displaystyle \frac{1}{2}}\right)\right].$$

(1)

Here, $\mathsf{\Gamma }=\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{5}{2}},...,\mathcal{N}-{\displaystyle \frac{1}{2}}\}$ and

$${\omega }_{1/2,q}=\sqrt{4h^2+{\left|\mathcal{Z}\right|}^2+{\left|\mathcal{W}\right|}^2\pm 2\sqrt{4h^2{\left|\mathcal{Z}\right|}^2+16{\gamma }^2{\delta }^2}},$$

(2)

where the following definitions have been used:

$$\begin{array}{cc}\hfill & \mathcal{Z}=-\left[(1+\delta )+(1-\delta )e^{-ik}\right]\hfill \\ \hfill & \mathcal{W}=-\gamma \left[(1+\delta )-(1-\delta )e^{-ik}\right]\hfill \end{array}$$

(3)

and $k={\displaystyle \frac{2\pi }{\mathcal{N}}}q$, with $\mathcal{N}=N/2$ the number of dimers. For $k=0$, the gap closes at

$$h^2=1-{\gamma }^2{\delta }^2,$$

(4)

while for $k=\pm \pi$, the gap closes at

$$h^2={\delta }^2-{\gamma }^2.$$

(5)

These equations represent the phase boundaries of the model, whose plots are reported in Figure 1. In [56], it has been shown that the charging process at zero external field shows signatures of the presence of the QPT lines of the model. Those lines can be obtained by setting $h=0$ in Equations (4) and (5), as shown by the blue and red dashed lines in Figure 2, respectively. The Figure also shows a very interesting and intricated phase diagram at finite h.

The goal of this work is to explore the charging process of a QB described by the Hamiltonian in Equation (1): the presence of a non-zero field enriches the previously studied quantum phase diagram, opening up the possibility of investigating different QPT scenarios.

3. Study of QPT Effects

The general approach of our analysis is as follows: we first select a plane in the ($\gamma$, $\delta$, h)-space by imposing a constraint on the model parameters. The QPT lines are then identified as the intersections of the phase boundaries, defined by Equations (4) and (5), with the chosen plane. Next, we perform a sudden quench of one or more model parameters, focusing on the asymptotic limit where the charging time $\tau$ approaches infinity, and finally, we plot the energy stored per dimer in the battery as a function of the pre-quench parameter. In the asymptotic regime we are studying, the energy stored in the QB assumes the form [56].

$$\Delta E(\gamma ,\delta ,h)=\sum _{q\in \mathsf{\Gamma }}\{2{\omega }_{1,q}|{\mathcal{M}}_{3,1}{|}^2|{\mathcal{M}}_{3,3}{|}^2+2{\omega }_{2,q}|{\mathcal{M}}_{4,2}{|}^2|{\mathcal{M}}_{4,4}{|}^2\}$$

(6)

Here, ${\mathcal{M}}_{i,j}$ represents the $(i,j)$-th element of the matrix $\mathcal{M}$, defined as $\mathcal{M}={\mathcal{V}}^{-1}\mathcal{U}$. In particular, $\mathcal{U}$ and $\mathcal{V}$ are both 4 × 4 matrices: the columns of $\mathcal{U}$ correspond to the eigenvectors of the QB Hamiltonian, which depends on the pre-quench parameters, and similarly, the columns of $\mathcal{V}$ correspond to the eigenvectors of the charging Hamiltonian, which depends on the post-quench parameters. Equation (6) provides the value of the stored energy given a generic sudden quench of the parameters. As mentioned, our aim here is to explore the effects of QPTs. In the following, we will provide paradigmatic examples in this direction which represent the whole phase space.

3.1. Quench of $\gamma$ at Given h and $\delta$

In this scenario, we perform a quench of the anisotropy parameter from an initial value ${\nu }_i$ to a final value ${\nu }_i+{\nu }_f$, with ${\nu }_i$ and ${\nu }_f$ positive constants, along the line given by the intersection of the $h=0.5$ plane and the $\delta =1.1$ one. The situation is sketched in Figure 3.

Figure 4 shows the energy stored per dimer (in units of $J=1$) as a function of ${\nu }_i$, with ${\nu }_f$ fixed at $0.3$ for both the zero-field scenario (black curve) and the non-zero constant external field one ($h=0.5$, green curve). The impact of the phase diagram is evident from the behavior of both curves: for the green curve, we can see that the energy initially rapidly increases until ${\nu }_i+{\nu }_f$ reaches the dashed red QPT line in Figure 3, while in the region between the two QPT lines, the curve rises more gradually and finally, after crossing the second QPT line, the curve begins to decrease. For the black curve, we observe analogous behavior, but other than gaining slightly more energy, the QPT lines are shifted with respect to the previous case since the intersections between the 3D phase diagram and the plane $h=0$ are different.

3.2. Quench of h at Given $\gamma$ and $\delta$

Now, we perform a quench of the external field, once again from an initial value ${\nu }_i$ to a final value ${\nu }_i+{\nu }_f$, along the line given by the intersection of the $\gamma =0.5$ and the $\delta =1.5$ planes, as shown in Figure 5.

As in the previous case, Figure 6 shows the energy stored per dimer as a function of ${\nu }_i$, with ${\nu }_f$ fixed at $0.41$. The observed trend is very similar to the one shown in Figure 4, with an initial rising until the first QPT line, an almost-linear rise until the second one, and then a decrease.

3.3. Quench of h and $\gamma$ at Given $\delta$

Lastly, we vary both the anisotropy and the external field. The plane on which we perform the time evolution has the equation $h=\gamma +\delta -1$ and the dotted black line has the following parametric equation

$$\left\{\begin{array}{cc}\hfill & \gamma ={\nu }_i\hfill \\ \hfill & \delta =1.5\hfill \\ \hfill & h={\nu }_i+0.5\hfill \end{array}\right.$$

(7)

so that the quench of ${\nu }_i$ has an impact on both $\gamma$ and h. The QPT lines are represented in Figure 7.

After fixing ${\nu }_f=0.28$, we obtain the plot reported in Figure 8: even in this more complex scenario, we can observe the same features also presented in the previous two cases.

4. Conclusions

In this work, we have investigated the interplay between QPTs and the energy storage properties of a dimerized XY spin chain subjected to an external transverse field used as a QB. By extending previous analyses carried out in the absence of an external field, we have demonstrated that the introduction of a non-zero field enriches the quantum phase diagram and significantly influences the charging process. We considered three different scenarios: a quench of the anisotropy parameter $\gamma$, a quench of the external field h and a simultaneous quench of both $\gamma$ and h. In all cases, we focused on the asymptotic regime of long charging times ($\tau \to \infty$), where the energy stored per dimer was analyzed as a function of the pre-quench parameter.

Our results reveal common patterns across all scenarios: first, the stored energy initially rises sharply until the first QPT line is crossed. In the region between two QPT lines, where a plateau was observed in the zero-field case ($h=0$) quenching the dimerization parameter [56], the energy now increases: this change reflects the role of both the type of quench and the external field in modifying the phenomenology of the plateau regime and suggests that QPT signatures at the level of the stored energy persist even for more complex models. Finally, after crossing the second QPT line, the stored energy begins to decrease. This behavior highlights the sensitivity of the system to the quantum phase diagram of the charging Hamiltonian. As an extension of this work, in order to have a more general understanding of the role of QPTs on quantum batteries based on integrable systems, it would be interesting to analyze scenarios in which the $\mathcal{M}$ matrix used to compute the energy stored in Equation (6) can be accessed analytically. For instance, by analyzing the so-called cluster Ising model [105,106], one could assess the role of multi-spin interactions, while considering free fermions on generic lattices the effects related to dimensionality and topology might be unveiled [107,108]. Additionally, the role of non-sudden quenches, initial thermal states, noise, and dissipation needs to be addressed to translate our analysis to real-world quantum batteries [109,110,111,112,113,114,115,116,117,118].