^{*}

Edited by: James Avery Sauls, Northwestern University, USA

Reviewed by: Byungchan Han, DGIST, South Korea; Takeshi Mizushima, Osaka University, Japan

*Correspondence: Thiago R. de Oliveira

This article was submitted to Condensed Matter Physics, a section of the journal Frontiers in Physics

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We present an example where spontaneous symmetry breaking (SSB) may affect not only the behavior of the entanglement at Quantum Phase Transitions (QPT), but also the origin of its non-analyticity. In particular, in the XXZ model, we study the non-analyticities in the concurrence between two spins, which was claimed to be accidental since it had its origin in the optimization involved in the concurrence definition. We show that when one takes into account the effect of the SSB, even though the values of the entanglement measure does not change, the origin of the non-analytical behavior changes. The non-analytical behavior is not due to the optimization process anymore and in this sense it is a “natural” non-analyticity. This is a much more subtle influence of the SSB not observed before. We also show that the value of entanglement between one site and the rest of the chain changes after taking into account the SSB.

It is now generally accepted that entanglement may help in finding and characterizing Quantum Phase Transitions (QPT), since it may inherit the non-analytic behavior of the ground state energy [

The use of entanglement measures to study QPT are also problematic because most of the measures are difficult to calculate and even more difficult to directly measure. Thus, even though the characterization of entanglement gives more information about the nature of the ground state and its correlation it may be easier to use the usual correlation functions and thermodynamic quantities to study the quantum phase transition. Besides such caveats, the study of the entanglement at QPT is in general more intricate than of thermodynamical quantities, because of the spontaneous symmetry breaking (SSB). At the critical point of a symmetry breaking QPT, the ground state becomes degenerate. In fact, this degenerescency is necessary for the spontaneous breaking of the Hamiltonian symmetry: the emergence of ground states without the Hamiltonian symmetry. However, as the ground state is degenerate, there are many possible ground states; some preserving the symmetry (equal superpositions of symmetry breaking states) others not. Furthermore, while thermodynamical quantities do not depend on which particular degenerate ground state one chooses, entanglement may. Therefore, one has to be careful when choosing the state. In general, states which preserve the symmetry are preferable, since they are simpler: have reduced density matrices with many null entries. But one has to be careful since there are examples where the entanglement depend on the particular ground state used [

Here, we show an example of a new and very subtle caveat in the relation between entanglement, SSB and QPT. More specifically, we show that the SSB may change not only the value of the entanglement but also the origin of its non-analytical behavior. Actually, in our example, SSB changes the origin of the non-analytical behavior without even changing the value of the entanglement; a much more subtle influence.

In order to be self-contained, we organize the article as follows: In the next section we make a brief discussion of the model studied. In the following we discuss the subtle SSB effect on the concurrence as a QPT measurement. Then we discuss the same effect on the Von Neumann entropy, which is another kind of entanglement measurement, we then finish with the conclusion.

We will describe the model closely following [

where _{2} symmetry over the plane ^{z} → −σ^{z}; (ii) a continuous _{2} symmetry implies that

Since we want to analyze the QPT at Δ = −1, the two important phases are:

Δ < −1: the system experiences a SSB, which lead it to a ferromagnetic phase, where all the spins point in the same direction creating a finite magnetization

−1 < Δ <1: the system is in a gapless phase, where the correlations decay polynomially and all the symmetries are preserved.

The Bethe ansatz solution gives the ground state energy [

where Δ = cosπν. For nearest neighbors we can obtain the correlation from _{0}(Δ):

For spins further apart, progress has been slow, but there are already some expressions available up to third neighbors [^{1}

We are interested in the subtleties of SSB in entanglement measurements for two spins in this chain. These entanglement measurements can be determined by the reduced density matrix of the two spins, which can be obtained from the magnetizations and correlations of the two spins. Applying the symmetries of the XXZ model, the state becomes

with _{2} symmetry is broken, the state becomes

with _{2} symmetry is broken _{z} and _{z} become finite and should appear in the reduced density matrix, as they do in Equation (6). Note also that the translational symmetry is still maintained, thus we have

The first formal and general relation between entanglement and QPT was given in Wu et al. [

The concurrence is a well known entanglement measure and for two spin 1/2 particles is given by

with

with

However if one consider the SSB, the expression for the concurrence is a little more complicated and given, as seen in Syljuasen [

with

The maximization in Equations (7), (8), and (10) appears because the entanglement measure involves an optimization procedure over all possible decompositions of the mixed state in a mixture over pure states. The expressions for the concurrence without taking into account the maximum operation are given by Equations (9) or (11), for symmetric and non-symmetric ground states, respectively. But note that such expression are not valid entanglement measures. Note also that, as expected, Equation (11) reduces to Equations (9) and (10) reduces to Equation (8), when there is no SSB (_{z} = _{z} =

In Figure

In Figure

This failure of concurrence to indicate the right order of the QPT was noted in Yang [

We now consider the effect of the SSB by taking into account that in the ferromagnetic phase, Δ < −1, all spins are aligned in the same direction: _{z} = _{z} =

So, we have two facts here:

Although the expressions for symmetric and non-symmetric state are different, the entanglement value is the same; something already noted by Syljuasen [

The origin of non-analyticity in

Therefore, even tough SSB does not change the behavior of the concurrence, it does change the origin of the non-analyticity: leading an accidental non-analyticity to “natural” one. Note also that SSB only changes the non-analyticity that correspond to a real QPT.

Unfortunately _{z}+_{z} in Equation (21). Thus, in some sense one could still argue that this is an accidental non-analytical behavior, but of different nature.

Another interesting fact is the raise of a discontinuity in the entanglement between one site and the rest of the chain given by the von Neumann entropy for one site when we take into account the SSB.

The von Neumann entropy for one site is given by the equation:

where

with _{i} = _{z}〉. Thus, the von Neumann entropy is:

Figure

We have studied the influence of SSB in the entanglement between two spins and between one spin and the rest of the chain in the one dimensional spin-

We first showed that, although SSB does not change the behavior of the concurrence at the first-order Quantum Phase Transition, as first noted by Syljuasen [

We also showed that the behavior of the entanglement between one site and the rest is affected by the SSB at the first-order Quantum Phase Transition. It only signals the phase transition when taking into account the SSB.

We thus give further evidence that the use of entanglement to study QPT may be much more intricate than at first glance. One has to be cautious at least about: (i) accidental non-analytic behavior due to optimizations in the entanglement measure adopted; (ii) effects of the SSB in the entanglement value and (iii) effects of spontaneous symmetry break in the origin of the non-analyticities.

LPJ did all the calculations. TRO participated in all the discussions and supervised the work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Both authors acknowledge financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). This work was performed as part of the Brazilian National Institute of Science and Technology for Quantum Information.

_{2}symmetry breaking

^{1}Note that there are typos in Equations (19) and (20) from Shiroishi and Takahashi [