Singularity in Entanglement Negativity Across Finite Temperature Phase Transitions
Abstract
Phase transitions at a finite (i.e. non-zero) temperature are typically dominated by classical correlations, in contrast to zero temperature transitions where quantum mechanics plays an essential role. Therefore, it is natural to ask if there are any signatures of a finite temperature phase transition in measures that are sensitive only to quantum correlations. Here we study one such measure, namely, entanglement negativity, across finite temperature phase transitions in several exactly solvable Hamiltonians and find that it is a singular function of temperature across the transition. As an aside, we also calculate the entanglement of formation exactly in a related, interacting model.
Interacting quantum systems with competing interactions can exhibit phase transitions at both zero and non-zero temperatures. Heuristically, the zero temperature phase transitions result due to quantum fluctutions while the finite temperature phase transitions typically result from thermal fluctuationsSachdev (2011). As an example, consider the 2+1-D transverse field Ising model on a square lattice: . Here the critical exponents associated with the zero temperature phase transition belong to the three dimensional Ising universality while those for the finite temperature phase transition belong to the two dimensional Ising universality Suzuki (1976); Sachdev (2011). That is, at any non-zero temperature, the universal critical exponents are identical to those corresponding to the purely classical Hamiltonian . Given this observation, it is natural to ask are there any singular correlations at a finite temperature transition that are intrinsically quantum-mechanical? For a pure state, von Neumann entanglement entropy is a good measure of quantum correlations, but since we are interested in finite temperature transitions, we need to consider measures of mixed state entanglement. With this motivation, in this paper we will introduce certain quantum models which exhibit finite temperature transitions, and we will analytically study mixed state entanglement measures in these models, with a particular focus on entanglement negativity Vidal and Werner (2002).
One way to motivate mixed state entanglement measures is via the notion of ‘separable’ states - these are states that can be prepared from any other state using only local operations and classical communications (LOCC), and therefore are not entangled. A bipartite mixed state is separable if it can be written as where while are valid density matrices Nielsen and Chuang (2002); Werner (1989). For pure states, the von Neumann entropy , where is the reduced density matrix on Hilbert space , is a faithful measure of quantum correlations. However, is rather ineffective at capturing mixed state quantum correlations. For example, even a thermal density matrix corresponding to a purely classical Hamiltonian will have a rather substantial von Neumann entropy that equals the thermal entropy for region . Several measures of mixed state entanglement have been proposed (see, e.g., Ref.Horodecki et al. (2009) for an overview) including entanglement of formation, entanglement of distillation, entanglement of purification, squashed entanglement and entanglement negativity. As yet, all of these measures, with the exception of entanglement negativity, require optimizing a function over all possible quantum states, making their calculation rather challenging. Therefore, below we will primarily focus on the entanglement negativity with one exception; for a specific many-body model we will also calculate the entanglement of formation.
The entanglement negativity (henceforth, just ‘negativity’ for brevity) is defined as follows Eisert and Plenio (1999); Vidal and Werner (2002): given a bipartite density matrix acting on the Hilbert space , one first performs a partial transpose only on the Hilbert space to obtain a matrix . Explicitly, if , then The matrix is Hermitian but is not necessarily positive semi-definite. The negativity is defined as . The utility of this procedure becomes apparent when one notices that negativity is zero for separable mixed states Werner (1989); Peres (1996); Horodecki et al. (1996); Simon (2000); Vidal and Werner (2002). This is because for separable states, is a valid density matrix, and therefore, . The main drawback of negativity is that it can be zero even for non-separable states Horodecki (1997). Heuristically, this means that although negativity is insensitive to classical correlations, it does not capture all quantum correlations. Since we will also briefly discuss entanglement of formation, denoted as , let us also recall its definition. for a bipartite mixed state is defined as follows Bennett et al. (1996): decomposing as a convex sum of pure states, where with , is given by where is the von Neumann entropy. Therefore, is the least possible entanglement of any ensemble of pure states that realizes a given mixed state. In contrast to negativity, is zero if and only if a state is separable.
To begin with, we note one feature of negativity shared by all Hamiltonians considered here, as well as in several other lattice models (see, e.g., Refs.Anders (2008); Ferraro et al. (2008); Sherman et al. (2016); Hart and Castelnovo (2018)) and continuum field theories Calabrese et al. (2012, 2015): above a certain temperature, the negativity for the corresponding thermal (Gibbs) state becomes exactly zero. This temperature is called ‘sudden death temperature’ denoted as . One of the central questions we will ask is the following. Consider an interacting system which exhibits spontaneous symmetry breaking below a critical temperature . Assuming that negativity is non-zero in the vicinity of the transition (i.e. the condition is satisfied), is a singular function of the tuning parameter (e.g. the temperature) across the transition?
We now state our main result. We find that in all models considered in this paper, whenever negativity is non-zero in the vicinity of the transition, it is always singular across the transition. This result is at variance with expectations from Ref.Sherman et al. (2016) where numerical calculations on finite sized systems for the 2+1-D quantum Ising model suggested that negativity is analytic across the corresponding . We will return to a comparison with Ref.Sherman et al. (2016) after discussing our results.
As a starting point, consider a single site mean-field Hamiltonian for the transverse field Ising model: , where is the coordination number. The corresponding thermal state is indeed separable, which might lead one to expect that perhaps negativity is always an analytic function across finite temperature transitions. However, a single site mean-field is too crude an approximation: within such a mean-field approximation, even the ground state is unentangled and shows no singularity in the quantum entanglement across a T = 0 quantum phase transition (QPT), in contrast to the known exact results (see, e.g., Refs.Osborne and Nielsen (2002); Metlitski et al. (2009); Kallin et al. (2013)). To improve upon this, we next consider a two-site mean-field theory:
(1) |
and study the negativity for a bipartition that runs across the two sites. A straightforward calculation shows that whenever , the critical temperature for the phase transition, the negativity is a singular function of the temperature across the transition, see Fig.1. Incidentally, since an analytical expression for entanglement of formation is available for any state acting on two qubits Wootters (1998), we calculate as well for this mean-field model, and find that it is also singular across the transition (Fig.1).
Motivated by the two-site mean-field result and the models studied in Ref.Assaad and Grover (2016), we next consider a Hamiltonian which exhibits a finite temperature transition, and where negativity is calculable exactly in the thermodynamic limit. The model is defined on a one-dimensional lattice with sites where each lattice site has four qubits:
(2) | |||||
The most notable feature of this Hamiltonian is that it is a sum of commuting terms, and it supports a finite temperature transition where the Ising symmetry corresponding to gets spontaneously broken. The first term in the Hamiltonian makes it non-local and leads to a finite temperature Ising transition in the mean-field universality class. Defining the order parameter , one finds that in the thermodynamic limit, the critical temperature is given by the solution of the equation while the order parameter is determined via , , as expected. Next, we calculate the negativity of this model for the bipartition that runs across the four qubits on a chosen site, i.e., and where we have chosen the cut across the site 0 for convention. One finds that for all , and for , the negativity is given by , where which implies that close to
(3) |
assuming sup ; otherwise negativity is zero which also yields an expression for by setting . Since the critical temperature depends only on , one can always tune , so that the sudden death temperature is higher than . Since is a singular function of temperature so is negativity. In fact simply inherits the cusp singularity of across the phase transition, i.e., 2 which also shows the temperature dependence of negativity for all temperatures including . , see Fig.
One drawback of the model just discussed is that it is non-local and relatedly, exhibits mean-field scaling exponents. Therefore, it would be worthwhile to study negativity in thermal states of local Hamiltonians that host a finite temperature transition. Before considering local models, we notice a property specific to commuting projector models that will simplify our subsequent discussion. Let’s decompose a commuting projector Hamiltonian as , and further denote the Hilbert space of spins on the boundary of region that interact with by . We also define i.e. spins strictly in the ‘bulk’ of . Note that acts only on the boundary Hilbert space of . One can show that sup
(4) |
where in the expression for ), and furthermore one can always find a basis in which , and can all be simultaneously diagonalized. is the reduced density matrix for the boundary spins. This property results from the fact that partial transpose affects operators only at the boundary (i.e. only in the factor
With the aforementioned property specific to commuting projector Hamiltonian , we now turn our attention to the negativity in a local Hamiltonian defined on a square lattice, with two species of spins, and , on each lattice site:
(5) |
This model exhibits a finite temperature phase transition in the 2D Ising universality class, and due to the commuting projector property, the corresponding is exactly same as the Onsager’s solution Onsager (1944) to the classical Ising model on the square lattice, , irrespective of the value of . Let us first consider the negativity between one spin on a single site, say, ‘a’ spin on site 0, and the rest of system. As just discussed, to calculate the negativity, we only need the reduced density matrix for spins at the boundary, which in this case are the spins on sites 0 and four neighbors of site 0. For simplicity, we present the result of the negativity only for a specific range of , namely, where the calculation is technically simpler, see supplemental material for details. This is sufficient to illustrate the singular nature of negativity across the finite temperature transition hinted above. One finds that the negativity is given by:
(6) | |||||
where
We can now argue rather generally that negativity will be singular across a phase transition in a commuting projector Hamiltonian for arbitrary bipartition scheme. Due to the property in Eq.4, one only needs to consider the reduced density matrix for the boundary spins, whose partial transpose takes the form
(7) |
where can be expressed as the tensor product of Pauli matrices acting on the Hilbert space . The associated coefficients are proportional to the expectation value of with respect to the bulk density matrix , and are therefore a singular function of the tuning parameter across , similar to the coefficients discussed above for the case of a single site negativity. From Eq.7, it follows that the negativity is
(8) |
where , and denote the value of the boundary spins in the basis where and all operators are simultaneously diagonalizable (this is always possible since the Hamiltonian is a sum of commuting projectors). In contrast to , the coefficients are determined only by the reduced density matrix on the boundary spins via the above expression, and are oblivious to the bulk criticality. Therefore, the negativity inherits the singularity associated with the bulk criticality due to its dependence on coefficients .
Finally, we consider a completely different class of models which are also exactly solvable and in which one again finds that the negativity is singular across the phase transition. In particular, consider the quantum spherical modelVojta (1996):
(9) |
where and satisfy the canonical commutation relation , while the constraint is imposed only on average via the Lagrange multiplier . The above model shows a phase transition associated with spontaneously breaking of the Ising symmetry at temperature determined via
So far we have showed that finite temperature transitions in quantum systems can show singular features in entanglement negativity, despite the fact that the universal critical exponents associated with these transitions are still given by classical statistical mechanics. Therefore, it is legitimate to ask whether negativity can at all distinguish the spontaneous symmetry breaking at finite temperature with spontaneous symmetry breaking at zero temperature? The answer is in the affirmative. For concreteness, again consider the exactly solvable model in Eq.2 although the argument is rather general. Below , and in the absence of an infinitesimal symmetry breaking field, the partition function gets equal contribution from both positive and negative values of the order parameter. On the other hand, in the thermodynamic limit, and in the presence of an infinitesimal symmetry breaking field, only one of the two sectors contribute, and therefore, the thermal entropy with and without field satisifies . This is why the spontaneous symmetry breaking at a finite temperature is an example of ergodicity breaking Goldenfeld (2018) or relatedly, a ‘self-correcting classical memory’ Bacon (2006). Since this is a classical phenomena, a faithful measure of quantum correlations should be insensitive to it. One may now explicitly calculate the negativity with and without infinitesimal symmetry breaking field for Hamiltonian in Eq.2, and show that (see supplemental material). Schematically, at a mean-field level, where is the mean-field value of the order parameter, and therefore . In strong contrast, for spontaneous symmetry breaking at , when , the ground state wavefunction (and not the density matrix) is a sum of the ground state wavefunctions corresponding to positive and negative order parameters (a ‘cat state’) while at , only one of the two sectors contribute. Therefore, all measures of quantum entanglement, including von Neumann entanglement entropy and in particular negativity satisfy .
The models introduced in this paper allowed for a rather straightforward evaluation of negativity while illustrating non-trivial features. It is natural to wonder whether one can calculate any other measures of mixed state entanglement for similar models. To that end, we now present a result on the entanglement of formation , a quantity which is generally rather hard to calculate since it requires optimization over all possible states. Consider the following Hamiltonian which is closely related to the Hamiltonians in Eqs.2 and 5:
(10) |
This Hamiltonian exhibits a finite temperature phase transition at . For defining the entanglement of formation , similar to our earlier discussion, we choose a bipartition that cuts through the two spins 1 and 2 on a chosen site . A straightforward analysis shows (see supplemental material) that in the thermodynamic limit, is exactly given by that corresponding to the mean-field density matrix defined as where and satisfies the mean-field equation . Using the exact result by Wooters on for two qubits (Ref.Wootters (1998)), this yields an analytical expression for . Unfortunately, in this model, the entanglement of formation exhibits a sudden death temperature which is lower than for all values of , and therefore, is zero in the vicinity of the transition.
To summarize, we analytically demonstrated that negativity is singular across finite temperature phase transitions for several models. This may seem counterintuitive since the universal properties associated with transitions are controlled by a purely classical Hamiltonian with the same symmetries. One way to resolve this apparent tension is to note that negativity is sensitive to short-distance quantum correlations close to the bipartition boundary. Since even local properties, such as magnetization or energy density, are singular across the transition, one expects that the area-law associated with negativity will generically pick up a singular contribution as well. In contrast to our results, Ref.Sherman et al. (2016), based on small scale numerics ( sites), concluded that negativity for the 2+1-D quantum Ising model has no singularity across the finite temperature transition. Although we don’t have any analytical results for the negativity of 2+1-D quantum Ising model, for the general reasons just mentioned, we suspect that negativity will be singular in this model as well. As is evident from the insets of Figs.1 and 3, it can be rather hard to detect the singularity in negativity unless one has access to an analytical expression, or precise numerical data on very large system sizes. We hope that our results will prompt further in-depth numerical and field-theoretic calculations of entanglement negativity in systems that exhibit finite temperature transitions.
The singularity in negativity for the local models discussed in this paper is somewhat analogous to the singular area-law contribution at a zero-temperature QPT discussed in Ref.Metlitski et al. (2009). At the same time, the absence of finite temperature topological order Hastings (2011) in our models suggests that unlike the zero temperature case, there is no additional subleading O(1) constant. If so, then one might be able to cancel out the singular contribution completely via an appropriate subtraction scheme, perhaps similar to that in Ref.Kitaev and Preskill (2006). Relatedly, it would be also interesting to find models where the singularity associated with negativity is universal and unrelated to the classical correlations. On a more practical front, it would be interesting to devise models where the singularity in negativity can be measured experimentally, using quantum state tomography Lanyon et al. (2017), or via swap-based methods on multiple copies of a system Gray et al. (2017); Islam et al. (2015); Kaufman et al. (2016).
Acknowledgements.
Acknowledgments: We thank John McGreevy and especially Tim Hsieh for helpful discussions and comments on the draft. TG is supported by an Alfred P. Sloan Research Fellowship. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) (see Ref.Towns et al. (2014)), which is supported by National Science Foundation grant number ACI1548562.References
- Sachdev (2011) S. Sachdev, Quantum phase transitions (Cambridge university press, 2011).
- Suzuki (1976) M. Suzuki, Progress of theoretical physics 56, 1454 (1976).
- Vidal and Werner (2002) G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).
- Nielsen and Chuang (2002) M. A. Nielsen and I. Chuang, Quantum computation and quantum information (AAPT, 2002).
- Werner (1989) R. F. Werner, Phys. Rev. A 40, 4277 (1989).
- Horodecki et al. (2009) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
- Eisert and Plenio (1999) J. Eisert and M. B. Plenio, Journal of Modern Optics 46, 145 (1999), https://www.tandfonline.com/doi/pdf/10.1080/09500349908231260 .
- Peres (1996) A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
- Horodecki et al. (1996) M. Horodecki, P. Horodecki, and R. Horodecki, Physics Letters A 223, 1 (1996).
- Simon (2000) R. Simon, Phys. Rev. Lett. 84, 2726 (2000).
- Horodecki (1997) P. Horodecki, Physics Letters A 232, 333 (1997).
- Bennett et al. (1996) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
- Anders (2008) J. Anders, Phys. Rev. A 77, 062102 (2008).
- Ferraro et al. (2008) A. Ferraro, D. Cavalcanti, A. García-Saez, and A. Acín, Phys. Rev. Lett. 100, 080502 (2008).
- Sherman et al. (2016) N. E. Sherman, T. Devakul, M. B. Hastings, and R. R. Singh, Physical Review E 93, 022128 (2016).
- Hart and Castelnovo (2018) O. Hart and C. Castelnovo, Phys. Rev. B 97, 144410 (2018).
- Calabrese et al. (2012) P. Calabrese, J. Cardy, and E. Tonni, Phys. Rev. Lett. 109, 130502 (2012).
- Calabrese et al. (2015) P. Calabrese, J. Cardy, and E. Tonni, Journal of Physics A: Mathematical and Theoretical 48, 015006 (2015).
- Osborne and Nielsen (2002) T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002).
- Metlitski et al. (2009) M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Phys. Rev. B 80, 115122 (2009).
- Kallin et al. (2013) A. B. Kallin, K. Hyatt, R. R. P. Singh, and R. G. Melko, Phys. Rev. Lett. 110, 135702 (2013).
- Wootters (1998) W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
- Assaad and Grover (2016) F. F. Assaad and T. Grover, Phys. Rev. X 6, 041049 (2016).
- (24) See supplemental material.
- Onsager (1944) L. Onsager, Phys. Rev. 65, 117 (1944).
- Vojta (1996) T. Vojta, Physical Review B 53, 710 (1996).
- Audenaert et al. (2002) K. Audenaert, J. Eisert, M. Plenio, and R. Werner, Physical Review A 66, 042327 (2002).
- Goldenfeld (2018) N. Goldenfeld, Lectures on phase transitions and the renormalization group (CRC Press, 2018).
- Bacon (2006) D. Bacon, Phys. Rev. A 73, 012340 (2006).
- Hastings (2011) M. B. Hastings, Phys. Rev. Lett. 107, 210501 (2011).
- Kitaev and Preskill (2006) A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
- Lanyon et al. (2017) B. P. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. F. Roos, Nature Physics 13, 1158 EP (2017).
- Gray et al. (2017) J. Gray, L. Banchi, A. Bayat, and S. Bose, ArXiv e-prints (2017), arXiv:1709.04923 [quant-ph] .
- Islam et al. (2015) R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature 528, 77 (2015).
- Kaufman et al. (2016) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016), http://science.sciencemag.org/content/353/6301/794.full.pdf .
- Towns et al. (2014) J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither, A. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D. Peterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr, Computing in Science and Engineering 16, 62 (2014).
Supplemental Material
I 1. General Results Regarding Commuting Projector Hamiltonians
i.1 1a. Partial Transposition Preserves the Set of Eigenvectors
Consider a commuting projector Hamiltonian , where and denote the part of with support only in real space region and , and denotes the interaction between and . Define as the set of local commuting operators, a commuting projector Hamiltonian can be written as . The thermal density matrix, with , can be expanded as: , where each is a tensor product of operators from the set . Since all operators in commute, , , and share the same eigenvectors. Under the partial transpose over the Hilbert space in , one obtains . If only acts on A or B, then . Only when the support of involves and simultaneously is it possible for to receive a minus sign under partial transpose. This implies that the operators basis for is still , and thus the eigenvectors of are exactly the same as those of , and the eigenvalues of can be obtained by replacing by their eigenvalues. In the argument above we implicitly assumed that all matrix elements of are real in the basis where we perform a partial transpose. If there exists complex matrix elements instead, might get a minus sign even when acts only on or . Nevertheless, one can check that is still generated by tensor products of , and therefore the conclusion remains the same.
i.2 1b. Partial Trace Preserves the Set of Eigenvectors
Here we show that for commuting projector Hamiltonians, the thermal density matrix and the reduced density matrix obtained by tracing out all the degrees of freedom in share the same set of eigenvectors. As discussed above, , where collects all possible operators from the product of commuting operators . By tracing out all the degrees of freedom in for , basis operators in which act non-trivially on vanish. This implies that the operator basis of reduced density matrix is generated by the those operators in which act on trivially, and thus commutes with all local commuting operators.
i.3 1c. Bipartite Negativity from Reduced Density Matrix on Boundary
Here we show that the negativity between two spatial regions of a thermal density matrix of a commuting projector Hamiltonian equals the negativity of the reduced density matrix localized on the boundary of the bipartition. Following the notation in the main text, we define as collection of spins on the boundary of that interacts with , and define as the collection of spins in the bulk of that only couples to spins in . We decompose a commuting projector Hamiltonian as , so that denotes the interaction between the bulk spins in , and denotes the interaction between the boundary spins in . For simplicity, we also assume that the system is time reversal invariant, so that for , the partial transpose over the Hilbert space in acts non-trivially only on :
(11) |
As discussed above, a partial transposed density matrix is still generated by local commuting operators which are present in . This implies that one can find a common eigenbasis for and , and the eigenvalues of can be obtained by replacing all local operators by their eigenvalues. Consequently, the eigenvalues of take the form:
(12) |
where denotes the spin configuration in the bulk of , and denotes the spin configuration on the boundary that interact with . The one-norm of can be obtained by summing all the absolute values of eigenvalues:
(13) |
The observation that partial transpose only affects the operators on the boundary motivates us to consider the reduced density matrix on the boundary:
(14) |
We take the partial transpose over
(15) |
where the commutative property of each local operator is used. As a result, the eigenvalue of is:
(16) |
By summing all absolute values of for and comparing it with Eq.13, one finds that
(17) |
which implies that the negativity of two spatial regions is given by the boundary of those two spatial regions. In fact with a similar calculation, one can show that the above equality also holds true for any commuting project Hamiltonian without time reversal symmetry.
Ii 2. Calculational details of negativity for various models discussed in the main text
ii.1 2a. Infinite-Range Commuting Projector Hamiltonian
Consider a one-dimensional lattice of size where each lattice site has four qubits, the model Hamiltonian is
(18) |
The density matrix at inverse temperature is with . Since every local term commutes, we can perform Hubbard-Stratonovich transformation for :
(19) |
where a local Hamiltonian for -site of four spins is defined as :
(20) |
Eq.19 implies that all sites are separable since manifestly takes the form where , is a local density matrix on -th site. As a result, to have non-zero negativity, an entanglement cut should be made across one of the sites (say -th site) such that four spins on -th site are not in the same subsystem. In the following calculation, comprises all the lattice sites with site index and two spins labelled by on -th site while comprises all the lattice sites with site index and two spins labelled by on -th site. The negativity can be calculated via a replica trick:
(21) |
Notice that is an even number as performing trace, but analytic continuation is taken in the end. First we calculate the thermal partition function:
(22) |
where
(23) |
The integral over is dominated by the saddle point , which satisfies :
(24) |
The critical behavior of can be determined by expanding Eq.24 to :
(25) |
where . Define , for , we can have non-zero solution for :
(26) |
while for , is the only allowed solution. Notice that the critical inverse temperature is determined by solving the transcendental equation:
(27) |
On the other hand, for the calculation of , since each site are separable, taking partial transpose over amounts to only taking the partial transpose on the two spins labelled by on the -th site:
(28) |
By introducing replicas, we have
(29) |
where
(30) |
This multi-dimensional integral is again dominated by saddle points , which can be obtained from :
(31) |
Assuming replica symmetry is preserved, we have with
(32) |
As , the above equation is exactly the saddle point equation for the thermal partition function (Eq.24). This implies . By plugging Eq.22 and Eq.29 into Eq.21, one finds
(33) |
where
(34) |
For , there is an unique saddle point , and
(35) |
For , there are two saddle points , and thus we arrive at
(36) |
Since = , we have
(37) |
This result implies that to calculate the bi-partite negativity between and , it is sufficient to calculate the reduced density matrix for -th site () where we made an entanglement cut. Incidentally, the above calculation explicitly demonstrates the claim mentioned in the main text where is the negativity in the absence of an infinitesimal symmetry breaking field (so that it receives contribution from both and ) while is the negativity in the presence of such a field so that it receives contribution only from one saddle point (say, ). From now on, we suppress lattice site index in the calculation since only four qubits on a single site is relevant. Meanwhile, will replace as the mean-field order parameter for brevity. The local density matrix is
(38) |
where the partition function is
(39) |
By taking partial transpose over , we have
(40) |
Due to the simple form of , we are able to obtain all the eigenvalues of , and exploit the following formula to calculate the negativity:
(41) |
where denotes eigenvalues of . Since commute with each other, the corresponding eigenvalues of these operators completely specify an eigenvector of , which takes the following form
(42) |
with for . With this observation, the eigenvalues of can be obtained by replacing operators by their eigenvalues:
(43) |
For