Non-Hermitian Chiral Magnetic Effect in Equilibrium
We analyze the Chiral Magnetic Effect for non-Hermitian fermionic systems using the biorthogonal formulation of quantum mechanics. In contrast to the Hermitian chiral counterparts, we show that the Chiral Magnetic Effect may take place in thermal equilibrium of an open non-Hermitian system with, generally, massive fermions. The key observation is that for non-Hermitian charged systems, there is no strict charge conservation as understood in the Hermitian case, so the Bloch theorem preventing currents in the thermodynamic limit in equilibrium does not apply.
The Chiral Magnetic Effect (CME) is the generation of an electric current in the presence of an external magnetic field in a system of massless (Weyl) fermions with a (chiral) imbalance in densities between left- and right-handed chiralities Fukushima et al. (2008):
The mismatch in populations of left- and right-handed Weyl nodes is encoded in the difference between their chemical potentials, .
The effect plays an essential role in a wide number of physical systems ranging from astrophysical and quark-gluon plasmas to chiral materials Kharzeev (2014). Its potential existence in a thermodynamic equilibrium is an important question due to the fact that the current (1) is a topologically protected and hence non-dissipative quantity even in the presence of strong interactions Kharzeev (2014).
In general, two ways have been proposed to get a nonzero chiral chemical potential : (i) Applying parallel electric and magnetic fields simultaneously so that the chiral anomaly creates a charge imbalance between the Weyl nodes; (ii) Splitting the positions of the left- and right-handed Weyl nodes in energies . In the former case the electric field drives system into a stationary but not equilibrium state while in the latter case the CME current is zero anyway because the “consistent” version of the CME current (1) gets cancelled by an extra term coming from the Bardeen polynomial Landsteiner (2016).
While the non-equilibrium situation has been explored extensively in the literature leading, for instance, to the celebrated negative quadratic magnetoresistivity in Weyl metals, the equilibrium scenario appears to be not possible, and to date there is consensus that the CME (1) does not exist in thermodynamic equilibriumMa and Pesin (2015); Yamamoto (2015); Zubkov (2016).
The statement of absence of the CME in equilibrium can be seen as an extension of a no-go theorem given by Bloch, concerning the existence of equilibrium currents in solids in the thermodynamic limit Bohm (1949). This theorem has been extended to chiral matter in Ref. Yamamoto (2015), and refined in Ref. Zubkov (2016) (the absence of CME in equilibrium using the chiral kinetic formalism has been obtained in Ref. Ma and Pesin (2015)).
There are three elements usually associated to this theorem in chiral matter: (i) the assumption that the system is in the equilibrium state, (ii) the existence of Weyl nodes that always come by pairs Nielsen and Ninomiya (1981a, b), and (iii) the Hermiticity of the Hamiltonian. As we have mentioned, it is known how to break the first condition and drive the system out of equilibrium. Recently, it has been proposed that the second assumption of having pairwise Weyl nodes can be broken in Weyl superconductors, where an external magnetic field induces a gap in one of the Weyl nodes (and its particle-hole conjugate), leaving effectively a single Weyl node O’Brien et al. (2017). However, we stress here that the presence of Weyl nodes is not a strict requirement to the absence (or presence) of CME Chang and Yang (2015); Zubkov (2016). In our paper we explore the third option by considering the CME in equilibrium non-Hermitian fermionic systems.
Nowadays there is a surge of interest in non-Hermitian systems for many different reasons, ranging from very fundamental questions in the quantum (and statistical) theory of fields and the role of topology in non-Hermitian systemsBender (2005); Chernodub (2017); Gong et al. (2018), to applied science. Among them, specially interesting are the non-Hermitian systems that display a real spectrum, as the -symmetric systems or the quasi-Hermitian systems. Although non-Hermitian, they display a unitary evolution, and it is possible to define a consistent thermodynamics for them Gardas et al. (2016).
The model we will study is a non-Hermitian extension of the massive Dirac model in dimensions, where, together with the usual mass term, an anti-Hermitian mass is introduced Bender et al. (2005); Jones-Smith and Mathur (2014); Alexandre and Bender (2015); Alexandre et al. (2015, 2017):
The advantage of this model is that the two first terms of the right hand side of Eq.(2) are Hermitian by themselves, so the only non-Hermitian term is .
It is already stated in the literature that non-Hermitian Hamiltonians are, in general, not gauge-invariant operators in a sense the vector (electric) current is not a conserved quantity. Physically, the situation corresponds to an open system possessing sinks and sources of electric charges. Mathematically, the non-conservation of electric current can be viewed as the fact that the Noether theorem relating continuous symmetries and conserved currents in field theories, does not hold in non-Hermitian systems Alexandre et al. (2015, 2017); Mannheim (2018). For this reason, there is some arbitrariness when defining a coupling to electromagnetic fields in the Hamiltonian (2). In the present work, we are interested to compare our results with the ones in Hermitian systems, so that we consider the coupling to electromagnetic fields to the model (2) with , which is Hermitian (and the principle of local gauge invariance holds), and later we switch on the non-Hermitian term .
The non-conservation of the vector current, caused by presence of sinks and sources in open non-Hermitian systems it is not directly related to explicit, spontaneous or anomalous mechanism of symmetry breaking known in Hermitian theories (see, for example, Ref. Alexandre et al. (2017)).
On general grounds, the time dependence of any operator can be constructed using the Heisenberg picture (please note that, now since , both and appear in the time evolution of ):
The time variation of is then as follows:
For Hermitian systems, and we recognize the commutation with as the condition any operator must satisfy to be a conserved quantity. For non-Hermitian systems, we immediately see that an operator is a conserved quantity if, instead of commuting with the Hamiltonian, it fulfills the quasi-Hermiticity condition: so . In the case of the charge symmetry, it is clear that the generator of this symmetry in Hermitian systems commutes with but does not satisfy the quasi-Hermiticity condition, so it is not a conserved quantity for non-Hermitian systems. As we will see below it is possible to find operators that, while not commuting with , satisfy the quasi-Hermiticity condition, thus defining conserved quantities.
Computation of the Chiral Magnetic Effect with biorthogonal quantum mechanics.-
Here we will tackle the problem of computing the CME for the non-Hermitian model in Eq.(2), using the biorthogonal quantum mechanics formalism. Within this formalism, we distinguish between the eigenstates of the Hamiltonian : , their complex conjugates: , the biorthogonal states : , and their complex conjugates, . The point is that, because is not Hermitian, , and . Also, for the same lack of Hermiticity, the states are not orthogonal , where is the standard scalar product in the corresponding Hilbert space. However, the state sets and form a biorthogonal basis: .
For the model (2) we can define a metric operator , that not only fulfills the quasi-Hermiticity condition, but it is also a positive definite quantity. The existence of such operator simplifies the construction of the biorthogonal basis sets, since these two bases are related to each other through :
with the normalization . For the Hamiltonian at hands (2), such metric operator turns out to be Alexandre et al. (2017). The existence of a metric operator allowing us to construct a well-defined inner product in the corresponding Hilbert space, defines a unitary time evolution of the states, as long as the energy spectrum is real. As we will see below, the latter condition is satisfied in the -unbroken phase of the model (2) in the mass strip similarly to its continuum counterpart Alexandre et al. (2017). Therefore, a consistent description of quantum mechanics is allowed for the non-Hermitian system.
Another relevant consequence of the existence of the metric operator is that is a conserved quantity, since, as we mentioned, the matrix fulfills the pseudo-Hermiticity condition, although does not commute with Simon et al. (2018) (and itself allows for a construction of a unitary evolution). There are thus two points to pay attention to.
First, as it is done in Hermitian statistical mechanics, we can define a Lagrange multiplier associated to the operator , viewed as a conserved quantity, that plays the role of the chemical potential. We thus can define the new Hamiltonian
The dispersion relation of for non-zero chemical potential can be seen in Fig. 1. Of course, now due to the non-Hermitian nature of the problem, conserved quantities as do not to commute with , but satisfy the aforementioned pseudo-Hermiticity condition. However, the possibility of finding a common basis between and any operator only exists if that operator commutes with the Hamiltonian, irrespective of the Hermiticity of . This means that, we will not be able to find a common basis for and in terms of the eigenstates of the number operator, as it happens in conventional Hermitian Quantum Mechanics, so we will have to build the biorthogonal basis by diagonalizing instead of .
Second, from the perspective of constructing a thermal equilibrium ensemble, the lack of Hermiticity in the system is not a problem, since the requirement the one-particle correlation function built from the biorthogonal basis must fulfill is to satisfy the KMS periodicity condition Haag et al. (1967), since it is known that states satisfying such boundary condition are thermal equilibrium states Kubo (1957); Martin and Schwinger (1959); Haag et al. (1967). The point is to notice that, for non-zero , the time evolution of any field operator is done through the exponential of (and not of ) so then the one-particle correlation function will satisfy the KMS boundary condition and we will be able to built an equilibrium ensemble. Also, it is important to notice that the electromagnetic field is not treated as a perturbation, but it is already included in the spectrum of (6)), despite the fact we have a non-Hermitian system Fetter and Walecka (1971). In the Supplementary Material we provide an explicit proof that the non-Hermitian system (2) and (6) satisfies the KMS condition. Also, this fact has been pointed out in the existing literature of non-Hermitian systems Jakubsky (2007).
The equilibrium thermal average of any observable ,
may be expressed, in our case, via the single-particle propagator in imaginary time:
where are the biorthogonal sets of single-particle eigenstates of the model (6) in the presence of an external magnetic field : and . The generic label comprises the band (particle/hole) label , the spin index , and the Landau level index .
For operators defined as , we can generalize the Hellmann–Feynman theorem to the biorthogonal basis (See Supplementary Material), if the eigenstates are real:
obtaining, after performing the Matsubara summation,
where is the Fermi distribution function in absence of the chemical potential. The chemical potential is part of the spectrum. Also, in Eq.(10) we have used the fact that the degeneracy of each Landau level is .
For the case of CME, , so
The dispersion relation for the LLL () sector is:
where is a particle/hole label. For , we have
For the Landau levels, the spin degree of freedom appears explicitly. In Fig. 2 we plot the Landau level spectrum for and . The all important difference between the eigenenergies for and is that, while with is an even function of for any value of , and , the energy with is not. That means that, when taking the derivative with respect to and integrating over a symmetric interval, the Landau levels, similarly to the CME in the Hermitian case, will not contribute to the integral in (11), but the Lowest Landau Level will do.
The result of the calculation (11) turns out to be:
This is the principal result of this Letter. For non-zero values of the non-Hermitian mass , which is the parameter that controls the non-Hermiticity of , there is a non-vanishing CME in equilibrium for massive fermions. The result (14) holds in the -unbroken phase with , where the energy spectrum, Eqs. (12) and (13), is real and the non-Hermitian model (2) is unitary.
Computation of the Chiral Separation Effect.-
The Chiral Separation Effect (CSE) generates the axial current given by the difference between the currents at right- and left-handed Weyl cones, Vilenkin (1980); Metlitski and Zhitnitsky (2005). The CSE is obtained by computing the average value of the chiral current, represented by the operator .
We follow the same route as in the case of the CME. We compute by adding a term to the Hamiltonian (6) and calculating the energy spectrum in presence of the parameter . Then, we apply the Hellmann–Feynman theorem to it, taking the derivative with respect to and constructing the expectation value for each Landau level. We send the parameter to zero after the calculation.
A lengthy but straightforward calculation shows that for the sector, is an odd function of in the limit for all values of , , and . Thus, the integral over is zero and the states do not contribute to the CSE. In contrast, for the sector, we simply have
so that , and
Performing the integral and renormalizing the result at the Fermi surface, we arrive to the non-Hermitian CSE:
where the mass of a quasiparticle is a real-valued quantity in the -unbroken region. This result coincides with the CSE current in the Hermitian QED with a fermion of mass Gorbar et al. (2013). At the exceptional point, , the fermion quasiparticle becomes massless, , and the non-Hermitian current (17) reduces to the known Hermitian result .
In the present Letter we have demonstrated that the CME in equilibrium is possible for open non-Hermitian systems (14). The key ingredient is to realize that the CME is zero if a vector charge conservation is imposed in the system. However, the charge conservation, associated to the symmetry, is not fulfilled in the non-Hermitian systems contrary to conventional Hermitian ones.
Another fact to pay attention is that the metric operator associated with the pseudo-Hermiticity condition () is not unique. While there is no practical consequence of this regarding the construction of a biorthogonal basis Brody (2016) (the average of observables do not depend on any particular choice of the metric operator), this observation is relevant as we can associate different chemical potentials to different metric operators (6) understood as conserved quantities in the non-Hermitian sense. Interestingly, all the metric operators are related to each other by a similarity transformation Mostafazadeh (2008), so we can generalize the results obtained here to other chemical potentials by modifying the spectrum correspondingly.
The CSE in the non-Hermitian system has the standard Hermitian form for a massive particle (17).
Finally, to the best of our knowledge, there are no experimental realizations of fermionic non-Hermitian systems possessing real spectrum to test our predictions. However, there are impressive experimental advances in the area of non-Hermitian symmetric photonic systems and other condensed matter analogs Zyablovsky et al. (2014); Hayata (2018). In fact, it has been recently proposed the experimental observation of the CME employing superconducting quantum circuit technology, and synthetic magnetic fields Tan et al. (2018). We suggest the same experimental setup to test our theory, by extending the experimental setup with equal gain-loss Quijandría et al. (2018). Besides, other topological equilibrium effects similar to the CME have been proposed to occur in electromagnetism Avkhadiev and Sadofyev (2017); Yamamoto (2017); Agullo et al. (2017); Chernodub et al. (2018), being the optical helicity and the optical chirality the electromagnetic symmetries that play the role of the chiral symmetry in ultrarelativistic fermionic systems. There, the biorthogonal formalism have probed to be useful to handle the effect of dissipation and loss in electromagnetism Alpeggiani et al. (2018); Vázquez-Lozano and Martínez (2018). The natural question is then to see how the topologically-related responses associated to these symmetries are modified by the presence of non-Hermitian effects.
We kindly acknowledge inspiring conversations with K. Landsteiner and P. Millington about non-Hermitian systems and the physics underlying the CME. The authors gratefully acknowledge financial support through the MINECO/AEI/FEDER, UE Grant No. FIS2015-73454-JIN, the Comunidad de Madrid MAD2D-CM Program (S2013/MIT-3007), Grant 3.6261.2017/8.9 of the Ministry of Science and Higher Education of Russia and Spanish–French mobility project PIC2016FR6/PICS07480.
- Fukushima et al. (2008) K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D 78, 074033 (2008), URL https://link.aps.org/doi/10.1103/PhysRevD.78.074033.
- Kharzeev (2014) D. E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133 (2014), eprint 1312.3348, URL https://www.sciencedirect.com/science/article/pii/S0146641014000039.
- Landsteiner (2016) K. Landsteiner, Acta Phys. Pol B 47, 2617 (2016), URL http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=47&page=2617.
- Ma and Pesin (2015) J. Ma and D. A. Pesin, Phys. Rev. B 92, 235205 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.235205.
- Yamamoto (2015) N. Yamamoto, Phys. Rev. D 92, 085011 (2015), URL https://link.aps.org/doi/10.1103/PhysRevD.92.085011.
- Zubkov (2016) M. A. Zubkov, Phys. Rev. D 93, 105036 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.93.105036.
- Bohm (1949) D. Bohm, Phys. Rev. 75, 502 (1949), URL https://link.aps.org/doi/10.1103/PhysRev.75.502.
- Nielsen and Ninomiya (1981a) H. Nielsen and M. Ninomiya, Nuclear Physics B 185, 20 (1981a), ISSN 0550-3213, URL http://www.sciencedirect.com/science/article/pii/0550321381903618.
- Nielsen and Ninomiya (1981b) H. Nielsen and M. Ninomiya, Nuclear Physics B 193, 173 (1981b), ISSN 0550-3213, URL http://www.sciencedirect.com/science/article/pii/0550321381905241.
- O’Brien et al. (2017) T. E. O’Brien, C. W. J. Beenakker, and I. Adagideli, Phys. Rev. Lett. 118, 207701 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.118.207701.
- Chang and Yang (2015) M.-C. Chang and M.-F. Yang, Phys. Rev. B 92, 205201 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.205201.
- Bender (2005) C. M. Bender, International Journal of Modern Physics A 20, 4646 (2005), URL https://doi.org/10.1142/S0217751X05028326.
- Chernodub (2017) M. N. Chernodub, Journal of Physics A: Mathematical and Theoretical 50, 385001 (2017), URL http://stacks.iop.org/1751-8121/50/i=38/a=385001.
- Gong et al. (2018) Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018), URL https://link.aps.org/doi/10.1103/PhysRevX.8.031079.
- Gardas et al. (2016) B. Gardas, S. Deffner, and A. Saxena, Scientific Reports 6, 23408 EP (2016), URL http://dx.doi.org/10.1038/srep23408.
- Bender et al. (2005) C. M. Bender, H. Jones, and R. Rivers, Physics Letters B 625, 333 (2005), ISSN 0370-2693, URL http://www.sciencedirect.com/science/article/pii/S0370269305012098.
- Jones-Smith and Mathur (2014) K. Jones-Smith and H. Mathur, Phys. Rev. D 89, 125014 (2014), URL https://link.aps.org/doi/10.1103/PhysRevD.89.125014.
- Alexandre and Bender (2015) J. Alexandre and C. M. Bender, Journal of Physics A: Mathematical and Theoretical 48, 185403 (2015), URL http://stacks.iop.org/1751-8121/48/i=18/a=185403.
- Alexandre et al. (2015) J. Alexandre, C. M. Bender, and P. Millington, Journal of High Energy Physics 2015, 111 (2015), ISSN 1029-8479, URL https://doi.org/10.1007/JHEP11(2015)111.
- Alexandre et al. (2017) J. Alexandre, P. Millington, and D. Seynaeve, Phys. Rev. D 96, 065027 (2017), URL https://link.aps.org/doi/10.1103/PhysRevD.96.065027.
- Mannheim (2018) P. D. Mannheim, arXiv e-prints arXiv:1808.00437 (2018), eprint 1808.00437, URL https://arxiv.org/abs/1808.00437.
- Simon et al. (2018) M. A. Simon, A. Buendia, and J. G. Muga, Mathematics 6 (2018), ISSN 2227-7390, URL http://www.mdpi.com/2227-7390/6/7/111.
- Haag et al. (1967) R. Haag, N. M. Hugenholtz, and M. Winnink, Communications in Mathematical Physics 5, 215 (1967), URL https://doi.org/10.1007/BF01646342.
- Kubo (1957) R. Kubo, Journal of the Physical Society of Japan 12, 570 (1957), URL https://doi.org/10.1143/JPSJ.12.570.
- Martin and Schwinger (1959) P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959), URL https://link.aps.org/doi/10.1103/PhysRev.115.1342.
- Fetter and Walecka (1971) A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971).
- Jakubsky (2007) V. Jakubsky, Mod. Phys. Lett. A 22, 1075 (2007), URL https://doi.org/10.1142/S0217732307023419.
- Vilenkin (1980) A. Vilenkin, Phys. Rev. D22, 3080 (1980), URL https://doi.org/10.1103/PhysRevD.88.025025.
- Metlitski and Zhitnitsky (2005) M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D72, 045011 (2005), eprint hep-ph/0505072, URL https://doi.org/10.1103/PhysRevD.88.025025.
- Gorbar et al. (2013) E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and X. Wang, Phys. Rev. D88, 025025 (2013), eprint 1304.4606, URL https://doi.org/10.1103/PhysRevD.88.025025.
- Brody (2016) D. C. Brody, Journal of Physics A: Mathematical and Theoretical 49, 10LT03 (2016), URL http://stacks.iop.org/1751-8121/49/i=10/a=10LT03.
- Mostafazadeh (2008) A. Mostafazadeh, Journal of Physics A: Mathematical and Theoretical 41, 244017 (2008), URL http://stacks.iop.org/1751-8121/41/i=24/a=244017.
- Zyablovsky et al. (2014) A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, Physics-Uspekhi 57, 1063 (2014), URL http://stacks.iop.org/1063-7869/57/i=11/a=1063.
- Hayata (2018) T. Hayata, Phys. Rev. B 97, 205102 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.205102.
- Tan et al. (2018) X. Tan, Y. Zhao, Q. Liu, G. Xue, H. Yu, Z. Wang, and Y. Yu, arXiv e-prints arXiv:1802.08371 (2018), eprint 1802.08371, URL https://arxiv.org/abs/1802.08371.
- Quijandría et al. (2018) F. Quijandría, U. Naether, S. K. Özdemir, F. Nori, and D. Zueco, Phys. Rev. A 97, 053846 (2018), URL https://link.aps.org/doi/10.1103/PhysRevA.97.053846.
- Avkhadiev and Sadofyev (2017) A. Avkhadiev and A. V. Sadofyev, Phys. Rev. D 96, 045015 (2017), URL https://link.aps.org/doi/10.1103/PhysRevD.96.045015.
- Yamamoto (2017) N. Yamamoto, Phys. Rev. D 96, 051902 (2017), URL https://link.aps.org/doi/10.1103/PhysRevD.96.051902.
- Agullo et al. (2017) I. Agullo, A. del Rio, and J. Navarro-Salas, Phys. Rev. Lett. 118, 111301 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.118.111301.
- Chernodub et al. (2018) M. N. Chernodub, A. Cortijo, and K. Landsteiner, Phys. Rev. D 98, 065016 (2018), URL https://link.aps.org/doi/10.1103/PhysRevD.98.065016.
- Alpeggiani et al. (2018) F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, Phys. Rev. Lett. 120, 243605 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.243605.
- Vázquez-Lozano and Martínez (2018) J. E. Vázquez-Lozano and A. Martínez, Phys. Rev. Lett. 121, 043901 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.121.043901.
Appendix A SUPPLEMENTARY MATERIAL OF “Non-Hermitian Chiral Magnetic Effect in Equilibrium”
a.1 The Hellmann–Feynman theorem
for biorthogonal systems
In this Appendix we give a proof of the extension of the Hellmann–Feynman theorem to biorthogonal systems with real spectrum.
As discussed in the main text, the biorthogonal basis is constructed with two set of states satisfying
together with the normalization .
Let us consider a Hamiltonian depending on some parameter . To ease notation, we will keep the dependence with the generic parameter implicit in the eigenstates and eigenvalues. We are interested in computing the averaged value
Then, we compute
We used here that and are eigenstates of and with the same eigenvalue . Simplifying a little, we finally get the result we wanted to prove:
a.2 Thermal equilibrium condition
in quasi-Hermitian systems
For Hermitian systems, the condition of thermal equilibrium can be formally established by showing that the Hermitian system satisfies the Kubo-Martin-Schwinger (KMS) boundary condition for the imaginary-time propagatorHaag et al. (1967). For quasi-Hermitian systems, it is possible to describe equilibrium in the same way, making use of the existing non-unitary mapping between the non-Hermitian and Hermitian Hamiltonians. In what follows, we will restrict ourselves to non-Hermitian systems described by Hamiltonian operators that do not depend on time.
Let us consider two operators and in the Heisenberg picture in the imaginary time formalism, described by the Hamiltonian . We consider that chemical potentials associated to symmetries of the problem are already included in as in Eq. (6). The KMS condition can be stated as the following identity:
If and are field operators that anti-commute, we have
As explained in Fetter and Walecka (1971), this means that the thermal averaged propagator ( refers to the Dyson time ordering) is an antiperiodic function of the imaginary time with the period . This allows the development of all the machinery of thermal field theory.
In order to show how this works for quasi-Hermitian systems, it is enough to show that, for a quasi-Hermitian Hamiltonian, it is possible to construct an Hermitian partner through a non-unitary mapping between them, so we map the statistical averages using the biorthogonal basis in the non-Hermitian case, map them to their Hermitian counterparts, establish the KMS condition in the latter, and going back to the non-Hermitian case inverting the mentioned mapping.
As demonstrated in Jakubsky (2007), the existence of a metric operator allows us to define the non-unitary mapping of some quasi-Hermitian Hamiltonian to an Hermitian partner , with (we will denote the Hermitian partners of operators with the hat symbol ):
We can define the Hermitian partner of any operator associated to the quasi-Hermitian system in the same way:
This includes the field operators and in the second quantization formalism. As discussed in the main text, the existence of the metric operator allows us to construct a well behaved scalar product in the Hilbert space and to construct biorthogonal basis sets, and . In this way, we can define the following statistical average (here we will use the suffix to denote the statistical average with the biorthogonal basis):
In the second line we have used ( in our particular case), and that the eigenstates of the non-Hermitian are related to the eigenstates of the Hermitian partner through . Also, we consider that the states of the Hermitian partner are conveniently normalized: .
To guarantee the proper normalization of (27), we need to relate the partition function in the quasi-Hermitian system and its Hermitian partner. This is a particular case of the previous identity, as we can choose and obtain the equality of the corresponding partition functions:
where we have denoted the partition function of the Hermitian partner by .
We can generalize (27) to any product of field operators. Then we obtain that
so we conclude that the averages performed with the biorthogonal basis and with the density matrix satisfy a KMS boundary condition and thus this state defines a thermal state in equilibrium, since it is trivial to modify the previous reasoning by including the Dyson time ordering operation. Also, this reasoning justifies the definition of the discrete-frequency Green function in Eq.(8) of the main text.