Abstract
Synthetic nonconservative systems with paritytime (PT) symmetric gain–loss structures can exhibit unusual spontaneous symmetry breaking that accompanies spectral singularity. Recent studies on PT symmetry in optics and weakly interacting open quantum systems have revealed intriguing physical properties, yet manybody correlations still play no role. Here by extending the idea of PT symmetry to strongly correlated manybody systems, we report that a combination of spectral singularity and quantum criticality yields an exotic universality class which has no counterpart in known critical phenomena. Moreover, we find unconventional lowdimensional quantum criticality, where superfluid correlation is anomalously enhanced owing to nonmonotonic renormalization group flows in a PTsymmetrybroken quantum critical phase, in stark contrast to the Berezinskii–Kosterlitz–Thouless paradigm. Our findings can be experimentally tested in ultracold atoms and predict critical phenomena beyond the Hermitian paradigm of quantum manybody physics.
Similar content being viewed by others
Introduction
Studies of phase transitions and critical behaviour in nonHermitian systems date back to the discovery of the LeeYang edge singularity^{1}, where an imaginary magnetic field in the hightemperature Ising model was demonstrated to trigger an exotic phase transition. More recently, the realtocomplex spectral phase transition has been found in a broad class of nonHermitian Hamiltonians that satisfy paritytime (PT) symmetry^{2}. While such systems were once considered to be of purely academic interest, related questions are now within experimental reach^{3,4,5,6,7}.
A Hamiltonian is said to be PTsymmetric if it commutes with the combined operator , but not necessarily with and separately. Here and are the parity and timereversal operators, respectively. PT symmetry is said to be unbroken if every eigenstate of is PTsymmetric; then, the entire spectrum is real even though is not Hermitian. PT symmetry is said to be spontaneously broken if some eigenstates of are not the eigenstates of the PT operator even though ; then, some pairs of eigenvalues of become complex conjugate to each other. PT symmetry breaking is typically accompanied by the coalescence of eigenstates and that of the corresponding eigenvalues at an exceptional point^{8} in the discrete spectrum or the spectral singularity^{9} in the continuum spectrum. While these features also hold for a certain class of antilinear symmetries^{10}, PT symmetry allows experimental implementations by spatial engineering of gain–loss structures, leading to a rich interplay between theory and experiment in optics^{4,5,6,7,11}, superconductors^{12}, atomic physics^{13} and optomechanics^{14}. In particular, the realtocomplex spectral transition (PT transition) has been observed in experiments of classical systems^{15}. In all these developments, however, manybody correlations still play no role.
Quantum critical phenomena, in contrast, arise from collective behaviour of strongly correlated systems and exhibit universal longdistance properties. In view of recent developments in designing open manybody systems in ultracold atoms^{16,17,18,19,20} and exciton–polariton condensates^{21}, it seems ripe to explore the role of PT symmetry in quantum critical phenomena and ask whether or not the concept of the universality need be extended in synthetic nonconservative systems.
Here we report that a combination of spectral singularity and quantum criticality yields an exotic critical point in the extended parameter space and that, in the PTbroken phase, a local gain–loss structure results in an anomalous enhancement of superfluid correlation owing to semicircular renormalization group (RG) flows. This contrasts sharply with the suppression of superfluid correlation due to hyperbolic RG flows in the Berezinskii–Kosterlitz–Thouless (BKT) paradigm. Our findings demonstrate that the interplay between manybody correlations and PT symmetry leads to the emergence of quantum critical phenomena beyond the Hermitian paradigm of quantum manybody physics.
Results
Paritytimesymmetric sineGordon model
We consider a class of onedimensional (1D) quantum systems described by the field theory Hamiltonian
where is a scalar field, is its conjugate momentum satisfying , and is a potential for the field . Without the potential term, equation (1) is known as the TomonagaLuttinger liquid (TLL) Hamiltonian, which gives a universal framework for describing 1D interacting bosons and fermions^{22}. Here, v is the sound velocity, the TLL parameter K characterizes the interaction strength, and and are related to the density and the Josephson phase, respectively. The introduction of the cosine potential results in the sineGordon model, which describes the BKT transition to a gapped phase. For bosons on a lattice, this corresponds to a superfluidtoMottinsulator (MI) transition^{23}. Here we consider a generalization to the PTsymmetric case by adding an imaginary contribution to the potential term as follows:
where α_{r} and α_{i} characterize the strengths of the real and imaginary parts of the potential. When the real part becomes relevant, it suppresses the fluctuations of , stabilizing a noncritical, gapped phase. In contrast, if the imaginary part is relevant, it facilitates the fluctuations of and enhances correlation in the conjugate field , as we will see later. The field theory (1) with the potential (2) satisfies PT symmetry since the field has odd parity. The PTsymmetric Hamiltonian can be implemented by a continuously monitored 1D interacting ultracold atoms (see Supplementary Note 1 and Supplementary Fig. 1).
We note that if α_{r}>α_{i}, has a real spectrum and thus PT symmetry is unbroken. This can be proved by the theorem^{24} which states that the spectrum is real if and only if there exists an operator satisfying , where is a Hermitian operator. Indeed, we can explicitly construct such an operator for α_{r}>α_{i} by the choice of , where is a constant part of and η ≡ arctanh(α_{i}/α_{r}). Then, the potential term in the effective field theory is transformed to and reduces to the sineGordon Hamiltonian^{25}. Divergence of η at α_{r}=α_{i} signals spontaneous breaking of PT symmetry.
Renormalization group analysis
To unravel the universal critical behaviour of the PTsymmetric Hamiltonian , we perform an RG analysis^{26} to obtain the following set of flow equations which are valid up to the third order in g_{r,i}:
Here l is the logarithmic RG scale and g_{r,i} ≡ α_{r,i}a^{2}/(ħv) are the dimensionless coupling constants with a being a shortdistance cutoff. The velocity v stays constant to all orders in g_{r,i} because of the Lorentz invariance of the theory. In contrast to the twodimensional phase diagram of the conventional sineGordon model, the PTsymmetric system has the threedimensional (3D) phase diagram (Fig. 1a). When PT symmetry is unbroken, that is, g_{i}<g_{r}, the spectrum is equivalent to that of the closed system as discussed above and the conventional RG flow diagram with hyperbolic flows is reproduced (Fig. 1b). Here the BKT boundary between the superfluid TLL phase and the MI phase extends over the curved surface. We note that the operator does not affect the critical properties of the ground state since it only modifies the zero modes associated with the field . Since the nonHermitian term can arise from the measurement backaction, the quantum phase transition induced by increasing g_{i} may be regarded as measurementinduced.
In the strongly correlated regime K<2, a new type of quantum phase transition appears on the PT threshold plane g_{i}=g_{r}. This phase transition is accompanied by spontaneous breaking of PT symmetry in eigenstates, contrary to the ordinary BKT transition exhibiting no symmetry breaking. The BKT and PT phase boundaries merge on the line defined by K=2 and g_{i}=g_{r} (Fig. 1c). In general, at the PT symmetrybreaking point, the spectral singularity^{9} arises where two or more eigenvalues as well as their eigenstates coalesce in the continuum spectrum. In optics, the spectral singularity leads to unidirectional wave phenomena^{5}. In contrast, in manybody systems, the coexistence of the spectral singularity and the quantum criticality at g_{i}=g_{r} and K=2 results in what we term a spectral singular critical point, which represents a unique universality class in nonconservative systems.
When the PT symmetry is broken, that is, g_{i}>g_{r}, unconventional RG flows emerge: starting from the K<2 side, g_{r,i} and K initially increase, and after entering the K>2 side, the flow winds and converges to the fixed line with g_{r,i}=0 (Fig. 1d). Physically, this significant increase in the TLL parameter K indicates that the superfluid correlation decays more slowly and is thus enhanced by the nonHermiticity of an imaginary potential. The enhancement is viewed as anomalous because, in the conventional BKT paradigm, a real potential suppresses the fluctuation of and stabilizes the gapped MI phase for K<2. Moreover, owing to the semicircular RG flows, the imaginary potential allows for a substantial increase of the TLL parameter K even if its strength g_{i} is initially very small. The PTbroken phase exhibits other observable consequences such as anomalous lasing and absorption as observed in optics^{27} (see Supplementary Note 2 for the experimental implementation in ultracold atoms).
Groundstate phase diagram of the lattice model
To numerically demonstrate these findings, we introduce a lattice Hamiltonian
whose lowenergy behaviour is described by the PTsymmetric effective field theory . Here are the spin–1/2 operators at site m and the parameters (−Δ, h_{s}, γ) are related to (K, g_{r}, g_{i}) in the field theory, where we set J=1. The nonHermitian term represents a periodic gain–loss structure and effectively strengthens the amplitude of the hopping term, leading to enhanced superfluid correlation. The determined phase diagram and a typical exact finitesize spectrum are shown in Fig. 2. The BKT transition is identified as a crossing point of appropriate energy levels^{28} and the PT threshold is determined as a coalescence point in lowenergy levels, as detailed in Methods section and Supplementary Methods. The coalescence point is found to be an exceptional point from the characteristic squareroot scaling^{8} of the energy gap (see the inset figure in Fig. 2b). We note that, above the PT threshold, some highly excited states turn out to have positive imaginary parts of eigenvalues and cause the instability in the longtime limit. The presence of such highenergy unstable modes is reminiscent of parametric instabilities in exciton–polariton systems^{29}, and can ultimately destroy the 1D coherence^{30}. In our setup, where the imaginary term is adiabatically ramped up, the amplitudes of these unstable modes can greatly be suppressed and the system can remain, with almost unit fidelity, in the ground state in which the critical behaviour is sustained (see Supplementary Note 3 and Supplementary Figs 2 and 3 for details).
Numerical demonstration of enhanced superfluid correlation
To demonstrate the anomalous enhancement of superfluid correlation in the PTbroken regime, we have performed numerical simulations using the infinite timeevolving block decimation (iTEBD) algorithm^{31}. The correlation function exhibits the critical decay with a varying critical exponent and the corresponding TLL parameter significantly increases, surpassing K=2 as shown in Fig. 3. Physically, this enhancement of superfluid correlation at long distances can be interpreted as follows. A local gain–loss structure introduced by the imaginary term causes locally equilibrated flows^{15} in the ground state. This results in the enhancement of fluctuations in the density, or equivalently, the suppression of fluctuations in the conjugate phase. It is this effect that increases the superfluid correlation. The numerical results are consistent with the analytical arguments given above, and demonstrate that the RG analysis is instrumental in studying critical properties of a nonHermitian manybody system.
Experimental realization in a onedimensional Bose gas
The PTsymmetric manybody Hamiltonian discussed above can be implemented in a 1D interacting ultracold bosonic atoms subject to a shallow PTsymmetric optical lattice V(x)=V_{r} cos(2πx/d)−iV_{i} sin(2πx/d), where V_{r} and V_{i} are the depths of the real and imaginary parts of a complex potential and d is the lattice constant. An imaginary optical potential can be realized by using a weak nearresonant standingwave light (Methods section). Since V(x) remains invariant under simultaneous parity operation (x→−x) and time reversal (that is, complex conjugation), the system satisfies the condition of PT symmetry (Fig. 4a). In open quantum systems, by postselecting null measurement outcomes, the time evolution is governed by an effective nonHermitian Hamiltonian^{32,33,34}. The achieved experimental fidelity has already been high enough to allow experimenters to implement various types of postselections^{35,36,37}. The lowenergy behaviour of this system is then described by the PTsymmetric effective field theory . We note that the lattice Hamiltonian (4) can also be realized in ultracold atoms by superimposing a deep lattice that does not influence the universal critical behaviour (Fig. 4b).
We stress that the dynamics considered here is different from the one described by a master equation, where dissipative processes, in general, tend to destroy correlations underlying quantum critical phenomena. In contrast, the postselections allow us to study the system free from the dissipative jump processes, while nontrivial effects due to measurement backaction still occur via the nonHermitian contributions in the effective Hamiltonian.
Discussion
The reported fixed points in the extended parameter space suggest that an interplay between spectral singularity and quantum criticality results in an exotic universality class beyond the conventional paradigm. It remains an open question how the universality accompanying spectral singularity found in this work is related to nonunitary conformal field theories (CFT) studied in various fields ranging from statistical mechanics^{38} to highenergy physics^{39}. It is particularly notable that a certain critical point of the integrable spin chain with PTsymmetric boundary fields corresponds to an exceptional point and is believed to be described by nonunitary CFT^{40}. This suggests an intimate connection between the spectral singular critical point and the nonunitary CFT. Given recent success in measuring entanglement entropy in ultracold atoms^{37}, it is of interest to study how quantum entanglement behaves in the presence of spectral singularity. In the PTbroken phase, we have shown that the ground state exhibits the enhanced superfluid correlation indicating the tighter binding of the topological excitations, in stark contrast to their proliferation as found in the BKT paradigm. In Hermitian systems, a relevant perturbation around RG fixed points has a tendency to suppress fluctuations of the concerned field and stabilize a noncritical, gapped phase. Our finding indicates that a relevant imaginary perturbation can realize the opposite situation of enhancing fluctuations of the concerned field and facilitating correlation in the conjugate field. An exploration of such unconventional quantum criticality in other synthetic, nonconservative manybody systems presents an interesting challenge. Further studies in these directions, together with their possible experimental realizations, could widen applications to future quantum metamaterials.
Methods
Details of numerical calculations
The phase diagram in Fig. 2a is determined from the exact diagonalization analysis of the lattice Hamiltonian (4). To identify the BKT transition point, we calculate the exact finitesize spectrum and find a crossing of lowenergy levels having appropriate quantum numbers^{28}. The PT transition point is identified as the first coalescence point in the lowenergy spectrum with increasing γ. The calculations are done for different system sizes and the final results are obtained through extrapolation of the data to the thermodynamic limit. Further details are given in Supplementary Methods and Supplementary Fig. 4. The correlation function and the associated variation of the TLL parameter K shown in Fig. 3 are calculated by applying the iTEBD algorithm^{31}. We emphasize that this method can be applied to study the groundstate properties of the nonHermitian system. The method can accurately calculate the imaginarytime evolution for an infinite system size, where τ is an imaginary time, is an initial state and denotes the norm of the state. In the limit of large τ, we obtain the quantum state, the real part of which eigenvalue is the lowest in the entire spectrum, that is, an effective ground state of a nonHermitian system. We note that the imaginary part of the eigenvalue does not affect the calculation since it only changes an overall phase of the wavefunction in the imaginarytime evolution. We then determine the TLL parameter K from the calculated correlation function by using the relation .
Derivation of the lowenergy field theory of ultracold atoms
Here we explain the derivation of the lowenergy effective field theory (1) of ultracold atoms. We start from the Hamiltonian in which the periodic potential V_{r} cos(2πx/d) is added to the Lieb–Liniger model^{41}. Then, we introduce an imaginary optical lattice potential by using a weak nearresonant standingwave light. This scheme is possible if the excited state of an atom has decay modes other than the initial ground state and its decay rate is faster than the spontaneous decay rate from to and the Rabi frequency^{42,43,44} (Fig. 4a). Such a condition can be satisfied by, for example, using appropriate atomic levels^{45} or lightinduced transitions^{16}. The difference between the wavelengths of the real and imaginary periodic potentials caused by different detunings of the lasers can be negligible. Using a secondorder perturbation theory^{8} for the Rabi coupling and adiabatically eliminating the excited state, we obtain an effective timeevolution equation for the groundstate atoms. We then assume that null measurement outcomes are postselected so that the dynamics is described by the nonHermitian Hamiltonian^{32,33,34}. In this situation, the overall imaginary constant in the eigenvalue spectrum does not affect the dynamics since it can be eliminated when we normalize the quantum state, leading to the imaginary potential iV_{i} sin(2πx/d). Finally, we follow the standard procedure^{22} of taking the lowenergy limit of the model and arrive at the Hamiltonian (1). The details of the calculations and experimental accessibility in ultracold atoms are described in Supplementary Notes 1 and 2.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Ashida, Y. et al. Paritytimesymmetric quantum critical phenomena. Nat. Commun. 8, 15791 doi: 10.1038/ncomms15791 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Fisher, M. E. YangLee edge singularity and φ^{3} field theory. Phys. Rev. Lett. 40, 1610–1613 (1978).
Bender, C. M. & Böttcher, S. Real spectra in nonHermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).
Rüter, C. E. et al. Observation of paritytime symmetry in optics. Nat. Phys. 6, 192–195 (2010).
Regensburger, A. et al. Paritytime synthetic photonic lattices. Nature 488, 167–171 (2012).
Feng, L. et al. Experimental demonstration of a unidirectional reflectionless paritytime metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013).
Peng, B. et al. Paritytimesymmetric whisperinggallery microcavities. Nat. Phys. 10, 394–398 (2014).
Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).
Kato, T. Perturbation Theory for Linear Operators Springer (1966).
Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009).
Bender, C. M., Berry, M. V. & Mandilara, A. Generalized PT symmetry and real spectra. J. Phys. A 35, L467 (2002).
Makris, K. G., ElGanainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2009).
Chtchelkatchev, N. M., Golubov, A. A., Baturina, T. I. & Vinokur, V. M. Stimulation of the fluctuation superconductivity by PT symmetry. Phys. Rev. Lett. 109, 150405 (2012).
Peng, P. et al. Antiparitytime symmetry with flying atoms. Nat. Phys. 12, 1139–1145 (2016).
Jing, J. et al. PTsymmetric phonon laser. Phys. Rev. Lett. 113, 053604 (2014).
Bender, C. M., Berntson, B. K., Parker, D. & Samuel, E. Observation of PT phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013).
Bakr, W. S. et al. A quantum gas microscope for detecting single atoms in a Hubbardregime optical lattice. Nature 462, 74–77 (2009).
Barontini, G. et al. Controlling the dynamics of an open manybody quantum system with localized dissipation. Phys. Rev. Lett. 110, 035302 (2013).
Brennecke, F. et al. Realtime observation of fluctuations at the drivendissipative Dicke phase transition. Proc. Natl Acad. Sci. USA 110, 11763–11767 (2013).
Patil, Y. S., Chakram, S. & Vengalattore, M. Measurementinduced localization of an ultracold lattice gas. Phys. Rev. Lett. 115, 140402 (2015).
Ashida, Y. & Ueda, M. Diffractionunlimited position measurement of ultracold atoms in an optical lattice. Phys. Rev. Lett. 115, 095301 (2015).
Gao, T. et al. Observation of nonHermitian degeneracies in a chaotic excitonpolariton billiard. Nature 526, 554–558 (2015).
Giamarchi, T. Quantum Physics in One Dimension Oxford University Press (2004).
Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluidinsulator transition. Phys. Rev. B 40, 546–570 (1989).
Mostafazadeh, A. PseudoHermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a nonHermitian Hamiltonian. J. Math. Phys. 43, 205–214 (2002).
Bender, C. M., Jones, H. F. & Rivers, R. J. Dual PTsymmetric quantum field theories. Phys. Lett. B 625, 333–340 (2005).
Amit, D. J., Goldschmidt, Y. Y. & Grinstein, S. Renormalisation group analysis of the phase transition in the 2D Coulomb gas, SineGordon theory and XYmodel. J. Phys. A 13, 585 (1980).
Peng, B. et al. Lossinduced suppression and revival of lasing. Science 346, 328–332 (2014).
Nomura, K. Correlation functions of the 2D sineGordon model. J. Phys. A 28, 5451 (1995).
Sarchi, D., Carusotto, I., Wouters, M. & Savona, V. Coherent dynamics and parametric instabilities of microcavity polaritons in doublewell systems. Phys. Rev. B 77, 125324 (2008).
Carusotto, I. & Ciuti, C. Spontaneous microcavitypolariton coherence across the parametric threshold: quantum Monte Carlo studies. Phys. Rev. B 72, 125335 (2005).
Vidal, G. Classical simulation of infinitesize quantum lattice systems in one spatial dimension. Phys. Rev. Lett. 98, 070201 (2007).
Carmichael, H. An Open System Approach to Quantum Optics Springer (1993).
Daley, A. J. Quantum trajectories and open manybody quantum systems. Adv. Phys 63, 77–149 (2014).
Ashida, Y., Furukawa, S. & Ueda, M. Quantum critical behaviour influenced by measurement backaction in ultracold gases. Phys. Rev. A 94, 053615 (2016).
Endres, M. et al. Observation of correlated particlehole pairs and string order in lowdimensional Mott insulators. Science 334, 200–203 (2011).
Fukuhara, T. et al. Spatially resolved detection of a spinentanglement wave in a BoseHubbard chain. Phys. Rev. Lett. 115, 035302 (2015).
Islam, R. et al. Measuring entanglement entropy in a quantum manybody system. Nature 528, 77–83 (2015).
Cardy, J. L. Conformal invariance and the YangLee edge singularity in two dimensions. Phys. Rev. Lett. 54, 1354–1356 (1985).
Seiberg, N. Notes on quantum Liouville theory and quantum gravity. Prog. Theor. Phys. Supp. 102, 319–349 (1990).
Pasquier, V. & Saleur, H. Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330, 521–556 (1990).
Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963).
Oberthaler, M. K. et al. Dynamical diffraction of atomic matter waves by crystals of light. Phys. Rev. A 60, 456–472 (1999).
Turlapov, A., Tonyushkin, A. & Sleator, T. Optical mask for lasercooled atoms. Phys. Rev. A 68, 023408 (2003).
Stützle, R. et al. Observation of nonspreading wave packets in an imaginary potential. Phys. Rev. Lett. 95, 110405 (2005).
Johnson, K. S. et al. Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit. Science 280, 1583–1586 (1998).
Acknowledgements
We acknowledge support from KAKENHI Grant Nos. JP25800225 and JP26287088 from the Japan Society for the Promotion of Science (JSPS), and a GrantinAid for Scientific Research on Innovative Areas ‘Topological Materials Science’ (KAKENHI Grant No. JP15H05855), and the Photon Frontier Network Program from MEXT of Japan, ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). We are grateful to Yusuke Horinouchi, Ryusuke Hamazaki, Zongping Gong, Shintaro Takayoshi, Yuya Nakagawa, Takeshi Fukuhara, Takashi Mori and Hosho Katsura for valuable discussions. Y.A. acknowledges support from JSPS (Grant No. JP16J03613).
Author information
Authors and Affiliations
Contributions
Y.A., S.F. and M.U. planned the project. Y.A. performed the analytical calculations. Y.A. and S.F. performed the numerical calculations. Y.A., S.F. and M.U. analysed and interpreted the results and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Notes, Supplementary Figures, Supplementary Methods, and Supplementary References (PDF 366 kb)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Ashida, Y., Furukawa, S. & Ueda, M. Paritytimesymmetric quantum critical phenomena. Nat Commun 8, 15791 (2017). https://doi.org/10.1038/ncomms15791
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms15791
This article is cited by

Theoretical investigation of dynamics and concurrence of entangled \({{\mathcal {P}}}{{\mathcal {T}}}\) and anti\({{\mathcal {P}}}{{\mathcal {T}}}\) symmetric polarized photons
Scientific Reports (2023)

Spin transport in nonHermitian quantum systems
Scientific Reports (2023)

Groundstate and thermal entanglements in nonHermitian XY system with real and imaginary magnetic fields
Quantum Information Processing (2023)

Entanglement entropy distinguishes PTsymmetry and topological phases in a class of nonunitary quantum walks
Quantum Information Processing (2023)

Extension of Noether’s theorem in \({\cal P}{\cal T}\)symmetry systems and its experimental demonstration in an optical setup
Science China Physics, Mechanics & Astronomy (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.