The DOI matches its Crossref record; 42 of the 42 references it cites were found in a registry or an index; none of the 21 sources its claims cite is checked for support yet.
Optical Realization of Relativistic Non-Hermitian Quantum Mechanics
Open accessAbstract
Light propagation in distributed-feedback optical structures with gain or loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and are related to well-known physical phenomena such as resonance narrowing and laser oscillation.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
Light propagation in distributed-feedback optical structures with gain/loss regions is proposed as a classical optical simulator of the one-dimensional non-Hermitian Dirac equation. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are mapped to observable DFB phenomena such as resonance narrowing and laser oscillation.
Key findings
- For a PT-non-invariant uniform-gain DFB structure, the critical normalized gain is λ_c L ≈ 1.755 for m₀L = 1; at λ_c two resonances cross the real energy axis and become spectral singularities, and above λ_c bound states emerge and the spectrum ceases to be real-valued.
- For the same structure with uniform loss, at λ_c two antiresonances cross the real axis; the spectral transmission does not diverge, and the bound states have Im(E) < 0, so perturbations are damped rather than amplified.
- For a PT-invariant DFB structure with symmetric gain and loss regions, PT symmetry breaking occurs at λ_c ≈ 4.46/L for m₀L = 1, with two pairs of complex-conjugate eigenvalues appearing from resonances and antiresonances crossing the real axis.
- The paper reports that spectral singularities and PT symmetry breaking of the Dirac Hamiltonian correspond to measurable DFB quantities, including resonance narrowing and the onset of lasing.
Methods
- Theoretical mapping of the 1D Dirac equation with real mass m(x) and complex vector potential V(x) to coupled-mode equations for counterpropagating optical envelopes in a DFB grating.
- Scattering theory and transfer-matrix analysis: the transfer matrix M(E), Green function/resolvent, and zeros of M₁₁ and M₂₂ are used to classify continuous spectrum, point spectrum, resonances, and spectral singularities.
- Two examples are analyzed numerically for m₀L = 1: a uniform index grating with homogeneous gain (PT-non-invariant) and a uniform grating with symmetric homogeneous gain/loss regions (PT-invariant).
Limitations
- The paper states no limitations.
- Evident from the text: this is a theoretical proposal for a classical optical simulator, not an experimental realization; no fabricated device or measured data are reported.
- Evident from the text: the analysis is restricted to the 1D Dirac equation and two specific DFB configurations with uniform grating and piecewise-constant gain/loss, with numerical examples given for m₀L = 1.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Figures
Verification
Claims the paper makes
13 No basis given · 8 Sources not found · 6 Own interpretation · 4 Not checked yet · 4 Own result · 1 No full text of the sources
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let us consider now the same DFB structure of Fig.1(a) but with γ ( x ) = λ , corresponding to a uniform grating with a homogeneous lossy region. The transfer matrix of the structure is given again by Eq.(7), but with σ = -E -iλ .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
None of the sources cited for this claim have been checked yet.
In the paper
“Note that the imaginary parts of the eigenvalues E for λ > λ c are precisely the growth rates of the two detuned unstable modes at the onset of lasing found in the theory of DFB lasers with a uniform grating [18, 19].”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“the envelopes ψ 1 and ψ 2 satisfy Eq.(1) with a real mass m and a complex-valued vector potential V defined by”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, for a PT -non-invariant Hamiltonian, a spectral singularity could arise from the vanishing of M 11 but not of M 22 , i.e. it might not correspond to a divergence of the spectral transmission or reflection coefficients.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The transfer matrix of the structure can be calculated as M = M 2 M 1 , where M 1 and M 2 are the transfer matrices of the uniform gain and lossy sections [see Eq.(7)].”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This work suggests that well-known phenomena occurring in DFB structures, like spectral narrowing of resonances and self-oscillation, are the measurable quantities associated to the onset of spectral singularities and PT symmetry breaking of Dirac Hamiltonians with certain complex couplings.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“As a first example, let us consider a PT -non-invariant potential corresponding to θ ( x ) = 0 and m ( x ) = m 0 , γ ( x ) = -λ in the interval | x | < L/ 2 [Fig.1(a)], where λ is the dimensionless gain coefficient. This case corresponds to the simplest version of a DFB laser with a uniform index grating and a uniform gain region [19].”
What the sources say
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The main motivation in the study of such quantum-optical analogs is that concepts like exceptional points, spectral singularities and PT symmetry breaking become measurable quantities in an optical experiment [8, 9].”
What the sources say
- Not matchedcites [8]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
In the paper
“An optical realization of the Dirac equation (1) is provided by propagation of light waves in an effective one-dimensional DFB structure [18].”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
None of the sources cited for this claim have been checked yet.
In the paper
“Light propagation in distributed feedback optical structures with gain/loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions.”
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1 · plain
“Here it is shown that light propagation in distributed-feedback (DFB) optical structures with gain and/or loss regions, which is at the heart of such important devices as DFB semiconductor lasers [18, 19], can provide a fertile ground to test the spectral properties of non-Hermitian Dirac Hamiltonians.”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1 · plain
- It may state two separate claims as one. 16
- Probably not a substantial claim. 4
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“At λ > λ c the resonances cross the real energy axis and bound states with Im( E ) > 0 emerge; correspondingly, the spectrum of H ceases to be real-valued.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As λ is increased, the spectrum remains real-valued, the resonances (antiresonances) rigidly shift parallel to the imaginary axis, until at a critical value λ = λ c two resonances cross the real energy axis, i.e. they become two spectral singularities [Fig.1(b)].”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the Hermitian limit λ = 0, the spectrum of H is purely continuous, and a number of resonances (i.e. zeros of M 22 in the Im( E ) < 0 half plane) as well as of anti-resonances (i.e. zeros of M 11 in the Im( E ) > 0 half plane) do exist [Fig.1(b)].”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this case, as the loss coefficient λ is increased from zero, the spectrum remains real-valued shifting parallel to the imaginary axis. At the critical value λ = λ c , two antiresonances (rather than resonances) now cross the real energy axis, i.e. they become two spectral singularities.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(ii) Point spectrum . The zeros of M 22 ( E ) on the Im( E ) > 0 half plane, together with the zeros of M 11 ( E ) on the Im( E ) < 0 half plane, define the point spectrum of H ; at such energies, the function φ (2) E ( x ) [for Im( E ) > 0] and φ (1) E ( x ) [for Im( E ) < 0] are bound states of H .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that, owing to the PT -invariance of H , M 11 ( E ) = M ∗ 22 ( E ∗ ), and thus the zeros of M 11 (not shown in the figure) are simply the complex conjugates of those of M 22 .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Such a transition, from λ < λ c to λ > λ c , is accompanied by a narrowing of the resonance widths in the transmission spectrum as λ → λ -c [see Fig.1(c)] and to the threshold for self-oscillation at λ = λ c .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 2(b) shows the loci of zeros of M 22 in the complex energy plane for increasing values of λL and for m 0 L = 1.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
In the paper
“Note that, for a PT -invariant Hamiltonian, M 11 ( E ) = M ∗ 22 ( E ∗ ) and hence at a spectral singularity both M 11 and M 22 vanish, the resolvent is unbounded for E = E 0 ± i 0 + , and the transmission and reflection spectral coefficients diverge according to Eq.(4) [22].”
What the sources say
- Not matchedcites [22] Unitarity of the scattering matrix, implying | t | 2 + | r ( l,r ) | 2 = 1 and thus bounded values of transmission and reflection coefficients, is satisfied for Hermitian Hamiltonians, however in non-Hermitian systems unitarity may be broken [see, for instance: F. Cannata, J.-P. Dedonder, and A. Ventura, Ann. Phys. 322 , 397 (2007)].
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
- Not matched
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The critical value λ c can be expressed in the form λ c = f ( m 0 L ) /L , where the function f ( m 0 L ) can be calculated numerically. For instance, for the case of Fig.1 ( m 0 L = 1) one has λ c L ≃ 1 . 755.”
plaincorrelationalSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(iii) Resonances . The zeros of M 22 ( E ) [ M 11 ( E )] on the Im( E ) < 0 [Im( E ) > 0] half plane, i.e. the poles of the analytic continuation of G + [ G -] on the Im( E ) < 0 [Im( E ) > 0] half plane, correspond to the resonances [antiresonances] of the scattering problem.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(i) Continuous spectrum . The continuous spectrum of H is the entire real energy axis ( -∞ < E < ∞ ), where G ( E ) has a branch cut.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As the point of PT symmetry breaking is approached, narrowing of the transmission resonances is observed [see Fig.2(c)], and the onset of lasing at λ = λ + c corresponds to the breaking of the PT symmetry.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
In the paper
“If every eigenfunction of a PT -invariant Hamiltonian is also an eigenfunction of the PT operator, the PT symmetry of H is said to be unbroken and its spectrum is real-valued [4].”
What the sources say
- Not matchedcites [4]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Let us consider the Dirac equation in one spatial dimension for a two-component spinor wave function ψ ( x, t ) = ( ψ 1 , ψ 2 ) T with time-independent vector ( V ) and scalar ( S ) couplings, which in natural units ( ℏ = c = 1) reads [11, 12]”
What the sources say
- Not matchedcites [11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [12]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“Similarities between light propagation in DFB structures and relativistic wave equations were noticed in early studies on gap solitons in connection with the massive Thirring model of field theory [20], however these previous studies did not consider non-Hermitian interactions.”
What the sources say
- Not checked yetcites [20] A.B. Aceves and S. Wabnitz, Phys. Lett. A 141 , 37 (1989); D.N. Christodoulides and R.I. Joseph, Phys. Rev. Lett. 62 , 1746 (1989).
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and related to well-known physical phenomena like resonance narrowing and laser oscillation.”
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4 · hedged
“This work suggests that well-known phenomena occurring in DFB structures, like spectral narrowing of resonances and self-oscillation, are the measurable quantities associated to the onset of spectral singularities and PT symmetry breaking of Dirac Hamiltonians with certain complex couplings.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4 · hedged
None of the sources cited for this claim have been checked yet.
In the paper
“Recently, optical structures in media with a complex refractive index have been proposed to test and visualize non-Hermitian features rooted in the non-relativistic Schr¨ odinger equation with a complex potential [8, 9, 14, 15].”
What the sources say
- Not matchedcites [8]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [14] A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38 , L171 (2005).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked yetcites [15] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32 , 2632 (2007); K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, Phys. Rev. Lett. 100 , 103904 (2008); S. Longhi, Phys. Rev. Lett. 103 , 123601 (2009).
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If either V or m assume complex values, the Dirac Hamiltonian H is non-Hermitian and its spectrum is, in general, complexvalued.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The authors state this as an assumption.
Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
“In the following, we will assume that both m ( x ) and V ( x ) have a limited support, i.e. m = V = 0 for | x | > L/ 2.”
Citations
Cites
(1998) · doi:10.1103/physrevlett.80.5243 · Cited by 6,280
Observation of parity–time symmetry in optics
Rüter, C. E., Makris, K. G., El‐Ganainy, R., Christodoulides, D. N., Segev, M., & Kip, D. (2010) · Nature Physics · doi:10.1038/nphys1515 · Cited by 3,564
Making sense of non-Hermitian Hamiltonians
Bender, C. M. (2007) · Reports on Progress in Physics · doi:10.1088/0034-4885/70/6/r03 · Cited by 2,984
Observation of P T -Symmetry Breaking in Complex Optical Potentials
Guo, A., Salamo, G. J., Duchesne, D., Morandotti, R., Volatier, M., Aimez, V., Siviloglou, G. A., & Christodoulides, D. N. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.093902 · Cited by 2,723
Coupled-Wave Theory of Distributed Feedback Lasers
Kogelnik, H., & Shank, C. V. (1972) · Journal of Applied Physics · doi:10.1063/1.1661499 · Cited by 2,176
Beam Dynamics in P T Symmetric Optical Lattices
Makris, K. G., El‐Ganainy, R., Christodoulides, D. N., & Musslimani, Z. H. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.103904 · Cited by 2,046
Naĭmark, M. A. (1967) · Frederick Ungar eBooks · Cited by 1,807
Complex Extension of Quantum Mechanics
Bender, C. M., Brody, D. C., & Jones, H. F. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.270401 · Cited by 1,803
Making Sense of Non-Hermitian Hamiltonians
Bender, C. M. (2007) · CERN Document Server (European Organization for Nuclear Research) · Cited by 1,546
Naimark, M. (1967) · doi:10.1017/cbo9780511627040.005 · Cited by 1,434
Cited by
Sorted by citations
Unidirectional Invisibility Induced by P T -Symmetric Periodic Structures
Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H. W., & Christodoulides, D. N. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.213901 · Cited by 1,905
Non-Hermitian photonics based on parity–time symmetry
Feng, L., El‐Ganainy, R., & Ge, L. (2017) · Nature Photonics · doi:10.1038/s41566-017-0031-1 · Cited by 1,501
The physics of exceptional points
Heiss, W. D. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/44/444016 · Cited by 1,449
Longhi, S. (2010) · Physical Review A · doi:10.1103/physreva.82.031801 · Cited by 950
Self-citationTopologically protected bound states in photonic parity–time-symmetric crystals
Weimann, S., Kremer, M., Plotnik, Y., Lumer, Y., Nolte, S., Makris, K. G., Segev, M., Rechtsman, M. C., & Szameit, A. (2016) · Nature Materials · doi:10.1038/nmat4811 · Cited by 864
Nonlinear switching and solitons in PT‐symmetric photonic systems
Suchkov, S. V., Sukhorukov, A. A., Huang, J., Dmitriev, S. V., Lee, C., & Kivshar, Y. S. (2016) · Laser & Photonics Review · doi:10.1002/lpor.201500227 · Cited by 349
“We also note that certain type of spectral singularities may not correspond to divergence of spectral reflection and transmission coefficients [161].”
Background$\mathcal{PT}$-symmetric electronics
Schindler, J., Lin, Z., Lee, J. M., Ramezani, H., Ellis, F. M., & Kottos, T. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/44/444029 · Cited by 308
Parity-time symmetry meets photonics: A new twist in non-Hermitian optics
Longhi, S. (2017) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/120/64001 · Cited by 309
Self-citationSpatial Kramers–Kronig relations and the reflection of waves
Horsley, S. A. R., Artoni, M., & Rocca, G. C. L. (2015) · Nature Photonics · doi:10.1038/nphoton.2015.106 · Cited by 248
Invisibility in $\mathcal {PT}$-symmetric complex crystals
Longhi, S. (2011) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/44/48/485302 · Cited by 252
Self-citationRamezani, H., Christodoulides, D. N., Kovanis, V. I., Vitebsky, I. M., & Kottos, T. (2012) · Physical Review Letters · doi:10.1103/physrevlett.109.033902 · Cited by 174
Experimental observation of the dual behavior of PT -symmetric scattering
Lin, Z., Schindler, J., Ellis, F. M., & Kottos, T. (2012) · Physical Review A · doi:10.1103/physreva.85.050101 · Cited by 138
Effective spontaneous PT -symmetry breaking in hybridized metamaterials
Kang, M., Fu, L., & Li, J. (2013) · Physical Review A · doi:10.1103/physreva.87.053824 · Cited by 130
Nonlinear modes in the harmonic PT -symmetric potential
Zezyulin, D. A., & Konotop, V. V. (2012) · Physical Review A · doi:10.1103/physreva.85.043840 · Cited by 126
Experimental Realization of Floquet P T -Symmetric Systems
Chitsazi, M., Li, H., Ellis, F. M., & Kottos, T. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.093901 · Cited by 128
Parity–time symmetric photonics
Zhao, H. X., & Feng, L. H. (2018) · National Science Review · doi:10.1093/nsr/nwy011 · Cited by 125
Yang, Z., Schnyder, A. P., Hu, J., & Chiu, C. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.086401 · Cited by 128
Bypassing the bandwidth theorem with PT symmetry
Ramezani, H., Schindler, J., Ellis, F. M., Günther, U., & Kottos, T. (2012) · Physical Review A · doi:10.1103/physreva.85.062122 · Cited by 116
Time-dependent $\mathcal {P}$$\mathcal {T}$-symmetric quantum mechanics
Gong, J., & Wang, Q.-H. (2013) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/46/48/485302 · Cited by 118
Entanglement entropy of non-unitary conformal field theory
Bianchini, D., Castro-Alvaredo, O. A., Doyon, B., Levi, E., & Ravanini, F. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/48/4/04ft01 · Cited by 130
Non-hermitian quantum thermodynamics
Gardas, B., Deffner, S., & Saxena, A. (2016) · Scientific Reports · doi:10.1038/srep23408 · Cited by 96
“Introduction More and more non-hermitian systems are becoming experimentally accessible [1].”
Disorder-driven exceptional lines and Fermi ribbons in tilted nodal-line semimetals
Moors, K., Zyuzin, A. A., Zyuzin, A. A., Tiwari, R. P., & Schmidt, T. L. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.041116 · Cited by 94
PT-symmetric microring laser-absorber
Longhi, S., & Feng, L. H. (2014) · Optics Letters · doi:10.1364/ol.39.005026 · Cited by 88
Self-citationBound states in the continuum in PT-symmetric optical lattices
Longhi, S. (2014) · Optics Letters · doi:10.1364/ol.39.001697 · Cited by 82
Self-citationOptical waveguide arrays: quantum effects and PT symmetry breaking
Joglekar, Y. N., Thompson, C., Scott, D. D., & Vemuri, G. (2013) · The European Physical Journal Applied Physics · doi:10.1051/epjap/2013130240 · Cited by 82
Bender, C. M. (2015) · Journal of Physics Conference Series · doi:10.1088/1742-6596/631/1/012002 · Cited by 81
“) Since its inception sixteen years ago there have been dozens of experimental confirmations published in Nature, Science, and Physical Review Letters in diverse research areas such as optics [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], NMR [15], superconductivity [16, 17], microwave cavities [18, 19], lasers [20, 21…”
BackgroundQuantum noise and mode nonorthogonality in non-Hermitian PT -symmetric optical resonators
Yoo, G., Sim, H.-S., & Schomerus, H. (2011) · Physical Review A · doi:10.1103/physreva.84.063833 · Cited by 74
Zero index metamaterials with PT symmetry in a waveguide system
Fu, Y., Xu, Y., & Chen, H. (2016) · Optics Express · doi:10.1364/oe.24.001648 · Cited by 73
Classical simulation of relativistic quantum mechanics in periodic optical structures
Longhi, S. (2011) · Applied Physics B · doi:10.1007/s00340-011-4628-7 · Cited by 72
Self-citationParity-time symmetry in optical microcavity systems
Wen, J., Jiang, X., Jiang, L., & Xiao, M. (2018) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/1361-6455/aae42f · Cited by 69
Parity-Time Symmetry and Exceptional Points [Electromagnetic Perspectives]
Krasnok, A., Nefedkin, N. E., & Alù, A. (2021) · IEEE Antennas and Propagation Magazine · doi:10.1109/map.2021.3115766 · Cited by 66
A parity-time symmetric coherent plasmonic absorber-amplifier
Baum, B. K., Alaeian, H., & Dionne, J. A. (2015) · Journal of Applied Physics · doi:10.1063/1.4907871 · Cited by 65
Parity-time symmetry breaking in magnetic systems
Galda, A., & Vinokur, V. M. (2016) · Physical review. B./Physical review. B · doi:10.1103/physrevb.94.020408 · Cited by 65
The Kibble-Zurek mechanism at exceptional points
Dóra, B., Heyl, M., & Moessner, R. (2019) · Nature Communications · doi:10.1038/s41467-019-10048-9 · Cited by 65
Backward lasing yields a perfect absorber
Longhi, S. (2010) · Physics · doi:10.1103/physics.3.61 · Cited by 62
Self-citationGoos-Hänchen shift in complex crystals
Longhi, S., Valle, G. D., & Staliūnas, K. (2011) · Physical Review A · doi:10.1103/physreva.84.042119 · Cited by 62
Self-citationBound states, scattering states, and resonant states in PT -symmetric open quantum systems
Garmon, S. S., Gianfreda, M., & Hatano, N. (2015) · Physical Review A · doi:10.1103/physreva.92.022125 · Cited by 59
Li, J., Yu, R., Ding, C., & Wu, Y. (2016) · Physical Review A · doi:10.1103/physreva.93.023814 · Cited by 59
Yang, Y., Jia, H., Bi, Y., Zhao, H., & Yang, J. (2019) · Physical Review Applied · doi:10.1103/physrevapplied.12.034040 · Cited by 59
“In recent years, non-Hermitian system with PT-symmetric potential has been widely applied in fields of classic waves because of the similarity between Schrodinger equation and classic wave equation [5-30].”
MethodLiu, X., Gupta, S. D., & Agarwal, G. S. (2014) · Physical Review A · doi:10.1103/physreva.89.013824 · Cited by 56
Robust PT -symmetric chain and properties of its Hermitian counterpart
Joglekar, Y. N., & Saxena, A. (2011) · Physical Review A · doi:10.1103/physreva.83.050101 · Cited by 54
Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures
Longhi, S. (2015) · Optics Letters · doi:10.1364/ol.40.005694 · Cited by 56
Self-citationRelativistic non-Hermitian quantum mechanics
Jones-Smith, K., & Mathur, H. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.89.125014 · Cited by 57
Two-dimensional solitons in PT linear lattice potentials
Zeng, J., & Lan, Y. (2012) · Physical Review E · doi:10.1103/physreve.85.047601 · Cited by 51
Ren, J., Franke, S., & Hughes, S. (2021) · Physical Review X · doi:10.1103/physrevx.11.041020 · Cited by 52
“[19] Stefano Longhi, “Optical realization of relativistic nonhermitian quantum mechanics,” Phys.”
BackgroundPT -symmetric slowing down of decoherence
Gardas, B., Deffner, S., & Saxena, A. (2016) · Physical Review A · doi:10.1103/physreva.94.040101 · Cited by 50
Gap solitons in parity–time symmetric moiré optical lattices
Liu, X., & Zeng, J. (2022) · Photonics Research · doi:10.1364/prj.474527 · Cited by 50
Jia, C., Li, X., & Zhang, L. (2011) · Few-Body Systems · doi:10.1007/s00601-011-0258-1 · Cited by 49
“In this regard, some authors investigated the exact solutions of the position-dependent effective mass Dirac equations [7–21] and spatially dependent mass Klein–Gordon equations for various potential models [22–31].”
BackgroundPerfect state transfer in PT -symmetric non-Hermitian networks
Zhang, X. Z., Jin, L., & Song, Z. (2012) · Physical Review A · doi:10.1103/physreva.85.012106 · Cited by 50
Sinha, D., & Ghosh, P. K. (2015) · Physical Review E · doi:10.1103/physreve.91.042908 · Cited by 46
Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories
Alexandre, J., Ellis, J. R., Millington, P., et al. (2019) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.99.075024 · Cited by 45
“[4] S.”
BackgroundNon-Hermitian anisotropic XY model with intrinsic rotation-time-reversal symmetry
Zhang, X. Z., & Song, Z. (2013) · Physical Review A · doi:10.1103/physreva.87.012114 · Cited by 45
Convective and absolute PT -symmetry breaking in tight-binding lattices
Longhi, S. (2013) · Physical Review A · doi:10.1103/physreva.88.052102 · Cited by 45
Self-citationInvited Article: Mitigation of dynamical instabilities in laser arrays via non-Hermitian coupling
Longhi, S., & Feng, L. (2018) · APL Photonics · doi:10.1063/1.5028453 · Cited by 42
Self-citationDisorder-induced exceptional points and nodal lines in Dirac superconductors
Zyuzin, A. A., & Simon, P. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.165145 · Cited by 44
“[7] S.”
BackgroundDissipative Floquet dynamical quantum phase transition
Naji, J., Jafari, M., Jafari, R., & Akbari, A. (2022) · Physical Review A · doi:10.1103/physreva.105.022220 · Cited by 44
Optical lattices with exceptional points in the continuum
Longhi, S., & Valle, G. D. (2014) · Physical Review A · doi:10.1103/physreva.89.052132 · Cited by 42
Self-citationThermalization Dynamics of Nonlinear Non-Hermitian Optical Lattices
Pyrialakos, G. G., Ren, H., Jung, P. S., Khajavikhan, M., & Christodoulides, D. N. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.213901 · Cited by 42
Spontaneously breaking non-Abelian gauge symmetry in non-Hermitian field theories
Alexandre, J., Ellis, J. R., Millington, P., et al. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.035008 · Cited by 40
“[4] S.”
BackgroundYang, W., Chen, A., Xie, X.-T., & Ni, L. (2017) · Physical Review A · doi:10.1103/physreva.96.013802 · Cited by 39
Optical lattices with higher-order exceptional points by non-Hermitian coupling
Zhou, X., Gupta, S. K., Huang, Z., Yan, Z., Zhan, P., Chen, Z., Lu, M., & Wang, Z. (2018) · Applied Physics Letters · doi:10.1063/1.5043279 · Cited by 39
Invisible defects in complex crystals
Longhi, S., & Valle, G. D. (2013) · Annals of Physics · doi:10.1016/j.aop.2013.03.011 · Cited by 37
Self-citationPerfect transmission scattering as a PT -symmetric spectral problem
Hernández-Coronado, H., Krejčiřı́k, D., & Siegl, P. (2011) · Physics Letters A · doi:10.1016/j.physleta.2011.04.021 · Cited by 37
“The relevance of PT -symmetric operators has been suggested in various domains of physics: nuclear physics [18], optics [6, 11, 19, 20], solid state [2], superconductivity [14, 15], and electromagnetism [16, 12].”
BackgroundPerturbation theory in the complex plane: exceptional points and where to find them
Marie, A., Burton, H. G. A., & Loos, P. (2021) · Journal of Physics Condensed Matter · doi:10.1088/1361-648x/abe795 · Cited by 34
Non-Hermitian Hamiltonian for a modulated Jaynes-Cummings model with PT symmetry
Bagarello, F., Lattuca, M., Passante, R., Rizzuto, L., & Spagnolo, S. (2015) · Physical Review A · doi:10.1103/physreva.91.042134 · Cited by 35
“In the case of the Lee model with imaginary coupling constant (ghost regime), it has been shown that it still has a PT (parity-time) symmetry [8, 9], and that it can be usefully used to describe the light transport in an infinite waveguide lattice with a time-dependent nonlinearity [9, 25].”
MethodLoss-induced super scattering and gain-induced absorption
Feng, S. (2016) · Optics Express · doi:10.1364/oe.24.001291 · Cited by 35
Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories
Alexandre, J., Ellis, J. R., & Millington, P. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.102.125030 · Cited by 35
“[4] S.”
Rafi‐Ul‐Islam, S. M., Siu, Z. B., & Jalil, M. B. A. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.103.035420 · Cited by 35
Self-sustained emission in semi-infinite non-Hermitian systems at the exceptional point
Zhang, X. Z., Jin, L., & Song, Z. (2013) · Physical Review A · doi:10.1103/physreva.87.042118 · Cited by 34
Classical simulation of relativistic quantum mechanics in periodic optical structuresPreprint
Longhi, S. (2011) · Cited by 47
Self-citationHuang, C., Zhang, R., Han, J., et al. (2014) · Physical Review A · doi:10.1103/physreva.89.023842 · Cited by 33
Reconfigurable Directional Lasing Modes in Cavities with Generalized PT˜ Symmetry
Lee, J. M., Factor, S. M., Lin, Z., Vitebsky, I. M., Ellis, F. M., & Kottos, T. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.253902 · Cited by 33
PT phase control in circular multi-core fibers
Longhi, S. (2016) · Optics Letters · doi:10.1364/ol.41.001897 · Cited by 30
Self-citationHermitian scattering behavior for a non-Hermitian scattering center
Jin, L., & Song, Z. (2012) · Physical Review A · doi:10.1103/physreva.85.012111 · Cited by 31
A note on the topological insulator phase in non-Hermitian quantum systems
Ghosh, P. K. (2012) · Journal of Physics Condensed Matter · doi:10.1088/0953-8984/24/14/145302 · Cited by 30
A physical interpretation for the non-Hermitian Hamiltonian
Jin, L., & Song, Z. (2011) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/44/37/375304 · Cited by 30
PT -symmetric non-Hermitian quantum field theories with supersymmetry
Alexandre, J., Ellis, J. R., & Millington, P. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.085015 · Cited by 29
“[4] S.”
BackgroundExceptional points and Bloch oscillations in non-Hermitian lattices with unidirectional hopping
Longhi, S. (2014) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/106/34001 · Cited by 29
Self-citationUnidirectional perfect absorber
Jin, L., Wang, P., & Song, Z. (2016) · Scientific Reports · doi:10.1038/srep32919 · Cited by 26
“φ β β f e f g J f k f ( / ) (2 cos ) (21) i 2 0 1”
Tight-binding lattices with an oscillating imaginary gauge field
Longhi, S. (2016) · Physical Review A · doi:10.1103/physreva.94.022102 · Cited by 23
Self-citationLonghi, S. (2015) · Journal of Optics · doi:10.1088/2040-8978/17/4/045803 · Cited by 25
Self-citationInformation and thermodynamic properties of a non‐Hermitian particle ensemble
Lima, F. C. E., Moreira, A. R. P., & Almeida, C. A. S. (2021) · International Journal of Quantum Chemistry · doi:10.1002/qua.26645 · Cited by 23
Tunable waveguide lattices with nonuniform parity-symmetric tunneling
Joglekar, Y. N., Thompson, C., & Vemuri, G. (2011) · Physical Review A · doi:10.1103/physreva.83.063817 · Cited by 23
Photonic realization of PT -symmetric quantum field theories
Longhi, S., & Valle, G. D. (2012) · Physical Review A · doi:10.1103/physreva.85.012112 · Cited by 23
Self-citationConfluent Crum-Darboux transformations in Dirac Hamiltonians with PT -symmetric Bragg gratings
Correa, F., & Jakubský, V. (2017) · Physical Review A · doi:10.1103/physreva.95.033807 · Cited by 23
Pseudo-Hermitian generalized Dirac oscillators
Dutta, D., Panella, O., & Roy, P. (2013) · Annals of Physics · doi:10.1016/j.aop.2013.01.001 · Cited by 22
Indifferent electromagnetic modes: Bound states and topology
Horsley, S. A. R. (2019) · Physical Review A · doi:10.1103/physreva.100.053819 · Cited by 22
Sensing Applications of PT‐Symmetry in Non‐Hermitian Photonic Systems
Wang, Z., Liang, Z., Hu, J., Zhou, P., Liu, L., Hu, G., Wang, W., & Ye, M. (2024) · Advanced Quantum Technologies · doi:10.1002/qute.202400349 · Cited by 22
Localized Single Frequency Lasing States in a Finite Parity-Time Symmetric Resonator Chain
Phang, S., Vuković, A., Creagh, S. C., Sewell, P., Gradoni, G., & Benson, T. M. (2016) · Scientific Reports · doi:10.1038/srep20499 · Cited by 22
Dynamical Phase Transitions in Quantum Systems
Rotter, I. (2010) · Journal of Modern Physics · doi:10.4236/jmp.2010.15043 · Cited by 21
“The optical realization of relativistic non-Hermitian quantum mechanics is considered in [43].”
BackgroundSong, Q., Dai, S., Han, D., Zhang, Z.-Q., Chan, C. T., & Zi, J. (2021) · Chinese Physics Letters · doi:10.1088/0256-307x/38/8/084203 · Cited by 21
A Precisely Frequency-Tunable Parity-Time-Symmetric Optoelectronic Oscillator
Ding, Q., Wang, M., Zhang, J., et al. (2020) · Journal of Lightwave Technology · doi:10.1109/jlt.2020.3014265 · Cited by 20
“The two supermodes tend to degenerate as Gn increases, then a PT phase transition, which is known as a non-Hermitian singularity [32], or an exceptional point [33], occurs at a critical point that the gain/loss coefficient equals to the coupling coefficient, i.”
BackgroundUniversal routes to spontaneous PT -symmetry breaking in non-Hermitian quantum systems
Schomerus, H. (2011) · Physical Review A · doi:10.1103/physreva.83.030101 · Cited by 18
Beyond the Petermann limit: Prospect of increasing sensor precision near exceptional points
Smith, D. D., Chang, H., Mikhailov, E. E., et al. (2022) · Physical Review A · doi:10.1103/physreva.106.013520 · Cited by 19
Geometric phase and phase diagram for a non-Hermitian quantum XY model
Zhang, X. Z., & Song, Z. (2013) · Physical Review A · doi:10.1103/physreva.88.042108 · Cited by 19
Light scattering in pseudopassive media with uniformly balanced gain and loss
Basiri, A., Vitebsky, I. M., & Kottos, T. (2015) · Physical Review A · doi:10.1103/physreva.91.063843 · Cited by 19
Complex Berry phase and imperfect non-Hermitian phase transitions
Longhi, S., & Feng, L. H. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.107.085122 · Cited by 18
Self-citationRepeatability of measurements: Non-Hermitian observables and quantum Coriolis force
Gardas, B., Deffner, S., & Saxena, A. (2016) · Physical Review A · doi:10.1103/physreva.94.022121 · Cited by 16
Accessing the Exceptional Points in Coupled Fabry–Perot Resonators through Hybrid Integration
Zhu, Y., & Zhu, L. (2018) · ACS Photonics · doi:10.1021/acsphotonics.8b01075 · Cited by 17
Topological corner states in non-Hermitian photonic crystals
Zhou, X., Wu, J., & Wu, Y. (2020) · Optics Communications · doi:10.1016/j.optcom.2020.125653 · Cited by 17
Metadata
- Authors
- Longhi, S.
- Affiliations
- Politecnico di Milano (IT)
- Corresponding author
- Longhi, Stefano
- ORCID
- Longhi, Stefano
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2010-06-30
- Volume
- 105
- Issue
- 1
- Pages
- 013903-013903
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1974418311
- Cited by
- 225
- References
- 41
- Citations by year
- 2026: 8 · 2025: 13 · 2024: 10 · 2023: 9 · 2022: 17 · 2021: 17 · 2020: 18 · 2019: 10 · 2018: 6 · 2017: 10 · 2016: 20 · 2015: 15 · 2014: 18 · 2013: 13 · 2012: 20
- FWCI
- 9.28
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Virtual Community of Pathological Anatomy (University of Castilla La Mancha) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- non-Hermitian quantum mechanics, Dirac equation, PT symmetry breaking, spectral singularities, optical gain and loss, laser oscillation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-23, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-23 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
