<i>PT</i>symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields

Korff, Christian

doi:10.1088/1751-8113/41/29/295206

Cited by 25 publications

(29 citation statements)

References 41 publications

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“…However, they serve to treat non-Hermitian Hamiltonians of a different kind, such as obvious modifications of H(λ, κ) and also to allow for alternative treatments of non-Hermitian spin chains, such as the XXZ-spin-chain in a magnetic field [19] H XXZ = 1 2 10) with ∆ ± = (q ± q −1 )/2 previously studied in [17,18]. Obviously when q / ∈ R this Hamiltonian is non-Hermitian, but we observe that it is PT -symmetric when using any of the parity operators defined in (2.8) and keeping T to be the usual complex conjugation…”

Section: Pt -Symmetry For Spin Chainsmentioning

confidence: 99%

“…A closed expression for the commutator [h 0 , q n ] in terms of commutators [q m , h 1 ] with m < n was derived in [18]. Perturbation theory has been carried out in the past for various non-Hermitian models, e.g.…”

Section: Perturbation Theorymentioning

confidence: 99%

“…In this case the Hamiltonian (1.1) acquires the simple form of a non-Hermitian 4 × 4-matrix. In order to make notations clear, we will write this matrix here in the various notations introduced so far, 18) where the first line shows the most explicit way of writing the Hamiltonian, the second line shows a simplified version, were the tensor products are omitted and absorbed into the σs as specified after (1.1). The last line uses the notation introduced in (4.1).…”

Section: The N = 2 Case: Two Sitesmentioning

confidence: 99%

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A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

Castro-Alvaredo¹,

Fring²

2009

J. Phys. A: Math. Theor.

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Citation: Castro-Alvaredo, O. and Fring, A. (2009). A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field. Journal of Physics A: Mathematical and Theoretical, 42(46), doi: 10.1088/1751-8113/42/46/465211 This is the unspecified version of the paper.This version of the publication may differ from the final published version. Castro-alvaredo@city.ac.uk, A.Fring@city.ac.uk Abstract: We investigate a lattice version of the Yang-Lee model which is characterized by a non-Hermitian quantum spin chain Hamiltonian. We propose a new way to implement PT -symmetry on the lattice, which serves to guarantee the reality of the spectrum in certain regions of values of the coupling constants. In that region of unbroken PT -symmetry we construct a Dyson map, a metric operator and find the Hermitian counterpart of the Hamiltonian for small values of the number of sites, both exactly and perturbatively. Besides the standard perturbation theory about the Hermitian part of the Hamiltonian, we also carry out an expansion in the second coupling constant of the model. PermanentOur constructions turns out to be unique with the sole assumption that the Dyson map is Hermitian. Finally we compute the magnetization of the chain in the z and x direction.

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Section: Pt -Symmetry For Spin Chainsmentioning

confidence: 99%

Section: Perturbation Theorymentioning

confidence: 99%

Section: The N = 2 Case: Two Sitesmentioning

confidence: 99%

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A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

Castro-Alvaredo¹,

Fring²

2009

J. Phys. A: Math. Theor.

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“…Some applications associated with PTsymmetric system have been explored including unidirectional transport [16,17] and single-mode lasers [18,19]. Also PT -symmetric non-Hermitian quantum spin models have been designed [20][21][22][23]25], which can well capture the essence of many discrete models. For a (or anti) PT -symmetric non-Hermitian Hamiltonian [24] which is invariant under the combined action of the P and T operations, (or anti) PT spontaneous symmetry breaking (SSB) occurs with adjustable parameters accompanied by a real-to-complex spectral phase transition.…”

Section: Introductionmentioning

confidence: 99%

Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with $\mathcal{RT}$ symmetry

Wang¹,

Yang²,

Guo³

et al. 2020

EPL

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In this paper, based on a one dimensional non-Hermitian spin model with RT -invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) PT -symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists spontaneous (anti) PT -symmetry breaking for the topological degenerate ground states with tunable parameters in external fields. We also found that even a very tiny imaginary external field applied will drive PT phase transition.

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“…[7][8][9] The understanding of such systems on physical bases seems to be crucial, given that the meaning of non-Hermitian Hamiltonians is rather unclear. Mostafazadeh showed that a necessary and sufficient condition for the reality of all the eigenvalues is the existence of an "equivalent Hermitian counterpart" 10 that can be built starting from the eigenfunctions of the Hermitian conjugate of the initial Hamiltonian.…”

mentioning

confidence: 99%

SpontaneousPTsymmetry breaking and quantum phase transitions in dimerized spin chains

Giorgi

2010

Phys. Rev. B

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The occurrence of parity-time reversal ͑PT͒ symmetry breaking is discussed in a non-Hermitian spin chain. The Hermiticity of the model is broken by the presence of an alternating, imaginary, transverse magnetic field. A full real spectrum, which occurs if and only if all the eigenvectors are PT symmetric, can appear only in presence of dimerization, i.e., only if the hopping amplitudes between nearest-neighbor spins assume alternate values along the chain. In order to make a connection between such system and the Hermitian world, we study the critical magnetic properties of the model and look for the conditions that would allow to observe the same phase diagram in the absence of the imaginary field. Such procedure amounts to renormalizing the spin-spin coupling amplitudes. DOI: 10.1103/PhysRevB.82.052404 PACS number͑s͒: 75.10.Jm, 03.65.Ϫw, 11.30.Er, 64.70.Tg Since the paper of Bender and Boettcher, 1 it is known that non-Hermitian Hamiltonians can display a real spectrum if they are invariant under the joint action of parity ͑P͒ and time reversal ͑T͒ symmetry. The parity operator P performs spatial reflection and its action consists of changing the sign of both position and momentum, whereas the antilinear timereversal operator T maps p in −p and the imaginary part i in −i. Non-Hermitian Hamiltonians that are PT symmetric generate a complex extension to the Hermitian quantum mechanics.2,3 The axiom that forces the Hamiltonian H to be Hermitian is in fact introduced in order to guarantee the existence of a stable ground state and unitary evolutions. However, these two basic requirements are still satisfied in the presence of PT symmetry. 4 On the other hand, because of the antilinear character of the parity-time reversal operator, the condition ͓H , PT͔ =0 is not sufficient to have a PT-symmetric system. It is in fact possible to observe spontaneous symmetry breaking in some of the eigenstates of H, associated to the appearance of complex conjugate eigenvalues. The prototypical model introduced in the literature 1 is that of system obeying the Hamiltonian p 2 − ͑ix͒ N . Its spectrum turns out to be real only if N Ն 2. As N decreases, the number of complex eigenvalues increases, and for N → 1 + , no real roots are detected. For N Ͻ 2, the PT symmetry is spontaneously broken.The search for non-Hermitian quantum models in discrete systems has led to study tight-binding particle models 5,6 and spin chains. [7][8][9] The understanding of such systems on physical bases seems to be crucial, given that the meaning of non-Hermitian Hamiltonians is rather unclear. Mostafazadeh showed that a necessary and sufficient condition for the reality of all the eigenvalues is the existence of an "equivalent Hermitian counterpart" 10 that can be built starting from the eigenfunctions of the Hermitian conjugate of the initial Hamiltonian.It is our aim to study spontaneous breaking of the PT symmetry in an exactly solvable model of dimerized spin chain. The non-Hermitian term introduced here is a staggered magnetic field affecting ...

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PTsymmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields

Cited by 25 publications

References 41 publications

A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with $\mathcal{RT}$ symmetry

SpontaneousPTsymmetry breaking and quantum phase transitions in dimerized spin chains

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