PT symmetry on the lattice: the quantum group invariant<i>XXZ</i>spin chain

Korff, Christian; Weston, Robert

doi:10.1088/1751-8113/40/30/016

Cited by 73 publications

(97 citation statements)

References 33 publications

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“…This Hamiltonian might look very different form the initial Hamiltonian (1) and there is a priori no reason to expect that it will have only nearest neighbour bulk interaction as the similarity transformation is highly nonlocal. Preliminary computations for the case when q = exp(iπ/2), not discussed in [1], show that (50) indeed contains nonlocal bulk interactions. Namely consider the non-Hermitian Hamiltonian …”

Section: Discussionmentioning

confidence: 97%

“…For r ∈ (N, ∞) such a reduction proved to be unnecessary. In both cases, (i) and (ii), we found in [1] explicit algebraic expressions for η utilising concepts related to P T -symmetry (see e.g. [7,8] for recent reviews) and the quantum analogue of Schur-Weyl duality [9].…”

Section: Introductionmentioning

confidence: 99%

“…In order that the reader can fully appreciate the main conjecture of this article we briefly review the results obtained for the quasi-Hermiticty operator η in the previous paper [1].…”

Section: Introductionmentioning

confidence: 99%

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Turning the quantum group invariantXXZspin-chain Hermitian: a conjecture on the invariant product

Korff¹

2008

J. Phys. A: Math. Theor.

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This is a continuation of a previous joint work with Robert Weston on the quantum group invariant XXZ spin-chain (math-ph/0703085). The previous results on quasiHermiticity of this integrable model are briefly reviewed and then connected with a new construction of an inner product with respect to which the Hamiltonian and the representation of the Temperley-Lieb algebra become Hermitian. The approach is purely algebraic, one starts with the definition of a positive functional over the Temperley-Lieb algebra whose values can be computed graphically. Employing the Gel'fand-NaimarkSegal (GNS) construction for C * -algebras a self-adjoint representation of the TemperleyLieb algebra is constructed when the deformation parameter q lies in a special section of the unit circle. The main conjecture of the paper is the unitary equivalence of this GNS representation with the representation obtained in the previous paper employing the ideas of PT-symmetry and quasi-Hermiticity. An explicit example is presented.c.korff@maths.gla.ac.uk

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Section: Discussionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Turning the quantum group invariantXXZspin-chain Hermitian: a conjecture on the invariant product

Korff¹

2008

J. Phys. A: Math. Theor.

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“…The P T -symmetry and quasi-Hermiticity of the quantum group invariant XXZ spin-chain for higher roots of unity q = exp(iπ/r), r > 2 have been discussed in another paper [9].…”

Section: Long Range Bulk Interaction From Non-hermitian Boundary Fieldsmentioning

confidence: 99%

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

Korff¹

2008

J. Phys. A: Math. Theor.

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The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by explicit construction of a new inner product employing a "quasi-fermion" algebra in momentum space where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c = 1, while the latter has been connected to a logarithmic CFT with central charge c = −2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry U (gl(1|1)) can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.

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“…Such continuum or lattice Hamiltonians [7][8][9][10] usually consist of a Hermitian kinetic energy part,K =K † , and a non-Hermitian, PT -symmetric potential part,V = PTV PT =V † . Although it is not Hermitian H PT has purely real eigenvalues E = E * over a range of parameters, and its eigenfunctions are simultaneous eigenfunctions of the combined PT -operation; this range is defined as the PT -symmetric region.…”

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confidence: 99%

Origin of maximal symmetry breaking in evenPT-symmetric lattices

Joglekar

Barnett

2011

Phys. Rev. A

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By investigating a parity and time-reversal (PT ) symmetric, N -site lattice with impurities ±iγ and hopping amplitudes t0(t b ) for regions outside (between) the impurity locations, we probe the origin of maximal PT -symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, γc = t b , and the simultaneous emergence of N complex eigenvalues is a robust feature of any PT -symmetric hopping profile. Our results show that the threshold strength γc can be widely tuned by a small change in the global profile of the lattice, and thus have experimental implications.Introduction: The discovery of "complex extension of quantum mechanics" by Bender and coworkers [1,2] set in motion extensive mathematical [3][4][5] and theoretical investigations [6] of non-Hermitian Hamiltonians H PT =K +V that are symmetric with respect to combined parity (P) and time-reversal (T ) operations. Such continuum or lattice Hamiltonians [7-10] usually consist of a Hermitian kinetic energy part,K =K † , and a non-Hermitian, PT -symmetric potential part,V = PTV PT =V † . Although it is not Hermitian H PT has purely real eigenvalues E = E * over a range of parameters, and its eigenfunctions are simultaneous eigenfunctions of the combined PT -operation; this range is defined as the PT -symmetric region. The breaking of PT -symmetry, along with the attendant non-reciprocal behavior, was recently observed in two coupled optical waveguides [11,12] and has ignited further interest in PT -symmetric lattice models. These evanescently coupled waveguides provide an excellent realization [13] of an ideal, one-dimensional lattice with tunable hopping [14], disorder [15], and non-Hermitian, on-site, impurity potentials [16,17].Recently nonuniform lattices with site-dependent hop-α/2 and a pair of imaginary impurities ±iγ at positions (m,m) have been extensively explored [17][18][19][20], wherem = N + 1 − m and N 1 is the number of lattice sites. The PT -symmetric phase in such a lattice is robust when α ≥ 0, the loss and gain impurities ±iγ are closest to each other, and γ ≤ γ c where the critical impurity strength is proportional to the bandwidth of the clean lattice, γ c ∝ 4t 0 (N/2) α . For a generic impurity position m, when the impurity strength γ > γ c (m) increases the number of complex eigenvalues increases sequentially from four to N − 1 when N is odd and to N when it is even. In an exceptional contrast, when m = N/2 -nearest neighbor impurities on an even lattice -all eigenvalues simultaneously become complex at the onset of PT -symmetry breaking. This maximal symmetry breaking is accompanied by unique signatures in the time-evolution of a wavepacket [20].

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PT symmetry on the lattice: the quantum group invariantXXZspin chain

Cited by 73 publications

References 33 publications

Turning the quantum group invariantXXZspin-chain Hermitian: a conjecture on the invariant product

Turning the quantum group invariantXXZspin-chain Hermitian: a conjecture on the invariant product

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

Origin of maximal symmetry breaking in evenPT-symmetric lattices

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