phase transition in multidimensional quantum systems

Non-Hermitian -symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken symmetry in which the eigenvalues are all real, and (ii) a region of broken symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled -symmetric Hamiltonians, , , , and are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at g ≈ 0.1, g ≈ 0.04, g ≈ 0.1 and g ≈ 0.05. These results suggest that the phase transition is a robust phenomenon not limited to systems having one degree of freedom.

[1]  H. Harney,et al.  PT symmetry and spontaneous symmetry breaking in a microwave billiard. , 2011, Physical review letters.

[2]  Tsampikos Kottos,et al.  Experimental study of active LRC circuits with PT symmetries , 2011, 1109.2913.

[3]  Yeshaiahu Fainman,et al.  Nonreciprocal Light Propagation in a Silicon Photonic Circuit , 2011, Science.

[4]  Hui Cao,et al.  Unidirectional invisibility induced by PT-symmetric periodic structures. , 2011, Physical review letters.

[5]  Claudio Bonati,et al.  Armchair graphene nanoribbons: PT-symmetry breaking and exceptional points without dissipation , 2011, 1102.2129.

[6]  Li Ge,et al.  PT-symmetry breaking and laser-absorber modes in optical scattering systems. , 2010, Physical review letters.

[7]  Oleg N. Kirillov,et al.  Exceptional points in a microwave billiard with time-reversal invariance violation. , 2010, Physical review letters.

[8]  Carl M. Bender,et al.  Extending $\mathcal{P}\mathcal{T}$ Symmetry from Heisenberg Algebra to E2 Algebra , 2010, 1009.3236.

[9]  K. Zhao,et al.  Enhanced magnetic resonance signal of spin-polarized Rb atoms near surfaces of coated cells , 2010 .

[10]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[11]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[12]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[13]  R. El-Ganainy,et al.  Optical solitons in PT periodic potentials , 2008, 2008 Conference on Lasers and Electro-Optics and 2008 Conference on Quantum Electronics and Laser Science.

[14]  Carl M. Bender,et al.  Interactions of Hermitian and non-Hermitian Hamiltonians , 2007, 0709.3605.

[15]  J. Rubinstein,et al.  Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. , 2007, Physical review letters.

[16]  Carl M. Bender,et al.  Making sense of non-Hermitian Hamiltonians , 2007, hep-th/0703096.

[17]  C. Bender,et al.  PT-symmetric quantum electrodynamics , 2005, hep-th/0501180.

[18]  C. Bender.  Introduction to 𝒫𝒯-symmetric quantum theory , 2005, quant-ph/0501052.

[19]  Carl M. Bender,et al.  The Script C operator in Script PScript T-symmetric quantum theories , 2004 .

[20]  W. Heiss,et al.  Exceptional Points – Their Universal Occurrence and Their Physical Significance , 2004 .

[21]  C. Bender,et al.  Scalar quantum field theory with a complex cubic interaction. , 2004, Physical review letters.

[22]  Dorje C. Brody,et al.  Must a Hamiltonian be Hermitian , 2003, hep-th/0303005.

[23]  C. Bender,et al.  Complex extension of quantum mechanics. , 2002, Physical review letters.

[24]  R. Tateo,et al.  Bethe Ansatz equations , and reality properties in PT-symmetric quantum mechanics , 2022 .

[25]  Michael V Berry,et al.  Complex WKB analysis of energy-level degeneracies of non-Hermitian Hamiltonians , 2001 .

[26]  R. Tateo,et al.  The ODE/IM correspondence , 2001, hep-th/0703066.

[27]  C. Bender,et al.  PT-symmetric quantum mechanics , 1998, 2312.17386.

[28]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[29]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[30]  Danny C. Sorensen,et al.  Deflation Techniques for an Implicitly Restarted Arnoldi Iteration , 1996, SIAM J. Matrix Anal. Appl..

[31]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[32]  Tai Tsun Wu,et al.  Analytic Structure of Energy Levels in a Field-Theory Model , 1968 .