Exact solution of a non-Hermitian PT\mathscr{PT}-symmetric Heisenberg spin chain

We construct the exact solution of a non-Hermitian $\mathscr{PT}$-symmetric isotropic Heisenberg spin chain with integrable boundary fields. We find that the system exhibits two types of phases we refer to as $A$ and $B$ phases. In the $B$ type phase, the $\mathscr{PT}$- symmetry remains unbroken and it consists of eigenstates with only real energies, whereas the $A$ type phase contains a $\mathscr{PT}$-symmetry broken sector comprised of eigenstates with only complex energies and a sector of unbroken $\mathscr{PT}$-symmetry with eigenstates of real energies. The $\mathscr{PT}$-symmetry broken sector consists of pairs of eigenstates whose energies are complex conjugates of each other. The existence of two sectors in the $A$ type phase is associated with the exponentially localized bound states at the edges with complex energies which are described by boundary strings. We find that both $A$ and $B$ type phases can be further divided into sub-phases which exhibit different ground states. We also compute the bound state wavefunction in one magnon sector and find that as the imaginary value of the boundary parameter is increased, the exponentially localized wavefunction broadens thereby protruding more into the bulk, which indicates that exponentially localized bound states may not be stabilized for large imaginary values of the boundary parameter.Comment: 28 pages and 4 figure

Dissipation driven phase transition in the non-Hermitian Kondo model

Non-Hermitian Hamiltonians capture several aspects of open quantum systems, such as dissipation of energy and non-unitary evolution. An example is an optical lattice where the inelastic scattering between the two orbital mobile atoms in their ground state and the atom in a metastable excited state trapped at a particular site and acting as an impurity, results in the two body losses. It was shown in \cite{nakagawa2018non} that this effect is captured by the non-Hermitian Kondo model. which was shown to exhibit two phases depending on the strength of losses. When the losses are weak, the system exhibits the Kondo phase and when the losses are stronger, the system was shown to exhibit the unscreened phase where the Kondo effect ceases to exist, and the impurity is left unscreened. We re-examined this model using the Bethe Ansatz and found that in addition to the above two phases, the system exhibits a novel $\widetilde{YSR}$ phase which is present between the Kondo and the unscreened phases. The model is characterized by two renormalization group invariants, a generalized Kondo temperature $T_K$ and a parameter `$\alpha$' that measures the strength of the loss. The Kondo phase occurs when the losses are weak which corresponds to $0<\alpha<\pi/2$. As $\alpha$ approaches $\pi/2$, the Kondo cloud shrinks resulting in the formation of a single particle bound state which screens the impurity in the ground state between $\pi/2<\alpha<\pi$. As $\alpha$ increases, the impurity is unscreened in the ground state but can be screened by the localized bound state for $\pi<\alpha<3\pi/2$. When $\alpha>3\pi/2$, one enters the unscreened phase where the impurity cannot be screened. We argue that in addition to the energetics, the system displays different time scales associated with the losses across $\alpha=\pi/2$, resulting in a phase transition driven by the dissipation in the system.Comment: 6 Pages, 2 Figures, 1 Appendix, due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil

Kondo effect in the isotropic Heisenberg spin chain

We investigate the boundary effects that arise when spin-$\frac{1}{2}$ impurities interact with the edges of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain through spin exchange interactions. We consider both cases when the couplings are ferromagnetic or anti-ferromagnetic. We find that in the case of antiferromagnetic interaction, when the impurity coupling strength is much weaker than that in the bulk, the impurity is screened in the ground state via the Kondo effect. The Kondo phase is characterized by the Lorentzian density of states and dynamically generated Kondo temperature $T_K$. As the impurity coupling strength increases, $T_K$ increases until it reaches its maximum value $T_0=2\pi J$ which is the maximum energy carried by a single spinon. When the impurity coupling strength is increased further, we enter another phase, the bound mode phase, where the impurity is screened in the ground state by a single particle bound mode exponentially localized at the edge to which the impurity is coupled. We find that the impurity can be unscreened by removing the bound mode. There exists a boundary eigenstate phase transition between the Kondo and the bound-mode phases, a transition which is characterized by the change in the number of towers of the Hilbert space. The transition also manifests itself in ground state quantities like local impurity density of states and the local impurity magnetization. When the impurity coupling is ferromagnetic, the impurity is unscreened in the ground state; however, when the absolute value of the ratio of the impurity and bulk coupling strengths is greater than $\frac{4}{5}$, the impurity can be screened by adding a bound mode that costs energy greater than $T_0$. When two impurities are considered, the phases exhibited by each impurity remain unchanged in the thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.Comment: 23 pages, 7 figures; due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil