arXiv:0806.3529v1 [quant-ph] 22 Jun 2008 Geometric phases and quantum phase transitions in open systems Alexander I. Nesterov ∗ Departamento de F´ısica, CUCEI, Universidad de Guadalajara, Av. Revoluci´on 1500, Guadalajara, CP 44420, Jalisco, M´exico S. G. Ovchinnikov † L. V. Kirensky Institute of Physics, SB RAS, 660036, Krasnoyarsk, Russia and Siberian Federal University, 660041, Krasnoyarsk, Russia (Dated: October 25, 2018) The relationship between quantum phase transition and complex geometric phase for open quan- tum system governed by the non-Hermitian effective Hamiltonian with the accidental crossing of the eigenvalues is established. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that related quantum phase transition is of the first order. PACS numbers: 03.65.Vf, 14.80.Hv, 03.65.-w, 03.67.-a Keywords: Berry phase, Dirac monopole, complex geometric phase, quantum phase transition, bifurcation Quantum phase transition (QPT) is characterized by qualitative changes of the ground state of many body system and occur at the zero temperature. QPT being purely quantum phenomena driven by quantum fluctu- ations is associated with the energy level crossing and implies the lost of analyticity in the energy spectrum at the critical points [1]. A first order QPT is determined by a discontinuity in the first derivative of the ground state energy. A second order QPT means that the first derivative is continuous, while the second derivative has either a finite discontinuity or divergence at the criti- cal point. Since QPT is accomplished by changing some parameter in the Hamiltonian of the system, but not the temperature, its description in the standard framework of the Landau-Ginzburg theory of phase transitions failed, and identification of an order parameter is still an open problem [2]. In this connection, an issue of a great in- terest is recently established relationship between geo- metric phases and quantum phase transitions [3, 4, 5, 6]. This relation is expected since the geometric phase as- sociated with the energy levels crossings has a peculiar behavior near the degeneracy point. It is supposed that the geometric phase, being a measure of the curvature of the Hilbert space, is able to capture drastic changes in the properties of the ground states in presence of QPT [4, 5, 6, 7]. In this Rapid Communication we analyze relation be- tween the geometric phase and QPT in an open quantum system governed by non-Hermitian Hamiltonian. We found that QPT is closely connected with the geometric phase and the latter may be considered as an universal order parameter for description of QPT. Studying the dissipative one-dimensional Ising model in a transverse magnetic field we demonstrated that the QPT being of * Electronic address: nesterov@cencar.udg.mx † Electronic address: sgo@iph.krasn.ru the second order in absence of dissipation is of the first order QPT for the open system. Degeneracy points and geometric phase.– We consider an open quantum mechanical system which together with its environment forms a closed system. The description of the such systems by effective non-Hermitian Hamilto- nian is well known beginning with the classical papers by Weisskopf and Wigner on the metastable states [8, 9][30]. For the Hermitian Hamiltonian coalescence of eigenval- ues results in different eigenvectors, and related degen- eracy referred to as ‘conical intersection’ is known also as ‘diabolic point’. However, in a quantum mechanical system governed by non-Hermitian Hamiltonian not only merging of eigenvalues of the Hamiltonian but the asso- ciated eigenvectors can be occurred as well. The point of coalescing is called an “exceptional point”. At the lat- ter the eigenvectors merge forming a Jordan block (for review and references see e.g. [10, 11]). In the context of the Berry phase the diabolic point is associated with ‘fictitious magnetic monopole’ as fol- lows. Assume that for adiabatic driving quantum system two energy levels may cross. Then the energy surfaces form the sheets of a double cone, and its apex is called a “diabolic point” [12]. Since for generic Hermitian Hamil- tonian the codimension of the diabolic point is three, it can be characterized by three parameters R =(X,Y,Z ). The eigenstates |n, R〉 give rise to the Berry’s connec- tion defined by A n (R)= i〈n, R|∇ R |n, R〉, and the cur- vature B n = ∇ R × A n associated with A n is the field strength of ‘magnetic’ monopole located at the diabolic point [13, 14]. The Berry phase γ n =  C A n · dR is interpreted as a holonomy associated with the parallel transport along a circuit C [15]. Similar treatment of the non-Hermitian Hamiltonian yields the ‘fictitious complex monopole’ located at the exceptional point [16]. For the first time, the extension of the Berry phase to the non-Hermitian systems has been done by Gar- rison and Wright as follows [17]. Let an adjoint pair {|Ψ(t)〉, 〈  Ψ(t)|} be a solution of the time dependent