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Quantum Phase Transition of the Sub-Ohmic Rotor Model Missouri University of Science and Technology Scholars' Mine Physics Faculty Research & Creative Works Physics 01 Nov 2011 Quantum Phase Transition of the Sub-Ohmic Rotor Model Manal Al-Ali Thomas Vojta Missouri University of Science and Technology, [email protected] Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork Part of the Physics Commons Recommended Citation M. Al-Ali and T. Vojta, "Quantum Phase Transition of the Sub-Ohmic Rotor Model," Physical review B: Condensed matter and materials physics, vol. 84, no. 19, pp. 195136-1-195136-6, American Physical Society (APS), Nov 2011. The definitive version is available at https://doi.org/10.1103/PhysRevB.84.195136 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. PHYSICAL REVIEW B 84, 195136 (2011) Quantum phase transition of the sub-Ohmic rotor model Manal Al-Ali and Thomas Vojta Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA (Received 18 October 2011; published 28 November 2011) We investigate the behavior of an N-component quantum rotor coupled to a bosonic dissipative bath having a sub-Ohmic spectral density J (ω) ∝ ωs with s<1. With increasing dissipation strength, this system undergoes a quantum phase transition from a delocalized phase to a localized phase. We determine the exact critical behavior of this transition in the large-N limit. For 1 >s>1/2, we find nontrivial critical behavior corresponding to an interacting renormalization group fixed point, while we find mean-field behavior for s<1/2. The results agree with those of the corresponding long-range interacting classical model. The quantum-to-classical mapping is therefore valid for the sub-Ohmic rotor model. DOI: 10.1103/PhysRevB.84.195136 PACS number(s): 05.10.Cc, 05.30.Jp I. INTRODUCTION model, the classical counterpart is a one-dimensional Ising model with long-range interactions that decay as 1/r1+s for Quantum phase transitions are abrupt changes in the ground large distances r. In recent years, the applicability of the state properties of a quantum many-particle system that occur quantum-to-classical mapping to the sub-Ohmic spin-boson when a nonthermal control parameter is varied.1 In analogy model has been controversially discussed after numerical to thermal phase transitions, they can be classified as either renormalization group results14 suggested that its critical first-order or continuous transitions. Continuous quantum behavior for s<1/2 deviates from that of the corresponding phase transitions, also called quantum-critical points, are Ising model. While there is now strong evidence15–18 that characterized by large-scale temporal and spatial fluctuations this conclusion is incorrect and that the quantum-to-classical that lead to unconventional behavior in systems ranging from mapping is actually valid, the issue appears to be still not strongly correlated electron materials to ultracold quantum fully settled.19 Moreover, possible failures of the quantum-to- gases (for reviews see, e.g., Refs. 2–7). classical mapping have also been reported for other impurity Impurity quantum phase transitions8 are an interesting models with both Ising20–22 and higher23,24 symmetries, and class of quantum phase transitions at which only the degrees the precise conditions under which it is supposed to hold are of freedom of a finite-size (zero-dimensional) subsystem not resolved. become critical at the transition point. The rest of the system In the present paper, we therefore investigate the large-N (the “bath”) does not undergo a transition. Impurity quantum limit of the sub-Ohmic quantum rotor model. Analogous phase transitions can occur, e.g., in systems composed of to the spin-boson model, this system undergoes a quantum a single quantum spin coupled to an infinite fermionic or phase transition with increasing dissipation strength from a bosonic bath. Fermionic examples include the anisotropic delocalized phase to a localized phase.25,26 We exactly solve Kondo model9 and the pseudogap Kondo model.10 the critical properties of this transition. Our analysis yields The prototypical system involving a bosonic bath is the nontrivial critical behavior corresponding to an interacting dissipative two-state system,11,12 also called the spin-boson renormalization group fixed point for 1 >s>1/2, while we model, which describes a two-level system coupled to a single find mean-field behavior for s<1/2. All critical exponents dissipative bath of harmonic oscillators. Its ground-state phase agree with those of the corresponding long-range interacting diagram depends on the behavior of the bath spectral density classical model,27 implying that the quantum-to-classical ∝ s J (ω) for small frequencies ω. Power-law spectra J (ω) ω mapping is valid. are of particular interest. In the super-Ohmic case (s>1), Our paper is organized as follows. We define the sub-Ohmic the system is in the delocalized (disordered) phase for any rotor model in Sec. II. In Sec. III, we derive its partition dissipation strength. In contrast, for sub-Ohmic dissipation function, and we solve the self-consistent large-N constraint (0 <s<1), there is a continuous quantum phase transition at zero and finite temperatures as well as with and without from a delocalized phase at weak dissipation to a localized an external field. Section IV is devoted to a discussion of 13 (ordered) phase at strong dissipation. In the marginal Ohmic observables and the resulting critical behavior. We conclude = case (s 1), a quantum phase transition exists too, but it is of in Sec. V. Kosterlitz-Thouless type.11,12 The sub-Ohmic spin-boson model has recently attracted II. SUB-OHMIC ROTOR MODEL considerable attention in the context of the so-called quantum- to-classical mapping. This concept relates the critical behavior A quantum rotor can be understood as a point moving of a quantum phase transition in d space dimensions to that on an N-dimensional hypersphere of radius N 1/2. It can be of a classical transition in d + 1 dimensions. The mapping represented by an N-component vector S satisfying S2 = N. is usually established by comparing the order-parameter field The rotor has a momentum P; the position and momentum theories of the transitions: Imaginary time in the quantum components fulfill the usual canonical commutation relations problem plays the role of the extra dimension in the cor- [Sα,Pβ ] = iδαβ . In the large-N limit N →∞, the hard con- responding classical system. In the case of the spin-boson straint S2 = N can be replaced by one for the thermodynamic 1098-0121/2011/84(19)/195136(6)195136-1 ©2011 American Physical Society MANAL AL-ALI AND THOMAS VOJTA PHYSICAL REVIEW B 84, 195136 (2011) average S2=N, because fluctuations of the magnitude of where the dot marks the derivative with respect to imaginary S are suppressed by the central limit theorem. The large-N time τ, and β = 1/T is the inverse temperature. quantum rotor is thus equivalent to the quantum spherical The bath action is quadratic in the qj ; we can thus exactly model of Ref. 28, which is given by the Hamiltonian integrate out the bath modes. After a Fourier transformation from imaginary time τ to Matsubara frequency ωn, this yields = 1 2 + 1 2 2 − + 2 − −A = 0 −A 0 HS 2 P 2 ω0S hS μ(S 1). (1) D[q˜i (ωn)] exp( B ) ZB exp( B ), where ZB is the partition function of the unperturbed bath and Here, S and P represent the position and momentum of one 2 2 rotor component, μ is a Lagrange multiplier enforcing the λj ω 2= A = T n S˜(ω )S˜(−ω ). (8) constraint S 1, and h is an external symmetry-breaking B 2 2 + 2 n n 29 2mj ωj ωn ωj field. ωn j We now couple (every component of) the rotor to a bath The sum over j can be turned into an integral over the spectral 30 of harmonic oscillators. In the conventional linear-coupling density J (ω). Carrying out this integral gives form, the Hamiltonian describing the bath and its coupling to 1 S reads A = 1−s | |s ˜ ˜ − B T αωc ωn S(ωn)S( ωn), (9) 2 2 2 p m λ ωn H = j + j ω2q2 + λ q S + j S2 , (2) B j j j j 2 = 2mj 2 2mj ω with the dimensionless coupling constant α j j A A 2πα¯ cosec(πs/2). Combining S and B yields the with qj , pj , and mj being the position, momentum, and mass effective action of the sub-Ohmic rotor model as of the jth oscillator. The ωj are the oscillator frequencies T 1−s s and λ the coupling strengths between the oscillators and S. Aeff =−βμ + + αω |ωn| S˜(ωn)S˜(−ωn) j 2 c The last term in the bracket is the usual counter term which ωn insures that the dissipation is invariant under translations in − T h˜(ω )S˜(−ω ), (10) S.11 The coupling between the rotor and the bath is completely n n ω characterized by the spectral density n where = ω2 + 2μ is the renormalized distance from quan- 2 0 π λj 2 A J (ω) = δ(ω − ω ), (3) tum criticality. The ωn term in S is subleading in the limit 2 m ω j ω → 0.
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