Classical critical behavior of spin models with long-range interactions

Luijten, Erik; Blöte, Henk W. J.

doi:10.1103/physrevb.56.8945

Cited by 196 publications

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“…We have proven that the naive quantumclassical mapping fails for the sub-ohmic spin-boson model: Using a novel RG expansion around the delocalized fixed point, we have shown that the quantum transition at 0 < s < 1/2 is controlled by an interacting fixed point, whereas the corresponding classical longrange Ising model shows mean-field behavior. Thus, the spin-boson problem for s < 1/2 is equivalent neither to the classical Ising model nor to the corresponding (quantum or classical) O(1) φ 4 theory [10]. In physical terms the inequivalence can be traced back to the different disordered (delocalized) fixed points in the two situations (expansions around these fixed points are suitable to access the critical behavior for small s): In the quantum model the transverse field fully polarizes the spin in x direction (which can be viewed as a "condensate" of spin flips), whereas the high-temperature limit of the classical Ising model is simply incoherently disordered.…”

Section: Spin-boson Model: Numerical Resultsmentioning

confidence: 99%

“…As proven by Dyson [14] this model displays a phase transition for 0 < s ≤ 1. Both analytical arguments, based on the equivalence to a O(1) φ 4 theory [9], and extensive numerical simulations [10] show that the upper-critical dimension for the d-dimensional long-range Ising model is d + c = 2s, i.e., in d = 1 the transition obeys non-trivial critical behavior for 1/2 < s < 1. In contrast, mean-field behavior obtains for 0 < s < 1/2, with exponents [9,10] …”

mentioning

confidence: 99%

“…This analytical result is supported by NRG calculations. In contrast, the transition in the classical Ising model is known to display mean-field behavior for 0 < s < 1/2 [9,10]. Scaling and critical exponents.…”

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confidence: 99%

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Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping

2005

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The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(ω) ∝ ω s . Using an ǫ expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for 0 < s < 1/2, controlled by a non-interacting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model. Low-energy theories for certain classes of quantum phase transitions in clean systems with d spatial dimensions are known to be equivalent to the ones of classical phase transitions in (d + z) dimensions, where z ist the dynamical exponent of the quantum transition [1]. This mapping is usually established in a path integral formulation of the effective action for the order parameter, where imaginary time in the quantum problem takes the role of z additional space dimensions in the classical counterpart. The tuning parameter for the phase transition, being the ratio of certain coupling constants in the quantum problem (where T is fixed to zero), becomes temperature for the classical transition. For the quantum Ising model, where the transverse field can drive the system into a disordered phase at T = 0, the quantumclassical equivalence in the scaling limit can be explicitly shown using transfer matrix techniques [1]. While this formal proof is only applicable for short-range interactions in time direction, it is believed that it also holds for long-range interactions, which can arise upon integrating out gapless degrees of freedom coupled to the order parameter. (Counter-examples are phase transitions in itinerant magnets, where the elimination of low-energy fermions produces non-analyticities in the resulting order parameter field theory [2].) A paradigmatic example is the spin-boson model [3,4], where an Ising spin (i.e. a generic two-level system) is coupled to a bath of harmonic oscillators: eliminating the bath variables leads to a retarded self-interaction for the local spin degree of freedom, which decays as 1/τ 2 in the well-studied case of ohmic damping. Interestingly, the same model is obtained as the low-energy limit of the anisotropic Kondo model which describes a spin-1/2 magnetic impurity coupled to a gas of conduction electrons [5,6].The purpose of this paper is to point out that the naive quantum-classical mapping can fail for long-ranged interactions in imaginary time even for the simplest case of (0 + 1) dimensions and Ising symmetry. We shall explicitly prove this failure for the sub-ohmic spin-boson model, by showing that the phase transitions in the quantum problem and in the corresponding classical long-range Ising model fall in di...

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Section: Spin-boson Model: Numerical Resultsmentioning

confidence: 99%

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confidence: 99%

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Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping

2005

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“…The functional form (4.5) also coincides with the large-L behaviour of the free energy for N = 4 spin models with long-range interactions [31] (see also [26,47]). …”

Section: A Modified Finite Size Scaling Hypothesismentioning

confidence: 99%

“…In particular, recent analyses of systems at the upper critical dimension have adopted a finite-sized approach [21,29,31,32,33,34,35,36].…”

Section: Introductionmentioning

confidence: 99%

Finite size scaling for O(N) φ4-theory at the upper critical dimension

Kenna

2004

Nuclear Physics B

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A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N ) φ 4 -theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ 4 4 -theories for all N .

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Localized magnetic moments in a Dirac semimetal as a spin model with long‐range interactions

Kogan

Kaveh

2015

Physica Status Solidi (b)

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Abstractauthoren We connect between the problem of thermodynamics of localized magnetic moments in a Dirac semimetal, the interaction with relativistic electrons leading to the effective ferromagnetic exchange between the moments, and the existing theories dealing with long‐range exchange interaction. We point out that the results of high‐temperature expansion for the free energy of a dilute ensemble of magnetic impurities in the semimetal performed by V. Cheianov et al. [Phys. Rev. B 86, 054424 (2012)] give an indication to the existence of a new disordered fixed point in such model.

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Classical critical behavior of spin models with long-range interactions

Cited by 196 publications

References 65 publications

Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping

Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping

Finite size scaling for O(N) φ4-theory at the upper critical dimension

Localized magnetic moments in a Dirac semimetal as a spin model with long‐range interactions

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