Thermodynamics of the classical spin triangle

Schmidt, Heinz‐Jürgen; Schröder, Christian

doi:10.1515/zna-2022-0034

Cited by 3 publications

(5 citation statements)

References 36 publications

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“…In this respect we follow [12], but we have to correct the equation (19) of this reference for the explicit form of s ∼ i (Ω). After some calculations we obtain…”

Section: Definitions and First Resultsmentioning

confidence: 99%

“…Therefore, at first sight this result would be sufficient to obtain the classical limit of thermodynamic functions expressible by derivatives of Z(β). However, although it seems plausible and has been confirmed for various examples, see, e.g., [18,19], it is not mathematically trivial that the limit of Z(β) can be carried over to its derivatives, e. g., to the specific heat c(β) = β 2 ∂ 2 ∂β 2 log Z. The present result opens an alternative way to treat the classical limit of thermodynamic functions that can be expressed by thermal expectation values, such as U (β) = H ∼ := Tr G ∼ (β) H ∼ or c(β) = β 2 U 2 − U 2 .…”

Section: Discussionmentioning

confidence: 97%

“…As these authors emphasize, the map F → A(F ) is many-to-one and hence the "contravariant symbol of A", denoted by G(A), is not unique, but has to be chosen in some way. In contrast, the covariant symbol g(A), defined in (19), is always unique. Following G. Ludwig [6], the common language to describe quantum and classical systems involve the notions of a "statistical duality" consisting of "states", "effects" and the "probability" that an effect will be triggered in a certain state of the system.…”

Section: B Case Of N Spinsmentioning

confidence: 99%

“…The next work that extends these results to the classical limit of time evolution is [12]. This brilliant work has a small flaw due to the wrong factors in the definition of spin-coherent states and in the following equation (19), but as far as I can see this has no consequences for the main results.…”

Section: Introductionmentioning

confidence: 96%

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Classical limit of Gibbs states for quantum spin systems

Schmidt¹

2022

Preprint

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We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number s. It is known that quantum states and observables can be represented by functions defined on the phase space S, which in our case is the N -fold product of unit spheres. Therefore, the classical limit s → ∞ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on S. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree n and thus have to deal with the problem of the limit of double sequences (depending on n and s) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.

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“…In this respect we follow [12], but we have to correct the equation (19) of this reference for the explicit form of s ∼ i (Ω). After some calculations we obtain…”

Section: Definitions and First Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 97%

Section: B Case Of N Spinsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Classical limit of Gibbs states for quantum spin systems

Schmidt¹

2022

Preprint

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“…There are 8 quadratic modes per square, minus two due to the global rotational symmetry of the antiferromagnetic moment. This leaves us with 6 independent quadratic modes, and a contribution to the specific heat per site as c V → [6•(1/2)]/6 = 1/2 as T → 0 [32]. (ii) At the isotropic point, there are three distinct regimes, which share the same physics as the isotropic kagome model at finite temperatures [8] (the similarity even extends to the spin-1/2 quantum case [33]).…”

Section: B Finite-temperature Physicsmentioning

confidence: 99%

Non-Coplanar Magnetic Orders in Classical Square-Kagome Antiferromagnets

Gembé¹,

Schmidt²,

Hickey³

et al. 2023

Preprint

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Motivated by the recent synthesis of a number of Mott insulating square-kagome materials, we explore the rich phenomenology of frustrated magnetism induced by this lattice geometry, also referred to as the squagome or shuriken lattice. On the classical level, square-kagome antiferromagnets are found to exhibit extensive degeneracies, order-by-disorder, and non-coplanar ordering tendencies, which we discuss for an elementary, classical Heisenberg model with nearest-neighbor and cross-plaquette interactions. Having in mind that upon introducing quantum fluctuations non-coplanar order can melt into chiral quantum spin liquids, we provide detailed information on the multitude of non-coplanar orders, including some which break rotational symmetry (possibly leading to nematic quantum orders), as well as a number of (incommensurate) spin spiral phases. Using extensive numerical simulations, we also discuss the thermodynamic signatures of these phases, which often show multi-step thermal ordering. Our comprehensive discussion of the classical square-kagome Heisenberg model, often drawing comparisons to the conventional kagome antiferromagnet, sets the stage for future explorations of quantum analogs of the various phases, either conceptually such as in quantum spin-1/2 generalizations of our model or experimentally such as in the Cu-based candidate materials.

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Thermodynamics of the Spin Square

Schmidt

Schröder

2023

Few-Body Syst

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Small spin systems at the interface between analytical studies and experimental application have been intensively studied in recent decades. The spin ring consisting of four spins with uniform antiferromagnetic Heisenberg interaction is an example of a completely integrable system, in the double sense: quantum mechanical and classical. However, this does not automatically imply that the thermodynamic quantities of the classical system can also be calculated explicitly. In this work, we derive analytical expressions for the density of states, the partition function, specific heat, entropy, and susceptibility. These theoretical results are confirmed by numerical tests. This allows us to compare the quantum mechanical quantities for increasing spin quantum numbers s with their classical counterparts in the classical limit $$s\rightarrow \infty $$ s → ∞ . As expected, a good agreement is obtained, except for the low temperature region. However, this region shrinks with increasing s, so that the classical state variables emerge as envelopes of the quantum mechanical ones.

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Thermodynamics of the classical spin triangle

Cited by 3 publications

References 36 publications

Classical limit of Gibbs states for quantum spin systems

Classical limit of Gibbs states for quantum spin systems

Non-Coplanar Magnetic Orders in Classical Square-Kagome Antiferromagnets

Thermodynamics of the Spin Square

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