Abstract We establish the exponential clustering of correlation functions for the high-temperature Gibbs states of Bose–Hubbard type models. To overcome the technical difficulties arising from the unboundedness of bosonic operators, we develop the interaction-picture cluster-expansion technique. This method also allows us to systematically bound the moments of the local particle number. This result provides an analytical justification for the low-boson-density condition frequently assumed in the study of bosonic many-body systems. As direct mathematical consequences of the clustering property, we derive a uniform upper bound on the specific heat density and establish a bosonic thermal area law with improved temperature dependence.
A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals.The main aspects of this approach are described on an introductory level.Then a mechanism of the stabilization of this model by quantum effects is suggested.In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given.It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough.It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.
A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.
Recent applications of the coupled-cluster method to the (${\ensuremath{\varphi}}^{4}$
)2 quantum field theory showed the necessity of a detailed analysis of its convergence behavior. The anharmonic oscillator has always served as a good test case for different approximation schemes. We treat the model both in the maximum-overlap condition and in the Hartree approximation. The ground-state energy is reproduced very well for all values of the coupling strength and already for a low-order truncation scheme. The expansion in correlation amplitudes can be carried out in extremely high order and shows a divergent tendency of the amplitudes. Introducing temperature dependence allows us to select the stable ground state out of a variety of solutions of the hierarchy of equations.
A model of quantum particles performing D -dimensional anharmonic oscillations \naround their equilibrium positions which form the d -dimensional \nsimple cubic lattice Zd is considered. The model undergoes a structural \nphase transition when the fluctuations of displacements of particles become \nmacroscopic. This phenomenon is described by susceptibilities depending \non Matsubara frequencies ωn , n ∈ Z . We prove two theorems \nconcerning the thermodynamic limits of these susceptibilities. The first theorem \nstates that the susceptibilities with nonzero ωn remain bounded at all \ntemperatures, which means that the macroscopic fluctuations in the model \nare always non-quantum. The second theorem gives a sufficient condition \nfor the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded \nat all temperatures. This condition involves the particle mass, the anharmonicity \nparameters and the interaction intensity. The physical meaning of \nthis result is that, for all D and all values of the temperature, strong quantum \neffects suppress critical points and the long range order. The proof \nis performed in the approach where the susceptibilities are represented as \nfunctional integrals. A brief description of the main features of this approach \nis delivered.
Preparing thermal (Gibbs) states is a common task in physics and computer science. Recent algorithms mimic cooling via system-bath coupling, where the cost is determined by mixing time, akin to classical Metropolis-like algorithms. However, few methods exist to demonstrate slow mixing in quantum systems, unlike the well-established classical tools for systems like the Ising model and constraint satisfaction problems. We present a quantum generalization of these tools through a generic bottleneck lemma that implies slow mixing in quantum systems. This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles and quantified either through Bohr spectrum jumps or operator locality. Using our bottleneck lemma, we establish unconditional lower bounds on the mixing times of Gibbs samplers for several families of Hamiltonians at low temperatures. For classical Hamiltonians with mixing time lower bounds
Tmix=exp[Ω(nα)], we prove that quantum Gibbs samplers also have
Tmix=exp[Ω(nα)]. This applies to models like random
K-SAT instances and spin glasses. For stabilizer Hamiltonians, we provide a concise proof of exponential lower bounds
Tmix=exp[Ω(n)] on mixing times of good
n-qubit stabilizer codes at low constant temperature. Finally, we consider constant-degree classical Hamiltonians and show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques. We show generic results for models with linear free energy barriers, and we demonstrate that our techniques extend to models with sublinear free energy barriers by proving
Tmix=exp[n1/2−o(1)] for the ferromagnetic 2D transverse field Ising model.