Gibbs States Research Articles

Hamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physics

We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a \(n\)-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval \(0.1\!-\!2\:\!\text{K}\), which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.

Dynamic phases induced by two-level system defects on driven qubits

Recent experimental evidences point to two-level defects, located in the oxides and on the interfaces of the Josephson junctions, as the major constituents of decoherence in superconducting qubits. How these defects affect the qubit evolution with the presence of external driving is less well understood since the semiclassical qubit-field coupling renders the Jaynes–Cummings model for qubit-defect coupling undiagonalizable. We analyze the decoherence dynamics in the continuous coherent state space induced by the driving and solve the master equation endowed with an extra decay-cladded driving term via a Fokker–Planck equation. The solutions for diffusion propagators as Gaussian distributions show four distinct dynamic phases: four types of convergence paths to limit cycles of varying radius by the distribution mean, which are determined by the competing external driving and the defect decays. The qubit trajectory resulted from these solutions is a super-Poissonian over displaced Fock states, which reduces to a Gibbs state of effective temperature decided by the defect at zero driving limit. Furthermore, the Poincare map shows the dependence of the rate of convergence on the initial state. In other words, the qubit evolution can serve as an indicator of the defect coupling strength through the variation of the driving strength as a parameter.

Thermal Area Law for Lattice Bosons

A physical system is said to satisfy a thermal area law if the mutual information between two adjacent regions in the Gibbs state is controlled by the area of their boundary. Lattice bosons have recently gained significant interest because they can be precisely tuned in experiments and bosonic codes can be employed in quantum error correction to circumvent classical no-go theorems. However, the proofs of many basic information-theoretic inequalities such as the thermal area law break down for bosons because their interactions are unbounded. Here, we rigorously derive a thermal area law for a class of bosonic Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard model. The main idea to go beyond bounded interactions is to introduce a quasi-free reference state with artificially decreased chemical potential by means of a double Peierls-Bogoliubov estimate.

A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in [1]. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.

DLR–KMS correspondence on lattice spin systems

The Dobrushin–Lanford–Ruelle condition (Dobrushin in Theory Prob Appl 17:582–600, 1970. https://doi.org/10.1137/1115049; Lanford and Ruelle in Commun Math Phys 13:194–215, 1969. https://doi.org/10.1007/BF01645487) and the classical Kubo–Martin–Schwinger (KMS) condition (Gallavotti and Verboven in Nuov Cim B 28:274–286, 1975. https://doi.org/10.1007/BF02722820) are considered in the context of classical lattice systems. In particular, we prove that these conditions are equivalent for the case of a lattice spin system with values in a compact symplectic manifold by showing that infinite-volume Gibbs states are in bijection with KMS states.

Violation of Detailed Balance in Quantum Open Systems.

We consider the dynamics of a quantum system immersed in a dilute gas at thermodynamic equilibrium using a quantum Markovian master equation derived by applying the low-density limit technique. It is shown that the Gibbs state at the bath temperature is always stationary while the detailed balance condition at this state can be violated beyond the Born approximation. This violation is generically related to the absence of time-reversal symmetry for the scattering T matrix, which produces a thermalization mechanism that allows the presence of persistent probability and heat currents at thermal equilibrium. This phenomenon is illustrated by a model of an electron hopping between three quantum dots in an external magnetic field.

Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with Markovian structure

Let [Formula: see text] be a compact metric space and [Formula: see text], we consider a set of admissible sequences [Formula: see text] determined by a continuous admissibility function [Formula: see text] and a compact set [Formula: see text]. Given a Lipschitz continuous potential [Formula: see text], we prove uniqueness of the Gibbs state [Formula: see text] and we show that it is a Gibbs–Bowen measure and satisfies a central limit theorem.

Level-2 Large Deviation Principle for Countable Markov Shifts Without Gibbs States

Level-2 Large Deviation Principle for Countable Markov Shifts Without Gibbs States

The singularities of the rate function of quantum coherent work in one-dimensional transverse field Ising model

Quantum coherence will undoubtedly play a fundamental role in understanding of the dynamics of quantum many-body systems, thereby to reveal its genuine contribution is of great importance. In this paper, we specialize our discussions to the one-dimensional transverse field quantum Ising model initialized in the coherent Gibbs state. After quenching the strength of the transverse field, the effects of quantum coherence are studied by the rate function of quantum work distribution. We find that quantum coherence not only recovers the quantum phase transition destroyed by thermal fluctuations, but also generates some entirely new singularities both in the static state and dynamics. It can be manifested that these singularities are rooted in spin flips causing the sudden change of the domain boundaries of spin polarization. This work sheds new light on the fundamental connection between quantum critical phenomena and quantum coherence.

Dynamics of mean-field bosons at positive temperature

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in Deuchert–Seiringer (2021) that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N , as N \rightarrow \infty , by that of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree–Fock–Bogoliubov equations to define a fluctuation dynamics.

A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperatureT=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperatureT>0in subpolynomial time, i.e., in timeO((1ε)c)for any constantc>0whereεis the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in1ε. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.

A gravitationally induced decoherence model using Ashtekar variables

We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.

Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps

For quadratic-like maps, we show a phenomenon of sensitive dependence of geometric Gibbs states: There are analytic families of quadratic-like maps for which an arbitrarily small perturbation of the parameter can have a definite effect on the low-temperature geometric Gibbs states. Furthermore, this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states. We introduce a geometric version of the Peierls condition for contour models ensuring that the low-temperature geometric Gibbs states are concentrated near the critical orbit.

Measure rigidity of Anosov flows via the factorization method

Using the factorization method, a method pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification over the moduli space of translation surfaces, we show that generalized $u$-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.

A Gibbsian Random Tree with Nearest Neighbour Interaction

We revisit the random tree model with nearest-neighbour interaction as described in Dunlop and Mardin (J Stat Phys 189:38, 2022), enhancing growth. When the underlying free Bienaymé–Galton–Watson (BGW) model is sub-critical, the (non-Markov) model with interaction exhibits a phase transition between sub- and super-critical regimes. In the critical regime, using tools from dynamical systems, we show that the partition function of the model approaches a limit at rate $$n^{-1}$$ in the generation number n. In the critical regime with almost sure extinction, we also prove that the mean number of external nodes in the tree at generation n decays like $$n^{-2}$$ . Finally, we give a spin representation of the random tree, opening the way to tools from the theory of Gibbs states, including FKG inequalities. We extend the construction in Dunlop and Mardin (J Stat Phys 189:38, 2022), when the law of the branching mechanism of the free BGW process has unbounded support.

Approach to Equilibrium in Translation-Invariant Quantum Systems: Some Structural Results

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase in entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase in the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

A Note on Adiabatic Time Evolution and Quasi-Static Processes in Translation-Invariant Quantum Systems

We study the slowly varying, non-autonomous quantum dynamics of a translation-invariant spin or fermion system on the lattice $$\mathbb {Z}^d$$ . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in Jakšić et al. (Approach to equilibrium in translation-invariant quantum systems: some structural results, in the present issue of Ann. H. Poincaré, 2023), we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

Proof of Single-Replica Equivalence in Short-Range Spin Glasses.

We consider short-range Ising spin glasses in equilibrium at infinite system size, and prove that, for fixed bond realization and a given Gibbs state drawn from a suitable metastate, each translation and locally invariant function (for example, self-overlaps) of a single pure state in the decomposition of the Gibbs state takes the same value for all the pure states in that Gibbs state. We describe several significant applications to spin glasses.

Infinite Volume Gibbs States and Metastates of the Random Field Mean-Field Spherical Model

For the discrete random field Curie–Weiss models, the infinite volume Gibbs states and metastates have been investigated and determined for specific instances of random external fields. In general, there are not many examples in the literature of non-trivial limiting metastates for discrete or continuous spin systems. We analyze the infinite volume Gibbs states of the mean-field spherical model, a model of continuous spins, in a general random external field with independent identically distributed components with finite moments of some order larger than four and non-vanishing variances of the second moments. Depending on the parameters of the model, we show that there exist three distinct phases: ordered ferromagnetic, ordered paramagnetic, and spin glass. In the ordered ferromagnetic and ordered paramagnetic phases, we show that there exists a unique infinite volume Gibbs state almost surely. In the spin glass phase, we show the existence of chaotic size dependence, provide a construction of the Aizenman–Wehr metastate, and consider both the convergence in distribution and almost sure convergence of the Newman–Stein metastates. The limiting metastates are non-trivial and their structure is universal due to the presence of Gaussian fluctuations and the spherical constraint.

Operational Definition of the Temperature of a Quantum State.

Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.