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arXiv:2402.04214 · Physical Review Letters

Symmetry Shapes Thermodynamics of Macroscopic Quantum Systems

Vasco Cavina, Ariane Soret, Timur Aslyamov, Krzysztof Ptaszyński, Massimiliano Esposito

Published Sep 23, 2024• 4 citations
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Abstract

We derive a systematic approach to the thermodynamics of quantum systems based on the underlying symmetry groups. We first show that the entropy of a system can be described in terms of group-theoretical quantities that are largely independent of the details of its density matrix. We then apply our technique to generic N identical interacting d-level quantum systems. Using permutation invariance, we find that, for large N, the entropy displays a universal asymptotic behavior in terms of a function s(x) that is completely independent of the microscopic details of the model, but depends only on the size of the irreducible representations of the permutation group S_{N}. In turn, the equilibrium state of the system and macroscopic fluctuations around it are shown to satisfy a large deviation principle with a rate function f(x)=e(x)-β^{-1}s(x), where e(x) only depends on the ground state energy of particular subspaces determined by group representation theory, and β is the inverse temperature. We apply our theory to the transverse-field Curie-Weiss model, a minimal model of phase transition exhibiting an interplay of thermal and quantum fluctuations.

DOI
10.1103/physrevlett.133.130401

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We derive a systematic approach to the thermodynamics of quantum systems based on the underlying symmetry groups. We first show that the entropy of a system can be described in terms of group-theoretical quantities that are largely independent of the details of its density matrix. We then apply our technique to generic N identical interacting d-level quantum systems. Using permutation invariance, we find that, for large N, the entropy displays a universal asymptotic behavior in terms of a function s(x) that is completely independent of the microscopic details of the model, but depends only on the size of the irreducible representations of the permutation group S_{N}. In turn, the equilibrium state of the system and macroscopic fluctuations around it are shown to satisfy a large deviation principle with a rate function f(x)=e(x)-β^{-1}s(x), where e(x) only depends on the ground state energy of particular subspaces determined by group representation theory, and β is the inverse temperature. We apply our theory to the transverse-field Curie-Weiss model, a minimal model of phase transition exhibiting an interplay of thermal and quantum fluctuations.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of Sz and with respect to the irreducible representations of the Sn group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.
Rare fluctuations in physical systems depend on the detailed microphysics responsible for the fluctuations. In classical statistical systems, the large deviation principle has elucidated the role of semi-classics in describing this regime, and has simultaneously provided a the mathematical foundation of statistical mechanics. Large deviation theory for quantum system is considerably less developed. As all physical systems are fundamentally quantum mechanical, this leaves a major gap in our understanding of rare fluctuations relevant to statistical physics, cosmology, and more. In this paper, we develop the practical aspects of the theory of large deviations relevant for calculating rare events in physical systems from quantum walks to cosmology. We first analyze the case of the anharmonic oscillator coupled to a bath, showing explicitly how the system evolves from dominantly statistical (e.g. thermal) to quantum fluctuations. We then generalize these results, showing that the dominant rare fluctuations minimize the measurement-induced relative entropy. This perspective provides a thermodynamic description of a wide range of open quantum systems. We apply these results to random walks that arise in cosmology through stochastic inflation. We show that the evolution of the density matrix of long wavelength fields on a fixed de Sitter background breaks the KMS symmetry, giving rise to a stationary density matrix that does not respect detailed balance.
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size NNN. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras Pk(N){P}_{k}(N)Pk​(N). The integer kkk is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.
Thermodynamics is based on a coarse-grained approach, from which its fundamental variables emerge, effectively erasing the complicated details of the microscopic dynamics within a macroscopic system. The strength of thermodynamics lies in the universality provided by this paradigm. In contrast, quantum mechanics focuses on describing the dynamics of microscopic systems, aiming to make predictions about experiments we perform, a goal shared by all fundamental physical theories, which are often framed as gauge theories in modern physics. Recently, a gauge theory for quantum thermodynamics was introduced, defining gauge-invariant work and heat, and exploring their connections to quantum phenomena. In this work, we extend that theory in two significant ways. First, we incorporate energy spectrum degeneracies, which were previously overlooked. Additionally, we define gauge-invariant entropy, exploring its properties and connections to other physical and informational quantities. This results in a complete framework for quantum thermodynamics grounded in the principle of gauge invariance. To demonstrate some implications of this theory, we apply it to well-known critical systems.
The macroscopic hydrodynamic equations are derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equations for the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.