Beyond the Kubo formula: unification of adiabatic and isothermal response in quantum systems

AI summary

Overview

Research area: Theoretical quantum many-body physics, specifically linear response theory, open quantum systems, and quantum statistical mechanics.

Technical level: Advanced. The paper assumes familiarity with the Kubo formula, retarded correlation functions, Keldysh Green's functions, and linear response in condensed matter systems, though its central conceptual point is stated in accessible terms.

Scope: The paper proposes a single dynamical framework in which the adiabatic and isothermal response of a quantum system appear as two opposite orderings of the static-frequency and vanishing-thermalization-rate limits, and demonstrates the protocol on the magnetic response of a free electron gas.

What This Paper Is About

The static limit of the Kubo formula gives the response of an isolated system in which the occupations of energy levels stay frozen, while the isothermal response measured in equilibrium allows those occupations to readjust thermally; these two generally differ, and the standard textbook fix relies on taking limits in a special order that only works in the thermodynamic limit with translational invariance. The authors ask which physical timescale controls the crossover between the two responses and build a generalized susceptibility that depends explicitly on a thermalization rate, so that both responses follow from one calculation. They then check the framework against the known magnetic susceptibility of a free electron gas, where the adiabatic response vanishes and the isothermal response is the Pauli result.

Key Contributions

1. Identification of the static Kubo limit as the adiabatic response. The authors show explicitly that the zero-frequency Kubo susceptibility equals the isolated/adiabatic susceptibility, in which level populations are frozen, and derive its eigenstate sum together with the separate isothermal expression. They show the two differ by a thermal fluctuation term, so the isothermal susceptibility cannot be reached by the zero-frequency limit alone.

2. A generalized susceptibility with an explicit thermalization rate. They introduce χ_AB(ω, κ), where κ = τ_th⁻¹ is a typical thermalization rate, and show that the adiabatic and isothermal susceptibilities are recovered by the two non-commuting limits χ_AB^S = lim_{ω→0} lim_{κ→0} χ_AB(ω, κ) and χ_AB^T = lim_{κ→0} lim_{ω→0} χ_AB(ω, κ). The same expression interpolates between the two responses at finite ω and κ.

3. A worked application to the free electron gas. Using a fermionic bath coupled by hopping (exchange of electrons at rate κ), they compute the magnetic susceptibility at zero temperature in closed form, χ_M(ω, κ) = χ_P · 2κ/(2κ − iω), and recover both the vanishing adiabatic response and the Pauli isothermal susceptibility from the two limit orderings.

4. A reassessment of the standard ω→0-before-q→0 prescription. They show that at finite momentum the two limits commute and the adiabatic and isothermal susceptibilities coincide for q ≠ 0, so the customary prescription is contingent on translational symmetry and a symmetry-breaking response operator rather than being a fundamental principle, and that their own protocol applies to systems where that prescription fails.

Main Findings

• The Kubo static limit is adiabatic, not isothermal. The adiabatic susceptibility is χ_AB^S = (1/Z) Σ_{m≠n} [(e^{−βE_m} − e^{−βE_n})/(E_m − E_n)] A_mn B_nm, and it equals lim_{ω→0} χ_AB(ω). The corresponding state ρ_S is diagonal but its populations do not satisfy detailed balance with respect to the shifted energies, so it is not a thermal state.

• The isothermal susceptibility contains an extra fluctuation term. It is given by χ_AB^T = χ_AB^S − β(⟨Â₀B̂₀⟩₀ − ⟨Â₀⟩₀⟨B̂₀⟩₀), where Â₀ and B̂₀ are the diagonal parts in the eigenbasis of the unperturbed Hamiltonian. This is the term the closed-system Kubo formula has no knowledge of.

• The quench response vanishes. If the perturbation is switched on much faster than the system can react, the density matrix is unchanged and χ^Q = 0, consistent with the ω → ∞ limit of the Kubo susceptibility.

• A hierarchy holds for self-response. For B̂ ∝ Â, the adiabatic self-susceptibility is a sum of non-positive terms and therefore χ_AA^S ≤ 0, while χ_AA^T − χ_AA^S = −β(⟨Â₀²⟩₀ − ⟨Â₀⟩₀²) ≤ 0 because the variance of Â₀ is positive. The result is χ_AA^T ≤ χ_AA^S ≤ χ_AA^Q = 0, consistent with previously derived bounds and with energy-balance and work arguments.

• The two limits do not commute. Taking ω → 0 first, then κ → 0, gives the isothermal response; taking κ → 0 first, then ω → 0, gives the adiabatic response. The paper presents this as Eq. (11) and illustrates it in Fig. 2, a surface plot of the real part of χ_M(ω, κ) normalized to χ_P in the zero-temperature limit, with arrows marking the two orderings.

• The electron gas reproduces the textbook results. Since Ŝ_z commutes with the Hamiltonian in the absence of a thermalizing process, the commutator vanishes and χ_M^S = 0. In the isothermal limit the framework returns the Pauli susceptibility χ_P = μ_B²(1/V) Σ_{kσ}(−∂f/∂ε_k), which at zero temperature becomes χ_P = μ_B² N(ε_F), with N(ε) the density of states. At generic temperatures χ_M(ω, κ = 0) = 0 and χ_M^T again matches the finite-temperature Pauli expression.

• The bath model is not essential. Any mechanism producing a self-energy with non-zero imaginary part, taken as κ = −Im Σ, yields the same structure; the paper names electron-electron, electron-phonon and electron-impurity interactions as alternatives, with the requirement that κ respect the system's conservation laws.

• Finite-momentum behavior is discontinuous at q = 0. At q ≠ 0 the ω → 0 and κ → 0 limits commute and χ_M^S(q) = χ_M^T(q), while the subsequent q → 0 limit gives the isothermal Pauli result. Hence χ_M(q) is discontinuous at q = 0. The equality at q ≠ 0 follows because the difference depends on expectation values of n̂_q on eigenstates of definite momentum, which vanish when q ≠ 0.

• A free-energy interpretation. The adiabatic susceptibility is the mixed derivative of the tangent-plane approximation to the free energy in the space of energy levels (fixed occupations), while the isothermal one also includes the curvature of the free energy, describing redistribution of thermal populations.

• No numerical benchmarks, datasets, or model training are reported. The paper is analytic; the only quantitative outputs are the closed-form expressions above and the illustrative plots in Figs. 1 and 2.

Methodology in Plain English

The authors begin with a closed many-body system in a thermal state, subject to a perturbation that is ramped up over a characteristic time τ_d, and compare τ_d with two other timescales: the internal dynamical time τ_s, set by the energy gaps, and the thermalization time τ_th. This comparison defines three regimes: a quench (τ_d → 0, state frozen), a quantum adiabatic regime (fast internal dynamics, populations frozen while eigenstates follow the perturbation), and an isothermal regime (τ_d ≫ τ_th, the system is instantaneously thermal).

Rather than treating the open system abstractly, they attach a fermionic bath through a hopping interaction that lets electrons enter and leave at rate κ, trace the bath out, and obtain retarded, advanced and Keldysh Green's functions with a constant imaginary self-energy κ. These feed into the standard diagrammatic expression for the spin-resolved density–density susceptibility, which is then combined into the magnetic susceptibility. The physics is extracted by sending ω and κ to zero in the two possible orders, and the resulting expression is cross-checked against the known adiabatic and Pauli results. Three appendices support the main text: a free-energy derivation of both susceptibilities, a detailed treatment of the magnetic case, and two side models — a Hamiltonian with a spin-flip term but no bath (which breaks the symmetry protecting Ŝ_z and displays phenomenology similar to adiabatic quantum annealing in the transverse-field Ising model) and a simple classical problem with distinct adiabatic and isothermal responses.

Why This Matters

Impact on research. The paper gives a physically motivated replacement for a textbook prescription that silently depends on the thermodynamic limit and translational invariance, and it makes the controlling timescale explicit. This matters for finite or inhomogeneous systems — quantum dots, qubits, trapped atomic gases — where the standard order-of-limits trick has no justification. It also connects linear response to the large body of work on thermalization and relaxation of many-body quantum systems and to recent Lindblad-based studies of linear response to perturbations of dissipative processes.

Real-world applications (as discussed in the paper):

• Ultracold atomic gases, which can be kept isolated from the environment over long times, where the adiabatic response is the relevant one.
• Solid-state quantum devices, where the probe frequency can be comparable to the system's relaxation rates, so the response is neither purely adiabatic nor purely isothermal.
• Quantum dots and circuits, whose response is described as having both an adiabatic and a thermalization contribution.
• Sensing, where the distinct responses could be exploited in devices including sensors of perturbation frequency or strength.

Industry relevance. The framework targets the regime in which quantum devices are actually probed — microwave frequencies comparable to relaxation rates — and provides a way to model or calibrate measured susceptibilities that mix adiabatic and thermal contributions. It is directly relevant to quantum device characterization and metrology, and to modeling magnetic response in electronic materials beyond the idealized uniform, translationally invariant limit.

Future Directions

• Understanding the q = 0 discontinuity. Both the adiabatic and isothermal susceptibilities at q ≠ 0 approach the uniform isothermal response rather than the uniform adiabatic one; the paper states that the origin of this connection deserves further study.
• Exploiting the two responses in quantum devices. The authors propose using the distinct adiabatic and isothermal behaviors in devices including sensors of perturbation frequency or strength.
• Corrections in intermediate regimes. Calculating susceptibility corrections when ω and κ are comparable, using adiabatic and quasistatic expansions, is identified as a next step.
• Classical thermodynamic adiabaticity with internal thermalization. The paper notes that in isolated systems satisfying the eigenstate thermalization hypothesis, internal thermalization can occur during quasistatic evolution at an effective temperature fixed by entropy conservation, giving a response that generally differs from the isothermal one; a detailed investigation is explicitly left for future work.

Target Audience

Theoretical researchers and graduate students in quantum many-body physics, condensed matter theory, and open quantum systems, particularly those working on linear response, thermalization, or quantum thermodynamics. It is also useful for experimentalists in quantum devices, quantum dots, and ultracold atoms who need to interpret measured susceptibilities when the probe frequency is not negligible compared with relaxation rates, and for anyone who has relied on the standard ω→0-before-q→0 prescription and wants to know when it is justified.

Authors’ abstract

The static limit of the Kubo susceptibility describes the adiabatic response of an isolated quantum system, in which energy-level occupations remain unchanged, and generally differs from the isothermal susceptibility, where these occupations readjust to thermal equilibrium. In this work, we introduce a single dynamical framework in which the adiabatic and isothermal responses arise naturally as two distinct limiting regimes. This can be done by considering explicitly an open system version of the Kubo approach in which the dissipation is governed by a typical thermalization rate of the system. By exchanging the order of the static limit and of the vanishing thermalization rate limit, the protocol yields either the adiabatic or isothermal response. This protocol has a solid physical motivation in terms of the thermalization dynamics and is valid for generic systems. We apply it to the case of the magnetic response of the electron gas and reproduce the expected results.