Ultrastrong coupling limit to quantum mean force Gibbs state for anharmonic environment

Prem Kumar
Optics and Quantum Information Group, The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India.
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

Sibasish Ghosh
Optics and Quantum Information Group, The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India.
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

The equilibrium state of a quantum system can deviate from the Gibbs state if the system-environment coupling is not weak. An analytical expression for this mean force Gibbs state (MFGS) is known in the ultrastrong coupling (USC) regime for the Caldeira-Leggett (CL) model that assumes a harmonic environment. Here, we derive analytical expressions for the MFGS in the USC regime for more general system-environment models. For all the generalized models considered here, we find the USC state to be diagonal in the basis set by the system-environment interaction, just like in the CL case. While for the generic model considered, the corresponding USC-MFGS state is found to alter from the CL-result, we do identify a class of models more general than the CL-model for which the CL-USC result remains unchanged. We also provide numerical verification for our results. These results provide key tools for the study of strong coupling thermodynamics under more realistic system-environment models, going beyond the CL-model.

§ INTRODUCTION

Non-negligible system-environment coupling can cause the equilibrium state of a system to deviate from the textbook Gibbs state <cit.>. The concept of Mean Force Gibbs State (MFGS) has been developed to capture the idea of this generalized equilibrium state, and has historically seen a lot of application in the field of chemistry <cit.> and, more recently, to the theory of strong coupling quantum thermodynamics <cit.>. Many of these approaches to calculate the quantum MFGS assume the Caldeira-Leggett (CL) model for the system and the environment, in which the environment is modelled as a set of harmonic oscillators coupled to the system.

The harmonic environment approximation has been widely successful in capturing the behaviour of a variety of physical systems <cit.>. Yet, advancement in experimental techniques and more detailed theoretical considerations have shown the inadequacy of the harmonic environment approximation, sparking  numerous studies on open quantum system with anharmonic environment <cit.>. For example, for an environment consisting of nonpolar liquids or for low-frequency intramolecular modes of polar liquids, the harmonic approximation is known to fail <cit.>.
It is hence worth investigating the MFGS result for system-environment models beyond the CL-model.

If $\hat{H}_{\text{SE}}$ denotes the full Hamiltonian of a system and environment, then the MFGS is formally defined as <cit.>

\begin{align}\label{eqn:definition of MFGS}
\hat{\rho} = Z^{-1}Tr_{E}\left[ e^{-\beta \hat{H}_{\text{SE}}}\right],
\end{align}

where $Z$ is an appropriate normalization constant.

For the CL-model, analytical expression for the MFGS in ultrastrong coupling (USC) regime has been determined for continuous variable (CV) systems <cit.> and, more recently, for a more general CL-model (consisting of an arbitrary system Hamiltonian and coupling-operator) <cit.>. The USC-MFGS acts as a good indicator for the deviation of the MFGS state from the textbook Gibbs state at large coupling. However, since this result is only valid for the CL-model, its applicability is significantly limited. It also remains uncertain as to which aspects of the USC result are artefacts of the specific structures of the CL-model, and which aspects might persist when the model is generalized. Here, we attempt to address this issue by studying the USC-MFGS for models beyond the CL-model.

We find that for all the system-environment models considered by us, one of the core features of the CL-USC result, that the MFGS is diagonal in the basis set by the system-environment interaction, remains intact, although the functional form of the MFGS does get affected. For example, the most general extension of the CL-model considered by us, which we call the GCL-model, shows such deviations from the CL-USC result. Within the GCL-model, we identify a subclass of models, which we call the GCL2-model, for which the CL-USC result remains exactly valid. Finally, we derive an analytical expression for the USC-MFGS for another class of model (distinct from the GCL-model), the so-called Zwanzig-model <cit.>, and find that the corresponding USC state, although still diagonal in the basis set by the system-environment interaction, has qualitative distinctions from the GCL2-USC result.

In <ref>, we rederive the known USC-MFGS result for the CL-model <cit.> using the path integral approach, to illustrate the method that we will be using to generalize the same result. <ref> contains the derivation of the USC result for the GCL-model, while <ref> acts as a special case of the same derivation for the more restricted GCL2-model. Finally, in <ref>, we provide the USC-MFGS derivation for the Zwanzig-model. <ref> provides some numerical verifications of the analytical results derived here, and <ref> provides conclusions and future directions.

§ USC-MFGS IN CL-MODEL

Consider a typical CL-model in which a generic quantum system, with free Hamiltonian $\hat{H}_S$, has a one-to-one interaction with a set of harmonic oscillators that are uncoupled from each other. The full system-environment Hamiltonian is hence given by

\begin{align}
\hat{H}_{\mathrm{SE}} &= \hat{H}_S + \hat{H}_I, \label{eqn:discrete hamiltonian partition}\\
\hat{H}_I &= \sum_k\left[\frac{\hat{p}_k^2}{2 m_k}+\frac{1}{2} m_k \omega_k^2\left(\hat{q}_k- \alpha_k \hat{A}\right)^2\right],\label{eqn:discrete CL_hamiltonia}
\end{align}

where $\hat{A}$ is a generic system operator through which the system couples to the environment (which from now onwards we will refer to as `system coupling-operator') and

\begin{align}
\alpha_k \equiv \frac{c_k}{m_k \omega_k^2}, \label{eqn:definition of alpha_k}
\end{align}

where $m_k \text{ and } \omega_k$ are the mass and frequency of the $k$th harmonic oscillator and $c_k$ is the corresponding system-environment coupling strength. For a CV system, typically we have $\hat{A} = \hat{q}$, where $\hat{q}$ is the position operator of the system; while for the spin-Boson model, $\hat{A}$ is taken to be a Pauli matrix <cit.>.

The MFGS (<ref>) can be calculated using the Feynman path integral in the basis of the operator $\hat{A}$ <cit.> by evolving the system and the environment in the `imaginary time' by an amount $\beta$ and then tracing over the environment degrees of freedom <cit.>.

Note a subtlety that arises when the system coupling-operator $\hat{A}$ has degeneracy, and hence each of its eigenstates is not uniquely labelled by an eigenvalue. To fix this problem, let us introduce an operator $\hat{\phi}$ that lifts the degeneracy in $\hat{A}$. That is, $[\hat{\phi}, \hat{A}] = 0$ and $\hat{A} + \hat{\phi}$ has no degeneracy. The final result will not depend upon the particular choice of $\hat{\phi}$. Without any loss of generality and for convenience, we will also assume that $\hat{\phi}$ is itself non-degenerate, so that its eigenvalues, $(a,b)$, can be used as a unique label to express the system's reduced density matrix elements as $\hat{\rho}(a,b)$.

Let a generic imaginary time path be denoted as $\{A(t),\phi(t), q_k(t)\}$, where $A(t) \in \{A_i\}, $
$\phi(t) \in \{\phi_i\}, $
$\text{and } q_k(t) \in \{{q_k}_i\}$
, where $\{A_i\},$
and $\{{q_k}_i\}$
are the sets of eigenvalues of operators $\hat{A},$
and $\hat{q}_k$, respectively.
We emphasise here that $A(t)$ denotes a path that the system can take in the path integral and should not be confused to mean that the system coupling-operator, $\hat{A}$, has a time dependence. In fact, $\hat{A}$ has been assumed to be time-independent throughout this paper.

Now, let $S_S[A(t), \phi(t)]$ and $S_I[A(t),q_k(t)]$ represent the imaginary time action corresponding to $\hat{H}_S$ and $\hat{H}_I$, respectively. Note how $S_I$ is not a function of $\phi(t)$ while $S_S$ is. This is because while $\hat{H}_S$ is an arbitrary system operator, $\hat{H}_I$ has system dependence only through the operator $\hat{A}$, and hence the corresponding action, $S_I$, cannot distinguish within the degenerate subspace of $\hat{A}$, which removes the $\phi(t)$ dependence.

Also note that the paths $A(t)$ and $\phi(t)$ are not independent of each other. That is, $A(t)$ and $\phi(t)$ denote some eigenvalues of the operators $\hat{A}$ and $\hat{\phi}$, respectively, such that one can find a common eigenvector $\ket{v(t)} \equiv \ket{A(t), \phi(t)}$ such that $\hat{A} \ket{v(t)} = A(t) \ket{v(t)}$ and $\hat{\phi}\ket{v(t)} = \phi(t) \ket{v(t)}$. But we can always treat $A(t)$ and $\phi(t)$ as independent of each other if we assume that the action $S_S[A(t), \phi(t)]$ is constructed in such a manner that, given some $A(t)$, it rules out all unphysical possibilities for the path $\phi(t)$ by assigning the corresponding action an infinite weight. For all the arguments in this paper, we don't need to go into any further details on how to construct such a functional $S_S[A(t), \phi(t)]$.

The full system-environment path integral is given as

\begin{align}
\hat{\rho}(a, b, a_k, b_k) &= \Tilde{Z}^{-1} \left( \prod_k \int_{a_k}^{b_k} \mathcal{D}q_k(t) \right)\int_{\Tilde{a}}^{\Tilde{b}} \mathcal{D}A(t)  \nonumber \\
&\quad \times\int_{a}^{b} \mathcal{D}\phi(t) e^{- S_S[A(t), \phi(t)] - S_I[A(t),q_k(t)]} \label{eqn: CL full pathe integral}.
\end{align}

Here $\Tilde{Z}$ is the overall normalization factor and $a$, $b$, $a_k$, $b_k$, $\Tilde{a}$, and $\Tilde{b}$ correspond to $\phi(0)$, $\phi(\beta)$, $q_k(0)$, $q_k(\beta)$, $A(0)$, and $A(\beta)$, respectively. Tracing over the environment will now give us the MFGS as

\begin{align}
\hat{\rho}_{\text{USC}}(a, &b) = \left( \prod_k \int_{-\infty}^{\infty} d a_k \right) \hat{\rho}(a, b, a_k, a_k)\\
&= \Tilde{Z}^{-1} \int_{\Tilde{a}}^{\Tilde{b}} \mathcal{D}A(t) \int_{a}^{b} \mathcal{D}\phi(t) e^{ -S_S[A(t), \phi(t)]} I[A(t)] \label{eqn:discrete full path integral}.
\end{align}

Here $I[A(t)] \equiv \prod_k I_k[A(t)]$ is called the influence functional <cit.>,
where $I_k[A(t)]$ encapsulates the effect of the $k$th environment particle only and is defined as

\begin{align}
I_k[A(t)] &= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}q_k(t) e^{- S_k\left[q_k(t), A(t)\right]}\label{eqn:environment side path integral}.
\end{align}

Here the action, $S_k$, is defined as

\begin{align}
S_k\left[q_k(t), A(t)\right] &\equiv \int_0^\beta dt \bigg\{ \frac{m_k}{2} \dot{q}_k(t)^2 \nonumber \\
&\quad + \frac{m_k\omega_k^2}{2} \left(q_k(t) - \alpha_k A(t)\right)^2\bigg\}\label{eqn:CL environment action}.
\end{align}

The explicit expression for $I[A(t)]$ for CL-model <cit.> can be used to derive the CL-USC result <cit.>. But here, since we need to eventually generalize this result beyond the CL-model, we will derive the expression for the USC-MFGS without explicitly evaluating $I[A(t)]$ for any specific model.

Let us choose an arbitrarily large but finite length scale $\Delta$ such that whenever $|q_k(t) - \alpha_k A(t)| > \Delta$ for any $t$, the potential energy (PE) cost of the corresponding path, $q_k(t)$, will be very high, and hence its contribution to the path integral will be negligible. We can hence write, for all paths $q_k^*(t)$ that do have significant contribution to the path integral,

\begin{align}
| q_k^*(t) - \alpha_k A(t) | < \Delta \label{eqn:PE minimizing path}.
\end{align}

Let us define

\begin{align}
\braket{A} &\equiv \int_0^\beta dt A(t) \label{eqn:mean definition},\\
\braket{A^2} &\equiv \int_0^\beta dt A(t)^2 \label{eqn:squared mean definition},\\
\sigma_A^2 &\equiv \braket{A^2} - \braket{A}^2 \label{eqn: variance definition}.
\end{align}

Then, in Appendix. <ref>, we have proven that if $\sigma_A^2 \gg \Delta^2/\alpha_k^2$, then

\begin{align}
\int_0^\beta dt \dot{q}_k^*(t)^2 &\geq \alpha_k^2 \mu \frac{\sigma_A^2}{\beta^2} \label{eqn:kinetic energy cost}.
\end{align}

Here $\mu > 0$ is some constant. Now, note that in the USC limit, defined as

\begin{align}
\lim c_k \to \infty , \label{eqn:USC limit}
\end{align}

we have  $\alpha_k \to \infty$ (<ref>). Hence, in this limit, unless $\sigma_A \lesssim O(\Delta/\alpha_k)$, the RHS in <ref> diverges, causing the kinetic term $\dot{q}_k^*(t)^2$ in the action $S_k\left[q_k(t), A(t)\right]$ (<ref>) to diverge. Hence, in the USC limit (<ref>), paths $q_k^*(t)$ (which have been defined to have convergent PE cost) will have divergent kinetic energy cost unless $\sigma_A$ vanishes, or in other words, unless we have

\begin{equation}
A(t) = A(0) = \Tilde{a}, \quad \text{for some constant} \; \Tilde{a} \label{eqn:frozen paths}.
\end{equation}

As we will see, <ref> will greatly simplify the evaluation of the MFGS path integral (<ref>). But before getting into that, we need to address a subtlety. If the system lives in a discrete Hilbert space, then $A(t)$ may not be continuous and differentiable everywhere, leading to potential mathematical irregularities. Although the results and proofs in this paper only require the paths \(A(t)\) to be square integrable (and not continuous and differentiable), we note that these mathematical irregularities can be addressed by substituting \(A(t)\) with a continuous and differentiable function \(X(t)\) that closely approximates it, such that the induced error, $|I[A(t)] - I[X(t)]|$, is arbitrarily small. For instance, <ref> illustrates how $X(t)$, using a cubic polynomial near $t=1$, smoothly approximates $A(t)$'s discrete jump.

Approximating a piecewise constant function, $A(t)$, with a continuous and differentiable function $X(t)$. The function $X(t)$ is equal to $A(t)$ everywhere except near the discontinuous jump in $A(t)$ (when $1 - \delta< t <1 + \delta$, for $\delta = 0.2$). Near $t=1$, $X(t)$ takes the form of a cubic function, $f(t') = a_0 + a_1 t' + a_3 t'^3$, with $t' = t- 1$, $a_0 = \frac{1}{2}$, $a_1 = \frac{3}{4 \delta}$ and $a_3 = -\frac{1}{4 \delta^3}$. This means that we have $\lim_{\delta \to 0} X(t) = A(t)$.

Now, given <ref>, the expression for the influence functional (<ref>) simplifies as

\begin{align}
I_k[\Tilde{a}] &= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}q_k(t) e^{-S_k\left[q_k(t), \Tilde{a}\right]}\\
&= \int_{-\infty}^{\infty} dx \int_{x-\alpha_k \Tilde{a}}^{x- \alpha_k \Tilde{a}} \mathcal{D}x_k(t) e^{-S_k\left[x_k(t), 0 \right]} \label{eqn:cl case I_k calc step 2}\\
&= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}x_k(t) e^{-S_k\left[x_k(t), 0 \right]}\\
&= I_k, \quad \text{which is independent of}\; \Tilde{a}.
\end{align}

Here in <ref>, we defined $x_k(t) \equiv q_k(t) - \alpha_k \Tilde{a}$ and we have used the relation $S_k\left[q_k(t), \Tilde{a}\right] = S_k\left[x_k(t), 0 \right]$ (<ref>). We hence have that $I[\Tilde{a}] \equiv I $ is independent of $\Tilde{a}$. Hence, the only contribution of the environment side path integral is to eliminate all the system side paths that do not satisfy <ref>. This means that jumps within a degenerate subspace are still allowed and with equal weight as far as the environment side path integral is concerned. Therefore, the full MFGS path integral (<ref>) simplifies as

\begin{align}
\hat{\rho}_{\text{USC}}(a,b) &= \Tilde{Z}^{-1} I \int_{a}^{b} \mathcal{D}\phi(t) \exp \left\{- S_S[\Tilde{a}, \phi(t)] \right\}\label{eqn:CL USC-MFGS integral form}.
\end{align}

Here $a \text{ and }b$ are the eigenvalues of the operator $\hat{\phi}$ and $\Tilde{a}$ is the eigenvalue of $\hat{A}$ <cit.>.
Then, the USC-MFGS is given as

\begin{align}
\hat{\rho}_{\text{USC}} &= Z^{-1} \exp \left\{ -\beta \Tilde{H} \right\} \label{eqn:general USC state},\\
\Tilde{H} &\equiv \sum_i \hat{P}_i \hat{H}_S \hat{P}_i.
\end{align}

Here $\hat{P}_i$ is the projection operator on the $i$th degenerate subspaces of the system coupling-operator, $\hat{A}$, and $Z$ is the normalization constant. <ref> is equivalent to the USC-MFGS derived by Cresser et al. <cit.> using perturbation theory, and reduces to the USC-MFGS derived earlier for a CV system <cit.>. Note that the derivation provided in this section is applicable for a generic CL-model, discrete or continuous.

§ USC-MFGS FOR GENERALIZED CALDEIRA-LEGGETT MODEL

The generalization of the CL-model has been considered by several authors <cit.> and can be characterized by an interaction Hamiltonian, $\hat{H}_I$ (<ref>), of the form

\begin{align}
\hat{H}_I &= \sum_k\left[\frac{\hat{p}_k^2}{2 m_k} + V_k(\hat{q}_k, c_k \hat{A}) \right]. \label{eqn:more general interaction hamiltonian}
\end{align}

Here $V_k(x,y)$ is a potential function bounded from below with a constraint that it should not renormalize the free effective system Hamiltonian $\hat{H}_S$ <cit.>. This is usually ensured by adding a so-called counter term in the potential, $V_k(x,y)$, which compensates for the shift in $\hat{H}_S$ because of the environment. Here, we assume that such a counter term has already been incorporated into $V_k(x,y)$, which translates into demanding that for a fixed value of $y$, if $x = \Tilde{x}_k(y)$ is the point at which $V_k(x,y)$ has a global minimum, then <cit.>

\begin{equation}
V_k(\Tilde{x}_k(y),y) = 0 \label{eqn:GCL minima point}.
\end{equation}

These generalized CL-models have been studied in various contexts. For instance, they have been used to explore the influence of an anharmonic environment on the non-Markovian dynamics of open systems <cit.>, to investigate certain condensed phase processes such as electron-transfer reactions <cit.>, and to examine the electronic absorption spectra of chromophores embedded in condensed phase environments <cit.>.

Here, we will generalize the derivation for the USC-MFGS provided in <ref> to such a larger class of model that we call the Generalized Caldeira-Leggett (GCL) model.
In order to proceed with the derivation of the USC-MFGS for such systems, we make two more physically motivated assumptions on the form of the potential $V_k(x,y)$ for our GCL-model, i.e.,

\begin{align}
\lim_{x\to\pm\infty} V_k(\Tilde{x}_k(y) + x, y) &\geq |O(x)| \quad \forall y \label{eqn:general CL-model interaction hamiltonian assumption 1},\\
\frac{d}{dy} \Tilde{x}_k(y) &\gtrsim O(1) \label{eqn:requirement on Tilde{x}_k(x)}.
\end{align}

Here $\Tilde{x}_k(y)$ is assumed to be continuous and differentiable everywhere. We remark that the continuity of the function $\Tilde{x}(y)$ is not affected by whether the system in question lives in a discrete dimensional Hilbert space or not because here we are treating $V_k(x,y)$ and $\Tilde{x}(y)$ as generic functions that can act on continuous variables as well as discrete operators.

Justification for <ref> is that keeping the system variable, $y$, fixed, if we move the environment variable $x$ far away from its global minima point $\Tilde{x}_k(y)$, then the value of the potential rises at least linearly with $|x-\Tilde{x}_k(y)|$. That is, we assume that the system and environment variables are attractively coupled to each other.

Note that we can relax the assumption <ref> to $\frac{d}{dy} \Tilde{x}_k(y) \geq \lambda$ for an arbitrarily small but finite $\lambda>0$, but we stick to the present form for simplicity.
We can motivate this assumption as following: if we instead assume that $\left|\frac{d}{dy} \Tilde{x}_k(y)\right| \approx 0$, then this would mean that for a large variation in $y$, $\Tilde{x}_k(y)$ would be almost constant (i.e., $\Tilde{x}_k(y)\approx x_0$) and for the corresponding value of the potential, we will have $V_k(x_0,y) \approx 0$ (<ref>). Hence, the assumption <ref> is applicable whenever the interaction potential is sensitive to a large change in the variable $y$ while $x$ is held constant.

The expression for the action, $S_k\left[q_k(t), A(t)\right]$ (<ref>) for the GCL-model now becomes,

\begin{align}
S_k\left[q_k(t), A(t)\right] &= \int_0^\beta dt \bigg\{\frac{m_k}{2} \dot{q}_k(t)^2 + V_k(q_k(t), c_k A(t)) \bigg\}\label{eqn:GCL action}.
\end{align}

<ref> implies that, for a fixed value of $y$, there is a (perhaps) large but essentially finite basin with length scale $\Delta$ about the position $x = \Tilde{x}_k(y)$, such that all paths $q_k(t)$ that explore the region $|x - \Tilde{x}_k(y)| > \Delta$ will have a PE cost so high that the contribution of these paths to the path integral will be negligible. Hence, for all paths $q_k^*(t)$ that do have a significant contribution to the path integral, we again have that

\begin{align}
| q_k^*(t) - \Tilde{x}_k(c_k A(t)) | < \Delta \label{eqn:valid bath paths for GCL model}.
\end{align}

Since $\Tilde{x}(c_k A(t))$ is ultimately a function of $t$, we can define $\braket{\Tilde{x}_A}$, $⟨x_A^2|$⟩ and $\sigma_{\Tilde{x}_A}^2$ analogous to <ref>, <ref> and <ref>.
Next, given <ref>, we prove in Appendix. <ref> that

\begin{align}
\sigma_{\Tilde{x}_A}^2 \gtrsim c_k^2 \sigma_A^2 \label{eqn: sigma x is greater than c_k sigma a}.
\end{align}

Next, we note that if $\sigma_{\Tilde{x}_A} \approx \Delta$, then $\sigma_A \lesssim O(\Delta / c_k)$, which vanishes in the USC limit (<ref>). Alternatively, if we instead assume that

\begin{align}
\sigma_{\Tilde{x}_A}^2 \gg \Delta^2 , \label{eqn: GCL model condition KE PE proof}
\end{align}

then, using <ref>, in Appendix. <ref> we prove that

\begin{align}
\int_0^\beta dt \dot{q}_k^*(t)^2 &\geq \mu \frac{\sigma_{\Tilde{x}_A}^2}{\beta^2}\\
&\gtrsim c_k^2 \mu \frac{\sigma_A^2}{\beta^2}, \quad \text{using \autoref{eqn: sigma x is greater than c_k sigma a}}\label{eqn: GCL variance conclusion}.
\end{align}

Here $\mu > 0$. We have already encountered an inequality similar to <ref> in the context of the CL-model (<ref>). Again, we are led to a similar conclusion that in the USC limit (<ref>), paths $q_k^*(t)$ (which have been defined to have convergent PE cost) will have divergent kinetic energy cost unless $\sigma_A$ vanishes. Therefore, again we have that $A(t) = A(0) = \Tilde{a}$, for some constant $\Tilde{a}$ (<ref>). The expression for the influence functional (<ref>) hence simplifies as

\begin{align}
I_k[\Tilde{a}] &= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}q_k(t) e^{-S_k\left[q_k(t), \Tilde{a}\right]}  \label{eqn:I_k path integral GCL}\\
\implies I[\Tilde{a}] &= \Tr \left[ e^{ -\beta \hat{H}_I(\Tilde{a})} \right].
\end{align}

Here $\hat{H}_I(\Tilde{a}) \equiv \sum_k\left[\frac{\hat{p}_k^2}{2 m_k} + V_k(\hat{q}_k, c_k \Tilde{a}) \right]$ (<ref>), with $\Tilde{a}$ being a number.
The expression for the full MFGS path integral (<ref>) then becomes

\begin{align}
\hat{\rho}_{\text{USC}}(a, b) &= \Tilde{Z}^{-1} I[\Tilde{a}] \int_{a}^{b} \mathcal{D}\phi(t) \exp\left\{ -S_S[\Tilde{a}, \phi(t)] \right\}.
\end{align}

Here $a \text{ and }b$ are eigenvalues of the operator $\hat{\phi}$ and $\Tilde{a}$ is eigenvalue of $\hat{A}$ <cit.>. The USC-MFGS for GCL-model is then given as

\begin{align}
\hat{\rho}_{\text{USC}} &= Z^{-1} \exp \left\{ -\beta \Tilde{H} \right\}\\
\Tilde{H} &\equiv \sum_i \hat{P}_i \left( \hat{H}_S + \hat{V}_0 \right) \hat{P}_i.
\end{align}

Here $Z$ is the normalization constant and $\hat{P}_i$ is the projection operator on the $i$th degenerate subspace of the operator $\hat{A}$ with eigenvalue $A_i$. Here, $\hat{V}_0$ is defined as

\begin{align}
\hat{V}_0 = \sum_i \log \left( \Tr\left( e^{-\beta \hat{H}_I(A_i)} \right) \right) \hat{P}_i.
\end{align}

The key takeaway here is that although the USC-MFGS for the GCL-model is still diagonal in the basis of the system coupling-operator $\hat{A}$, the actual expression does deviate from the CL-USC result. Determining the actual expression for the GCL-USC result would involve calculating the Gibbs partition function corresponding to the free environment Hamiltonian, $\hat{H}_I(A_i)$, as a function of different values of $A_i$, and hence a closed form expression is not possible without further information on the form of $\hat{H}_I(A_i)$.

§.§ Special Case

Since the CL-USC result gets modified for the GCL case, we will now identify a subclass within the GCL-model, which we will call the GCL2-model, for which the USC-MFGS result remains the same as that for the CL case, and hence can also be determined in closed form.
To this end, let us assume a special form of the GCL-model potential $V_k(x,y)$ (<ref>) as

\begin{align}
V_k(x,y) &= U_k(x-y).
\end{align}

Generalization of the CL-model of this form has been considered before in literature <cit.> and physically reflects a symmetry in the potential such that we have $V_k(x + w, y + w) = V_k(x,y)$. The function $\Tilde{x}_k(y)$, defined earlier, can now be expressed as

\begin{align}
\Tilde{x}_k(y) = y + x_k ,\label{eqn: x tilde for GCL2 model}
\end{align}

where $x_k$ is some constant. Note how $\frac{d}{dy} \Tilde{x}_k(y) = 1$ and hence the corresponding GCL-model assumption (<ref>) is automatically satisfied here. The requirement that $U_k(x)$ does not renormalize the free system Hamiltonian (<ref>), on the other hand, translates into

\begin{align}
U_k(x_k) &= 0 \label{eqn:x=y gives global minima}.
\end{align}

Similarly, the final GCL-model assumption (<ref>) translates here as

\begin{align}
\lim_{x\to\pm\infty} U_k(x_k + x) &\geq |O(x)| \label{eqn:diverging interaction potential for large distance}.
\end{align}

The remaining argument goes through just as before. The modified environment side action (<ref>),

\begin{align}
S_k[q_k(t), A(t)] \equiv \int_0^\beta dt \bigg\{ \frac{m_k}{2} \dot{q}_k(t)^2 \nonumber \\
\quad + U_k(q_k(t) - c_k A(t)) \bigg\},\label{eqn:GCL2 bath side action definition}
\end{align}

gives the following expression for the influence functional, analogous to <ref>,

\begin{align}
I_k[\Tilde{a}] &= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}q_k(t) e^{-S_k[q_k(t), \Tilde{a}]}\\
&= \int_{-\infty}^{\infty} dx \int_{x-c_k\Tilde{a}}^{x-c_k\Tilde{a}} \mathcal{D}x_k(t) e^{-S_k[ x_k(t), 0]} \label{eqn:GCL2 case I_k calc step 2}\\
&= \int_{-\infty}^{\infty} dx \int_{x}^{x} \mathcal{D}x_k(t) e^{-S_k[x_k(t), 0]}\\
&\equiv I_k, \quad \text{which is independent of}\; \Tilde{a}.
\end{align}

Here, in <ref>, we defined $x_k(t) \equiv q_k(t) - c_k \Tilde{a}$ and we have used the relation $S_k\left[q_k(t), \Tilde{a}\right] = S_k\left[x_k(t), 0 \right]$ (<ref>).

Hence, just like in the case of the CL-model, we again have that $I[\Tilde{a}] \equiv I $ is independent of $\Tilde{a}$. This again leads to the familiar CL-USC result (<ref>). This means that CL-USC result is valid for a larger class of open system model, namely for GCL2-model, as defined here.

§ USC-MFGS FOR ZWANZIG-MODEL

We will now study the USC-MFGS for the so-called Zwanzig-model <cit.>, which has been previously studied in the context of Brownian motion and barrier crossing problems. Let the full system-environment Hamiltonian for the Zwanzig-model be given as $\hat{H}_{SE} = \hat{H}_S + \hat{H}_I$, where $\hat{H}_S$ is the free system Hamiltonian and $\hat{H}_I$ is given as

\begin{align}
\hat{H}_I &= \sum_k^N\left[ \frac{\hat{p}_k^2}{2 m_k} + U_k^{\text{free}}(\hat{q}_k) + \frac{c_k}{2} U_k(\hat{q}_k - \hat{A}) \right] \label{eqn:new spring model}.
\end{align}

Here $U_k^{\text{free}}(q_k)$ is an arbitrary free potential of the $k$th environment particle and $U_k(x)$ is as defined in <ref>, but with just one extra condition that $U_k(x)$ now has a unique global minima point at $x=x_k$ and if there is any other local minima at a point $x=y_k$, then we have, for some non-infinitesimal positive number $\lambda$,

\begin{align}
U_k(y_k) - U_k(x_k) \geq \lambda > 0 \label{eqn:unique global minima}.
\end{align}

In other words, we now assume that not only does $U_k(x)$ have a unique global minimum, but also that if the potential has any other local minima, then there is a non-infinitesimal energy difference between the global minimum and the local minima. The significance of this assumption will become clear shortly.

Note that in <ref>, the interaction potential was of the form $U_k(\hat{q}_k - c_k \hat{A})$ while here, it is of the form $c_k U_k(\hat{q}_k - \hat{A})$. Hence, the Zwanzig-model is microscopically motivated to have a spring like interaction between the system and the environment particles. In fact, if $U_k(x)$ is a harmonic potential, then $c_k$ can be interpreted as the stiffness constant of the corresponding spring. Also note that here, we have deliberately not added a counter term in the interaction Hamiltonian because any renormalization of the system's free Hamiltonian will be assumed here to be of physical origin.

Now, the path integral expression for the MFGS will be given exactly as in CL-model (<ref> and <ref>), where the action associated with $I_k[A(t)]$ will now be given as

\begin{align}
S_k[q_k(t), A(t)] \equiv \int_0^\beta dt \bigg\{ \frac{m_k}{2} \dot{q}_k(t)^2 + U_k^{\text{free}}(q_k(t))\nonumber \\
\quad + \frac{c_k}{2} U_k(q_k(t) - A(t))\bigg\}  \label{eqn:new model A(t)}.
\end{align}

In the USC limit (<ref>), for paths that contribute significantly to the path integral, the condition

\begin{align}
q_k(t) = A(t) + x_k \label{eqn:q_k = A new model}
\end{align}

will get strictly imposed here. This is because as discussed in the context of the GCL2-model, $U_k(x_k)=0$ is defined as the global minima of the potential $U_k(x)$ (<ref>) and we have further assumed that for Zwanzig-model, this global minimum is unique and has a non-infinitesimal energy difference from any other local minima that might be there (<ref>). Hence, in the USC limit, we will have

\begin{align}
\lim_{c_k \to \infty} \exp\left\{ - c_k U_k(x)\right\} =
\begin{cases}
0 & \text{for } x \neq x_k \\
1 & \text{for } x = x_k
\end{cases}.
\end{align}

Replacing <ref> in <ref>, the corresponding influence functional (<ref>) becomes

\begin{align}
I_k[A(t)] &= e^{-\int_0^\beta dt \left\{ \frac{m_k}{2} \dot{A}(t)^2 + U_k^{\text{free}}(A(t) + x_k) \right\}} \label{eqn:zwanzig influence functional}.
\end{align}

Since the influence functional $I_k[A(t)]$ (<ref>) arises when we take partial trace over the environment degrees of freedom, the environment side paths, $q_k(t)$, come with a constraint that $q_k(0) = q_k(\beta)$. In the USC limit, since $q_k(t)$ gets further constrained as $q_k(t) = A(t) + x_k$ (<ref>), this in turn puts a constraint that for paths that contribute to the path integral, $A(0) = A(\beta)$. This effectively diagonalizes the final USC-MFGS in the basis of system coupling operator $\hat{A}$, but through a mathematical mechanism that is different from the one encountered in the GCL-model.

§.§ Special Case 1: Discrete system

If the system lives in a discrete Hilbert space, $A(t)$ generally has instantaneous jumps, in which case the kinetic term in the action in $I_k[A(t)]$ will blow up (<ref>), preventing the jump. Hence, for a discrete system, paths with an instantaneous jump will be suppressed and we will have $\dot{A}(t) = 0$.

Next, for $A(t) = \Tilde{a}$, the influence functional (<ref>) simplifies as

\begin{align}
\implies  I_k[\Tilde{a}] &= \exp \left\{-\int_0^\beta dt U_k^{\text{free}}(\Tilde{a} + x_k)  \right\}\\
&= \exp \left\{-\beta U_k^{\text{free}}(\Tilde{a} + x_k)  \right\}.
\end{align}

Replacing this in the full MFGS path integral expression (<ref>) gives

\begin{align}
\hat{\rho}_{\text{USC}}(a,b) &= \Tilde{Z}^{-1} e^{-\beta U_{\text{eff}}(\Tilde{a})   }\int_{a}^{b} \mathcal{D}\phi(t) \exp \left\{-S_S[\Tilde{a}, \phi(t)] \right\}.
\end{align}

Here $a \text{ and }b$ are the eigenvalues of the operator $\hat{\phi}$ and $\Tilde{a}$ is the eigenvalue of $\hat{A}$ <cit.> and $U_{\text{eff}}(x)$ is defined as

\begin{align}
U_{\text{eff}}(x) \equiv \sum_{k=1}^N U_k^{\text{free}}(x + x_k).
\end{align}

Finally, the USC-MFGS is given as,

\begin{align}
\implies \hat{\rho}_{\text{USC}} &= Z^{-1} \exp \left\{ -\beta \Tilde{H} \right\} \label{eqn:Zwanzig-model discrete case USC-MFGS}\\
\Tilde{H} &\equiv \sum_i \hat{P}_i (\hat{H}_S + \hat{V}_0) \hat{P}_i \label{eqn:beyond cl hmf}.
\end{align}

Here $\hat{P}_i$ is the projection operator on the $i$th degenerate subspaces of the operator $\hat{A}$ with eigenvalue $A_i$. Here $\hat{V}_0$ is defined as

\begin{align}
\hat{V}_0 = \sum_i U_{\text{eff}}(A_i) \hat{P}_i.
\end{align}

Hence, effectively, this is just the USC-MFGS result for GCL-model, but with a renormalized free system Hamiltonian (<ref>).

The trace distance between the USC-MFGS for the GCL2-model and numerically evaluated MFGS as a function of the coupling strength $c$, for different choices of the interaction potential (<ref>).

§.§ Special Case 2: Continuous Variable system

Now, assume that we are dealing with a CV system, where we have a particle with mass $m$ in a potential $V(q)$. Then, the free Hamiltonian $\hat{H}_S$ is given as

\begin{align}
\hat{H}_S &= \frac{\hat{p}^2}{2m} + V(\hat{q}).
\end{align}

Also, for simplicity, assume that the system coupling-operator is given as $\hat{A} = \hat{q}$. Then, using the simplified expression for the influence functional for Zwanzig-model (<ref>) in the full MFGS path integral (<ref>) gives the USC-MFGS as

\begin{align}
\hat{\rho}_{\text{USC}}(x,y) &= \Tilde{Z}^{-1} \delta(x,y) \int_{x}^{x} \mathcal{D}q(t)\nonumber \\
& \quad e^{-\int_0^\beta dt  \left( \frac{1}{2}M_{\text{eff}} \dot{q}(t)^2  + V_{\text{eff}}(q(t)) \right) }
\end{align}

\begin{align}
M_{\text{eff}} &= m + \sum_k m_k\\
V_{\text{eff}}(x) &= V(x) + U_{\text{eff}}(x)
\end{align}

The USC-MFGS can now be determined as

\begin{align}
\braket{ q | \rho_{\text{USC}} | q} &= Z^{-1} \braket{ q |e^{-\beta \hat{H}_{\text{eff}}} | q} \ket{q}\bra{q} \label{eqn:zwanzig USC result},\\
\hat{H}_{\text{eff}} &= \frac{\hat{p}^2}{2 M_{\text{eff}}} + V_{\text{eff}}(\hat{q}).
\end{align}

Note one key distinction here from the GCL2-USC result, in that although both the states are diagonal in the basis set by the system coupling-operator, the corresponding GCL2-USC result for CV system would be of the form, $\rho_{\text{USC-GCL2}} = Z^{-1} e^{-\beta V_{\text{eff}}(q)}$ <cit.>. The Zwanzig-USC result (<ref>) will converge to the GCL2 result only when $M_{\text{eff}} \to \infty$, which will suppress paths of the type $\dot{q}(t) \neq 0$ in the path integral. This will typically not be the case for a system interacting with a few environment particles or particles that are not very massive, for example.

We note that there has been a disagreement in the literature regarding the form of the USC equilibrium state under the CL-model <cit.>. For the CL-model, Kawai et al. conjectured the system's dynamical state at long times to be $\rho^{\text{conj}} = \sum_i \hat{P}_i e^{-\beta \hat{H}_0} \hat{P}_i$ <cit.> and we note that, coincidentally, the Zwanzig-USC result for CV systems (<ref>) is similar to this conjectured form.

§ NUMERICAL VERIFICATION

The trace distance between the USC-MFGS for the Zwanzig-model and the numerically calculated MFGS as a function of the coupling strength $c$.

In this section, we conduct numerical tests to validate the analytical results presented in this paper. Specifically, we verify the USC-MFGS results for both the GCL2 (<ref>) and the Zwanzig model (<ref>). Additionally, we examine how the convergence rate of the MFGS towards the USC results varies with an increase in the system-environment coupling, across different model parameters.

For both the GCL2 and Zwanzig model, we consider a simplified scenario involving a qutrit system interacting with a single environmental particle. This allows for the convenient numerical evaluation of the exact MFGS, denoted by $\hat{\rho}_{\text{num}}$.
By increasing the coupling strength, we can then compare the approach of $\hat{\rho}_{\text{num}}$ to the corresponding $\hat{\rho}_{\text{USC}}$ derived in this paper.
We note that the system MFGS will get closer to the USC-result if multiple environment particles are simultaneously made to interact it.

The system free Hamiltonian, $\hat{H}_S$, and the system coupling-operator, $\hat{A}$, are fixed throughout to be

\begin{align}
\hat{H}_S &= \begin{pmatrix}
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & -1
\end{pmatrix}, \quad
\hat{A} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -0.5
\end{pmatrix}.
\end{align}

We set $\beta = 5 E_h$ throughout, where we are working in the Hartree atomic units.
For the GCL-model, the interaction Hamiltonian, $\hat{H}_I$, (<ref>) simplifies as

\begin{align}
\hat{H}_I &= \frac{\hat{P}^2}{2 M} + V(\hat{Q} - c \hat{A}).
\end{align}

Here $M=1$ and $V(x)$ is assumed to be of the form

\begin{align}
V(x) &= \sum_{n=1}^3 a_{2n} x^{2n} \label{eqn:GCL2 interaction potential}.
\end{align}

<ref> plots the trace distance between $\hat{\rho}_{\text{num}}$ and $\hat{\rho}_{\text{USC}}$ (<ref>) for this model as we increase the coupling parameter, $c$, for different values of the positive semi-definite coefficients $a_{2n}$. We generically find that, as expected, the trace distance drops quickly as the value of $c$ is increased. We also find that higher order system-environment coupling causes the MFGS to approach the USC state faster as a function of the coupling parameter $c$.

This behaviour can be explained for a generic GCL-model if we note that a path $A(t)$ will contribute significantly to the path integral only if we have $\sigma_A \lesssim O(\Delta/c_k)$. For USC limit, we therefore require that $\Delta/c_k \ll 1$. Hence, the USC limit is approached based on the interplay between the coupling parameter $c_k$ and $\Delta$ <cit.>. So, if $\Delta$ is small for some coupling potential $V_k(x,y)$, like in the case of the potentials with $a_4 = 1$ or $a_6 = 1$ in <ref>, then it will facilitate the approach to the USC limit.

Next, for Zwanzig-model, the interaction Hamiltonian $\hat{H}_I$ (<ref>) is assumed to be

\begin{align}
\hat{H}_I &=  \frac{\hat{P}^2}{2 M} +  U(\hat{Q}) + \frac{c}{2} (\hat{Q}-\hat{q})^2.
\end{align}

Here, again, $M=1$ and $U(x)$ is a Morse potential

\begin{align}
U(x) \equiv (1 - e^{-x})^2.
\end{align}

<ref> plots the trace distance between $\hat{\rho}_{\text{num}}$ and $\hat{\rho}_{\text{USC}}$ (<ref>) for this model as we increase the value of the coupling parameter, $c$. We again observe that, as expected, the trace distance drops quickly as the value of $c$ is increased, though the rate of approach to the USC result is much slower than the GCL case.

§ CONCLUSION

The MFGS result, defined in <ref>, has been studied in USC limit for a system coupled to an anharmonic environment, hence going beyond the known USC result valid for the CL-model. For all the generalized system-environment models studied, the USC-MFGS has been found to always be diagonal in the basis set by the system-environment interaction, although the exact form of the state differs from the CL-USC result, like in the case of a quite general class of models that we call the GCL-model (<ref>). We also identify a subclass of the GCL-model, which we call the GCL2-model (<ref>), for which the USC-MFGS remains equivalent to the CL-USC result. Furthermore, for a so-called Zwanzig-model (distinct from the GCL-model) (<ref>), analytical expression for the USC state has been derived and is found to have qualitative differences from the GCL-USC result (<ref>).

Since the USC-MFGS acts as a good indicator for the deviation of the MFGS state from the textbook Gibbs state at large coupling, these results have immediate application in several aspects of strong coupling quantum thermodynamics as well as chemical and biological systems, where the effect of the anharmonicity of the environment becomes relevant. These results also demonstrate how accurate the CL-USC result is when the different assumptions of the model for which it was derived are relaxed one by one.

Our results across a broad range of models indicate that the USC-MFGS invariably becomes diagonal in the basis of the system coupling-operator. Looking forward, it would be interesting to investigate whether this diagonalization is a universal feature of the USC-limit, independent of the further details of the system-environment model. Additionally, exploring the MFGS in models beyond the CL-model at intermediate system-environment coupling would extend the applicability of the MFGS formalism to more realistic scenarios.

§ APPENDIX

§ PROPOSITION 1

Let $f(t)$ be a continuous, differentiable and square integrable function such that its derivative, $\dot{f}(t)$, is also square integrable, such that

\begin{align}
f(t) \equiv g(t) + \epsilon(t) \label{eqn:defining eqn}.
\end{align}

Here $g(t)$ is also square integrable. Next, for some $\Delta > 0$, we have the constraint that

\begin{align}
| \epsilon(t) | < \Delta \label{eqn: limit on epsilon}.
\end{align}

Next, let us define

\begin{align}
\braket{g} &\equiv \int_0^\beta dt g(t),\\
\braket{g^2} &\equiv \int_0^\beta dt g(t)^2,\\
\sigma_g^2 &\equiv \braket{g^2} - \braket{g}^2 \label{eqn: definition of sigma_g}.
\end{align}

Then, in the limit $\sigma_g^2 \gg \Delta^2$, we will prove that

\begin{align}
\int_0^\beta dt \dot{f}(t)^2 &\geq \mu \frac{\sigma_g^2}{\beta^2} \label{eqn: to prove},
\end{align}

where $\mu > 0$ is a constant.

The intuitive idea behind the proof is as following. Part 1 of the proof consists of proving that

\begin{align}
\lim_{\sigma_g^2 \gg \Delta^2}\sigma_f^2 \approx \sigma_g^2 \label{eqn:proof1 part 1}.
\end{align}

Part 2 then consists of proving that given $\sigma_f$,

\begin{align}
\int_0^\beta dt \dot{f}(t)^2 \geq \mu \sigma_f^2/\beta^2, \quad \text{where }\mu >0\label{eqn:proof1 part 2}.
\end{align}

Replacing <ref> into <ref> then gives the final result (<ref>).

Both these claims are intuitive, as the first claim states that if the absolute difference between two functions is much smaller than the standard deviation of one of the functions, then their standard deviations are also approximately equal. The second claim states that, for a function $f(t)$, if $\sigma_f^2>0$, then $\braket{\dot{f}^2} >0$ and also that $\braket{\dot{f}^2} \propto \sigma_f^2$, which makes sense because if you scale a function as $f(t) \to 2 f(t)$, then we have $\sigma_f^2 \to 4 \sigma_f^2$ and $\braket{\dot{f}^2} \to 4 \braket{\dot{f}^2}$.

\textbf{Part 1 of the proof:} Let us calculate $σ_f^2$ (using \autoref{eqn:defining eqn}) as
\begin{align}
\sigma_f^2 &= \braket{(g + \epsilon)^2} - \braket{g + \epsilon}^2\\
&= \braket{g^2} + 2 \braket{ g \epsilon} + \braket{\epsilon^2} - \left( \braket{g}^2 + 2 \braket{g} \braket{\epsilon} + \braket{\epsilon}^2 \right)\\
&= \sigma_g^2 + \sigma_\epsilon^2 + 2 \left( \braket{g \epsilon} - \braket{g} \braket{\epsilon} \right)\\
&\leq \sigma_g^2 + O(\Delta^2) + 2 \sigma_g O(\Delta).
\end{align}
Here, we used $⟨g ϵ|-⟩ ⟨g|⟨%s|%s⟩⟩ϵ ≤σ_g σ_ϵ$ that can be derived using the Cauchy-Schwarz inequality. Finally, in the limit $σ_g^2 ≫Δ^2$, we get the required result (\autoref{eqn:proof1 part 1}).

\textbf{Part 2 of the proof:} Proof of \autoref{eqn:proof1 part 2} follows directly from Poincaré inequality \cite{leoni2024first, evans2022partial} that can be formally stated as following \cite{ stover_poincare}.
\begin{mybox}[Poincaré inequality]
Let $\Omega$ be an open, bounded, and connected subset of $\mathbb{R}^d$ for some $d$ and let $d \mathbf{x}$ denote $d$-dimensional Lebesgue measure on $\mathbb{R}^d$. Then the Poincaré inequality says that there exist constants $C_1$ and $C_2$ such that
\begin{align}
\int_\Omega h^2(\mathbf{x}) d \mathbf{x} &\leq C_1 \int_\Omega | \nabla h(\mathbf{x})|^2 d \mathbf{x} \nonumber\\
& \quad + C_2 \left[ \int_\Omega h(\mathbf{x}) d \mathbf{x} \right]^2,
\end{align}
for all functions $h$ in the Sobolev Space $H^1(\Omega)$ consisting of all functions in $L^2(\Omega)$ whose generalized derivatives are all also square integrable.
\end{mybox}

For 1-dimensional Euclidean space, for simplicity and without loss of any generality, let us redefine $f(t)$ such that $⟨f|=⟩ 0$. In other words, we shift $f(t) →f(t) - ⟨f|$⟩. Then, the Poincaré inequality implies that

\begin{align}
\braket{f^2} &\leq C_1 \braket{\dot{f}^2} + C_2 \braket{f}^2\\
\implies \sigma_f^2 &\leq C_1 \braket{\dot{f}^2}.
\end{align}

Since $\sigma_f^2 \geq 0$ and $ \braket{\dot{f}^2} \geq 0$, we have that $C_1$ is positive. We therefore have

\begin{align}
\braket{\dot{f}^2} \geq \frac{\beta^2}{C_1} \frac{\sigma_g^2}{\beta^2}.
\end{align}

This proves <ref> for $\mu = \beta^2/C_1$ and concludes the proof of the main result (<ref>) as well.

§.§ Proof of special case of Poincaré Inequality

Here, for completeness, we provide a proof for the special case of Poincaré inequality (<ref>), as per the present requirement. To simplify the proof, other than assuming that $\braket{f}=0$, we also shift the range of the imaginary time from the conventional $0 \leq t \leq \beta$ to $-\beta/2 \leq t \leq \beta/2$. Now, given a constraint

\begin{align}
\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} f(t)^2 dt = \sigma_f^2 \label{eqn:constraint equation},
\end{align}

we need to minimize over the action

\begin{align}
S = \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} dt \dot{f}(t)^2 \label{eqn: kinetic action},
\end{align}

We do this by perturbing the optimum path, $f(t) \to f(t) + \delta(t)$, with $\delta(-\beta/2) = \delta(\beta/2) = 0$. Then <ref> demands that

\begin{align}
\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} f(t) \delta(t) dt = 0 \label{eqn:contrained perturbation}.
\end{align}

Now, the condition for the minimum action is $\delta S = 0$, where

\begin{align}
\frac{\delta S}{2} &= \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} dt \dot{f}(t)\dot{\delta}(t)\\
&= \dot{f}(\beta/2) \delta(\beta/2) - \dot{f}(-\beta/2) \delta(-\beta/2)\nonumber\\
& \quad \quad - \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} dt \ddot{f}(t)\delta(t)\\
&= -\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} dt \ddot{f}(t)\delta(t) = 0\label{eqn:equation for minimum action}.
\end{align}

Comparing <ref> with <ref>, for some constant $\lambda$, we get

\begin{align}
\ddot{f}(t) &= \pm \lambda^2 f(t).
\end{align}

The solution of $f(t)$ can be of either sinusoidal or hyperbolic type. For the sinusoidal solution, we have

\begin{align}
f(t) = A \sin(\lambda t + \phi).
\end{align}

Then the condition, $\braket{f} = 0$, demands that

\begin{align}
\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} A \sin(\lambda t + \phi) &= 0\\
\implies -\frac{A}{\lambda} \left(\cos\left(\frac{\lambda \beta}{2} + \phi\right) - \cos\left(-\frac{\lambda \beta}{2} + \phi\right)\right) &= 0\\
\implies \cos\left(\frac{\lambda \beta}{2} + \phi\right) = \cos\left(-\frac{\lambda \beta}{2} + \phi\right)&.
\end{align}

The solution is either, for a generic $\lambda$, $\phi = 0$ or for a generic $\phi$, $\lambda = 2 \pi n/\beta$ for $n\in \mathbb{Z}$. Hence, we have

\begin{align}
f(t) =
\begin{cases}
A \sin(\lambda t) \quad \text{ for generic } \lambda\\
A \sin(\lambda t + \phi) \quad \text{ for }\lambda = 2 \pi n/\beta; \quad n \in \mathbb{Z}\\
\end{cases}.
\end{align}

For hyperbolic solution, $f(t)$ is given as

\begin{align}
f(t) = A \sinh(\lambda t).
\end{align}

Here, we do not have a $\cosh$ kind of term because of the requirement that $\braket{f}= \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} dt f(t) = 0$.

§.§.§ Sinusoidal solution

Let us first consider the solution for a generic $\lambda$ and $\phi = 0$

\begin{align}
f(t) &\equiv A \sin(\lambda t)\label{eqn:sinusoidal phi=0 case}\\
\implies \braket{f^2} &= A^2 \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} \sin^2(\lambda t) dt\\
&= \frac{A^2}{2} \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} 1 - \cos(2\lambda t) dt\label{eqn: sinusoidal 3rd line}\\
&= \frac{A^2}{2} \left( \beta - \frac{2 \sin(\lambda \beta)}{2 \lambda} \right) \\
&= \frac{A^2 \beta}{2} \left( 1 - \frac{\sin(x)}{x} \right) \quad ;x \equiv \lambda \beta\\
\implies A^2 &= \frac{ \braket{f^2} }{\beta} \frac{ 2x}{x- \sin(x)}\label{eqn:sinusoidal A^2}.
\end{align}

This expression for $A^2$ is crucial, so let us also evaluate it for the case when, in <ref>, $\phi$ is present and $\lambda = 2 \pi n/\beta$ for $n\in \mathbb{Z}$.

\begin{align}
f(t) &= A \sin\left(\frac{2 \pi n t}{\beta} + \phi\right)\\
\implies \braket{f^2} &= A^2 \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} \sin^2\left(\frac{2 \pi n t}{\beta} + \phi\right) dt\\
&= \frac{A^2 \beta}{2}\\
\implies A^2 &= 2 \frac{\braket{f^2}}{\beta}\\
&= \frac{\braket{f^2}}{\beta} \frac{ 2x}{x- \sin(x)} \bigg{|}_{x = 2 \pi n}
\end{align}

This is consistent with <ref>. Hence, <ref> holds generically for the full sinusoidal case. Next, we have that

\begin{align}
f(t)^2 + \frac{\dot{f}(t)^2}{\lambda^2} &= A^2\\
\implies \braket{f^2} + \frac{\braket{\dot{f}^2}}{\lambda^2} &= A^2 \beta
\end{align}

\begin{align}
\implies \braket{\dot{f}^2} &= \lambda^2 \left( A^2 \beta - \braket{f^2} \right) \label{eqn:sinusoidal step 3}\\
&= \frac{x^2}{\beta^2} \left( \braket{f^2} \frac{2 x}{x- \sin(x)}- \braket{f^2} \right) \label{eqn:sinusoidal step 4}\\
&= \frac{\braket{f^2} }{\beta^2} x^2 \left( \frac{x + \sin(x)}{x - \sin(x)}\right)\\
&\geq \left(\min_x h_{\text{sin}}(x) \right) \frac{\braket{f^2} }{\beta^2} \label{eqn:sinusoidal solution}.
\end{align}

\begin{align}
h_{\text{sin}}(x) &\equiv x^2 \left( \frac{x + \sin(x)}{x - \sin(x)}\right) \label{eqn: h sin x}.
\end{align}

Note that $\min_{x} h_{\text{sin}}(x) > 0$, which can also be seen from <ref>. <ref> is the desired result for the sinusoidal case.

Plot of $h_{\text{sin}}(x)$ (<ref>) and $h_{\text{hyp}}(x)$ (<ref>), which demonstrates that the minima of both these functions are larger than zero.

§.§.§ Hyperbolic solution

\begin{align}
f(t) &\equiv A \sinh(\lambda t)\\
\braket{f^2} &\equiv A^2 \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} \sinh^2(\lambda t) dt\\
&= - \frac{A^2}{2} \int_{-\frac{\beta}{2}}^{\frac{\beta}{2}} 1 - \cosh(2\lambda t) dt\\
\implies A^2 &= -\frac{\braket{f^2}}{\beta} \frac{2x}{x - \sinh(x)}.
\end{align}

Here, in the last step, we used the analogous derivation from <ref> to <ref>, but with an extra minus sign. Now, note that we have

\begin{align}
\frac{\dot{f}(t)^2}{\lambda^2} - f(t)^2 &= A^2\\
\implies \frac{\braket{\dot{f}^2}}{\lambda^2} - \braket{f^2} &= A^2 \beta
\end{align}

\begin{align}
\implies \braket{\dot{f}^2} &= -\lambda^2 \left( -A^2 \beta - \braket{f^2} \right) \label{eqn:hyperbolic step 3}\\
&= -\frac{x^2}{\beta^2} \left( \braket{f^2} \frac{2x}{x - \sinh(x)} - \braket{f^2} \right)\label{eqn:hyperbolic step 4}\\
&\geq \left(\min_x h_{\text{hyp}}(x) \right) \frac{\braket{f^2} }{\beta^2} \label{eqn:hyperbolic solution}.
\end{align}

In the last step, we used the analogous calculation from <ref> to <ref>, with just an extra minus sign. Here, $h_{\text{hyp}}(x)$ is defined as

\begin{align}
h_{\text{hyp}}(x) &\equiv x^2 \left( \frac{\sinh(x) + x}{\sinh(x) - x}\right)\label{eqn: h hyp x}.
\end{align}

Note that $\min_{x} h_{\text{hyp}}(x) > 0$, which can also be seen from <ref>. <ref> is the desired result for the hyperbolic case.

§ PROPOSITION 2

Let $f(x)$ be a continuous, differentiable and square integrable function and let us define $h(x) \equiv \frac{d}{dx} f(x)$ such that

\begin{align}
h(x)  \gtrsim O(1) \label{eqn: constraint on the derivation of h(x)}.
\end{align}

Now, let $g(t)$ be another square integrable function such that

\begin{align}
\braket{f \circ g} &\equiv \int_0^\beta dt f(g(t))\\
\braket{(f \circ g)^2} &\equiv \int_0^\beta dt f(g(t))^2\\
\sigma_{f \circ g}^2 &\equiv \braket{(f \circ g)^2} - \braket{f \circ g}^2.
\end{align}

Then, we will prove that

\begin{align}
\sigma_{f \circ g}^2 \gtrsim \sigma_g^2,
\end{align}

where $\sigma_g^2$ is defined analogously (<ref>).

We first write

\begin{align}
f(x_1) &= f(x_0) + \int_{x_0}^{x_1} h(x') dx'.
\end{align}

Now, from <ref>, if $x_1 > x_0$, then

\begin{align}
\int_{x_0}^{x_1} h(x') dx' &\gtrsim x_1 - x_0 > 0\\
\implies \int_{x_1}^{x_0} h(x') dx' &\lesssim x_0 - x_1 < 0\\
\implies \left| \int_{a}^{b} h(x') dx' \right| &\gtrsim |b - a| \quad \forall a,b\\
\implies |f(b) - f(a)| &\gtrsim |b - a|.
\end{align}

Then we have

\begin{align}
|(f \circ g)(t_1) - (f \circ g)(t_2)| &\gtrsim |g(t_1) - g(t_2)| \quad \forall t_1, t_2.
\end{align}

Now, let $t'$ be the time at which we have $(f \circ g)(t') = \braket{(f \circ g)}$. Then we have
\begin{align}
\sigma_{f \circ g}^2 &= \int_0^\beta dt ((f \circ g)(t) - (f \circ g)(t'))^2\\
&\gtrsim \int_0^\beta dt (g(t) - g(t'))^2\\
&\geq \int_0^\beta dt (g(t) - \braket{g})^2\\
\implies \sigma_{f \circ g}^2 &\gtrsim \sigma_g^2.
\end{align}

\end{appendices}
\end{document}