July | 2023 | Quantum Frontiers

Mark Srednicki doesn’t look like a high priest. He’s a professor of physics at the University of California, Santa Barbara (UCSB); and you’ll sooner find him in khakis than in sacred vestments. Humor suits his round face better than channeling divine wrath would; and I’ve never heard him speak in tongues—although, when an idea excites him, his hands rise to shoulder height of their own accord, as though halfway toward a priestly blessing. Mark belongs less on a ziggurat than in front of a chalkboard. Nevertheless, he called himself a high priest.

Specifically, Mark jokingly called himself a high priest of the eigenstate thermalization hypothesis, a framework for understanding how quantum many-body systems thermalize internally. The eigenstate thermalization hypothesis has an unfortunate number of syllables, so I’ll call it the ETH. The ETH illuminates closed quantum many-body systems, such as a clump of $N$ ultracold atoms. The clump can begin in a pure product state $| \psi(0) \rangle$ , then evolve under a chaotic¹ Hamiltonian $H$ . The time- $t$ state $| \psi(t) \rangle$ will remain pure; its von Neumann entropy will always vanish. Yet entropy grows according to the second law of thermodynamics. Breaking the second law amounts almost to a enacting a miracle, according to physicists. Does the clump of atoms deserve consideration for sainthood?

No—although the clump’s state remains pure, a small subsystem’s state does not. A subsystem consists of, for example, a few atoms. They’ll entangle with the other atoms, which serve as an effective environment. The entanglement will mix the few atoms’ state, whose von Neumann entropy will grow.

The ETH predicts this growth. The ETH is an ansatz about $H$ and an operator $O$ —say, an observable of the few-atom subsystem. We can represent $O$ as a matrix relative to the energy eigenbasis. The matrix elements have a certain structure, if $O$ and $H$ satisfy the ETH. Suppose that the operators do and that $H$ lacks degeneracies—that no two energy eigenvalues equal each other. We can prove that $O$ thermalizes: Imagine measuring the expectation value $\langle \psi(t) | O | \psi(t) \rangle$ at each of many instants $t$ . Averaging over instants produces the time-averaged expectation value $\overline{ \langle O \rangle_t }$ .

Another average is the thermal average—the expectation value of $O$ in the appropriate thermal state. If $H$ conserves just itself,² the appropriate thermal state is the canonical state, $\rho_{\rm can} := e^{-\beta H}/ Z$ . The average energy $\langle \psi(0) | H | \psi(0) \rangle$ defines the inverse temperature $\beta$ , and $Z$ normalizes the state. Hence the thermal average is $\langle O \rangle_{\rm th}$ $:=$ ${\rm Tr} ( O \rho_{\rm can} )$ .

The time average approximately equals the thermal average, according to the ETH: $\overline{ \langle O \rangle_t }$ $= \langle O \rangle_{\rm th} + O \big( N^{-1} \big)$ . The correction is small in the total number $N$ of atoms. Through the lens of $O$ , the atoms thermalize internally. Local observables tend to satisfy the ETH, and we can easily observe only local observables. We therefore usually observe thermalization, consistently with the second law of thermodynamics.

I agree that Mark Srednicki deserves the title high priest of the ETH. He and Joshua Deutsch independently dreamed up the ETH in 1994 and 1991. Since numericists reexamined it in 2008, studies and applications of the ETH have exploded like a desert religion. Yet Mark had never encountered the question I posed about it in 2021. Next month’s blog post will share the good news about that question.

¹Nonintegrable.

²Apart from trivial quantities, such as projectors onto eigenspaces of $H$ .