Locally controlled arrested thermalization

Letter

Locally controlled arrested thermalization

Ken K. W. Ma and Hitesh J. Changlani

Phys. Rev. B 109, L180301 – Published 9 May 2024

Abstract

The long-time dynamics of quantum systems, typically, but not always, results in a thermal steady state. The microscopic processes that lead to or circumvent this fate are of interest, since everyday experience tells us that not all spatial regions of a system heat up or cool down uniformly. This motivates the question: Under what conditions can one slow down or completely arrest thermalization locally? Is it possible to construct realistic Hamiltonians and initial states such that a local region is effectively insulated from the rest, or acts as a barrier between two or more regions? We answer this in the affirmative by outlining the conditions that govern the flow of energy and entropy between subsystems. Using these ideas we provide a representative example for how simple quantum few-body states can be used to engineer a “thermal switch” between interacting regions.

Received 19 June 2023
Accepted 22 March 2024

DOI:https://doi.org/10.1103/PhysRevB.109.L180301

Physics Subject Headings (PhySH)

Eigenstate thermalization Quantum thermodynamics Spin dynamics

Magnetic systems Nonequilibrium systems

Exact diagonalization Quantum spin chains

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Ken K. W. Ma ¹ and Hitesh J. Changlani ^1,2

¹National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
²Department of Physics, Florida State University, Tallahassee, Florida 32306, USA

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Vol. 109, Iss. 18 — 1 May 2024

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Images

Figure 1
Panel (a) shows the bipartite one-dimensional Ising chain under an external magnetic field, with $g_{j}$ being the field strength at site $j$ . To achieve commuting subsystem Hamiltonians, the magnetic field at the boundary (i.e., the fifth and sixth spins) is set to zero. Panel (b) shows the Heisenberg chain with a thermal switch (formed by the fifth and sixth spins). Here, dots, squares (left panel), and bonds (right panel) in red, blue, and orange are in subsystems $A$ , $B$ , and their intervening region $A B$ , respectively. Note that $A B$ has support in both $A$ and $B$ .
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Figure 2
Time evolution of the half-cut entanglement entropy between the two subsystems [(a)] for the tilted field Ising model described by Eqs. (6) and (7), and the expectation values of energy for the subsystems [(b)]. The spin chain consists of ten spins. Each subsystem has five spins. Three different scenarios (the spin-spin interaction for the two spins at the boundary) are considered, which are explained by the legends in the figure. The initial states for all three cases are ${| ↑ ↑ ↑ ↑ ↑ 〉}_{A} \otimes {| ran 〉}_{B}$ , where ${| ran 〉}_{B}$ stands for a random state for the five spins in subsystem $B$ . The inset of (a) shows the half-cut EE of the system when $A$ is prepared in the initial state ${| x 〉}^{\otimes 5}$ , where $| x 〉 = (| ↑ 〉 + | ↓ 〉) / \sqrt{2}$ .
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Figure 3
Spatiotemporal profile of the estimated energy at each bond [(a)–(c)], spin magnetization $〈 S_{j}^{z} 〉$ for all individual spins [(d)–(f)], and their corresponding von Neumann entropy [(g)–(i)] in the tilted field Ising model. Three different kinds of boundary are considered. For the left, middle, and right columns, the interaction terms are $H_{A B} = - S_{5}^{x} S_{6}^{x}$ , $H_{A B} = - S_{5}^{z} S_{6}^{z}$ , and $H_{A B} = - S_{5}^{z} S_{6}^{x}$ , respectively. In all cases, the initial state of the system is ${| ↑ ↑ ↑ ↑ ↑ 〉}_{A} \otimes {| ran 〉}_{B}$ . Here, we simulate the dynamics of the system for $0 \leq t \leq 5 \times 10^{4}$ . The insets of the left column show the results in the short-time interval $0 \leq t \leq 100$ . See the main text for more details.
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Figure 4
Time evolution of the bipartite entanglement entropy with subsystem $A$ defined as the first five spins [(a)], and the expected energy for the left and right subsystems [(b)]. The left subsystem is defined as the spins $1 -- 4$ and the “left triangle” formed by spins 4,5,6, whereas the right subsystem is defined as the “right triangle” formed by spins 5,6,7 and the spins $7 -- 10$ (see Fig. 1). (c) and (e) show respectively the time evolution of $〈 S_{j}^{z} (t) 〉$ and the von Neumann entropy for every individual spin. Here, the initial state of the bond (spins 5 and 6) is set to the singlet state $| S 〉 = (| ↑ ↓ 〉 - | ↓ ↑ 〉) / \sqrt{2}$ . In (d) and (f), the corresponding results for a triplet initial state of the bond $| T 〉 = (| ↑ ↓ 〉 + | ↓ ↑ 〉) / \sqrt{2}$ are illustrated. The insets show the short-time behavior from $t = 0$ to $t = 50$ .
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