Shankar Balasubramanian, Sarang Gopalakrishnan, Alexey Khudorozhkov, Ethan Lake
{"title":"玻璃文字问题:超低松弛、希尔伯特空间干扰和计算复杂性","authors":"Shankar Balasubramanian, Sarang Gopalakrishnan, Alexey Khudorozhkov, Ethan Lake","doi":"10.1103/physrevx.14.021034","DOIUrl":null,"url":null,"abstract":"We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> exhibit almost no relaxation until times <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>O</mi><mo mathvariant=\"bold\" stretchy=\"false\">(</mo><mi>exp</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo stretchy=\"false\">)</mo><mo mathvariant=\"bold\" stretchy=\"false\">)</mo></mrow></math>, at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions.","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":null,"pages":null},"PeriodicalIF":11.6000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity\",\"authors\":\"Shankar Balasubramanian, Sarang Gopalakrishnan, Alexey Khudorozhkov, Ethan Lake\",\"doi\":\"10.1103/physrevx.14.021034\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>q</mi></math> exhibit almost no relaxation until times <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><mi>O</mi><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">(</mo><mi>exp</mi><mo stretchy=\\\"false\\\">(</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo stretchy=\\\"false\\\">)</mo><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">)</mo></mrow></math>, at which point they abruptly collapse. 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Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity
We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector exhibit almost no relaxation until times , at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions.
期刊介绍:
Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.