1
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Bagchi D, Kethepalli J, Bulchandani VB, Dhar A, Huse DA, Kulkarni M, Kundu A. Unusual ergodic and chaotic properties of trapped hard rods. Phys Rev E 2023; 108:064130. [PMID: 38243434 DOI: 10.1103/physreve.108.064130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Accepted: 11/22/2023] [Indexed: 01/21/2024]
Abstract
We investigate ergodicity, chaos, and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly nonergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.
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Affiliation(s)
- Debarshee Bagchi
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Jitendra Kethepalli
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Vir B Bulchandani
- Department of Physics, Princeton University, New Jersey 08544, USA
- Princeton Center for Theoretical Science, Princeton University, New Jersey 08544, USA
| | - Abhishek Dhar
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - David A Huse
- Department of Physics, Princeton University, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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2
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Roy D, Huse DA, Kulkarni M. Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations. Phys Rev E 2023; 108:054112. [PMID: 38115452 DOI: 10.1103/physreve.108.054112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Accepted: 10/11/2023] [Indexed: 12/21/2023]
Abstract
The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its "light cone" in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equation exhibits a qualitatively similar OTOC and light cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ, which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity, and dissipation in partial differential equations (deterministic or stochastic) through the lens of OTOC.
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Affiliation(s)
- Dipankar Roy
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
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3
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Ha H, Morningstar A, Huse DA. Many-Body Resonances in the Avalanche Instability of Many-Body Localization. Phys Rev Lett 2023; 130:250405. [PMID: 37418717 DOI: 10.1103/physrevlett.130.250405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2023] [Accepted: 06/03/2023] [Indexed: 07/09/2023]
Abstract
Many-body localized (MBL) systems fail to reach thermal equilibrium under their own dynamics, even though they are interacting, nonintegrable, and in an extensively excited state. One instability toward thermalization of MBL systems is the so-called "avalanche," where a locally thermalizing rare region is able to spread thermalization through the full system. The spreading of the avalanche may be modeled and numerically studied in finite one-dimensional MBL systems by weakly coupling an infinite-temperature bath to one end of the system. We find that the avalanche spreads primarily via strong many-body resonances between rare near-resonant eigenstates of the closed system. Thus we find and explore a detailed connection between many-body resonances and avalanches in MBL systems.
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Affiliation(s)
- Hyunsoo Ha
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Alan Morningstar
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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4
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Shivam S, De Luca A, Huse DA, Chan A. Many-Body Quantum Chaos and Emergence of Ginibre Ensemble. Phys Rev Lett 2023; 130:140403. [PMID: 37084451 DOI: 10.1103/physrevlett.130.140403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2022] [Revised: 03/06/2023] [Accepted: 03/08/2023] [Indexed: 05/03/2023]
Abstract
We show that non-Hermitian Ginibre random matrix behaviors emerge in spatially extended many-body quantum chaotic systems in the space direction, just as Hermitian random matrix behaviors emerge in chaotic systems in the time direction. Starting with translational invariant models, which can be associated with dual transfer matrices with complex-valued spectra, we show that the linear ramp of the spectral form factor necessitates that the dual spectra have nontrivial correlations, which in fact fall under the universality class of the Ginibre ensemble, demonstrated by computing the level spacing distribution and the dissipative spectral form factor. As a result of this connection, the exact spectral form factor for the Ginibre ensemble can be used to universally describe the spectral form factor for translational invariant many-body quantum chaotic systems in the scaling limit where t and L are large, while the ratio between L and L_{Th}, the many-body Thouless length is fixed. With appropriate variations of Ginibre models, we analytically demonstrate that our claim generalizes to models without translational invariance as well. The emergence of the Ginibre ensemble is a genuine consequence of the strongly interacting and spatially extended nature of the quantum chaotic systems we consider, unlike the traditional emergence of Hermitian random matrix ensembles.
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Affiliation(s)
- Saumya Shivam
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Andrea De Luca
- Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, F-95302 Cergy-Pontoise, France
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Amos Chan
- Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
- Physics Department, Lancaster University, Lancaster, LA1 4YW, United Kingdom
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5
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Sommers GM, Gullans MJ, Huse DA. Self-dual quasiperiodic percolation. Phys Rev E 2023; 107:024137. [PMID: 36932570 DOI: 10.1103/physreve.107.024137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2022] [Accepted: 01/20/2023] [Indexed: 06/18/2023]
Abstract
How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of D_{f}=1.911943(1) and D_{f}=1.707234(40) for the two models, significantly different from D_{f}=91/48=1.89583... of random percolation. The critical exponents ν, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the ν=4/3 of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior.
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Affiliation(s)
- Grace M Sommers
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Michael J Gullans
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Institute for Advanced Study, Princeton, New Jersey 08540, USA
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6
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Christakis L, Rosenberg JS, Raj R, Chi S, Morningstar A, Huse DA, Yan ZZ, Bakr WS. Probing site-resolved correlations in a spin system of ultracold molecules. Nature 2023; 614:64-69. [PMID: 36725998 DOI: 10.1038/s41586-022-05558-4] [Citation(s) in RCA: 9] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 11/11/2022] [Indexed: 02/03/2023]
Abstract
Synthetic quantum systems with interacting constituents play an important role in quantum information processing and in explaining fundamental phenomena in many-body physics. Following impressive advances in cooling and trapping techniques, ensembles of ultracold polar molecules have emerged as a promising platform that combines several advantageous properties1-11. These include a large set of internal states with long coherence times12-17 and long-range, anisotropic interactions. These features could enable the exploration of intriguing phases of correlated quantum matter, such as topological superfluids18, quantum spin liquids19, fractional Chern insulators20 and quantum magnets21,22. Probing correlations in these phases is crucial to understanding their properties, necessitating the development of new experimental techniques. Here we use quantum gas microscopy23 to measure the site-resolved dynamics of quantum correlations of polar 23Na87Rb molecules confined in a two-dimensional optical lattice. By using two rotational states of the molecules, we realize a spin-1/2 system with dipolar interactions between particles, producing a quantum spin-exchange model21,22,24,25. We study the evolution of correlations during the thermalization process of an out-of-equilibrium spin system for both spatially isotropic and anisotropic interactions. Furthermore, we examine the correlation dynamics of a spin-anisotropic Heisenberg model engineered from the native spin-exchange model by using periodic microwave pulses26-28. These experiments push the frontier of probing and controlling interacting systems of ultracold molecules, with prospects for exploring new regimes of quantum matter and characterizing entangled states that are useful for quantum computation29,30 and metrology31.
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Affiliation(s)
| | | | - Ravin Raj
- Department of Physics, Princeton University, Princeton, NJ, USA
| | - Sungjae Chi
- Department of Physics, Princeton University, Princeton, NJ, USA
| | | | - David A Huse
- Department of Physics, Princeton University, Princeton, NJ, USA
| | - Zoe Z Yan
- Department of Physics, Princeton University, Princeton, NJ, USA
| | - Waseem S Bakr
- Department of Physics, Princeton University, Princeton, NJ, USA.
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7
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Chan A, Shivam S, Huse DA, De Luca A. Many-body quantum chaos and space-time translational invariance. Nat Commun 2022; 13:7484. [PMID: 36470877 PMCID: PMC9722696 DOI: 10.1038/s41467-022-34318-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2021] [Accepted: 10/18/2022] [Indexed: 12/12/2022] Open
Abstract
We study the consequences of having translational invariance in space and time in many-body quantum chaotic systems. We consider ensembles of random quantum circuits as minimal models of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor as a sum over many-body Feynman diagrams in the limit of large local Hilbert space dimension q. At sufficiently large t, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) behaviour. At finite t, we show that translational invariance introduces additional mechanisms via two novel Feynman diagrams which delay the emergence of RMT. Our analytics suggests the existence of exact scaling forms which describe the approach to RMT behavior in the scaling limit where both t and L are large while the ratio between L and LTh(t), the many-body Thouless length, is fixed. We numerically demonstrate, with simulations of two distinct circuit models, that the resulting scaling functions are universal in the scaling limit.
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Affiliation(s)
- Amos Chan
- grid.16750.350000 0001 2097 5006Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544 USA ,Physics Department, Lancaster University, Lancaster, LA1 4YW USA
| | - Saumya Shivam
- grid.16750.350000 0001 2097 5006Department of Physics, Princeton University, Princeton, NJ 08544 USA
| | - David A. Huse
- grid.16750.350000 0001 2097 5006Department of Physics, Princeton University, Princeton, NJ 08544 USA
| | - Andrea De Luca
- grid.507676.5Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, F-95302 Cergy-Pontoise, France
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8
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Micklitz T, Morningstar A, Altland A, Huse DA. Emergence of Fermi's Golden Rule. Phys Rev Lett 2022; 129:140402. [PMID: 36240392 DOI: 10.1103/physrevlett.129.140402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 08/18/2022] [Indexed: 06/16/2023]
Abstract
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.
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Affiliation(s)
- Tobias Micklitz
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
| | - Alan Morningstar
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Alexander Altland
- Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Cologne, Germany
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Institute for Advanced Study, Princeton, New Jersey 08540, USA
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9
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Barratt F, Agrawal U, Gopalakrishnan S, Huse DA, Vasseur R, Potter AC. Field Theory of Charge Sharpening in Symmetric Monitored Quantum Circuits. Phys Rev Lett 2022; 129:120604. [PMID: 36179163 DOI: 10.1103/physrevlett.129.120604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2021] [Accepted: 08/17/2022] [Indexed: 06/16/2023]
Abstract
Monitored quantum circuits (MRCs) exhibit a measurement-induced phase transition between area-law and volume-law entanglement scaling. MRCs with a conserved charge additionally exhibit two distinct volume-law entangled phases that cannot be characterized by equilibrium notions of symmetry-breaking or topological order, but rather by the nonequilibrium dynamics and steady-state distribution of charge fluctuations. These include a charge-fuzzy phase in which charge information is rapidly scrambled leading to slowly decaying spatial fluctuations of charge in the steady state, and a charge-sharp phase in which measurements collapse quantum fluctuations of charge without destroying the volume-law entanglement of neutral degrees of freedom. By taking a continuous-time, weak-measurement limit, we construct a controlled replica field theory description of these phases and their intervening charge-sharpening transition in one spatial dimension. We find that the charge fuzzy phase is a critical phase with continuously evolving critical exponents that terminates in a modified Kosterlitz-Thouless transition to the short-range correlated charge-sharp phase. We numerically corroborate these scaling predictions also hold for discrete-time projective-measurement circuit models using large-scale matrix-product state simulations, and discuss generalizations to higher dimensions.
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Affiliation(s)
- Fergus Barratt
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Utkarsh Agrawal
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Sarang Gopalakrishnan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Institute for Advanced Study, Princeton, New Jersey 08540, USA
| | - Romain Vasseur
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Andrew C Potter
- Department of Physics and Astronomy, and Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
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10
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Zabalo A, Gullans MJ, Wilson JH, Vasseur R, Ludwig AWW, Gopalakrishnan S, Huse DA, Pixley JH. Operator Scaling Dimensions and Multifractality at Measurement-Induced Transitions. Phys Rev Lett 2022; 128:050602. [PMID: 35179942 DOI: 10.1103/physrevlett.128.050602] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2021] [Accepted: 01/05/2022] [Indexed: 06/14/2023]
Abstract
Repeated local measurements of quantum many-body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar critical exponents, making it unclear how many distinct universality classes are present. Here, we probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points for (1+1)-dimensional systems. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large on-site Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.
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Affiliation(s)
- A Zabalo
- Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
| | - M J Gullans
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - J H Wilson
- Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
- Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
- Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
| | - R Vasseur
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - A W W Ludwig
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - S Gopalakrishnan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
- Department of Physics and Astronomy, CUNY College of Staten Island, Staten Island, New York 10314, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - J H Pixley
- Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
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11
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S BK, Huse DA, Kulkarni M. Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators. Phys Rev E 2021; 104:044117. [PMID: 34781511 DOI: 10.1103/physreve.104.044117] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2020] [Accepted: 09/23/2021] [Indexed: 11/07/2022]
Abstract
We present exact results for the classical version of the out-of-time-order commutator (OTOC) for a family of power-law models consisting of N particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form ∝∑_{i,j=1(i≠j)}^{N}|x_{i}-x_{j}|^{-k}∀k>1 where x_{i} is the position of the ith particle. We present numerical results for the OTOC for finite N at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many-body ground state. In the large-N limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone." We find remarkably distinct features in OTOC below and above k=3 in terms of going from non-Airy behavior (1<k<3) to an Airy universality class (k>3). We present certain additional rich features for the case k=2 that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatiotemporal spread of perturbations in long-range interacting systems.
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Affiliation(s)
- Bhanu Kiran S
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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12
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Monteiro F, Tezuka M, Altland A, Huse DA, Micklitz T. Quantum Ergodicity in the Many-Body Localization Problem. Phys Rev Lett 2021; 127:030601. [PMID: 34328752 DOI: 10.1103/physrevlett.127.030601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2020] [Revised: 04/09/2021] [Accepted: 05/28/2021] [Indexed: 06/13/2023]
Abstract
We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long-range interactions. This extension leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "nonergodic extended states" in many-body systems discussed in the recent literature.
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Affiliation(s)
- Felipe Monteiro
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
| | - Masaki Tezuka
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
| | - Alexander Altland
- Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Cologne, Germany
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Tobias Micklitz
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
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13
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Kumar M, Kundu A, Kulkarni M, Huse DA, Dhar A. Transport, correlations, and chaos in a classical disordered anharmonic chain. Phys Rev E 2020; 102:022130. [PMID: 32942452 DOI: 10.1103/physreve.102.022130] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2020] [Accepted: 07/28/2020] [Indexed: 11/07/2022]
Abstract
We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators, and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size N, disorder strength Δ, and temperature T. The conductivity κ_{N}, obtained for finite length (N), saturates to a value κ_{∞}>0 in the large N limit, for all values of disorder strength Δ and temperature T>0. We show evidence that for any Δ>0 the conductivity goes to zero faster than any power of T in the (T/Δ)→0 limit, and find that the form κ_{∞}∼e^{-B|ln(CΔ/T)|^{3}} fits our data. This form has earlier been suggested by a theory based on the dynamics of multioscillator chaotic islands. The finite-size effect can be κ_{N}<κ_{∞} due to boundary resistance when the bulk conductivity is high (the weak disorder case), or κ_{N}>κ_{∞} due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the out-of-time-ordered commutator.
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Affiliation(s)
- Manoj Kumar
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India.,Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA.,Institute for Advanced Study, Princeton, New Jersey 08540, USA
| | - Abhishek Dhar
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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14
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Abstract
We uncover a local order parameter for measurement-induced phase transitions: the average entropy of a single reference qubit initially entangled with the system. Using this order parameter, we identify scalable probes of measurement-induced criticality that are immediately applicable to advanced quantum computing platforms. We test our proposal on a 1+1 dimensional stabilizer circuit model that can be classically simulated in polynomial time. We introduce the concept of a "decoding light cone" to establish the local and efficiently measurable nature of this probe. We also estimate bulk and surface critical exponents for the transition. Developing scalable probes of measurement-induced criticality in more general models may be a useful application of noisy intermediate scale quantum devices, as well as point to more efficient realizations of fault-tolerant quantum computation.
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Affiliation(s)
- Michael J Gullans
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Institute for Advanced Study, Princeton, New Jersey 08540, USA
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15
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Gullans MJ, Huse DA. Localization as an Entanglement Phase Transition in Boundary-Driven Anderson Models. Phys Rev Lett 2019; 123:110601. [PMID: 31573240 DOI: 10.1103/physrevlett.123.110601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2019] [Indexed: 06/10/2023]
Abstract
The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.
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Affiliation(s)
- Michael J Gullans
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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16
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Brown PT, Mitra D, Guardado-Sanchez E, Nourafkan R, Reymbaut A, Hébert CD, Bergeron S, Tremblay AMS, Kokalj J, Huse DA, Schauß P, Bakr WS. Bad metallic transport in a cold atom Fermi-Hubbard system. Science 2018; 363:379-382. [PMID: 30523078 DOI: 10.1126/science.aat4134] [Citation(s) in RCA: 55] [Impact Index Per Article: 9.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2018] [Accepted: 10/30/2018] [Indexed: 11/03/2022]
Abstract
Strong interactions in many-body quantum systems complicate the interpretation of charge transport in such materials. To shed light on this problem, we study transport in a clean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testing ground for strong interaction physics in the Fermi-Hubbard model. We determine the diffusion constant by measuring the relaxation of an imposed density modulation and modeling its decay hydrodynamically. The diffusion constant is converted to a resistivity by using the Nernst-Einstein relation. That resistivity exhibits a linear temperature dependence and shows no evidence of saturation, two characteristic signatures of a bad metal. The techniques we developed in this study may be applied to measurements of other transport quantities, including the optical conductivity and thermopower.
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Affiliation(s)
- Peter T Brown
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Debayan Mitra
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | | | - Reza Nourafkan
- Département de Physique, Institut Quantique, and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
| | - Alexis Reymbaut
- Département de Physique, Institut Quantique, and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
| | - Charles-David Hébert
- Département de Physique, Institut Quantique, and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
| | - Simon Bergeron
- Département de Physique, Institut Quantique, and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
| | - A-M S Tremblay
- Département de Physique, Institut Quantique, and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada.,Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
| | - Jure Kokalj
- Faculty of Civil and Geodetic Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia.,Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
| | - David A Huse
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Peter Schauß
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Waseem S Bakr
- Department of Physics, Princeton University, Princeton, NJ 08544, USA.
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17
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Das A, Chakrabarty S, Dhar A, Kundu A, Huse DA, Moessner R, Ray SS, Bhattacharjee S. Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain. Phys Rev Lett 2018; 121:024101. [PMID: 30085710 DOI: 10.1103/physrevlett.121.024101] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2018] [Indexed: 06/08/2023]
Abstract
We find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.
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Affiliation(s)
- Avijit Das
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Saurish Chakrabarty
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Abhishek Dhar
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Roderich Moessner
- Max-Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Samriddhi Sankar Ray
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Subhro Bhattacharjee
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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18
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Pixley JH, Wilson JH, Huse DA, Gopalakrishnan S. Weyl Semimetal to Metal Phase Transitions Driven by Quasiperiodic Potentials. Phys Rev Lett 2018; 120:207604. [PMID: 29864319 DOI: 10.1103/physrevlett.120.207604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2018] [Indexed: 06/08/2023]
Abstract
We explore the stability of three-dimensional Weyl and Dirac semimetals subject to quasiperiodic potentials. We present numerical evidence that the semimetal is stable for weak quasiperiodic potentials, despite being unstable for weak random potentials. As the quasiperiodic potential strength increases, the semimetal transitions to a metal, then to an "inverted" semimetal, and then finally to a metal again. The semimetal and metal are distinguished by the density of states at the Weyl point, as well as by level statistics, transport, and the momentum-space structure of eigenstates near the Weyl point. The critical properties of the transitions in quasiperiodic systems differ from those in random systems: we do not find a clear critical scaling regime in energy; instead, at the quasiperiodic transitions, the density of states appears to jump abruptly (and discontinuously to within our resolution).
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Affiliation(s)
- J H Pixley
- Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
- Condensed Matter Theory Center and the Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
| | - Justin H Wilson
- Institute of Quantum Information and Matter and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Sarang Gopalakrishnan
- Department of Engineering Science and Physics, CUNY College of Staten Island, Staten Island, New York 10314, USA and Initiative for the Theoretical Sciences, CUNY Graduate Center, New York, New York 10016 USA
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19
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Devakul T, Khemani V, Pollmann F, Huse DA, Sondhi SL. Obtaining highly excited eigenstates of the localized XX chain via DMRG-X. Philos Trans A Math Phys Eng Sci 2017; 375:20160431. [PMID: 29084883 PMCID: PMC5665784 DOI: 10.1098/rsta.2016.0431] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 08/14/2017] [Indexed: 06/07/2023]
Abstract
We benchmark a variant of the recently introduced density matrix renormalization group (DMRG)-X algorithm against exact results for the localized random field XX chain. We find that the eigenstates obtained via DMRG-X exhibit a highly accurate l-bit description for system sizes much bigger than the direct, many-body, exact diagonalization in the spin variables is able to access. We take advantage of the underlying free fermion description of the XX model to accurately test the strengths and limitations of this algorithm for large system sizes. We discuss the theoretical constraints on the performance of the algorithm from the entanglement properties of the eigenstates, and its actual performance at different values of disorder. A small but significant improvement to the algorithm is also presented, which helps significantly with convergence. We find that, at high entanglement, DMRG-X shows a bias towards eigenstates with low entanglement, but can be improved with increased bond dimension. This result suggests that one must be careful when applying the algorithm for interacting many-body localized spin models near a transition.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.
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Affiliation(s)
- Trithep Devakul
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Vedika Khemani
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
| | - Frank Pollmann
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Street 38, 01187 Dresden, Germany
- Department of Physics, Technical University of Munich, 85748 Garching, Germany
| | - David A Huse
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - S L Sondhi
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
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20
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Ponte P, Laumann CR, Huse DA, Chandran A. Thermal inclusions: how one spin can destroy a many-body localized phase. Philos Trans A Math Phys Eng Sci 2017; 375:20160428. [PMID: 29084891 PMCID: PMC5665782 DOI: 10.1098/rsta.2016.0428] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/11/2017] [Indexed: 06/07/2023]
Abstract
Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N≫1 localized bits subject to random fields. On increasing J, the system transitions from an MBL to a delocalized phase on the vanishing scale Jc(N)∼1/N, up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either 'on' (strongly entangled) or 'off' (weakly entangled) in eigenstates. The clusters of 'on' bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.
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Affiliation(s)
- Pedro Ponte
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
- Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1
| | - C R Laumann
- Department of Physics, Boston University, Boston, MA 02215, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - A Chandran
- Department of Physics, Boston University, Boston, MA 02215, USA
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21
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Brown PT, Mitra D, Guardado-Sanchez E, Schauß P, Kondov SS, Khatami E, Paiva T, Trivedi N, Huse DA, Bakr WS. Spin-imbalance in a 2D Fermi-Hubbard system. Science 2017; 357:1385-1388. [DOI: 10.1126/science.aam7838] [Citation(s) in RCA: 97] [Impact Index Per Article: 13.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2017] [Accepted: 08/24/2017] [Indexed: 11/02/2022]
Abstract
The interplay of strong interactions and magnetic fields gives rise to unusual forms of superconductivity and magnetism in quantum many-body systems. Here, we present an experimental study of the two-dimensional Fermi-Hubbard model—a paradigm for strongly correlated fermions on a lattice—in the presence of a Zeeman field and varying doping. Using site-resolved measurements, we revealed anisotropic antiferromagnetic correlations, a precursor to long-range canted order. We observed nonmonotonic behavior of the local polarization with doping for strong interactions, which we attribute to the evolution from an antiferromagnetic insulator to a metallic phase. Our results pave the way to experimentally mapping the low-temperature phase diagram of the Fermi-Hubbard model as a function of both doping and spin polarization, for which many open questions remain.
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22
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Abstract
We provide a systematic comparison of the many-body localization (MBL) transition in spin chains with nonrandom quasiperiodic versus random fields. We find evidence suggesting that these belong to two separate universality classes: the first dominated by "intrinsic" intrasample randomness, and the second dominated by external intersample quenched randomness. We show that the effects of intersample quenched randomness are strongly growing, but not yet dominant, at the system sizes probed by exact-diagonalization studies on random models. Thus, the observed finite-size critical scaling collapses in such studies appear to be in a preasymptotic regime near the nonrandom universality class, but showing signs of the initial crossover towards the external-randomness-dominated universality class. Our results provide an explanation for why exact-diagonalization studies on random models see an apparent scaling near the transition while also obtaining finite-size scaling exponents that strongly violate Harris-Chayes bounds that apply to disorder-driven transitions. We also show that the MBL phase is more stable for the quasiperiodic model as compared to the random one, and the transition in the quasiperiodic model suffers less from certain finite-size effects.
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Affiliation(s)
- Vedika Khemani
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - D N Sheng
- Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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23
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24
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Goldstein S, Huse DA, Lebowitz JL, Tumulka R. Thermal Equilibrium of a Macroscopic Quantum System in a Pure State. Phys Rev Lett 2015; 115:100402. [PMID: 26382669 DOI: 10.1103/physrevlett.115.100402] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Indexed: 06/05/2023]
Abstract
We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may, however, not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note, however, that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.
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Affiliation(s)
- Sheldon Goldstein
- Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
| | - David A Huse
- Department of Physics, Princeton University, Jadwin Hall, Washington Road, Princeton, New Jersey 08544-0708, USA
| | - Joel L Lebowitz
- Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
| | - Roderich Tumulka
- Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
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25
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Kim H, Bañuls MC, Cirac JI, Hastings MB, Huse DA. Slowest local operators in quantum spin chains. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 92:012128. [PMID: 26274145 DOI: 10.1103/physreve.92.012128] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2014] [Indexed: 06/04/2023]
Abstract
We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures. We find operators with significantly slower relaxation than the slowest simple "hydrodynamic" mode due to energy diffusion. Then we examine some properties of the nontrivial slow operators. Using both exhaustive search and tensor network techniques, we find similar slowly relaxing operators for a Floquet spin chain; this system is hydrodynamically "trivial," with no conservation laws restricting their dynamics. We argue that such slow relaxation may be a generic feature following from locality and unitarity.
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Affiliation(s)
- Hyungwon Kim
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
- Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
| | - J Ignacio Cirac
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
| | - Matthew B Hastings
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
- Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, USA
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
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26
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Abstract
We explore the dynamics of the entanglement entropy near equilibrium in highly entangled pure states of two quantum-chaotic spin chains undergoing unitary time evolution. We examine the relaxation to equilibrium from initial states with either less or more entanglement entropy than the equilibrium value, as well as the dynamics of the spontaneous fluctuations of the entanglement that occur in equilibrium. For the spin chain with a time-independent Hamiltonian and thus an extensive conserved energy, we find slow relaxation of the entanglement entropy near equilibration. Such slow relaxation is absent in a Floquet spin chain with a Hamiltonian that is periodic in time and thus has no local conservation law. Therefore, we argue that slow diffusive energy transport is responsible for the slow relaxation of the entanglement entropy in the Hamiltonian system.
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Affiliation(s)
- Liangsheng Zhang
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Hyungwon Kim
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
- Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
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27
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Kim H, Ikeda TN, Huse DA. Testing whether all eigenstates obey the eigenstate thermalization hypothesis. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 90:052105. [PMID: 25493738 DOI: 10.1103/physreve.90.052105] [Citation(s) in RCA: 94] [Impact Index Per Article: 9.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2014] [Indexed: 05/16/2023]
Abstract
We ask whether the eigenstate thermalization hypothesis (ETH) is valid in a strong sense: in the limit of an infinite system, every eigenstate is thermal. We examine expectation values of few-body operators in highly excited many-body eigenstates and search for "outliers," the eigenstates that deviate the most from ETH. We use exact diagonalization of two one-dimensional nonintegrable models: a quantum Ising chain with transverse and longitudinal fields, and hard-core bosons at half-filling with nearest- and next-nearest-neighbor hopping and interaction. We show that even the most extreme outliers appear to obey ETH as the system size increases and thus provide numerical evidences that support ETH in this strong sense. Finally, periodically driving the Ising Hamiltonian, we show that the eigenstates of the corresponding Floquet operator obey ETH even more closely. We attribute this better thermalization to removing the constraint of conservation of the total energy.
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Affiliation(s)
- Hyungwon Kim
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Tatsuhiko N Ikeda
- Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
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28
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Zhao B, Kerridge MC, Huse DA. Three species of Schrödinger cat states in an infinite-range spin model. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 90:022104. [PMID: 25215686 DOI: 10.1103/physreve.90.022104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2013] [Indexed: 06/03/2023]
Abstract
We explore a transverse-field Ising model that exhibits both spontaneous symmetry breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrödinger cat states: superpositions of states with "up" and "down" spontaneous magnetization. This model exhibits two dynamical phase transitions within its ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels.
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Affiliation(s)
- Bo Zhao
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Merritt C Kerridge
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
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29
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Zhu Z, Huse DA, White SR. Unexpected z-direction Ising antiferromagnetic order in a frustrated spin-1/2 J1-J2 XY model on the honeycomb lattice. Phys Rev Lett 2013; 111:257201. [PMID: 24483752 DOI: 10.1103/physrevlett.111.257201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2013] [Revised: 11/24/2013] [Indexed: 06/03/2023]
Abstract
Using the density matrix renormalization group on wide cylinders, we study the phase diagram of the spin-1/2 XY model on the honeycomb lattice, with first-neighbor (J1=1) and frustrating second-neighbor (J2>0) interactions. For the intermediate frustration regime 0.22≲J2≲0.36, we find a surprising antiferromagnetic Ising phase, with ordered moments pointing along the z axis, despite the absence of any S(z)S(z) interactions in the Hamiltonian. Surrounding this phase as a function of J2 are antiferromagnetic phases with the moments pointing in the x-y plane for small J2 and a close competition between an x-y plane magnetic collinear phase and a dimer phase for large values of J2. We do not find any spin-liquid phases in this model.
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Affiliation(s)
- Zhenyue Zhu
- Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Steven R White
- Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
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30
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Abstract
We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.
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Affiliation(s)
- Hyungwon Kim
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
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31
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Zhu Z, Huse DA, White SR. Weak plaquette valence bond order in the S = 1/2 honeycomb J1 - J2 Heisenberg model. Phys Rev Lett 2013; 110:127205. [PMID: 25166840 DOI: 10.1103/physrevlett.110.127205] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2013] [Indexed: 06/03/2023]
Abstract
Using the density matrix renormalization group, we investigate the S = 1/2 Heisenberg model on the honeycomb lattice with first (J(1)) and second (J(2)) neighbor interactions. We are able to study long open cylinders with widths up to 12 lattice spacings. For J(2)/J(1) near 0.3, we find an apparently paramagnetic phase, bordered by an antiferromagnetic phase for J(2) ≲ 0.26 and by a valence bond crystal for J(2) ≳ 0.36. The longest correlation length that we find in this intermediate phase is for plaquette valence bond order. This correlation length grows strongly with cylinder circumference, indicating either quantum criticality or weak plaquette valence bond order.
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Affiliation(s)
- Zhenyue Zhu
- Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Steven R White
- Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
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32
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Abstract
We solve for the time-dependent finite-size scaling functions of the one-dimensional transverse-field Ising chain during a linear-in-time ramp of the field through the quantum critical point. We then simulate Mott-insulating bosons in a tilted potential, an experimentally studied system in the same equilibrium universality class, and demonstrate that universality holds for the dynamics as well. We find qualitatively athermal features of the scaling functions, such as negative spin correlations, and we show that they should be robustly observable within present cold atom experiments.
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Affiliation(s)
- Michael Kolodrubetz
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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33
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Thomas CK, Huse DA, Middleton AA. Zero- and low-temperature behavior of the two-dimensional ±J Ising spin glass. Phys Rev Lett 2011; 107:047203. [PMID: 21867036 DOI: 10.1103/physrevlett.107.047203] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2011] [Revised: 05/03/2011] [Indexed: 05/31/2023]
Abstract
Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.
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Affiliation(s)
- Creighton K Thomas
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
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34
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Abstract
We use the density matrix renormalization group to perform accurate calculations of the ground state of the nearest-neighbor quantum spin S = 1/2 Heisenberg antiferromagnet on the kagome lattice. We study this model on numerous long cylinders with circumferences up to 12 lattice spacings. Through a combination of very-low-energy and small finite-size effects, our results provide strong evidence that, for the infinite two-dimensional system, the ground state of this model is a fully gapped spin liquid.
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Affiliation(s)
- Simeng Yan
- Department of Physics and Astronomy, University of California, Irvine, CA 92617, USA
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35
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Mathy CJM, Parish MM, Huse DA. Trimers, molecules, and polarons in mass-imbalanced atomic Fermi gases. Phys Rev Lett 2011; 106:166404. [PMID: 21599393 DOI: 10.1103/physrevlett.106.166404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2010] [Indexed: 05/30/2023]
Abstract
We consider the ground state of a single "spin-down" impurity atom interacting attractively with a "spin-up" atomic Fermi gas. By constructing variational wave functions for polarons, molecules, and trimers, we perform a detailed study of the transitions between these dressed bound states as a function of mass ratio r=m↑/m↓ and interaction strength. Crucially, we find that the presence of a Fermi sea enhances the stability of the p-wave trimer, which can be viewed as a Fulde-Ferrell-Larkin-Ovchinnikov molecule that has bound an additional majority atom. For sufficiently large r, we find that the transitions lie outside the region of phase separation of the imbalanced Fermi gas and should thus be observable in experiment, unlike the well-studied equal-mass case.
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Affiliation(s)
- Charles J M Mathy
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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36
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Richardella A, Roushan P, Mack S, Zhou B, Huse DA, Awschalom DD, Yazdani A. Visualizing Critical Correlations Near the Metal-Insulator Transition in Ga
1-
x
Mn
x
As. Science 2010; 327:665-9. [DOI: 10.1126/science.1183640] [Citation(s) in RCA: 200] [Impact Index Per Article: 14.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- Anthony Richardella
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
| | - Pedram Roushan
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Shawn Mack
- Center for Spintronics and Quantum Computation, University of California, Santa Barbara, CA 93106, USA
| | - Brian Zhou
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - David A. Huse
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - David D. Awschalom
- Center for Spintronics and Quantum Computation, University of California, Santa Barbara, CA 93106, USA
| | - Ali Yazdani
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
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37
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Trebst S, Ardonne E, Feiguin A, Huse DA, Ludwig AWW, Troyer M. Collective states of interacting Fibonacci anyons. Phys Rev Lett 2008; 101:050401. [PMID: 18764375 DOI: 10.1103/physrevlett.101.050401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2008] [Indexed: 05/26/2023]
Abstract
We show that chains of interacting Fibonacci anyons can support a wide variety of collective ground states ranging from extended critical, gapless phases to gapped phases with ground-state degeneracy and quasiparticle excitations. In particular, we generalize the Majumdar-Ghosh Hamiltonian to anyonic degrees of freedom by extending recently studied pairwise anyonic interactions to three-anyon exchanges. The energetic competition between two- and three-anyon interactions leads to a rich phase diagram that harbors multiple critical and gapped phases. For the critical phases and their higher symmetry end points we numerically establish descriptions in terms of two-dimensional conformal field theories. A topological symmetry protects the critical phases and determines the nature of gapped phases.
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Affiliation(s)
- Simon Trebst
- Microsoft Research, Station Q, University of California-Santa Barbara, CA 93106, USA
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38
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Abstract
We calculate the zero-temperature (T=0) phase diagram of a polarized two-component Fermi gas in an array of weakly coupled parallel one-dimensional (1D) "tubes" produced by a two-dimensional optical lattice. Increasing the lattice strength drives a crossover from three-dimensional (3D) to 1D behavior, stabilizing the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated superfluid phase. We argue that the most promising regime for observing the FFLO phase is in the quasi-1D regime, where the atomic motion is largely 1D but there is weak tunneling in the other directions that stabilizes long-range order. In the FFLO phase, we describe a phase transition where the quasiparticle spectrum changes from gapless near the 3D regime to gapped in quasi-1D.
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Affiliation(s)
- Meera M Parish
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.
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39
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40
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Abstract
We present a theory of the thermodynamics of an incommensurate quantum solid. The ground state of the solid is assumed to be an incommensurate crystal, with quantum zero-point vacancies and interstitials and thus a non-integer number of atoms per unit cell. We show that at low temperature T, the variation of the net vacancy concentration should be as T4 and that the first correction to the specific heat due to this varies as T7; these are quite consistent with experiments on solid helium-4. We also make some observations about the recent experimental reports of "supersolidity" in solid helium-4 that motivate a renewed interest in quantum crystals.
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Affiliation(s)
- P W Anderson
- Department of Physics, Princeton University, Princeton, NJ 08544, USA
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41
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Vega DA, Harrison CK, Angelescu DE, Trawick ML, Huse DA, Chaikin PM, Register RA. Ordering mechanisms in two-dimensional sphere-forming block copolymers. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:061803. [PMID: 16089757 DOI: 10.1103/physreve.71.061803] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2005] [Indexed: 05/03/2023]
Abstract
We study the coarsening dynamics of two-dimensional hexagonal patterns formed by single microdomain layers of block copolymers, using numerical simulations. Our study is focused on the temporal evolution of the orientational correlation length, the interactions between topological defects, and the mechanisms of coarsening. We find no free disclinations in the system; rather, they are located on large-angle grain boundaries, commonly where such boundaries bifurcate. The correlation lengths determined from the scattering function, from the density of dislocations, and from the density of disclinations exhibit similar behavior and grow with time according to a power law. The orientational correlation length also grows following a power law, but with a higher exponent than the other correlation lengths. The orientational correlation length grows via annihilation of dislocations, through preferential annihilation of small-angle grain boundaries due to poor screening of the strain field around dislocations located on small-angle grain boundaries. Consequently, the patterns are characterized by large-angle grain boundaries. The most commonly observed mechanism of coarsening is the collapse of smaller grains residing on the boundary of two larger grains delimited by large-angle grain boundaries. Simulations agree remarkably well with experimental results recently obtained.
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Affiliation(s)
- Daniel A Vega
- Department of Physics, Universidad Nacional del Sur, B8000CPB Bahía Blanca, Argentina
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42
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Trebst S, Huse DA, Troyer M. Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 70:046701. [PMID: 15600559 DOI: 10.1103/physreve.70.046701] [Citation(s) in RCA: 69] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2004] [Revised: 07/08/2004] [Indexed: 05/24/2023]
Abstract
We present an adaptive algorithm which optimizes the statistical-mechanical ensemble in a generalized broad-histogram Monte Carlo simulation to maximize the system's rate of round trips in total energy. The scaling of the mean round-trip time from the ground state to the maximum entropy state for this local-update method is found to be O ( [N ln N](2) ) for both the ferromagnetic and the fully frustrated two-dimensional Ising model with N spins. Our algorithm thereby substantially outperforms flat-histogram methods such as the Wang-Landau algorithm.
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Affiliation(s)
- Simon Trebst
- Theoretische Physik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland
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43
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Tutuc E, Shayegan M, Huse DA. Counterflow measurements in strongly correlated GaAs hole bilayers: evidence for electron-hole pairing. Phys Rev Lett 2004; 93:036802. [PMID: 15323852 DOI: 10.1103/physrevlett.93.036802] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2004] [Indexed: 05/24/2023]
Abstract
We study interacting GaAs bilayer hole systems, with very small interlayer tunneling, in a counterflow geometry where equal currents are passed in opposite directions in the two, independently contacted layers. At low temperatures, both the longitudinal and Hall counterflow resistances tend to vanish in the quantum Hall state at total bilayer filling nu=1, demonstrating the pairing of oppositely charged carriers in opposite layers. The counterflow Hall resistance decreases much more strongly than the longitudinal resistances as the temperature is reduced.
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Affiliation(s)
- E Tutuc
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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44
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Abstract
We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.
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Affiliation(s)
- David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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45
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Ussishkin I, Sondhi SL, Huse DA. Gaussian superconducting fluctuations, thermal transport, and the nernst effect. Phys Rev Lett 2002; 89:287001. [PMID: 12513173 DOI: 10.1103/physrevlett.89.287001] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2002] [Indexed: 05/24/2023]
Abstract
We calculate the contribution of superconducting fluctuations to thermal transport in the normal state, at low magnetic fields. We do so in the Gaussian approximation to their critical dynamics which is also the Aslamazov-Larkin approximation in the microscopics. Our results for the thermal conductivity tensor and the transverse thermoelectric response are new. The latter compare favorably with the data of Ong and collaborators on the Nernst effect in the cuprates.
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Affiliation(s)
- Iddo Ussishkin
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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46
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Abstract
We present a general theory of a class of multicritical points in the phase diagrams of random antiferromagnetic spin chains. We show that low-energy properties of these points are almost completely determined by a permutation symmetry of the effective theory not shared by the microscopic Hamiltonian. One case provides an analytic theory of the quantum critical point in the random spin-3/2 chain, studied in a recent work by Refael, Kehrein, and Fisher.
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Affiliation(s)
- Kedar Damle
- Physics Department, Harvard University, Cambridge, MA 02138, USA
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47
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Ramirez AP, Shastry BS, Hayashi A, Krajewski JJ, Huse DA, Cava RJ. Multiple field-induced phase transitions in the geometrically frustrated dipolar magnet: Gd(2) Ti(2)O(7). Phys Rev Lett 2002; 89:067202. [PMID: 12190608 DOI: 10.1103/physrevlett.89.067202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2001] [Indexed: 05/23/2023]
Abstract
Field-driven phase transitions generally arise from competition between Zeeman energy and exchange or crystal-field anisotropy. Here we present the phase diagram of a frustrated pyrochlore magnet Gd(2)Ti(2)O(7), where crystal-field splitting is small compared to the dipolar energy. We find good agreement between zero-temperature critical fields and those obtained from a mean-field model. Here, dipolar interactions couple real space and spin space, so the transitions in Gd(2)Ti(2)O(7) arise from field-induced "cooperative anisotropy," reflecting the broken spatial symmetries of the pyrochlore lattice.
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Affiliation(s)
- A P Ramirez
- Los Alamos National Laboratory, K764, New Mexico 87545, USA
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48
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Harrison C, Cheng Z, Sethuraman S, Huse DA, Chaikin PM, Vega DA, Sebastian JM, Register RA, Adamson DH. Dynamics of pattern coarsening in a two-dimensional smectic system. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:011706. [PMID: 12241374 DOI: 10.1103/physreve.66.011706] [Citation(s) in RCA: 84] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2001] [Indexed: 05/23/2023]
Abstract
We have followed the coarsening dynamics of a single layer of cylindrical block copolymer microdomains in a thin film. This system has the symmetry of a two-dimensional smectic. The orientational correlation length of the microdomains was measured by scanning electron microscopy and found to grow with the average spacing between +/-1/2 disclinations, following a power law xi2(t) approximately t(1/4). By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe dominant mechanisms of disclination annihilation involving tripoles and quadrupoles (three and four disclinations, respectively). We describe how annihilation events involving multiple disclinations result in similarly reduced kinetic exponents as observed here. These results map onto a wide variety of physical systems that exhibit similarly striped patterns.
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49
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50
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Abstract
We have studied the ordering dynamics of the striped patterns of a single layer of cylindrical block copolymer microdomains in a thin film. By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe a dominant mechanism of disclination annihilation involving three or four disclinations (quadrupoles). Pairwise disclination annihilation events are suppressed as a result of the topological constraints in this system. The kinetic scaling laws with exponents observed here are consistent with topologically allowed annihilation events involving multiple disclinations. The results provide insight into two-dimensional pattern formation and may lead to the successful application of block copolymer lithography.
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Affiliation(s)
- C Harrison
- Department of Physics, Princeton Materials Institute, Princeton University, Princeton, NJ 08544, USA
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