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Effective Field Theory of Random Quantum Circuits. ENTROPY 2022; 24:e24060823. [PMID: 35741543 PMCID: PMC9223178 DOI: 10.3390/e24060823] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Revised: 06/09/2022] [Accepted: 06/10/2022] [Indexed: 11/23/2022]
Abstract
Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, and a hallmark of quantum chaos—universal Wigner–Dyson level statistics—has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive the universal random matrix behavior of a large family of random circuits. In particular, we rederive the Wigner–Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method of evaluating integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the study of quantum circuits. The effective field theory derived here provides both a method to quantitatively characterize the quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and a path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems.
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García-García AM, Jia Y, Rosa D, Verbaarschot JJM. Dominance of Replica Off-Diagonal Configurations and Phase Transitions in a PT Symmetric Sachdev-Ye-Kitaev Model. PHYSICAL REVIEW LETTERS 2022; 128:081601. [PMID: 35275644 DOI: 10.1103/physrevlett.128.081601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2021] [Revised: 09/12/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
We show that, after ensemble averaging, the low temperature phase of a conjugate pair of uncoupled, quantum chaotic, non-Hermitian systems such as the Sachdev-Ye-Kitaev (SYK) model or the Ginibre ensemble of random matrices is dominated by saddle points that couple replicas and conjugate replicas. This results in a nearly flat free energy that terminates in a first-order phase transition. In the case of the SYK model, we show explicitly that the spectrum of the effective replica theory has a gap. These features are strikingly similar to those induced by wormholes in the gravity path integral which suggests a close relation between both configurations. For a nonchaotic SYK, the results are qualitatively different: the spectrum is gapless in the low temperature phase and there is an infinite number of second order phase transitions unrelated to the restoration of replica symmetry.
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Affiliation(s)
- Antonio M García-García
- Shanghai Center for Complex Physics, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Yiyang Jia
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
| | - Dario Rosa
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science(IBS), Daejeon 34126, Korea
- Department of Physics, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
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Liao Y, Vikram A, Galitski V. Many-Body Level Statistics of Single-Particle Quantum Chaos. PHYSICAL REVIEW LETTERS 2020; 125:250601. [PMID: 33416379 DOI: 10.1103/physrevlett.125.250601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2020] [Accepted: 10/26/2020] [Indexed: 06/12/2023]
Abstract
We consider a noninteracting many-fermion system populating levels of a unitary random matrix ensemble (equivalent to the q=2 complex Sachdev-Ye-Kitaev model)-a generic model of single-particle quantum chaos. We study the corresponding many-particle level statistics by calculating the spectral form factor analytically using algebraic methods of random matrix theory, and match it with an exact numerical simulation. Despite the integrability of the theory, the many-body spectral rigidity is found to have a surprisingly rich landscape. In particular, we find a residual repulsion of distant many-body levels stemming from single-particle chaos, together with islands of level attraction. These results are encoded in an exponential ramp in the spectral form factor, which we show to be a universal feature of nonergodic many-fermion systems embedded in a chaotic medium.
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Affiliation(s)
- Yunxiang Liao
- Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Amit Vikram
- Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Victor Galitski
- Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
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Vidal P, Kanzieper E. Integrability of zero-dimensional replica field theories at β=1. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:030101. [PMID: 24125199 DOI: 10.1103/physreve.88.030101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
Building on insights from the theory of integrable lattices, the integrability is claimed for nonlinear replica σ models derived in the context of real symmetric random matrices. Specifically, the fermionic and the bosonic replica partition functions are proven to form a single (supersymmetric) Pfaff-KP hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density-density eigenvalue correlation function in the infinite-dimensional Gaussian orthogonal ensemble. Implications of the formalism outlined are briefly discussed.
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Affiliation(s)
- Pedro Vidal
- Department of Applied Mathematics, H.I.T.-Holon Institute of Technology, Holon 5810201, Israel and Fakultät für Physik, Universität Bielefeld, Bielefeld 33615, Germany
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Slanina F. Equivalence of replica and cavity methods for computing spectra of sparse random matrices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011118. [PMID: 21405672 DOI: 10.1103/physreve.83.011118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2010] [Revised: 12/01/2010] [Indexed: 05/30/2023]
Abstract
We show by direct calculation that the replica and cavity methods are exactly equivalent for the spectrum of an Erdős-Rényi random graph. We introduce a variational formulation based on the cavity method and use it to find approximate solutions for the density of eigenvalues. We also use this variational method for calculating spectra of sparse covariance matrices.
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Affiliation(s)
- František Slanina
- Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic.
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Osipov VA, Kanzieper E. Are bosonic replicas faulty? PHYSICAL REVIEW LETTERS 2007; 99:050602. [PMID: 17930738 DOI: 10.1103/physrevlett.99.050602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2007] [Indexed: 05/25/2023]
Abstract
Motivated by the ongoing discussion about a seeming asymmetry in the performance of fermionic and bosonic replicas, we present an exact, nonperturbative approach to both fermionic and bosonic zero-dimensional replica field theories belonging to the broadly interpreted beta=2 Dyson symmetry class. We then utilize the formalism developed to demonstrate that the bosonic replicas do correctly reproduce the microscopic spectral density in the QCD-inspired chiral Gaussian unitary ensemble. This disproves the myth that the bosonic replica field theories are intrinsically faulty.
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Affiliation(s)
- Vladimir Al Osipov
- Department of Applied Mathematics, H.I.T.-Holon Institute of Technology, Holon 58102, Israel.
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Cicuta GM, Orland H. Real symmetric random matrices and replicas. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051120. [PMID: 17279890 DOI: 10.1103/physreve.74.051120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2006] [Indexed: 05/13/2023]
Abstract
Various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large- N limit. A result on the Laplacian random matrix with Wigner-rescaling is generalized to arbitrary probability distribution.
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Affiliation(s)
- Giovanni M Cicuta
- Dipartimento di Fisica, Università di Parma, Parco Area delle Scienze 7A, 43100 Parma, Italy.
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Altland A, Kamenev A, Tian C. Anderson localization from the replica formalism. PHYSICAL REVIEW LETTERS 2005; 95:206601. [PMID: 16384077 DOI: 10.1103/physrevlett.95.206601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2005] [Indexed: 05/05/2023]
Abstract
We study Anderson localization in quasi-one-dimensional disordered wires within the framework of the replica sigma model. Applying a semiclassical approach (geodesic action plus Gaussian fluctuations) recently introduced within the context of supersymmetry by Lamacraft, Simons, and Zirnbauer, we compute the exact density of transmission matrix eigenvalues of superconducting wires (of symmetry class CI.) For the unitary class of metallic systems (class A) we are able to obtain the density function, save for its large transmission tail.
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Affiliation(s)
- Alexander Altland
- Institut für Theoretische Physik, Universtät zu Köln, Köln, 50937, Germany
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Takahashi K. Nonlinear sigma model approach for level correlations in chiral disordered systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:066147. [PMID: 15697474 DOI: 10.1103/physreve.70.066147] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2004] [Revised: 10/11/2004] [Indexed: 05/24/2023]
Abstract
We study level correlations of disordered systems with chiral unitary symmetry (AIII symmetry). We use a random matrix model with a finite correlation length to derive a supersymmetric nonlinear sigma model. The result is compared with existing results based on other models. Using the methods of Sov. Phys. JETP 60, 656 (1994)] and Phys. Rev. Lett. 75, 902 (1995)], we calculate the density of states and two-level correlation function. The result is expressed using the spectral determinant as in traditional nonchiral systems. We discuss the renormalization of the mean level spacing which is not present in the traditional systems.
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Nishigaki SM, Gangardt DM, Kamenev A. Correlation functions of the BC Calogero Sutherland model. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/12/316] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Splittorff K, Verbaarschot JJM. Replica limit of the toda lattice equation. PHYSICAL REVIEW LETTERS 2003; 90:041601. [PMID: 12570409 DOI: 10.1103/physrevlett.90.041601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2002] [Indexed: 05/24/2023]
Abstract
In a recent breakthrough Kanzieper showed that it is possible to obtain exact nonperturbative random matrix results from the replica limit of the corresponding Painlevé equation. In this article we analyze the replica limit of the Toda lattice equation and obtain exact expressions for the two-point function of the Gaussian unitary ensemble and the resolvent of the chiral unitary ensemble. In the latter case both the fully quenched and the partially quenched limit are considered. This derivation explains in a natural way the appearance of both compact and noncompact integrals, the hallmark of the supersymmetric method, in the replica limit of the expression for the resolvent.
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Kanzieper E. Replica field theories, painlevé transcendents, and exact correlation functions. PHYSICAL REVIEW LETTERS 2002; 89:250201. [PMID: 12484867 DOI: 10.1103/physrevlett.89.250201] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2002] [Indexed: 05/24/2023]
Abstract
Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices. Further applications are briefly discussed.
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Affiliation(s)
- Eugene Kanzieper
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel.
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Altland A, Kamenev A. Wigner-Dyson statistics from the Keldysh sigma-model. PHYSICAL REVIEW LETTERS 2000; 85:5615-5618. [PMID: 11136060 DOI: 10.1103/physrevlett.85.5615] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2000] [Indexed: 05/23/2023]
Abstract
The level statistics of disordered metallic grains with broken time reversal invariance is obtained from the Keldysh nonlinear sigma-model. The basic strategy of the approach--a nonperturbative integration over all saddle points of the sigma-model manifold--might prove useful in the analysis of problems involving disorder and interactions.
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Affiliation(s)
- A Altland
- Theoretische Physik III, Ruhr-Universität Bochum, 44780, Germany
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Kamenev A. Weak charge quantization as an instanton of the interacting sigma model. PHYSICAL REVIEW LETTERS 2000; 85:4160-4163. [PMID: 11056649 DOI: 10.1103/physrevlett.85.4160] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2000] [Indexed: 05/23/2023]
Abstract
Coulomb blockade in a quantum dot attached to a diffusive conductor is considered in the framework of the nonlinear sigma model. It is shown that the weak charge quantization on the dot is associated with instanton configurations of the Q field in the conductor. The instantons have a finite action and are replica nonsymmetric. It is argued that such instantons may play a role in the transition regime to the interacting insulator.
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Affiliation(s)
- A Kamenev
- Department of Physics, Technion, Haifa 32000, Israel
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