Level-spacing distributions and the Airy kernel

Tracy, Craig A.; Widom, Harold

doi:10.1016/0370-2693(93)91114-3

Cited by 787 publications

(1,206 citation statements)

References 25 publications

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“…First, it is interesting to look at what happens close to the edge of the front x/t ±1. For the DW limit it has been shown [12] that suitably rescaled correlations are described by the Airy kernel [26]. Such a result follows readily from our .…”

supporting

confidence: 77%

Inhomogeneous quenches in a free fermionic chain: Exact results

Viti¹,

Stéphan²,

Dubail³

et al. 2016

EPL

114

189

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We consider the non-equilibrium physics induced by joining together two tight binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be the same or different. We present a number of exact results for the correlation functions in the non-interacting case. We present a short-time expansion, which can sometimes be fully resummed, and which reproduces the so-called 'light cone' effect or wavefront behavior of the correlators. For large times, we show how all interesting physical regimes may be obtained by stationary phase approximation techniques. In particular, we derive semiclassical formulas in the case when both time and positions are large, and show that these are exact in the thermodynamic limit. We present subleading corrections to the large-time behavior, including the corrections near the edges of the wavefront. We also provide results for the return probability or Loschmidt echo. In the maximally inhomogeneous limit, we prove that it is exactly gaussian at all times. The effects of interactions on the Loschmidt echo are also discussed.PACS numbers: 75.10. Fd, 75.45.Gm, 05.60.Gg Introduction -Local quantum quenches are a particularly neat setup to address theoretical questions about non-equilibrium current-carrying stationary states as well as transport properties of many-body isolated quantum systems. The characterization of the long-time behavior of local correlation functions unveils universal features of the quantum dynamics [1][2][3][4][5] and paves the way to the construction of effective field theories capable of capturing them. Analytic results are relevant as benchmarks for cold atom experiments that very recently [6][7][8] started to investigate particle and energy transport under a unitary dynamics.In this work, we study a quench which, although injecting an extensive amount of energy into the system, has mostly local effects. We take two uniformly filled long tight binding chains in their respective ground-states, but with a different number of fermions. At time t = 0 the two edges are connected so that it turns into a single chainThe hopping and interactions between sites j = 0 and j = 1 are initially absent; the initial state |ψ 0 = |ψ l |ψ r is the tensor product of the ground states (with specified occupancies) of the two decoupled chains. For unequal fillings, one expects some particle current at time t > 0. We denote by k l F (k r F ) the Fermi momenta on the left (right), so that the particle number is2 ) on the left (right). For simplicity we focus on the symmetric case k l F + k r F = π, but extension to other values is straightforward. It is useful to keep two limits in mind. When the fillings are equal k l F = k r F = π/2 there is no particle current. This particular quench was studied in [1][2][3]9], using low-energy field theory, and belongs to the class of Fermi-edge problems [10,11]. The other simple limit is k The motivation for the present study is two-fold. First, ou...

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supporting

confidence: 77%

Inhomogeneous quenches in a free fermionic chain: Exact results

Viti¹,

Stéphan²,

Dubail³

et al. 2016

EPL

114

189

View full text Add to dashboard Cite

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“…Johnstone in a key paper [22] showed that suitably normalized, and for m, n, large, the distribution of the largest eigenvalue λ 1 is approximately a distribution discovered by Tracy and Widom [23], which in this paper we call TW. Set Now set Then the distribution of x is approximately TW.…”

Section: Resultsmentioning

confidence: 79%

Population Structure and Eigenanalysis

2006

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Current methods for inferring population structure from genetic data do not provide formal significance tests for population differentiation. We discuss an approach to studying population structure (principal components analysis) that was first applied to genetic data by Cavalli-Sforza and colleagues. We place the method on a solid statistical footing, using results from modern statistics to develop formal significance tests. We also uncover a general “phase change” phenomenon about the ability to detect structure in genetic data, which emerges from the statistical theory we use, and has an important implication for the ability to discover structure in genetic data: for a fixed but large dataset size, divergence between two populations (as measured, for example, by a statistic like FST) below a threshold is essentially undetectable, but a little above threshold, detection will be easy. This means that we can predict the dataset size needed to detect structure.

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“…There, the time evolution of a domain-wall state under a free fermionic Hamiltonian (XX model) was considered. The authors were able to establish a connection between the probability distributions of particular observables close to the light cone and the distribution functions of the largest eigenvalues of the Gaussian unitary random matrix ensemble [86]. In particular, the two-point functions of the fermions lying in a region, around the edge, scaling as t 1 3 have corrections that decay as t − 1 3 , and they can be written in terms of the so-called Airy kernel…”

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confidence: 99%

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

2017

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We derive a Moyal dynamical equation that describes exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a hydrodynamic theory. The question at hand is whether some large-time corrections are captured by higher-order hydrodynamics. We consider in particular the dynamics after that two chains, prepared in different conditions, are joined together. In these situations a light cone, separating regions with macroscopically different properties, emerges from the junction. In free fermionic systems some observables close to the light cone follow a universal behavior, known as Tracy-Widom scaling. Universality means weak dependence on the system's details, so this is the perfect setting where hydrodynamics could emerge. For the transverse-field Ising chain and the XX model, we show that hydrodynamics captures the scaling behavior close to the light cone. On the other hand, our numerical analysis suggests that hydrodynamics fails in more general models, whenever a condition is not satisfied.Over the past few years, we are experiencing an increasing interest in the physics behind the nonequilibrium time evolution of inhomogeneous states. An example is the time evolution of two semi-infinite chains that are joined together after having been prepared in different equilibrium conditions [1,2]. This kind of settings allows one to investigate the transport properties of quantum many-body systems even if the system is isolated from the environment.The first analytic results in this context were obtained in noninteracting models . There, under the assumption of quasistationarity, a semiclassical picture applies where the information about the initial state is carried by free stable quasiparticles moving throughout the system. Similar results were obtained in the framework of conformal field theory and Luttinger liquid descriptions [25][26][27][28][29][30][31][32][33][34][35][36][37]. In the presence of interactions the situation was less clear [38][39][40][41][42][43][44][45][46][47], but, eventually, Refs [48,49] have shown that the continuity equations satisfied by the (quasi)local conserved quantities are sufficient to characterize the late-time behavior. The framework developed in Refs [48,49] is now known as generalized hydrodynamics [48], where "generalized" is used to emphasize that integrable models have infinitely many (quasi)local charges [50]. We will generally omit "generalized" and refer to the system of equations derived in [48,49] as first-order hydrodynamics, 1 st GHD, to emphasize that it is a system of first-order partial differential equations.Within 1 st GHD, it was possible to compute the profiles of local observables [48,49,[51][52][53][54][55][56][57], to conjecture an expression for the time evolution of the entanglement entropy [58], and to efficiently calculate Drude weights [59][60][61][62][63]. There are however fundamental questions that can not be addressed within 1 st GHD; diffusive transport [64][65][66][67][68][69] and large-tim...

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Level-spacing distributions and the Airy kernel

Cited by 787 publications

References 25 publications

Inhomogeneous quenches in a free fermionic chain: Exact results

Inhomogeneous quenches in a free fermionic chain: Exact results

Population Structure and Eigenanalysis

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

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