Full counting statistics in the transverse field Ising chain

Groha, Stefan; Eßler, Fabian H. L.; Calabrese, Pasquale

doi:10.21468/scipostphys.4.6.043

Cited by 71 publications

(59 citation statements)

References 89 publications

(188 reference statements)

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“…Following Ref. 42 we can derive a determinant representation for F (θ, t) that can be efficiently evaluated. We start by introducing Majorana fermion operators by…”

Section: Full Counting Statisticsmentioning

confidence: 99%

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How order melts after quantum quenches

Collura

Eßler

2020

Phys. Rev. B

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Injecting a sufficiently large energy density into an isolated many-particle system prepared in a state with long-range order will lead to the melting of the order over time. Detailed information about this process can be derived from the quantum mechanical probability distribution of the order parameter. We study this process for the paradigmatic case of the spin-1/2 Heisenberg XXZ chain. We determine the full quantum mechanical distribution function of the staggered subsystem magnetization as a function of time after a quantum quench from the classical Néel state. This provides a detailed picture of how the Néel order melts and reveals the existence of an interesting regime at intermediate times that is characterized by a very broad probability distribution.Introduction. -A fundamental objective of quantum theory is to determine probability distribution functions of observables in given quantum states. In fewparticle systems the time evolution of such probability distributions provides a lot of useful information beyond what is contained in the corresponding expectation values. Recent advances in cold-atom experiments have made possible not only the study of non-equilibrium time evolution of (almost) isolated many-particle systems [1][2][3][4][5][6][7][8][9][10][11][12][13], but given access to the full quantum mechanical probability distributions of certain observables [14][15][16][17][18]. This provides an opportunity to gain new insights about the coherent dynamics of many-particle quantum systems. One intriguing question one may ask is how order melts, or forms, when an isolated many-particle system is driven across a phase transition. Related questions have been studied in solids, but there one essentially deals with open quantum systems and has access to very different observables, see e.g. [19,20]. The basic setup we have in mind is as follows. Let us consider a system of quantum spins with Hamiltonian H that is initially prepared in a state with density matrix ρ(0). In this state there is long-range order characterized by a local order parameter O =

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“…Following Ref. 42 we can derive a determinant representation for F (θ, t) that can be efficiently evaluated. We start by introducing Majorana fermion operators by…”

Section: Full Counting Statisticsmentioning

confidence: 99%

“…Following Ref. 42 we can express the characteristic function of P (m, t) as a determinant of a 2 × 2 matrix (see Supplementary material), which is easily evaluated numerically. This provides us with exact results at ∆ = 0 for all times, cf.…”

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confidence: 99%

How order melts after quantum quenches

Collura

Eßler

2020

Phys. Rev. B

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“…Yet, no theoretical prediction for this quantity, not even approximate, was available in the existing literature for the Lieb-Liniger model. More generally, the full counting statistics of local observables in and out of equilibrium have been considered in many studies [46][47][48][49][50][51][52][53][54][55][56][57][58][59], even though analytical results in integrable systems have been provided only in a handful of cases [60][61][62][63][64][65][66].Recently, important progress on the problem of computing one-point functions in the one-dimensional Bose gas has been made, boosted by the results of Ref. [67], where a novel field-theoretical approach was introduced: the latter is based on the observation that the Lieb-Liniger model can be obtained as an appropriate non-relativistic (NR) limit of the sinh-Gordon (shG) field theory.…”

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confidence: 99%

From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

Bastianello

Piroli

2018

J. Stat. Mech.

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We derive exact formulas for the expectation value of local observables in a one-dimensional gas of bosons with point-wise repulsive interactions (Lieb-Liniger model). Starting from a recently conjectured expression for the expectation value of vertex operators in the sinh-Gordon field theory, we derive explicit analytic expressions for the one-point K-body correlation functions (Ψ † ) K (Ψ) K in the Lieb-Liniger gas, for arbitrary integer K. These are valid for all excited states in the thermodynamic limit, including thermal states, generalized Gibbs ensembles and non-equilibrium steady states arising in transport settings. Our formulas display several physically interesting applications: most prominently, they allow us to compute the full counting statistics for the particle-number fluctuations in a short interval. Furthermore, combining our findings with the recently introduced generalized hydrodynamics, we are able to study multi-point correlation functions at the Eulerian scale in non-homogeneous settings. Our results complement previous studies in the literature and provide a full solution to the problem of computing one-point functions in the Lieb-Liniger model. I. INTRODUCTIONCorrelation functions encode all of the information which can be experimentally extracted from a many-body quantum system. At the same time, the problem of their computation is extremely complicated from the theoretical point of view, restricting us, in general, to rely uniquely on perturbative or purely numerical methods.An outstanding exception to this picture are integrable systems [1], characterized by the existence of an extensive number of local conservation laws, which provide an ideal theoretical laboratory to deepen our knowledge of many-body physics. This is especially true due to the possibility of obtaining exact, unambiguous predictions for several quantities of interest, allowing us, for instance, to test the validity of approximate or numerical methods which can be applied to more general cases. While integrability directly provides the tools for diagonalizing the Hamiltonian, the computation of correlation functions constitute a remarkable challenge, which has attracted a constant theoretical effort over the past fifty years [2][3][4][5]. Classical studies have in particular focused on ground-state and thermal correlations, and joint efforts have led to spectacular results, for example in the case of prototypical interacting spin models such as the well-known Heisenberg chain [6][7][8][9][10][11][12].More recently, new energy has been pumped into the study of integrable models, also due to the new experimental possibilities offered by cold-atom physics. Nearly ideal integrable systems can now be realized in cold-atom experiments both in and out equilibrium [13][14][15], elevating the relevance of existing works beyond the purely theoretical interest, and motivating further advances in the framework of non-equilibrium physics (see [16] for a collection of recent reviews on this topic).From the experimental point of vie...

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“…Being a very natural concept FCS has been studied for a long time in different communities. It has been studied in the context of charge fluctuations 45,46 , Bose gases [47][48][49][50][51] , particle number fluctuations [52][53][54][55][56] , quantum spin chains [57][58][59][60][61][62][63][64] and out of equilibrium quantum systems 51,65,66 .…”

Section: Introductionmentioning

confidence: 99%

Formation probabilities and statistics of observables as defect problems in free fermions and quantum spin chains

Najafi

Rajabpour

2020

Phys. Rev. B

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We show that the computation of formation probabilities (FP) in the configuration basis and the full counting statistics (FCS) of observables in the quadratic fermionic Hamiltonians are equivalent to the calculation of emptiness formation probability (EFP) in the Hamiltonian with a defect. In particular, we first show that the FP of finding a particular configuration in the ground state is equivalent to the EFP of the ground state of the quadratic Hamiltonian with a defect. Then, we show that the probability of finding a particular value for any quadratic observable is equivalent to a FP problem and ultimately leads to the calculation of EFP in the ground state of a Hamiltonian with a defect. We provide new exact determinant formulas for the FP in the generic quadratic fermionic Hamiltonians. As applications of our formalism we study the statistics of the number of particles and kinks. Our conclusions can be extended also to the quantum spin chains that can be mapped to the free fermions via Jordan-Wigner (J-W) transformtion. In particular, we provide an exact solution to the problem of the transverse field XY chain with a staggered line defect. We also study the distribution of magnetization and kinks in the transverse field XY chain and show how the dual nature of these quantities manifest itself in the distributions.

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Full counting statistics in the transverse field Ising chain

Cited by 71 publications

References 89 publications

How order melts after quantum quenches

How order melts after quantum quenches

From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

Formation probabilities and statistics of observables as defect problems in free fermions and quantum spin chains

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