Full counting statistics of the subsystem energy for free fermions and quantum spin chains

Najafi, Khadijeh; Rajabpour, M. A.

doi:10.1103/physrevb.96.235109

Cited by 26 publications

(20 citation statements)

References 52 publications

(63 reference statements)

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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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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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“…An interesting feature of these experiments is that in certain limits, density measurements after matter-wave interference [13,25] correspond to projective von Neumann measurements of the relative phase field [26]. This allows for the reconstruction of full distribution functions of quantum mechanical observables [21][22][23], which is of considerable theoretical interest [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] in general. In the case at hand, situations without tunnel-coupling can be modelled by a two-component Luttinger liquid [42][43][44].…”

Section: Contentsmentioning

confidence: 99%

Report on scipost_202005_00003v1

2020

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We consider a model of two tunnel-coupled one-dimensional Bose gases with hard-wall boundary conditions. Bosonizing the model and retaining only the most relevant interactions leads to a decoupled theory consisting of a quantum sine-Gordon model and a free boson, describing respectively the antisymmetric and symmetric combinations of the phase fields. We go beyond this description by retaining the perturbation with the next smallest scaling dimension. This perturbation carries conformal spin and couples the two sectors. We carry out a detailed investigation of the effects of this coupling on the non-equilibrium dynamics of the model. We focus in particular on the role played by spatial inhomogeneities in the initial state in a quantum quench setup.

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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 of the subsystem energy for free fermions and quantum spin chains

Cited by 26 publications

References 52 publications

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

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

Report on scipost_202005_00003v1

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

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