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arXiv:2606.04086

Quantum String Interactions Revealed by Full Counting Statistics

Chang-Yan Wang, Xue-Feng Zhang

cond-mat.str-elcond-mat.quant-gasquant-phSubmitted Jun 2, 2026• Announced Jun 4, 2026• 0 citations
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Abstract

How quantum strings interact is a basic question for extended objects in quantum many-body physics. Even the simplest hard-core constraint (no crossing), can generate a nontrivial effective potential, whose microscopic form is difficult to determine because the relative distance between the strings is intrinsically nonlocal. Here we show that this nonlocality is naturally captured by full counting statistics (FCS). For two hard-core quantum strings, we derive an analytic FCS expression for the emergent interaction by identifying the virtual process in which the two strings touch and hop back. Using the FCS--entanglement relation, we find the effective potential has the entanglement-controlled asymptotic form $\lnΔE(r)\sim -π^2 r^2/(12 S_\ell)$ up to subleading terms, where $S_\ell$ is the entanglement entropy between the two halves of a quantum string. We confirm the theory using high-precision numerical calculations and finite-size FCS estimates. Our results reveal FCS as a direct route to effective interactions between quantum topological line-defects, which may also be extended to higher-form charge.

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Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.

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Performing quantum measurements produces not only the expectation value of a physical observable $O$ but also the probability distribution $P(o)$ of all possible outcomes $o$. The full counting statistics (FCS) $Z(ϕ, O)\equiv \sum_o e^{iϕo}P(o)$, a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of $Q_A$, the charge operator in subsystem $A$, for 1D systems described by non-Hermitian SYK-like models, which are solvable in the large-$N$ limit. In both the volume-law entangled phase for interacting systems and the critical phase for non-interacting systems, the conformal symmetry emerges, which gives $F(ϕ, Q_A)\equiv \log Z(ϕ, Q_A)\sim ϕ^2\log |A|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(ϕ, Q_A)\sim (1-\cosϕ) |\partial A|$ for $ζ\gg J$, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.

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Performing quantum measurements produces not only the expectation value of a physical observable $O$ but also the probability distribution $P(o)$ of all possible outcomes $o$. The full counting statistics (FCS) $Z(\ensuremath{\phi},O)\ensuremath{\equiv}{\ensuremath{\sum}}_{o}{e}^{i\ensuremath{\phi}o}P(o)$, a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of ${Q}_{A}$, the charge operator in subsystem $A$, for one-dimensional systems described by non-Hermitian Sachdev-Ye-Kitaev-like models, which are solvable in the large-$N$ limit. In both the volume-law entangled phase for interacting systems and the critical phase for noninteracting systems, the conformal symmetry emerges, which gives $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\equiv}lnZ(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}{\ensuremath{\phi}}^{2}ln|A|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}(1\ensuremath{-}cos\ensuremath{\phi})|\ensuremath{\partial}A|$ for $\ensuremath{\zeta}\ensuremath{\gg}J$, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequences of discrete symmetry, long-range hopping, and generalizations to higher dimensions.

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Full counting statistics for superconducting quantum point contacts

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The calculation of the full counting statistics (FCS) for quantum mechanical systems has attracted much attention in recent years. In this thesis the FCS for superconducting quantum point contacts have been calculated. Such quantum point contacts frequently occur in modern microelectronic devices, which exploit the special features of superconductors. The FCS for normal metal-superconductor and superconductor-superconductor contacts have been calculated using a generalized Keldysh formalism and the results have been compared to both experimental and theoretical results. The calculation of the FCS allows a precise understanding of the physical nature of charge transfer. Particularly doubled shot noise due to Andreev reflections was explained and the behaviour of the Josephson current for long measurement times was calculated. The obtained results are also valuable for the understanding of transport through quantum dots with superconducting leads.

AV Akademikerverlag eBooks•Oct 1, 2013•0 citations
How quantum strings interact is a basic question for extended objects in quantum many-body physics. Even the simplest hard-core constraint (no crossing), can generate a nontrivial effective potential, whose microscopic form is difficult to determine because the relative distance between the strings is intrinsically nonlocal. Here we show that this nonlocality is naturally captured by full counting statistics (FCS). For two hard-core quantum strings, we derive an analytic FCS expression for the emergent interaction by identifying the virtual process in which the two strings touch and hop back. Using the FCS--entanglement relation, we find the effective potential has the entanglement-controlled asymptotic form ln⁡ΔE(r)∼−π2r2/(12Sℓ)\lnΔE(r)\sim -π^2 r^2/(12 S_\ell)lnΔE(r)∼−π2r2/(12Sℓ​) up to subleading terms, where SℓS_\ellSℓ​ is the entanglement entropy between the two halves of a quantum string. We confirm the theory using high-precision numerical calculations and finite-size FCS estimates. Our results reveal FCS as a direct route to effective interactions between quantum topological line-defects, which may also be extended to higher-form charge.
Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.
Performing quantum measurements produces not only the expectation value of a physical observable OOO but also the probability distribution P(o)P(o)P(o) of all possible outcomes ooo. The full counting statistics (FCS) Z(ϕ,O)≡∑oeiϕoP(o)Z(ϕ, O)\equiv \sum_o e^{iϕo}P(o)Z(ϕ,O)≡∑o​eiϕoP(o), a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of QAQ_AQA​, the charge operator in subsystem AAA, for 1D systems described by non-Hermitian SYK-like models, which are solvable in the large-NNN limit. In both the volume-law entangled phase for interacting systems and the critical phase for non-interacting systems, the conformal symmetry emerges, which gives F(ϕ,QA)≡log⁡Z(ϕ,QA)∼ϕ2log⁡∣A∣F(ϕ, Q_A)\equiv \log Z(ϕ, Q_A)\sim ϕ^2\log |A|F(ϕ,QA​)≡logZ(ϕ,QA​)∼ϕ2log∣A∣. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as F(ϕ,QA)∼(1−cos⁡ϕ)∣∂A∣F(ϕ, Q_A)\sim (1-\cosϕ) |\partial A|F(ϕ,QA​)∼(1−cosϕ)∣∂A∣ for ζ≫Jζ\gg Jζ≫J, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.
Performing quantum measurements produces not only the expectation value of a physical observable OOO but also the probability distribution P(o)P(o)P(o) of all possible outcomes ooo. The full counting statistics (FCS) $Z(\ensuremath{\phi},O)\ensuremath{\equiv}{\ensuremath{\sum}}_{o}{e}^{i\ensuremath{\phi}o}P(o)$, a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of QA{Q}_{A}QA​, the charge operator in subsystem AAA, for one-dimensional systems described by non-Hermitian Sachdev-Ye-Kitaev-like models, which are solvable in the large-NNN limit. In both the volume-law entangled phase for interacting systems and the critical phase for noninteracting systems, the conformal symmetry emerges, which gives $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\equiv}lnZ(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}{\ensuremath{\phi}}^{2}ln|A|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}(1\ensuremath{-}cos\ensuremath{\phi})|\ensuremath{\partial}A|$ for $\ensuremath{\zeta}\ensuremath{\gg}J$, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequences of discrete symmetry, long-range hopping, and generalizations to higher dimensions.
This thesis investigates the dynamics of quantum many-body systems in isolation, in three different setups. We start by considering a paradigmatic quantum chaotic model of magnetism, the Heisenberg antiferromagnet in 2 and 3 spatial dimensions. The focus is on obtaining approximations for the full counting statistics (FCS) of a subsystem order parameter, the staggered magnetization, both in and out of equilibrium (following quantum quenches). We employ mappings of spin variables to bosonic ones, together with self-consistent time-dependent mean-field schemes, to compute the FCS via Gaussian techniques. In particular, we introduce a formula for the FCS of any subsystem observable that is at most quadratic in the bosons. We obtain results both in the presence and absence of long-range magnetic order. We then turn to the first of two ergodicity-breaking scenarios, by considering Hilbert space fragmentation (HSF) in charge- and dipole-conserving quantum chains. We determine the critical charge density at which a "freezing" phase transition from weak to strong fragmentation takes place, for any on-site Hilbert space dimension. To characterize the degree of dynamical connectivity in the system, we introduce a novel framework that enables us to obtain analytic results for the strongly fragmented phase, and numerically explore the weakly fragmented one. We derive some critical exponents, and show that the entanglement entropy of typical eigenstates in the strongly fragmented phase obeys an area law. We also uncover the existence of rare eigenstates whose entropy appears to scale beyond any area law. Finally, we investigate dynamical correlators in the Lieb-Liniger gas, one of the simplest interacting integrable theories solvable by Bethe ansatz. We employ Markov chain Monte Carlo methods to sample matrix elements (form factors) that appear in the Lehmann representation of the finite-temperature single-particle bosonic Green’s function. In the physical regimes we consider, sampling is necessary because a full summation of the relevant form factors is computationally intractable. We benchmark our results against exact solutions available in specific limits, and find them to be very accurate.
We study the full counting statistics (FCS) and symmetry-resolved entanglement entropies of integer and fractional quantum Hall states. For the filled lowest Landau level of spin-polarized electrons on an infinite cylinder, we compute exactly the charged moments associated with a cut orthogonal to the cylinder's axis. This yields the behavior of FCS and entropies in the limit of large perimeters: in a suitable range of fluctuations, FCS is Gaussian and entanglement spreads evenly among different charge sectors. Subleading charge-dependent corrections to equipartition are also derived. We then extend the analysis to Laughlin wave functions, where entanglement spectroscopy is carried out assuming the Li-Haldane conjecture. The results confirm equipartition up to small charge-dependent terms, and are then matched with numerical computations based on exact matrix product states.
The calculation of the full counting statistics (FCS) for quantum mechanical systems has attracted much attention in recent years. In this thesis the FCS for superconducting quantum point contacts have been calculated. Such quantum point contacts frequently occur in modern microelectronic devices, which exploit the special features of superconductors. The FCS for normal metal-superconductor and superconductor-superconductor contacts have been calculated using a generalized Keldysh formalism and the results have been compared to both experimental and theoretical results. The calculation of the FCS allows a precise understanding of the physical nature of charge transfer. Particularly doubled shot noise due to Andreev reflections was explained and the behaviour of the Josephson current for long measurement times was calculated. The obtained results are also valuable for the understanding of transport through quantum dots with superconducting leads.