arXiv subjects

M. A. Rajabpour

Publications and source records attributed to M. A. Rajabpour.

At least 37 records · Page 2Linked to original sources

arXiv2024

Efficient Representation of Gaussian Fermionic Pure States in Non-Computational Bases

This paper introduces an innovative approach for representing Gaussian fermionic states, pivotal in quantum spin systems and fermionic models, within a range of alternative quantum bases. We focus on transitioning these states from the conventional computational (σ^z) basis to more complex bases, such as ((ϕ, \fracπ{2}, α)), which are essential for accurately calculating critical quantities like formation probabilities and Shannon entropy. We present a novel algorithm that not only simplifies the basis transformation but also reduces computational complexity, making it feasible to calculate amplitudes of large systems efficiently. Our key contribution is a technique that translates amplitude calculations into the Pfaffian computation of submatrices from an antisymmetric matrix, a process facilitated by understanding domain wall relationships across different bases. As an application, we will determine the formation probabilities for various bases and configurations within the critical transverse field Ising chain, considering both periodic and open boundary conditions. We aim to categorize the configurations and bases by examining the universal constant term that characterizes the scaling of the logarithm of the formation probability in the periodic system, as well as the coefficient of the logarithmic term in the case of open systems. In the open system scenario, this coefficient is influenced by the central charge and the conformal weight of the boundary condition-changing operator. This work is set to expand the toolkit available for researchers in quantum information theory and many-body physics, providing a more efficient and elegant solution for exploring Gaussian fermionic states in non-standard quantum bases.

quant-ph↗

arXiv2024

Statistical physics of principal minors: Cavity approach

Determinants are useful to represent the state of an interacting system of (effectively) repulsive and independent elements, like fermions in a quantum system and training samples in a learning problem. A computationally challenging problem is to compute the sum of powers of principal minors of a matrix which is relevant to the study of critical behaviors in quantum fermionic systems and finding a subset of maximally informative training data for a learning algorithm. Specifically, principal minors of positive square matrices can be considered as statistical weights of a random point process on the set of the matrix indices. The probability of each subset of the indices is in general proportional to a positive power of the determinant of the associated sub-matrix. We use Gaussian representation of the determinants for symmetric and positive matrices to estimate the partition function (or free energy) and the entropy of principal minors within the Bethe approximation. The results are expected to be asymptotically exact for diagonally dominant matrices with locally tree-like structures. We consider the Laplacian matrix of random regular graphs of degree $K=2,3,4$ and exactly characterize the structure of the relevant minors in a mean-field model of such matrices. No (finite-temperature) phase transition is observed in this class of diagonally dominant matrices by increasing the positive power of the principal minors, which here plays the role of an inverse temperature.

cond-mat.stat-mech↗

arXiv2024

Generalization of Balian-Brezin decomposition for exponentials with linear fermionic part

Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick's theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian-Brezin decomposition (BBD) to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick's theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the $\mathfrak{so}(N)$ Lie algebra.

math-ph↗

arXiv2024

A field theory representation of sum of powers of principal minors and physical applications

We introduce a novel field theory representation for the Sum of Powers of Principal Minors (SPPM), a mathematical construct with profound implications in quantum mechanics and statistical physics. We begin by establishing a Berezin integral formulation of the SPPM problem, showcasing its versatility through various symmetries including $SU(n)$, its subgroups, and particle-hole symmetry. This representation not only facilitates new analytical approaches but also offers deeper insights into the symmetries of complex quantum systems. For instance, it enables the representation of the Hubbard model's partition function in terms of the SPPM problem. We further develop three mean field techniques to approximate SPPM, each providing unique perspectives and utilities: the first method focuses on the evolution of symmetries post-mean field approximation, the second, based on the bosonic representation, enhances our understanding of the stability of mean field results, and the third employs a variational approach to establish a lower bound for SPPM. These methods converge to identical consistency relations and values for SPPM, illustrating their robustness. The practical applications of our theoretical advancements are demonstrated through two compelling case studies. First, we exactly solve the SPPM problem for the Laplacian matrix of a chain, a symmetric tridiagonal matrix, allowing for precise benchmarking of mean-field theory results. Second, we present the first analytical calculation of the Shannon-Rényi entropy for the transverse field Ising chain, revealing critical insights into phase transitions and symmetry breaking in the ferromagnetic phase. This work not only bridges theoretical gaps in understanding principal minors within quantum systems but also sets the stage for future explorations in more complex quantum and statistical physics models.

quant-ph↗

arXiv2023

Identifying quantum many-body integrability and chaos using eigenstates trace distances

While the concepts of quantum many-body integrability and chaos are of fundamental importance for the understanding of quantum matter, their precise definition has so far remained an open question. In this work, we introduce an alternative indicator for quantum many-body integrability and chaos, which is based on the statistics of eigenstates by means of nearest-neighbor subsystem trace distances. We show that this provides us with a faithful classification through extensive numerical simulations for a large variety of paradigmatic model systems including random matrix theories, free fermions, Bethe-ansatz solvable systems, and models of many-body localization. While existing indicators, such as those obtained from level-spacing statistics, have already been utilized with great success, they also face limitations. This concerns for instance the quantum many-body kicked top, which is exactly solvable but classified as chaotic in certain regimes based on the level-spacing statistics, while our introduced indicator signals the expected quantum many-body integrability. We discuss the universal behaviors we observe for the nearest-neighbor trace distances and point out that our indicator might be useful also in other contexts such as for the many-body localization transition.

cond-mat.stat-mech↗

arXiv2023

Trace distance between fermionic Gaussian states from a truncation method

In this paper, we propose a novel truncation method for determining the trace distance between two Gaussian states in fermionic systems. For two fermionic Gaussian states, characterized by their correlation matrices, we consider the von Neumann entropies and dissimilarities between their correlation matrices and truncate the correlation matrices to facilitate trace distance calculations. Our method exhibits notable efficacy in two distinct scenarios. In the first scenario, the states have small von Neumann entropies, indicating finite or logarithmic-law entropy, while their correlation matrices display near-commuting behavior, characterized by a finite or gradual nonlinear increase in the trace norm of the correlation matrix commutator relative to the system size. The second scenario encompasses situations where the two states are nearly orthogonal, with a maximal canonical value difference approaching 2. To evaluate the performance of our method, we apply it to various compelling examples. Notably, we successfully compute the subsystem trace distances between low lying eigenstates of Ising and XX spin chains, even for significantly large subsystem sizes. This is in stark contrast to existing literature, where subsystem trace distances are limited to subsystems of approximately ten sites. With our truncation method, we extend the analysis to subsystems comprising several hundred sites, thus expanding the scope of research in this field.

cond-mat.str-el↗

arXiv2022

Subsystem distances between quasiparticle excited states

We investigate the subsystem Schatten distance, trace distance and fidelity between the quasiparticle excited states of the free and the nearest-neighbor coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain. The results support the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. From the quasiparticle picture, we get the universal subsystem distances that are valid when both the large energy condition and the large momentum difference condition are satisfied, by which we mean each of the excited quasiparticles has a large energy and the momentum difference of each pair of the excited quasiparticles is large. In the free fermionic and bosonic chains, we use the subsystem mode method and get efficiently the subsystem distances, which are also valid in the coupled fermionic and bosonic chains if the large energy condition is satisfied. Moreover, under certain limit the subsystem distances from the subsystem mode method are even valid in the XXX chain. We expect that the results can be also generalized for other integrable models.

cond-mat.stat-mech↗

arXiv2022

Universal logarithmic correction to Rényi (Shannon) entropy in generic systems of critical quadratic fermions

The Rényi (Shannon) entropy, i.e. $Re_α(Sh)$, of the ground state of quantum systems in local bases normally show a volume-law behavior. For a subsystem of quantum chains at critical point there is an extra logarithmic subleading term with a coefficient which is universal. In this paper we study this coefficient for generic time-reversal translational invariant quadratic critical free fermions. These models can be parameterized by a complex function which has zeros on the unit circle. When the zeros on the unit circle do not have degeneracy and there is no zero outside of the unit circle we are able to classify the coefficient of the logarithm. In particular, we numerically calculate the Rényi (Shannon) entropy in configuration basis for wide variety of these models and show that there are two distinct classes. For systems with $U(1)$ symmetry the coefficient is proportional to the central charge, i.e. one half of the number of points that one can linearize the dispersion relation of the system; for all the values of $α$ with transition point at $α=4$. For systems without this symmetry, when $α>1$ this coefficient is again proportional to the central charge. However, the coefficient for $α\leq 1$ is a new universal number. Finally, by using the discrete version of Bisognano-Wichmann modular Hamiltonian of the Ising chain we show that these coefficients are universal and dependent on the underlying CFT.

cond-mat.stat-mech↗

arXiv2022

Entanglement entropy in quantum spin chains with broken parity number symmetry

Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in-depth the properties of the eigenstates of the enlarged Hamiltonian. In particular, we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.

cond-mat.str-el↗

arXiv2022

Entanglement of magnon excitations in spin chains

We calculate exactly the entanglement content of magnon excited states in the integrable spin-1/2 XXX and XXZ chains in the scaling limit. In particular, we show that as far as the number of excited magnons with respect to the size of the system is small one can decompose the entanglement content, in the scaling limit, to the sum of the entanglement of particular excited states of free fermionic or bosonic theories. In addition we conjecture that the entanglement content of the generic translational invariant free fermionic and bosonic Hamiltonians can be also classified, in the scaling limit, with respect to the entanglement content of the fermionic and bosonic chains with the number operator as the Hamiltonian in certain circumstances. Our results effectively classify the entanglement content of wide range of integrable spin chains in the scaling limit.

cond-mat.stat-mech↗

arXiv2021

Universal Rényi Entropy of Quasiparticle Excitations

The Rényi entropies of quasiparticle excitations in the many-body gapped systems show a remarkable universal picture which can be understood partially by combination of a semiclassical argument with the quantum effect of (in)distinguishability. The universal Rényi entropies are independent of the model, the quasiparticle momenta, and the connectedness of the subsystem. In this letter we calculate exactly the single-interval and double-interval Rényi entropies of quasiparticle excitations in the many-body gapped fermions, bosons, and XY chains. We find additional contributions to the universal Rényi entropy in the excited states with quasiparticles of different momenta. The additional terms are different in the fermionic and bosonic chains, depend on the momentum differences of the quasiparticles, and are different for the single interval and the double interval. We derive the analytical Rényi entropy in the extremely gapped limit, matching perfectly the numerical results as long as either the intrinsic correlation length of the model or all the de Broglie wavelengths of the quasiparticles are small. When the momentum difference of any pair of distinct quasiparticles is small, the additional terms are non-negligible. On the contrary, when the difference of the momenta of each pair of distinct quasiparticles is large, the additional terms could be neglected. The universal single-interval Rényi entropy and its additional terms in the XY chain are the same as those in the fermionic chain, while the universal Rényi entropy of the double intervals and its additional terms are different, due to the fact that the local degrees of freedom of the XY chain are the Pauli matrices not the spinless fermions. We argue that the derived formulas have universal properties and can be applied for a wider range of models than those discussed here.

cond-mat.stat-mech↗

arXiv2021

Corrections to universal Rényi entropy in quasiparticle excited states of quantum chains

We investigate the energy eigenstate Rényi entropy of generic bipartition in the fermionic, bosonic, and spin-1/2 XY chains. When the gap of the theory is large or all the momenta of the excited quasiparticles are large, the Rényi entropy takes a universal form, which is independent of the model, the quasiparticle momenta, and the subsystem connectedness. We calculate analytically the Rényi entropy in the extremely gapped limit and find different additional contributions to the universal Rényi entropy in various models. The corrections to the universal Rényi entropy cannot be neglected when the momentum differences of the excited quasiparticles are small. The Rényi entropy derived in the extremely gapped limit is still valid in the slightly gapped and even critical chains as long as all the momenta of the excited quasiparticles are large. In the case of double interval in the XY chain we find new universal results and their corrections. We call the result universal even though it is only valid for double interval in the spin-1/2 XY chain. In the case of the bosonic chain in the extremely massive limit we find analytically a novel formula for the Rényi entropy written as the permanent of a certain matrix. We support all of our analytical results with numerical calculations.

cond-mat.stat-mech↗

arXiv2021

Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part II. Multi-particle states

We study the excited state Rényi entropy and subsystem Schatten distance in the two-dimensional free massless non-compact bosonic field theory, which is a conformal field theory. The discretization of the free non-compact bosonic theory gives the harmonic chain with local couplings. We consider the field theory excited states that correspond to the harmonic chain states with excitations of more than one quasiparticle, which we call multi-particle states. This extends the previous work by the same authors to more general excited states. In the field theory we obtain the exact Rényi entropy and subsystem Schatten distance for several low-lying states. We also obtain the short interval expansion of the Rényi entropy and subsystem Schatten distance for general excited states. In the locally coupled harmonic chain we calculate numerically the excited state Rényi entropy and subsystem Schatten distance using the wave function method. We find excellent matches of the analytical results in the field theory and numerical results in the gapless limit of the harmonic chain. We also make some preliminary investigations of the Rényi entropy and the subsystem Schatten distance in the extremely gapped limit of the harmonic chain.

hep-th↗

arXiv2021

Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain

We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a byproduct, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both $σ^z$ and $σ^x$ basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a new conjecture.

cond-mat.stat-mech↗

arXiv2020

Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory -- I. Single-particle states

We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless harmonic chain. We calculate analytically the second Rényi entropy and the second Schatten distance in the massless bosonic theory. We then use the wave functions of the excited states and calculate the second Rényi entropy and the second Schatten distance in the gapless limit of the harmonic chain, which match perfectly with the analytical results in the massless bosonic theory. We verify that in the large momentum limit the single-particle state Rényi entropy takes a universal form. We also show that in the limit of large momenta and large momentum difference the subsystem Schatten distance takes a universal form but it is replaced by a corrected form when the momentum difference is small. Finally we also comment on the mutual Rényi entropy of two disjoint intervals in the excited states of the two-dimensional free non-compact bosonic theory.

hep-th↗

arXiv2020

Scaling of the Formation Probabilities and Universal Boundary Entropies in the Quantum XY Spin Chain

We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse $σ^z$-basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal Boundary Entropy at the critical point. The latter is a benchmark of the underlying Boundary Conformal Field Theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the Boundary Entropies are computed by exploiting a generalization of the Euler-MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.

cond-mat.stat-mech↗

arXiv2020

Entanglement and boundary entropy in quantum spin chains with arbitrary direction of the boundary magnetic fields

We calculate the entanglement and the universal boundary entropy (BE) in the critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory (BCFT) depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, that apply to the ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group (DMRG). The transverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem {\it{integrable}}. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.

cond-mat.str-el↗

arXiv2020

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

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) transformation. 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.

cond-mat.stat-mech↗