Many-body density matrices for free fermions

Cheong, Siew Ann; Henley, Christopher L.

doi:10.1103/physrevb.69.075111

Cited by 143 publications

(160 citation statements)

References 14 publications

(17 reference statements)

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“…Thus H and C have common eigenfunctions and their eigenvalues ε k and ζ k are related according to (7). The ζ k lie between 0 and 1 while the ε k vary between −∞ and ∞.…”

Section: Basic Formulaementioning

confidence: 99%

“…For large N , the spectrum of C must approach that ofĈ which is given by the step function n(q). For this reason, most of the ζ k lie exponentially close to 0 and 1 [11,8] and C is not easy to handle numerically.…”

Section: Basic Formulaementioning

confidence: 99%

“…For this reason, they have been investigated for a number of model systems in the last years [4][5][6][7][8]. It was found that for free electrons or bosons the reduced density matrices have exponential form exp(−H) and thus look like thermal operators.…”

mentioning

confidence: 99%

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On the reduced density matrix for a chain of free electrons

Peschel¹

2004

J. Stat. Mech.: Theor. Exp.

144

215

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The properties of the reduced density matrix describing an interval of N sites in an infinite chain of free electrons are investigated. A commuting operator is found for arbitrary filling and also for open chains. For a half filled periodic chain it is used to determine the eigenfunctions for the dominant eigenvalues analytically in the continuum limit. Relations to the critical six-vertex model are discussed.Reduced density matrices for some portion of a larger system play a central role in the density-matrix renormalization group (DMRG) method [1][2][3]. For this reason, they have been investigated for a number of model systems in the last years [4][5][6][7][8]. It was found that for free electrons or bosons the reduced density matrices have exponential form exp(−H) and thus look like thermal operators. It was also realized that one can base the determination of H on the one-particle correlation functions in the state one is studying [7,[9][10][11]. However, for critical lattice systems, which are particularly interesting in view of DMRG applications, the properties of H have only been studied numerically so far. For free electrons hopping on a chain one can see in this way the typical finite-size effects on the spectra and the concentration of the eigenfunctions near the boundaries. Nevertheless, a more explicit solution, which also makes contact with some results found in field theory [12] in calculations of entanglement entropies, would be desirable.In the following we present such a solution by looking at the hopping model in a somewhat different way. We show that there exists a relatively simple operator T which commutes with the reduced density matrix for the ground state and thus has the same eigenfunctions. This holds for arbitrary filling and also for a subsystem at the end of a semi-infinite chain. The case of a half-filled periodic chain is then studied in detail. Working with T and taking a proper continuum limit allows to determine the general character of the single-particle eigenstates. For the low-lying states it is found that the eigenvalues of H and T even coincide up to a scale factor. We also discuss the connection with the critical six-vertex model. This allows to derive the spectrum of H by a conformal mapping. It also allows to view the density matrix as a particular transfer matrix and the commutation relation as a special case of similar relations which occur in the treatment of integrable two-dimensional models.

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“…Thus H and C have common eigenfunctions and their eigenvalues ε k and ζ k are related according to (7). The ζ k lie between 0 and 1 while the ε k vary between −∞ and ∞.…”

Section: Basic Formulaementioning

confidence: 99%

Section: Basic Formulaementioning

confidence: 99%

See 1 more Smart Citation

On the reduced density matrix for a chain of free electrons

Peschel¹

2004

J. Stat. Mech.: Theor. Exp.

144

215

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“…In a manner similar to generating natural orbitals, we suggest a good basis set for such an expansion of the spatial RDM can be generated with the correlation method [28,29]. The correlation method was developed originally to calculate the spatial entanglement for single determinant wave functions.…”

mentioning

confidence: 99%

Calculating the entanglement spectrum in quantum Monte Carlo with application toab initioHamiltonians

Tubman

Yang

2014

Phys. Rev. B

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Several algorithms have been proposed to calculate the spatial entanglement spectrum from high order Renyi entropies. In this work we present an alternative approach for computing the entanglement spectrum with quantum Monte Carlo for both continuum and lattice Hamiltonians. This method provides direct access to the matrix elements of the spatially reduced density matrix and we determine an estimator that can be used in variational Monte Carlo as well as other Monte Carlo methods. The algorithm is based on using a generalization of the Swap operator, which can be extended to calculate a general class of density matrices that can include combinations of spin, space, particle and even momentum coordinates. We demonstrate the method by applying it to the Hydrogen and Nitrogen molecules and describe for the first time how the spatial entanglement spectrum encodes a covalent bond that includes all the many body correlations.Density matrices traced out in real space are becoming a fundamental tool in characterizing different states of matter in condensed matter systems [1][2][3][4][5][6][7]. While the interest in spatial reduced density matrices (RDM) is quite recent, the use of density matrices in general is quite ubiquitous [8]. The calculation and usage of particle RDMs in quantum Monte Carlo (QMC) simulations are quite extensive and many techniques have been developed to calculate such density matrices [9][10][11].Recent QMC entanglement studies have focused on determining the Renyi entropies [12,13]. Calculations of spatial Renyi entanglement with QMC include lattice calculations of topological systems [14] and continuum calculations of Fermi liquids [15,16] and molecules [17]. The Renyi entropies calculated in these works are generally determined with the Swap operator which can be applied to interacting systems. In the community of ab initio research, there has also been much recent work on using QMC to make highly accurate calculations of the momentum distribution of realistic materials [10]. It turns out the estimators used to calculate the momentum distribution can be seen as a form of the Swap operator. In this work we use the best techniques that have been developed from both of these communities to introduce a generalization of the Swap operator, making it an efficient tool to calculate the entanglement spectrum of spatial RDMs.The spatial entanglement spectrum is derived from a density matrix ρ A in which a system is split into two regions (A and B), and the degrees of freedom in region B are traced out. The matrix elements of such a density matrix can be expanded in any basis that is complete in region A. However, in our numerical calculations we in general can only consider a finite number of basis elements. Therefore practical calculations of the entanglement spectrum are basis dependent, and a carefully selected basis is required. The algorithm presented here is different from recent proposals [18][19][20][21] for calculating the entanglement spectrum in several ways. First, there is no calculation o...

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“…The entanglement Hamiltonian can be constructed as a single-particle operator in a quadratic matrix [36,46,47], as it is completely determined by any correlation matrix of operators acting on the remaining part after the subsystem has been traced out. Our system consists of two subsystems, A and B.…”

Section: Entanglement Spectramentioning

confidence: 99%

Entanglement spectra of superconductivity ground states on the honeycomb lattice

Predin

Schliemann

2017

Eur. Phys. J. B

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We analytically evaluate the entanglement spectra of the superconductivity states in graphene, primarily focusing on the s-wave and chiral d x 2 −y 2 + idxy superconductivity states. We demonstrate that the topology of the entanglement Hamiltonian can differ from that of the subsystem Hamiltonian. In particular, the topological properties of the entanglement Hamiltonian of the chiral d x 2 −y 2 + idxy superconductivity state obtained by tracing out one spin direction clearly differ from those of the time-reversal invariant Hamiltonian of noninteracting fermions on the honeycomb lattice.

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Many-body density matrices for free fermions

Cited by 143 publications

References 14 publications

On the reduced density matrix for a chain of free electrons

On the reduced density matrix for a chain of free electrons

Calculating the entanglement spectrum in quantum Monte Carlo with application toab initioHamiltonians

Entanglement spectra of superconductivity ground states on the honeycomb lattice

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