S. De Baerdemacker

Method for making 2-electron response reduced density matrices approximately N-representable

C. Lanssens, Paul W. Ayers, D. Van Neck, S. De Baerdemacker, K. Gunst, P. Bultinck
Journal of Chemical Physics
148, 8, 084104
2018
A1

Abstract 

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schrödinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not $N$-representable. That is, the response 2-RDM does not correspond to an actual physical $N$-electron wave function. We present a new algorithm for making these non-$N$-representable 2-RDMs approximately $N$-representable, i.e., it has the right symmetry and normalization and it fulfills the $P$-, $Q$-, and $G$-conditions. Next to an algorithm which can be applied to any 2-RDM, we have also developed a 2-RDM optimization procedure specifically for seniority-zero 2-RDMs. We aim to find the 2-RDM with the right properties which is the closest (in the sense of the Frobenius norm) to the non-$N$-representable 2-RDM by minimizing the square norm of the difference between this initial response 2-RDM and the targeted 2-RDM under the constraint that the trace is normalized and the 2-RDM, $Q$-matrix, and $G$-matrix are positive semidefinite, i.e., their eigenvalues are non-negative. Our method is suitable for fixing non-$N$-representable 2-RDMs which are close to being $N$-representable. Through the $N$-representability optimization algorithm we add a small correction to the initial 2-RDM such that it fulfills the most important $N$-representability conditions.

Variational method for integrability-breaking Richardson-Gaudin models

P. Claeys, J.-S. Caux, D. Van Neck, S. De Baerdemacker
Physical Review B
96, 155149
2017
A1

Abstract 

We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is shown to return exact results for integrable models and improve substantially on perturbation theory for models close to integrability. For large integrability-breaking interactions, it is shown how (avoided) level crossings necessitate the use of excited states of integrable Hamiltonians in order to accurately describe the ground states of general nonintegrable models.

Green Open Access

Inner products in integrable Richardson-Gaudin models

P. Claeys, D. Van Neck, S. De Baerdemacker
Scipost Physics
3, 028
2017
A1

Abstract 

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.
Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

Open Access version available at UGent repository
Gold Open Access

Block product density matrix embedding theory for strongly correlated spin systems

K. Gunst, S. Wouters, S. De Baerdemacker, D. Van Neck
Physical Review B
95, 195127
2017
A1

Abstract 

Density matrix embedding theory (DMET) is a relatively new technique for the calculation of strongly correlated systems. Recently, block product DMET (BPDMET) was introduced for the study of spin systems such as the antiferromagnetic J1−J2 model on the square lattice. In this paper, we extend the variational Ansatz of BPDMET using spin-state optimization, yielding improved results. We apply the same techniques to the Kitaev-Heisenberg model on the honeycomb lattice, comparing the results when using several types of clusters. Energy profiles and correlation functions are investigated. A diagonalization in the tangent space of the variational approach yields information on the excited states and the corresponding spectral functions.

Open Access version available at UGent repository
Green Open Access

Block-ZXZ synthesis of an arbitrary quantum circuit

A. De Vos (Alexis), S. De Baerdemacker
Physical Review A
94, 052317
2016
A1

Abstract 

Given an arbitrary 2w×2w unitary matrix U, a powerful matrix decomposition can be applied, leading to four different syntheses of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w=1) to gates with arbitrary value of w. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that U is a permutation matrix.

The Birkhoff theorem for unitary matrices of arbitrary dimensions

S. De Baerdemacker, A. De Vos (Alexis), L. Chen, L. Yu
Linear Algebra and its Applications
514,151–164
2016
A1

Abstract 

It was shown recently that Birkho's theorem for doubly stochastic matrices
can be extended to unitary matrices with equal line sums whenever the dimension
of the matrices is prime. We prove a generalization of the Birkho
theorem for unitary matrices with equal line sums for arbitrary dimension.

Read-Green resonances in a topological superconductor coupled to a bath

P. Claeys, S. De Baerdemacker, D. Van Neck
Physical Review B
93 (22), 220503
2016
A1

Abstract 

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchanges, which can be observed through resonances in low-energy and low-momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.

Open Access version available at UGent repository

Read-Green resonances in a topological superconductor coupled to a bath

P. Claeys, S. De Baerdemacker, D. Van Neck
Physical Review B
93 (22) 220503(R)
2016
A1

Abstract 

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchanges, which can be observed through resonances in low-energy and low-momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.

Maximum probability domains for Hubbard models

G. Acke, S. De Baerdemacker, P. Claeys, M. Van Raemdonck, W. Poelmans, D. Van Neck, P. Bultinck
Molecular Physics
114 (7-8), 1392-1405
2016
A1

Abstract 

The theory of maximum probability domains (MPDs) is formulated for the Hubbard model in terms of projection operators and generating functions for both exact eigenstates as well as Slater determinants. A fast MPD analysis procedure is proposed, which is subsequently used to analyse numerical results for the Hubbard model. It is shown that the essential physics behind the considered Hubbard models can be exposed using MPDs. Furthermore, the MPDs appear to be in line with what is expected from Valence Bond (VB) Theory-based knowledge.

Open Access version available at UGent repository

The Birkhoff theorem for unitary matrices of prime dimension

A. De Vos (Alexis), S. De Baerdemacker
Linear Algebra and its Applications
493 (2016), 455-468
2016
A1

Abstract 

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. We prove that any unitary matrix with equal line sums can also be written as a sum of permutation matrices (with sum of weights equal 1). Furthermore, when the matrix dimension is prime, we prove that the unitary matrix lies inside a convex complex Birkhoff polytope.

Open Access version available at UGent repository

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