arXiv subjects

Laurens Vanderstraeten

Publications and source records attributed to Laurens Vanderstraeten.

At least 37 records · Page 2Linked to original sources

arXiv2024

Generating function for projected entangled-pair states

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating function approach for tensor network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin-$1/2$ transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin-$1/2$ $J_1$-$J_2$ model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multi-particle excitations.

cond-mat.str-el↗

arXiv2024

Emergent conformal boundaries from finite-entanglement scaling in matrix product states

The use of finite entanglement scaling with matrix product states (MPS) has become a crucial tool for studying 1+1d critical lattice theories, especially those with emergent conformal symmetry. We argue that finite entanglement introduces a relevant deformation in the critical theory. As a result, the bipartite entanglement Hamiltonian defined from the MPS can be understood as a boundary conformal field theory with a physical and an entanglement boundary. We are able to exploit the symmetry properties of the MPS to engineer the physical conformal boundary condition. The entanglement boundary, on the other hand, is related to the concrete lattice model and remains invariant under this relevant perturbation. Using critical lattice models described by the Ising, Potts, and free compact boson CFTs, we illustrate the influence of the symmetry and the relevant deformation on the conformal boundaries in the entanglement spectrum.

cond-mat.str-el↗

arXiv2024

Extracting the Speed of Light from Matrix Product States

We provide evidence that the spectrum of the local effective Hamiltonian and the transfer operator in infinite-system matrix product state simulations are identical up to a global rescaling factor, i.e.~the speed of light of the system, when the underlying system is described by a 1+1 dimensional CFT. We provide arguments for this correspondence based on a path integral point of view. This observation turns out to yield very precise estimates for the speed of light in practice, confirming exact results to high precision where available, but also allowing us to finally determine the speed of light of the non-integrable, critical $SU(2)$ Heisenberg chains with half-integer spin $S>1/2$ with unprecedented accuracy. We also show that the same technology applied to doped Hubbard ladders provides highly accurate velocities for a range of dopings. Combined with measurements of compressibilities we present new results for the Luttinger liquid parameter in the Luther-Emery regime of doped Hubbard ladders, outperforming earlier approaches based on the fitting of real-space correlation functions.

cond-mat.str-el↗

arXiv2023

Dynamical Signatures of Symmetry Broken and Liquid Phases in an $S=1/2$ Heisenberg Antiferromagnet on the Triangular Lattice

We present the dynamical spin structure factor of the antiferromagnetic spin-$\frac{1}{2}$ $J_1-J_2$ Heisenberg model on a triangular lattice obtained from large-scale matrix-product state simulations. The high frustration due to the combination of antiferromagnetic nearest and next-to-nearest neighbour interactions yields a rich phase diagram. We resolve the low-energy excitations both in the $120^{\circ}$-ordered phase and in the putative spin liquid phase at $J_2/J_1 = 0.125$. In the ordered phase, we observe an avoided decay of the lowest magnon-branch, demonstrating the robustness of this phenomenon in the presence of gapless excitations. Our findings in the spin-liquid phase chime with the field-theoretical predictions for a gapless Dirac spin liquid, in particular the picture of low-lying monopole excitations at the corners of the Brillouin zone. We comment on possible practical difficulties of distinguishing proximate liquid and solid phases based on the dynamical structure factor.

cond-mat.str-el↗

arXiv2023

Phase diagram of the chiral SU(3) antiferromagnet on the kagome lattice

Motivated by the search for chiral spin liquids (CSL), we consider a simple model defined on the kagome lattice of interacting SU(3) spins (in the fundamental representation) including two-site and three-site permutations between nearest neighbor sites and on triangles, respectively. By combining analytical developments and various numerical techniques, namely exact Lanczos diagonalizations and tensor network variational approaches, we find a rich phase diagram with non-topological (``trivial") and topological (possibly chiral) gapped spin liquids (SLs). Trivial spin liquids include an Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase and a trimerized phase, the latter breaking the inversion center between the up and down triangles of the kagome lattice. A topological SL is stabilized in a restricted part of the phase diagram by the time-reversal symmetry breaking (complex) 3-site permutation term. Analyzing the chiral edge modes of this topological SL on long cylinders or on finite disks, we have come up with two competing scenarios, either a CSL or a double Chern-Simon SL characterized by a single or by two counter-propagating Wess-Zumino-Witten SU(3)$_1$ chiral mode(s), respectively. In the vicinity of the extended ferromagnetic region we have found a magnetic phase corresponding either to a modulated canted ferromagnet or to a uniform partially magnetized ferromagnet.

cond-mat.str-el↗

arXiv2023

Efficient higher-order matrix product operators for time evolution

We introduce a systematic construction of higher-order matrix product operator (MPO) approximations of the time evolution operator for generic (short and long range) one-dimensional Hamiltonians. We demonstrate the utility of our construction, by showing an order of magnitude speedup in simulation cost compared to conventional first-order MPO time evolution schemes.

cond-mat.str-el↗

arXiv2023

Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and $\mathrm{RP}^2$ models: zero-temperature phase transition versus finite-temperature crossover

Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and $\mathrm{RP}^2$ models. We demonstrate that uniform matrix product states (MPS) with explicit $\mathrm{SO}(3)$ symmetry can probe correlation lengths up to $\mathcal{O}(10^3)$ sites accurately, and we study the scaling of entanglement entropy and universal features of MPS entanglement spectra. For the Heisenberg model, we find no signs of a finite-temperature phase transition, supporting the scenario of asymptotic freedom. For the $\mathrm{RP}^2$ model we observe an abrupt onset of scaling behaviour, consistent with hints of a finite-temperature phase transition reported in previous studies. A careful analysis of the softening of the correlation length divergence, the scaling of the entanglement entropy and the MPS entanglement spectra shows that our results are inconsistent with true criticality, but are rather in agreement with the scenario of a crossover to a pseudo-critical region which exhibits strong signatures of nematic quasi-long-range order at length scales below the true correlation length. Our results reveal a fundamental difference in scaling behaviour between the Heisenberg and $\mathrm{RP}^2$ models: Whereas the emergence of scaling in the former shifts to zero temperature if the bond dimension is increased, it occurs at a finite bond-dimension independent crossover temperature in the latter.

cond-mat.stat-mech↗

arXiv2022

Simulating chiral spin liquids with projected entangled-pair states

Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i) the rapid decrease of spurious features -- SU(2) symmetry breaking and long-range tails in correlations -- together with (ii) a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.

cond-mat.str-el↗

arXiv2022

A scaling hypothesis for projected entangled-pair states

We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly-correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions $D$. The key ingredient consists of using the effective correlation length $χ$ for inducing a collapse of data points, $f(D,χ)=f(ξ(D,χ))$, for arbitrary values of $D$ and the environment bond dimension $χ$. As such we circumvent the need for extrapolations in $χ$ and can use many distinct data points for a fixed value of $D$. Here, we need that the PEPS has been optimized using a fixed-$χ$ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2-D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3-D dimer model, the 3-D classical Ising model, and the 2-D quantum Heisenberg model.

quant-ph↗

arXiv2022

Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice

Motivated by dipolar-coupled artificial spin systems, we present a theoretical study of the classical $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice. We establish the ground-state phase diagram of this model for $J_1 > |J_2|, |J_3|$ based on exact results for the ground-state energies. When all the couplings are antiferromagnetic, the model has three macroscopically degenerate ground-state phases, and using tensor networks, we can calculate the entropies of these phases and of their boundaries very accurately. In two cases, the entropy appears to be a fraction of that of the triangular lattice Ising antiferromagnet, and we provide analytical arguments to support this observation. We also notice that, surprisingly enough, the dipolar ground state is not a ground state of the truncated model, but of the model with smaller $J_3$ interactions, an indication of a very strong competition between low-energy states in this model.

cond-mat.str-el↗

arXiv2022

Hole Spectral Function of a Chiral Spin Liquid in the Triangular Lattice Hubbard Model

Quantum spin liquids are fascinating phases of matter, hosting fractionalized spin excitations and unconventional long-range quantum entanglement. These exotic properties, however, also render their experimental characterization challenging, and finding ways to diagnose quantum spin liquids is therefore a pertinent challenge. Here, we numerically compute the spectral function of a single hole doped into the half-filled Hubbard model on the triangular lattice using techniques based on matrix product states. At half-filling the system has been proposed to realize a chiral spin liquid at intermediate interaction strength, surrounded by a magnetically ordered phase at strong interactions and a superconducting/metallic phase at weak interactions. We find that the spectra of these phases exhibit distinct signatures. By developing appropriate parton mean-field descriptions, we gain insight into the relevant low-energy features. While the magnetic phase is characterized by a dressed hole moving through the ordered spin background, we find indications of spinon dynamics in the chiral spin liquid. Our results suggest that the hole spectral function, as measured by angle-resolved photoemission spectroscopy, provides a useful tool to characterize quantum spin liquids.

cond-mat.str-el↗

arXiv2022

Direct sampling of projected entangled-pair states

Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers on long-standing questions such as the nature of the phases in the two-dimensional $J_1 - J_2$ model. The sampling in these Monte Carlo algorithms is typically performed with Markov Chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.

cond-mat.str-el↗

arXiv2022

Spinon confinement and deconfinement in spin-1 chains

Motivated by the rich phase diagrams recently unveiled in frustrated spin-1 chains with competing next-nearest neighbour and four-spin interactions, we investigate the nature of the elementary excitations of spin-1 chains in the vicinity of phase transitions using the matrix-product state (MPS) representation of elementary excitations. First, we show that both spinons and magnons naturally arise in SU(2) invariant spin chains when describing ground states and elementary excitations using MPS. Then we investigate the nature of the elementary excitations across the first-oder transition between the Haldane phase and the topologically trivial next-nearest neighbor Haldane phase. We show explicitly that spinons deconfine at the transition, and we calculate the dispersion of these deconfined spinons with MPS. We also show that, immediately away from the transition line, spinons confine, forming dispersive spinon/anti-spinon bound states on both sides of the transition. Finally, we show that deconfined spinons also appear at the transition between the Haldane phase and the spontaneously dimerized phase when this transition is first order.

cond-mat.str-el↗

arXiv2022

Efficient MPS methods for extracting spectral information on rings and cylinders

Based on the MPS formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing DMRG or MPS ground-state code. Although this approach is built on open-boundary MPS, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number, and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2-D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2-D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

cond-mat.str-el↗

arXiv2022

Variational methods for contracting projected entangled-pair states

The norms or expectation values of infinite projected entangled-pair states (PEPS) cannot be computed exactly, and approximation algorithms have to be applied. In the last years, many efficient algorithms have been devised -- the corner transfer matrix renormalization group (CTMRG) and variational uniform matrix product state (VUMPS) algorithm are the most common -- but it remains unclear whether they always lead to the same results. In this paper, we identify a subclass of PEPS for which we can reformulate the contraction as a variational problem that is algorithm independent. We use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions. Moreover, we devise a new variational contraction scheme, which we can extend to compute general N-point correlation functions.

cond-mat.str-el↗

arXiv2022

Momentum-resolved time evolution with matrix product states

We introduce a method based on matrix product states (MPS) for computing spectral functions of (quasi) one-dimensional spin chains, working directly in momentum space in the thermodynamic limit. We simulate the time evolution after applying a momentum operator to an MPS ground state by working with the momentum superposition of a window MPS. We show explicitly for the spin-1 Heisenberg chain that the growth of entanglement is smaller in momentum space, even inside a two-particle continuum, such that we can attain very accurate spectral functions with relatively small bond dimension. We apply our method to compute spectral lineshapes of the gapless XXZ chain and the square-lattice J1-J2 Heisenberg model on a six-leg cylinder.

cond-mat.str-el↗

arXiv2022

The Haldane gap in the SU(3) [3 0 0] Heisenberg chain

We calculate the Haldane gap of the $\mathrm{SU}(3)$ spin $[3~0~0]$ Heisenberg model using variational uniform fully symmetric $\mathrm{SU}(3)$ matrix product states, and find that the minimal gap $Δ/J = 0.0263 $ is obtained in the $[2~1~0]$ sector at momentum $2π/3$. We also discuss the symmetry protected topological order of the ground state, and determine the full dispersion relation of the elementary excitations and the correlation lengths of the system.

cond-mat.str-el↗

arXiv2022

Generative modeling with projected entangled-pair states

We argue and demonstrate that projected entangled-pair states (PEPS) outperform matrix product states significantly for the task of generative modeling of datasets with an intrinsic two-dimensional structure such as images. Our approach builds on a recently introduced algorithm for sampling PEPS, which allows for the efficient optimization and sampling of the distributions.

quant-ph↗