arXiv subjects

Roman Orus

Publications and source records attributed to Roman Orus.

At least 73 records · Page 4Linked to original sources

arXiv2018

Infinite Projected Entangled-Pair State algorithm for ruby and triangle-honeycomb lattices

The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and entanglement entropy (EE) on an infinite cylinder. As a benchmark, we apply the iPEPS method to the ruby model with anisotropic interactions and explore the ground-state properties of the system. We further extract the phase diagram of the model in different regimes of the couplings by measuring two-point correlators, ground state fidelity and EE on an infinite cylinder. Our phase diagram is in agreement with previous studies of the model by exact diagonalization.

cond-mat.str-el↗

arXiv2017

Holographic encoding of universality in corner spectra

In numerical simulations of classical and quantum lattice systems, 2d corner transfer matrices (CTMs) and 3d corner tensors (CTs) are a useful tool to compute approximate contractions of infinite-size tensor networks. In this paper we show how the numerical CTMs and CTs can be used, {\it additionally\/}, to extract universal information from their spectra. We provide examples of this for classical and quantum systems, in 1d, 2d and 3d. Our results provide, in particular, practical evidence for a wide variety of models of the correspondence between $d$-dimensional quantum and $(d+1)$-dimensional classical spin systems. We show also how corner properties can be used to pinpoint quantum phase transitions, topological or not, without the need for observables. Moreover, for a chiral topological PEPS we show by examples that corner tensors can be used to extract the entanglement spectrum of half a system, with the expected symmetries of the $SU(2)_k$ Wess-Zumino-Witten model describing its gapless edge for $k=1,2$. We also review the theory behind the quantum-classical correspondence for spin systems, and provide a new numerical scheme for quantum state renormalization in 2d using CTs. Our results show that bulk information of a lattice system is encoded holographically in efficiently-computable properties of its corners.

cond-mat.str-el↗

arXiv2017

Tensor network simulation of QED on infinite lattices: learning from (1+1)d, and prospects for (2+1)d

The simulation of lattice gauge theories with tensor network (TN) methods is becoming increasingly fruitful. The vision is that such methods will, eventually, be used to simulate theories in $(3+1)$ dimensions in regimes difficult for other methods. So far, however, TN methods have mostly simulated lattice gauge theories in $(1+1)$ dimensions. The aim of this paper is to explore the simulation of quantum electrodynamics (QED) on infinite lattices with TNs, i.e., fermionic matter fields coupled to a $U(1)$ gauge field, directly in the thermodynamic limit. With this idea in mind we first consider a gauge-invariant iDMRG simulation of the Schwinger model -i.e., QED in $(1+1)d$-. After giving a precise description of the numerical method, we benchmark our simulations by computing the substracted chiral condensate in the continuum, in good agreement with other approaches. Our simulations of the Schwinger model allow us to build intuition about how a simulation should proceed in $(2+1)$ dimensions. Based on this, we propose a variational ansatz using infinite Projected Entangled Pair States (PEPS) to describe the ground state of $(2+1)d$ QED. The ansatz includes $U(1)$ gauge symmetry at the level of the tensors, as well as fermionic (matter) and bosonic (gauge) degrees of freedom both at the physical and virtual levels. We argue that all the necessary ingredients for the simulation of $(2+1)d$ QED are, a priori, already in place, paving the way for future upcoming results.

hep-lat↗

arXiv2017

A simple tensor network algorithm for two-dimensional steady states

Understanding dissipation in 2D quantum many-body systems is a remarkably difficult open challenge. Here we show how numerical simulations for this problem are possible by means of a tensor network algorithm that approximates steady-states of 2D quantum lattice dissipative systems in the thermodynamic limit. Our method is based on the intuition that strong dissipation kills quantum entanglement before it gets too large to handle. We test its validity by simulating a dissipative quantum Ising model, relevant for dissipative systems of interacting Rydberg atoms, and benchmark our simulations with a variational algorithm based on product and correlated states. Our results support the existence of a first order transition in this model, with no bistable region. We also simulate a dissipative spin-1/2 XYZ model, showing that there is no re-entrance of the ferromagnetic phase. Our method enables the computation of steady states in 2D quantum lattice systems.

cond-mat.str-el↗

arXiv2017

Kitaev honeycomb tensor networks: exact unitary circuits and applications

The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivial physical properties such as topological quantum order, abelian and non-abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different "solution ingredients" of the Kitaev honeycomb model can be accounted for in the TN language, namely: Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. We also show how the properties of two-point correlation functions follow easily. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d Projected Entangled Pair State (PEPS) with finite bond dimension. The results in this paper can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.

cond-mat.str-el↗

arXiv2016

The spin-1/2 Kagome XXZ model in a field: competition between lattice nematic and solid orders

We study numerically the spin-1/2 XXZ model in a field on an infinite Kagome lattice. We use different algorithms based on infinite Projected Entangled Pair States (iPEPS) for this, namely: (i) with simplex tensors and 9-site unit cell, and (ii) coarse-graining three spins in the Kagome lattice and mapping it to a square-lattice model with nearest-neighbor interactions, with usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at $m_z = \frac{1}{3}$ using 6- 9- and 12-site PEPS unit cells, and at $m_z = \frac{1}{9}, \frac{5}{9}$ and $\frac{7}{9}$ using a 9-site PEPS unit cell, the later set-up being able to accommodate $\sqrt{3} \times \sqrt{3}$ solid order. We also find that, at $m_z = \frac{1}{3}$, (lattice) nematic and $\sqrt{3} \times \sqrt{3}$ VBC-order states are degenerate within the accuracy of the 9-site simplex-method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1 \times 2$ VBC-Solid orders. Within our accuracy, the 6-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even when the PEPS unit-cell is not commensurate with the expected ground state. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ spin chain in a field, and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor networks.

cond-mat.str-el↗

arXiv2016

Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry

We elaborate a simple classification scheme of all rank-5 SU(2)-spin rotational symmetric tensors according to i) the on-site physical spin-$S$, (ii) the local Hilbert space $V^{\otimes 4}$ of the four virtual (composite) spins attached to each site and (iii) the irreducible representations of the $C_{4v}$ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally-invariant Projected Entangled Pair States (PEPS) with bond dimension $D\leqslant 6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $({\cal D}-1)$-dimensional manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of ${\cal D}$ independent tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time reversal symmetry (chiral spin liquids) or (iii) SU(2)-spin rotation symmetry down to $U(1)$ (spin nematics or Néel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)$_2$ Wess-Zumino-Witten model.

cond-mat.str-el↗

arXiv2016

Symmetry-protected intermediate trivial phases in quantum spin chains

Symmetry-protected trivial (SPt) phases of matter are the product-state analogue of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to non-trivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses Matrix Product States (MPS) and the infinite Time-Evolving Block Decimation (iTEBD) method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by Conformal Field Theories (CFT) with central charge c=1. Our results are in agreement, and further generalize, those in [Y. Fuji, F. Pollmann, M. Oshikawa, Phys. Rev. Lett. 114, 177204 (2015)].

cond-mat.str-el↗

arXiv2016

Spin-S Kagome quantum antiferromagnets in a field with tensor networks

Spin-$S$ Heisenberg quantum antiferromagnets on the Kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero temperature) phase diagrams up to $S=2$ directly in the thermodynamic limit thanks to infinite Projected Entangled Pair States (iPEPS), a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau vs field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semi-classical, as the plateaus at $\frac{1}{3}$th, $(1-\frac{2}{9S})$th and $(1-\frac{1}{9S})$th of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin-$\frac{1}{2}$ $\frac{1}{9}$-plateau exhibiting coexistence of charge and bond orders. Upon restoration of the spin rotation $U(1)$ symmetry a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough field, such as the $S=2$ (topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.

cond-mat.str-el↗

arXiv2015

The iPEPS algorithm, improved: fast full update and gauge fixing

The infinite Projected Entangled Pair States (iPEPS) algorithm [J. Jordan et al, PRL 101, 250602 (2008)] has become a useful tool in the calculation of ground state properties of 2d quantum lattice systems in the thermodynamic limit. Despite its many successful implementations, the method has some limitations in its present formulation which hinder its application to some highly-entangled systems. The purpose of this paper is to unravel some of these issues, in turn enhancing the stability and efficiency of iPEPS methods. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. As we shall show, this implies two crucial advantages: (i) dramatic computational savings, and (ii) improved overall stability. Besides, we extend the application of the \emph{local gauge fixing}, successfully implemented for finite-size PEPS [M. Lubasch, J. Ignacio Cirac, M.-C. Bañuls, PRB 90, 064425 (2014)], to the iPEPS algorithm. We see that the gauge fixing not only further improves the stability of the method, but also accelerates the convergence of the alternating least squares sweeping in the (either "full" or "fast full") tensor update scheme. The improvement in terms of computational cost and stability of the resulting "improved" iPEPS algorithm is benchmarked by studying the ground state properties of the quantum Heisenberg and transverse-field Ising models on an infinite square lattice.

cond-mat.str-el↗

arXiv2015

All spin-1 topological phases in a single spin-2 chain

Here we study the emergence of different Symmetry-Protected Topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $(\mathbb{Z}_2 \times \mathbb{Z}_2) + T$ symmetry group, where $\mathbb{Z}_2$ is some $π$-rotation in the spin internal space and $T$ is time-reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous, and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using Matrix Product States via the infinite Time Evolving Block Decimation method, as well as by effective field theory considerations. Our findings show, for the first time, that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.

cond-mat.str-el↗

arXiv2014

Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization

Topological order in a 2d quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here we show how topological phase transitions in 2d systems can be much better assessed by multipartite entanglement, as measured by the topological geometric entanglement of blocks. Specifically, we present an efficient tensor network algorithm based on Projected Entangled Pair States to compute this quantity for a torus partitioned into cylinders, and then use this method to find sharp evidence of topological phase transitions in 2d systems with a string-tension perturbation. When compared to tensor network methods for Rényi entropies, our approach produces almost perfect accuracies close to criticality and, on top, is orders of magnitude faster. The method can be adapted to deal with any topological state of the system, including minimally entangled ground states. It also allows to extract the critical exponent of the correlation length, and shows that there is no continuous entanglement-loss along renormalization group flows in topological phases.

cond-mat.str-el↗

arXiv2014

Topological Minimally Entangled States via Geometric Measure

Here we show how the Minimally Entangled States (MES) of a 2d system with topological order can be identified using the geometric measure of entanglement. We show this by minimizing this measure for the doubled semion, doubled Fibonacci and toric code models on a torus with non-trivial topological partitions. Our calculations are done either quasi-exactly for small system sizes, or using the tensor network approach in [R. Orus, T.-C. Wei, O. Buerschaper, A. Garcia-Saez, arXiv:1406.0585] for large sizes. As a byproduct of our methods, we see that the minimisation of the geometric entanglement can also determine the number of Abelian quasiparticle excitations in a given model. The results in this paper provide a very efficient and accurate way of extracting the full topological information of a 2d quantum lattice model from the multipartite entanglement structure of its ground states.

cond-mat.str-el↗

arXiv2014

Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography

This is a short review on selected theory developments on Tensor Network (TN) states for strongly correlated systems. Specifically, we briefly review the effect of symmetries in TN states, fermionic TNs, the calculation of entanglement Hamiltonians from Projected Entangled Pair States (PEPS), and the relation between the Multi-scale Entanglement Renormalization Ansatz (MERA) and the AdS/CFT or gauge/gravity duality. We stress the role played by entanglement in the emergence of several physical properties and objects through the TN language. Some recent results along these lines are also discussed.

cond-mat.str-el↗

arXiv2014

A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States

This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject. It should be a good place for newcomers to get familiarized with some of the key ideas in the field, specially regarding the numerics. After a very general introduction we motivate the concept of tensor network and provide several examples. We then move on to explain some basics about Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed.

cond-mat.str-el↗

arXiv2013

Geometric Entanglement in Topologically Ordered States

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically-ordered systems such as the toric code, double semion, color code, and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks, and a contribution quantifying the underlying pattern of long-range entanglement of the topologically-ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterisation of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the 2D color code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments. Motivated by this, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian symmetry, and which we use to analyse quantum double models. Furthermore, we also analyse the robustness of the topological contribution using renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of our results rely on the ability to disentangle single sites from the system, which is always possible in our framework. Aditionally, we relate our results to the relative entropy of entanglement in topological systems, and discuss some tensor network numerical approaches that could be used to extract the topological contribution for large systems beyond exactly-solvable models.

cond-mat.str-el↗

arXiv2012

Topological Geometric Entanglement

Here we show the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks, for the topological universality class of the toric code model. As happens for the entanglement entropy, we find that for large block sizes the geometric entanglement is, up to possible subleading corrections, the sum of two contributions: a non-universal bulk contribution obeying a boundary law times the number of blocks, and a universal contribution quantifying the underlying pattern of long-range entanglement of a topologically-ordered state.

cond-mat.str-el↗

arXiv2012

Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems

In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical simulation of quantum lattice systems of infinite size. This exploration is done mainly in one and two spatial dimensions (1d and 2d). We describe a number of numerical algorithms based on corner matri- ces and tensors to approximate different ground state properties of these systems. The proposed methods make also use of matrix product operators and projected entangled pair operators, and naturally preserve spatial symmetries of the system such as translation invariance. In order to assess the validity of our algorithms, we provide preliminary benchmarking calculations for the spin-1/2 quantum Ising model in a transverse field in both 1d and 2d. Our methods are a plausible alternative to other well-established tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in 2d. The computational complexity of the proposed algorithms is also considered and, in 2d, important differences are found depending on the chosen simulation scheme. We also discuss further possibilities, such as 3d quantum lattice systems, periodic boundary conditions, and real time evolution. This discussion leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of corner transfer matrices and corner tensors. Our paper also offers a perspective on many properties of the corner transfer matrix and its higher-dimensional generalizations in the light of novel tensor network methods.

cond-mat.str-el↗