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)].Comment: 7 pages, 5 figures, 1 table, revised version. Accepted in PR

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 $\pi$-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.Comment: 7 pages, 6 figures, revised version. To appear in PR

Entanglement Estimation in Tensor Network States via Sampling

We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions. Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory. In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames. Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components. We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry. Although the cost is exponential in the subsystem size, it is sufficiently moderate so that—in contrast with other approaches—accurate results can be obtained on a personal computer for relatively large subsystem sizes

Pinwheel valence bond crystal ground state of the spin-1/ 2 Heisenberg antiferromagnet on the shuriken lattice

We investigate the nature of the ground state of the spin-12 Heisenberg antiferromagnet on the shuriken lattice by complementary state-of-the-art numerical techniques, such as variational Monte Carlo (VMC) with versatile Gutzwiller-projected Jastrow wave functions, unconstrained multivariable variational Monte Carlo (mVMC), and pseudofermion/pseudo-Majorana functional renormalization group (PFFRG/PMFRG) methods. We establish the presence of a quantum paramagnetic ground state and investigate its nature, by classifying symmetric and chiral quantum spin liquids, and inspecting their instabilities towards competing valence bond crystal (VBC) orders. Our VMC analysis reveals that a VBC with a pinwheel structure emerges as the lowest-energy variational ground state, and it is obtained as an instability of the U(1) Dirac spin liquid. Analogous conclusions are drawn from mVMC calculations employing accurate BCS pairing states supplemented by symmetry projectors, which confirm the presence of pinwheel VBC order by a thorough analysis of dimer-dimer correlation functions. Our work highlights the nontrivial role of quantum fluctuations via the Gutzwiller projector in resolving the subtle interplay between competing orders

Quantum many-body systems and Tensor Network algorithms

Theory of quantum many-body systems plays a key role in understanding the properties of phases of matter found in nature. Due to the exponential growth of the dimensions of the Hilbert space with the number of particles, quantum many-body problems continue to be one of the greatest challenges in physics and most of these systems are impossible to study exactly. We therefore need efficient and accurate numerical algorithms to understand them. In this thesis, we exploit a new numerical technique known as Tensor Network algorithms to study exotic phases of matter in three different investigations in one and two spatial dimensions. In the first part, we use Matrix Product States which is a one-dimensional ansatz of the Tensor Network family to study trivial and topological phases of matter protected by symmetries in a spin-2 quantum chain. For this, we investigate a Heisenberg-like model with bilinear, biquadratic, bicubic and biquartic interactions with an additional uniaxial anisotropy term. We also add a staggered magnetic field afterwards to break the symmetries protecting the topological phases and study their ground state properties. In the second part of the thesis, we use Tensor Network States in 2D known as Projected Entangled Pair States to study frustrated quantum systems in a kagome lattice, more specifically, the XXZ model. We study the emergence of different magnetization plateaus by adding an external magnetic field and show the delicate interplay between the number of unit cells and the symmetry of the ground state. Finally, we propose an algorithm based on Tensor Networks to study open dissipative quantum systems in 2D. We then use it to investigate the spin-1/2 Ising and the XYZ model in a square lattice, both of which can be realized experimentally using cold Rydberg atoms

A simple tensor network algorithm for two-dimensional steady states

Our understanding of open quantum many-body systems is limited because it is difficult to perform a theoretical treatment of both quantum and dissipative effects in large systems. Here the authors present a tensor network method that can find the steady state of 2D driven-dissipative many-body models

Simulation methods for open quantum many-body systems

Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics toward the steady state. However, new computational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. Several approaches to simulating open many-body systems are reviewed, and advances made in recent years toward the simulation of large system sizes are pointed out

Stark time crystals: Symmetry breaking in space and time

The compelling original idea of a time crystal has referred to a structure that repeats in time as well as in space, an idea that has attracted significant interest recently. While obstructions to realize such structures became apparent early on, focus has shifted to seeing a symmetry breaking in time in periodically driven systems, a property of systems referred to as discrete time crystals. In this work, we introduce Stark time crystals based on a type of localization that is created in the absence of any spatial disorder. We argue that Stark time crystals constitute a phase of matter coming very close to the original idea and exhibit a symmetry breaking in space and time. Complementing a comprehensive discussion of the physics of the problem, we move on to elaborating on possible practical applications, and we argue that the physical demands of witnessing genuine signatures of many-body localization in large systems may be lessened in such physical systems

Spin-1/2 kagome XXZ model in a field: Competition between lattice nematic and solid orders

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