arXiv subjects

Erik S. Sørensen

Publications and source records attributed to Erik S. Sørensen.

At least 19 recordsLinked to original sources

arXiv2025

Multiparty entanglement loops in quantum spin liquids

Quantum spin liquids (QSLs) give rise to exotic emergent particles by weaving intricate entanglement patterns in the underlying electrons. Bipartite measures between subregions can detect the presence of anyons, but little is known about the full entanglement structure of QSLs. Here, we study the multiparty entanglement of QSLs via entanglement microscopy. We find that in contrast to conventional matter, the genuine multiparty entanglement (GME) between spins is absent in the smallest subregions, a phenomenon we call "entanglement frustration". Instead, GME is more collective, and arises solely in loops. By exploiting exact results and large-scale numerics, we confirm these properties in various gapped and gapless QSLs realised in physically motivated Hamiltonians, as well as with string-net wavefunctions hosting abelian or non-abelian anyons. Our results shed new light on the phase diagram of Kitaev's honeycomb model in a Zeeman field, and the Kagome Heisenberg model under various perturbations. Going beyond QSLs, we provide evidence that entanglement loops are a universal property of quantum gauge theories. This leads to a new understanding of fractionalization, and the means by which gauge bosons encode quantum information.

cond-mat.str-el↗

arXiv2025

Ace-TN: GPU-Accelerated Corner-Transfer-Matrix Renormalization of Infinite Projected Entangled-Pair States

The infinite projected entangled-pair state (iPEPS) ansatz is a powerful tensor-network approximation of an infinite two-dimensional quantum many-body state. Tensor-based calculations are particularly well-suited to utilize the high parallel efficiency of modern GPUs. We present Ace-TN, a modular and easily extendable open-source library developed to address the current need for an iPEPS framework focused on GPU acceleration. We demonstrate the advantage of using GPUs for the core iPEPS simulation methods and present a simple parallelization scheme for efficient multi-GPU execution. The latest distribution of Ace-TN can be obtained at https://github.com/ace-tn/ace-tn.

cond-mat.str-el↗

arXiv2024

Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder

Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma ($Γ$) interaction, causes magnetic frustration and its interplay with the Kitaev ($K$) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined $KΓ$ model, quasi-one dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic $Γ$ limit (denoted by A$Γ$ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the A$Γ$ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of $KΓ$ ladder is also reported.

cond-mat.str-el↗

arXiv2023

Kondo Screening Cloud Scaling: Impurity Entanglement and Magnetization

The screening of an impurity spin in the Kondo model occurs over a characteristic length scale $ξ_K$, that defines the size of the Kondo screening cloud or ``mist". The presence of such a length sc A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations depending on $r$, in terms of the single variable $r/ξ_K$ rather than depending on $r$ and $ξ_K$ separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength $J_K'$. Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, $g$. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling $J_K'$ is varied, a precise determination of $ξ_K$ can then be obtained. The length scale, $ξ_K$, is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain with.

cond-mat.str-el↗

arXiv2023

Exact Ground States and Phase Diagram of the Quantum Compass Model under an in-plane Field

We consider the square lattice $S$=1/2 quantum compass model (QCM) parameterized by $J_x, J_z$, under a field, $\mathbf{h}$, in the $x$-$z$ plane. At the special field value, $(h_x^\star,h_z^\star)$=$2S(J_x,J_z)$, we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground-states, below a gap. Exact excited states can also be found. The exact product states are characterized by a staggered vector chirality, attaining a non-zero value in the surrounding phase. The resulting gapped phase, which we denote by $SVC$ occupies most of the in-plane field phase diagram. For some values of $h_x>h_z$ and $h_z>h_x$ at the edges of the phase diagram, we have found transitions between the $SVC$ phase and phases of weakly-coupled Ising-chain states, $Z$ and $X$. In zero field, the QCM is known to have an emergent sub-extensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, $L$ and $R$, which are each separated from one another and the $SVC$ phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally-dependent Ising interactions.

cond-mat.str-el↗

arXiv2023

Eleven Competing Phases in the Heisenberg-Gamma (J$Γ$) Ladder

The spin-orbit generated $Γ$ interaction is known to induce strong frustration and to be significant in realistic models of materials. To gain an understanding of the possible phases that can arise from this interaction, it is of considerable interest to focus on a limited part of parameter space in a quasi one-dimensional model where high precision numerical results can be obtained. Here we study the Heisenberg-Gamma (J$Γ$) ladder, determining the complete zero temperature phase diagram by analyzing the entanglement spectrum (ES) and energy susceptibility. A total of 11 different phases can be identified. Two of the phases, the antiferromagnetic Gamma (A$Γ$) and ferromagnetic Gamma (F$Γ$) phases, have previously been observed in the Kitaev-Gamma ladder, demonstrating that the A$Γ$-phase is a symmetry protected topological phase (SPT) protected by $TR\times \mathcal{R}_{b}$ symmetry, the product of time-reversal ($TR$) and $π$ rotation around the $b$-axis ($\mathcal{R}_{b}$), while the F$Γ$-phase is related to a rung-singlet phase through a local unitary transformation. Three other phases, $Υ$, $Ω$ and $δ$ show no conventional order, a doubling of the entanglement spectrum and for the $Υ$ and $Ω$-phases a gap is clearly present. The $δ$-phase has a significantly smaller gap and displays incommensurate correlations, with a peak in the static structure factor, $S(k)$ continuously shifting from $k/π\mathord{=}2/3$ to $k\mathord{=}π$. In the $Ω$-phase we find pronounced edge-states consistent with a SPT phase protected by the same $TR\times \mathcal{R}_{b}$ symmetry as the A$Γ$-phase. The precise nature of the $Υ$ and $δ$-phases is less clear.

cond-mat.str-el↗

arXiv2023

Scaling at the OTOC Wavefront: free versus chaotic models

Out of time ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at $x=v_{B}t$, where $v_{B}$ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that the leading edge of a propagating Gaussian with the argument $-m(x)\left( x-v_B t \right)^2 +b(x)t$ gives an excellent fit to the region around $x=v_{B}t$ for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients $m(x)$ and $b(x)$ have an inverse power law dependence on $x$ whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously, and on the other hand an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.

cond-mat.stat-mech↗

arXiv2023

State Space Geometry of the Spin-1 Antiferromagnetic Heisenberg Chain

We study the phase diagram of the spin-1 antiferromagnetic Heisenberg chain with uniaxial anisotropy and applied magnetic field in terms of the genuine multipartite entanglement as witnessed by the mean quantum Fisher information density. By generalizing the manifold studied in [1, 2] to the many body case for spin 1, we connect the state space curvature in the vicinity of the ground state of the Heisenberg chain to the genuine multipartite entanglement. Our analysis demonstrates that the quantum critical points and symmetry protected topological (SPT) phase exhibit large state space curvature, while the separable phases are completely flat, offering insight into the physical interpretation of state space curvature. We further show that the entanglement in the SPT phase is enhanced by the presence of uniaxial anisotropy, and undiminished in the presence of uniform magnetic fields. The magnon condensate phase induced by large fields is shown to emanate from the Gaussian critical point, and exhibits massive multipartite entanglement over a robust region of the parameter space.

cond-mat.str-el↗

arXiv2023

Islands of Chiral Solitons in Integer Spin Kitaev Chains

An intriguing chiral soliton phase has recently been identified in the $S$=1/2 Kitaev spin chain. Here we show that for $S$=1,2,3,4,5 an analogous phase can be identified, but contrary to the $S$=1/2 case the chiral soliton phases appear as islands within the sea of the polarized phase. In fact, a small field applied in a general direction will adiabatically connect the integer spin Kitaev chain to the polarized phase. Only at sizable intermediate fields along symmetry directions does the soliton phase appear centered around the special point $h^\star_x$=$h^\star_y$=$S$ where two exact product ground-states can be identified. The large $S$ limit can be understood from a semi-classical analysis, and variational calculations provide a detailed picture of the $S$=1 soliton phase. Under open boundary conditions, the chain has a single soliton in the ground-state which can be excited, leading to a proliferation of in-gap states. In contrast, even length periodic chains exhibit a gap above a twice degenerate ground-state. The presence of solitons leaves a distinct imprint on the low temperature specific heat.

cond-mat.str-el↗

arXiv2022

Field Induced Chiral Soliton Phase in the Kitaev Spin Chain

The bond-dependent Ising interaction present in the Kitaev model has attracted considerable attention. The appearance of an unexpected intermediate phase under a magnetic field is particularly intriguing, and one may wonder if a similar phase occurs in the Kitaev spin chain with alternating $x$- and $y$-bond Ising interactions. Previous studies have focused on a transverse field, $h_z$, and reported a direct transition to the polarized state. Here, we investigate phases with arbitrary angle of two longitudinal fields, $h_x$ and $h_y$. For a magnetic field applied along the diagonal, $h_x$=$h_y$, the chain remains gapless up to a critical field $h^{c_1}_{xy}$. Surprisingly, above $h^{c1}_{xy}$ it enters an unusual intermediate phase before reaching the polarized state at $h^{c_2}_{xy}$. This phase is characterized by a staggered vector chirality and for periodic boundary conditions, a two-fold degeneracy with a finite gap. For open boundary systems the ground-state exhibits a single soliton, lowering the energy, and gapless excitations. However, the corresponding anti-soliton raises the energy sufficiently that a gap appears for soliton and anti-soliton pairs in periodic systems. An intuitive variational picture is developed describing the soliton phase.

cond-mat.str-el↗

arXiv2022

Chiral Broken Symmetry Descendants of the Kagomé Lattice Chiral Spin Liquid

The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. In particular, the breaking of chiral (mirror) symmetry in quantum materials has been shown to have important technological applications. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids on the kagomé lattice. Such chiral spin liquids are closely tied to bosonic fractional quantum Hall states and host anyonic quasiparticles; however, their connection to nearby magnetically ordered states has remained a mystery. Here, we show that two distinct non-coplanar magnetic orders with uniform spin chirality, the XYZ umbrella state and the Octahedral spin crystal, emerge as competing orders in close proximity to the kagomé chiral spin liquid. Our results highlight the intimate link between a many-body topologically ordered liquid and broken symmetry states with nontrivial real-space topology.

cond-mat.str-el↗

arXiv2021

Classical spin order near antiferromagnetic Kitaev point in the spin-1/2 Kitaev-Gamma chain

A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent SU(2)$_1$ phase and a rank-1 spin ordered phase with $O_h\rightarrow D_4$ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a rank-1 ordered phase with an $O_h\rightarrow D_3$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the $O_h\rightarrow D_3$ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking $O_h\rightarrow D_{3d}\rightarrow D_3$ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with $O_h\rightarrow D_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the $O_h\rightarrow D_3$ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.

cond-mat.str-el↗

arXiv2020

Revealing divergent length scales using quantum Fisher information in the Kitaev honeycomb model

We compute the quantum Fisher information (QFI) associated with two different local operators in the Kitaev honeycomb model, and find divergent behaviour in the second derivatives of these quantities with respect to the driving parameter at the quantum phase transition between the gapped and gapless phases for both fully anti-ferromagnetic and fully ferromagnetic exchange couplings. The QFI associated with a local magnetization operator behaves differently from that associated with a local bond operator depending on whether the critical point is approached from the gapped or gapless side. We show how the behaviour of the second derivative of the QFI at the critical point can be understood in terms of diverging length scales in the correlators of the local generators.

cond-mat.str-el↗

arXiv2020

The Heart of Entanglement: Chiral, Nematic, and Incommensurate Phases in the Kitaev-Gamma Ladder in a Field

The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma (KG) model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive and one reason may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically. We therefore focus on the KG model defined on a two-leg ladder which is much more amenable to a complete study, and determine the entire phase diagram in the presence of a magnetic field along [111]-direction. Due to the competition between the interactions and the field, an extremely rich phase diagram emerges with fifteen distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outline a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a C3 symmetric cluster of the two-dimensional honeycomb lattice, and offer insights into possible spin liquids in the two-dimensional limit.

cond-mat.str-el↗

arXiv2019

Out of Time Order Correlations in the Quasi-Periodic Aubry-André model

We study out of time ordered correlators (OTOC) in a free fermionic model with a quasi-periodic potential. This model is equivalent to the Aubry-André model and features a phase transition from an extended phase to a localized phase at a non-zero value of the strength of the quasi-periodic potential. We investigate five different time-regimes of interest for out of time ordered correlators; early, wavefront, $x=v_Bt$, late time equilibration and infinite time. For the early time regime we observe a power law for all potential strengths. For the time regime preceding the wavefront we confirm a recently proposed universal form and use it to extract the characteristic velocity of the wavefront for the present model. A Gaussian waveform is observed to work well in the time regime surrounding $x=v_Bt$. Our main result is for the late time equilibration regime where we derive a finite time equilibration bound for the OTOC, bounding the correlator's distance from its late time value. The bound impose strict limits on equilibration of the OTOC in the extended regime ans is valid not only for the Aubry-André model but for any quadratic model model. Finally, momentum out of time ordered correlators for the Aubry-André model are studied where large values of the OTOC are observed at late times at the critical point.

cond-mat.stat-mech↗

arXiv2019

Theory of the field-revealed Kitaev spin liquid

Elementary excitations in highly entangled states such as quantum spin liquids may exhibit exotic statistics, different from those obeyed by fundamental bosons and fermions. Excitations called non-Abelian anyons are predicted to exist in a Kitaev spin liquid - the ground state of an exactly solvable model proposed by Kitaev almost a decade ago. A smoking-gun signature of such non-Abelian anyons, namely a half-integer quantized thermal Hall conductivity, was recently reported in $α$-RuCl$_3$. While fascinating, a microscopic theory for this phenomenon in $α$-RuCl$_3$ remains elusive because the pure Kitaev phase cannot capture these anyons appearing in an intermediate magnetic field. Here we present a microscopic theory of the Kitaev spin liquid emerging between the low- and high-field states. Essential to this result is an antiferromagnetic off-diagonal symmetric interaction that permits the Kitaev spin liquid to protrude from the pure ferromagnetic Kitaev limit under a magnetic field. This generic model captures a field-revealed Kitaev spin liquid, and displays strong anisotropy of field effects. A wide regime of non-Abelian anyon Kitaev spin liquid is predicted when the magnetic field is perpendicular to the honeycomb plane.

cond-mat.str-el↗

arXiv2018

Non-local String Order Parameter in the $S = 1/2$ Kitaev-Heisenberg Ladder

We study the spin-$\frac{1}{2}$ Kitaev-Heisenberg (KJ) model in a two-leg ladder. Without a Heisenberg interaction, the Kitaev phase in the ladder model has Majorana fermions with local Z$_2$ gauge fields, and is usually described as a disordered phase without any order parameter. Here we prove the existence of a non-local string order parameter (SOP) in the Kitaev phase which survives with a finite Heisenberg interaction. The SOP is obtained by relating the Kitaev ladder, through a non-local unitary transformation, to a one-dimensional $XY$ chain with an Ising coupling to a dangling spin at every site. This differentiates the Kitaev phases from other nearby phases including a rung singlet. Two phases with non-zero SOP corresponding to ferromagnetic and antiferromagnetic Kitaev interactions are identified. The full phase diagram of the KJ ladder is determined using exact diagonalization and density matrix renormalization group methods, which shows a striking similarity to the KJ model on a two-dimensional honeycomb lattice.

cond-mat.str-el↗

arXiv2016

Entanglement structure of the two-channel Kondo model

Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels we confirm field-theory predictions for the boundary entropy difference, $\ln (g_{UV}/g_{IR})=\ln(2)/2$, between the ultraviolet and infrared limits and the leading $\ln(x)/x$ impurity correction to the block entropy. The impurity entanglement, $S_{\text{imp}}$, is shown to scale with the characteristic length $ξ_{2CK}$. We show that both the Schmidt gap and the entanglement of the impurity with one of the channels $-$ as measured by the negativity$-$ faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, $ν\!\approx\! 2$ and $β\!\approx\! 0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.

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