arXiv subjects

Matthieu Mambrini

Publications and source records attributed to Matthieu Mambrini.

At least 19 recordsLinked to original sources

arXiv2025

Detuning the Floquet anomalous chiral spin liquid

At high-frequency a periodically-driven quantum spin-1/2 system can emulate a chiral spin liquid (CSL) described by an effective static local chiral Hamiltonian. In contrast, at low-frequency these settings realize "Swap" models exhibiting {\it anomalous} CSL phases, in which one-way spin transport occurs at the edge although the bulk time-evolution operator over one period is trivial. In this work we explicitly construct a family of Floquet quantum spin-1/2 models on the square lattice implementing Swap models to investigate the stability of the anomalous CSL under frequency detuning and the transition to the high-frequency regime. We used the average-energy spectrum on finite-size torus and cylinders to unfold the Floquet quasi-energy spectrum over the whole frequency range and obtain the geometrical Berry phases. This enabled us to identify three regimes upon increasing detuning: i) a finite-size regime (with no folding of the Floquet spectrum), ii) an intermediate (narrow) regime with folding and very few resonances and iii) a regime with an increased density of resonances suggesting heating. At small detuning, edge modes are revealed by spectroscopic tools and from the Streda response of the system giving access to the anomalous winding number. The analysis of all the data suggests that the anomalous CSL is not continuously connected to the high-frequency CSL. We also discuss the possible occurrence of a long-lived prethermal anomalous CSL.

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arXiv2024

Floquet dynamical chiral spin liquid at finite frequency

Chiral Spin Liquids (CSL) are quantum spin analogs of electronic Fractional Chern Insulators. Their realizations on ultracold-atom or Rydberg-atom platforms remain very challenging. Recently, a setup of time-periodic modulations of nearest-neighbor Heisenberg couplings applied on an initial genuine spin liquid state on the square lattice has been proposed to stabilize a (Abelian) $\mathbb{Z}_2$ CSL phase. In the high-frequency limit, it was shown that time evolution can be described in terms of a static effective chiral Hamiltonian. Here we revisit this proposal and consider drives at lower frequency in a regime where the high-frequency Magnus expansion fails. We show that a Dynamical CSL (DCSL) is nevertheless stabilized in a finite range of frequency. The topological nature of this dynamical phase, as well as its instability below a critical frequency, is connected to specific features of the Floquet pseudo-energy spectrum. We also show that the DCSL can be represented faithfully by a two-dimensional time-periodic tensor network and, as in the static case, topological order is associated to a tensor gauge symmetry ($\mathbb{Z}_2$ in that case).

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arXiv2024

Quantum state preparation of topological chiral spin liquids via Floquet engineering

In condensed matter, Chiral Spin Liquids (CSL) are quantum spin analogs of electronic Fractional Quantum Hall states (in the continuum) or Fractional Chern Insulators (on the lattice). As the latter, CSL are remarquable states of matter, exhibiting topological order and chiral edge modes. Preparing CSL on quantum simulators like cold atom platforms is still an open challenge. Here we propose a simple setup on a finite cluster of spin-1/2 located at the sites of a square lattice. Using a Resonating Valence Bond (RVB) non-chiral spin liquid as initial state on which fast time-modulations of strong nearest-neighbor Heisenberg couplings are applied, following different protocols (out-of-equilibrium quench or semi-adiabatic ramping of the drive), we show the slow emergence of such a CSL phase. An effective Floquet dynamics, obtained from a high-frequency Magnus expansion of the drive Hamiltonian, provides a very accurate and simple framework fully capturing the out-of-equilibrium dynamics. An analysis of the resulting prepared states in term of Projected Entangled Pair states gives further insights on the topological nature of the chiral phase. Finally, we discuss possible applications to quantum computing.

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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.

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arXiv2023

Tensor network variational optimizations for real-time dynamics: application to the time-evolution of spin liquids

Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to bedescribed by the same Conformal Field Theory of central charge c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.

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arXiv2022

Real-time dynamics of a critical Resonating Valence Bond spin liquid

Implementation of the hardcore-dimer Hilbert space in cold Rydberg-atom simulators opens a new route of investigating real-time dynamics of dimer liquids under Hamiltonian quench. Here, we consider an initial Resonating Valence Bond (RVB) state on the square lattice realizing a critical Coulomb phase with algebraic and dipolar correlations. Using its representation as a special point of a broad manifold of SU($2$)-symmetric, translationnally invariant, Projected Entangled Pair States (PEPS), we compute its non-equilibrium dynamics upon turning on inter-site Heisenberg interactions. We show that projecting the time-evolution onto the PEPS manifold remains accurate at small time scales. We also find that the state evolves within a PEPS sub-manifold characterized by a U($1$) gauge symmetry, suggesting that the Coulomb phase is stable under such unitary evolution.

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arXiv2022

Weakly first-order quantum phase transition between Spin Nematic and Valence Bond Crystal Order in a square lattice SU(4) fermionic model

We consider a model Hamiltonian with two SU(4) fermions per site on a square lattice, showing a competition between bilinear and biquadratic interactions. This model has generated interest due to possible realizations in ultracold atom experiments and existence of spin liquid ground states. Using a basis transformation, we show that part of the phase diagram is amenable to quantum Monte Carlo simulations without a sign problem. We find evidence for spin nematic and valence bond crystalline phases, which are separated by a weak first order phase transition. A U(1) symmetry is found to emerge in the valence bond crystal histograms, suggesting proximity to a deconfined quantum critical point. Our results are obtained with the help of a loop algorithm which allows large-scale simulations of bilinear-biquadratic SO(N ) models on arbitrary lattices in a certain parameter regime.

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arXiv2021

Abelian SU$(N)_1$ Chiral Spin Liquids on the Square Lattice

In the physics of the Fractional Quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin-like systems, we propose a family of SU($N$)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary $N$ of an SU($3$) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU($2$)$_1$ and SU($3$)$_1$ chiral spin liquids (CSL) on similar models we search for topological SU($N$)$_1$ CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to $N=10$ and an innovative SU($N$)-symmetric version of iDMRG to compute entanglement spectra on (infinitely-long) cylinders in all topological sectors provide unambiguous signatures of the SU($N$)$_1$ character of the chiral liquids. An SU($4$)-symmetric chiral PEPS, constructed in a manner similar to its SU($2$) and SU($3$) analogs, is shown to give a good variational ansatz of the $N=4$ ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying $N$ provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.

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arXiv2020

Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice

Within the tensor network framework, the (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operator (iPEPO) coupled via ancilla degrees of freedom. To investigate the thermal properties of the spin-1/2 Heisenberg model on the square lattice, we introduce a family of fully spin-$SU(2)$ and lattice-$C_{4v}$ symmetric on-site tensors (of bond dimensions $D=4$ or $D=7$) and a plaquette-based Trotter-Suzuki decomposition of the imaginary-time evolution operator. A variational optimization is performed on the plaquettes, using a full (for $D=4$) or simple (for $D=7$) environment obtained from the single-site Corner Transfer Matrix Renormalization Group fixed point. The method is benchmarked by a comparison to quantum Monte Carlo in the thermodynamic limit. Although the iPEPO spin correlation length starts to deviate from the exact exponential growth for inverse-temperature $β\gtrsim 2$, the behavior of various observables turns out to be quite accurate once plotted w.r.t the inverse correlation length. We also find that a direct $T=0$ variational energy optimization provides results in full agreement with the $β\rightarrow\infty$ limit of finite-temperature data, hence validating the imaginary-time evolution procedure. Extension of the method to frustrated models is described and preliminary results are shown.

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arXiv2020

Quantum spin liquid phases in the bilinear-biquadratic two-SU(4)-fermion Hamiltonian on the square lattice

We consider the phase diagram of the most general SU(4)-symmetric two-site Hamiltonian for a system of two fermions per site (ie self-conjugate $\bf 6$ representation) on the square lattice. It is known that this model hosts magnetic phases breaking SU(4) symmetry and quantum disordered dimer-like phases breaking lattice translation symmetry. Motivated by a previous work [O. Gauthé, S. Capponi and D. Poilblanc, Phys. Rev. B $\textbf{99}$, 241112(R) (2019)], we investigate the possibility of the existence of SU(4) quantum spin liquid phases in this model, using SU(4)-symmetric Projected Entangled Pair States (PEPS) of small bond dimensions, which can be classified according to point group and charge (C) symmetries. Among several (disconnected) families of SU(4)-symmetric PEPS, breaking or not C-symmetry, we identify critical or topological spin liquids which may be stable in some regions of the phase diagram. These results are confronted to exact diagonalisation (ED) and density matrix renormalisation group (DMRG) calculations.

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arXiv2020

SU$(3)_1$ Chiral Spin Liquid on the Square Lattice: a View from Symmetric PEPS

Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of Projected Entangled Pair States (PEPS). Guided by extensive exact diagonalization and density matrix renormalization group calculations, we construct an optimized symmetric PEPS for a SU$(3)_1$ chiral spin liquid on the square lattice. Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. In all three $\mathbb{Z}_3$ sectors, the level counting of the linear dispersing modes is in full agreement with SU$(3)_1$ Wess-Zumino-Witten conformal field theory prediction. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence. Possible universal properties of topological SU$(N)_k$ chiral PEPS are discussed.

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arXiv2019

Critical colored-RVB states in the frustrated quantum Heisenberg model on the square lattice

We consider a family of SU(2)-symmetric Projected Entangled Paired States (PEPS) on the square lattice, defining colored-Resonating Valence Bond (RVB) states, to describe the quantum disordered phase of the $J_1-J_2$ frustrated Heisenberg model.For $J_2/J_1\sim 0.55$ we show the emergence of critical (algebraic) dimer-dimer correlations -- typical of Rokhsar-Kivelson (RK) points of quantum dimer models on bipartite lattices -- while, simultaneously, the spin-spin correlation length remains short. Our findings are consistent with a spin liquid or a weak Valence Bond Crystal in the neighborhood of an RK point.

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arXiv2017

Quantum critical phase with infinite projected entangled paired states

A classification of SU(2)-invariant Projected Entangled Paired States (PEPS) on the square lattice, based on a unique site tensor, has been recently introduced by Mambrini et al.~\cite{Mambrini2016}. It is not clear whether such SU(2)-invariant PEPS can either i) exhibit long-range magnetic order (like in the Néel phase) or ii) describe a genuine quantum critical point (QCP) or quantum critical phase (QCPh) separating two ordered phases. Here, we identify a specific family of SU(2)-invariant PEPS of the classification which provides excellent variational energies for the $J_1-J_2$ frustrated Heisenberg model, especially at $J_2=0.5$, corresponding to the approximate location of the QCP or QCPh separating the Néel phase from a dimerized phase. The PEPS are build from virtual states belonging to the $\frac{1}{2}^{\otimes N} \oplus 0$ SU(2)-representation, i.e. with $N$ "colors" of virtual \hbox{spin-$\frac{1}{2}$}. Using a full update infinite-PEPS approach directly in the thermodynamic limit, based on the Corner Transfer Matrix renormalization algorithm supplemented by a Conjugate Gradient optimization scheme, we provide evidence of i) the absence of magnetic order and of ii) diverging correlation lengths (i.e. showing no sign of saturation with increasing environment dimension) in both the singlet and triplet channels, when the number of colors $N\ge 3$. We argue that such a PEPS gives a qualitative description of the QCP or QCPh of the $J_1-J_2$ model.

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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.

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arXiv2015

Engineering SU(2) invariant spin models to mimic quantum dimer physics on the square lattice

We consider the spin-1/2 hamiltonians proposed by Cano and Fendley [J. Cano and P. Fendley, Phys. Rev. Lett. 105, 067205 (2010)] which were built to promote the well-known Rokshar-Kivelson (RK) point of quantum dimer models to spin-1/2 wavefunctions. We first show that these models, besides the exact degeneracy of RK point, support gapless spinless excitations as well as a spin gap in the thermodynamic limit, signatures of an unusual spin liquid. We then extend the original construction to create a continuous family of SU(2) invariant spin models that reproduces the phase diagram of the quantum dimer model, and in particular show explicit evidences for existence of columnar and staggered phases. The original models thus appear as multicritical points in an extended phase diagram. Our results are based on the use of a combination of numerical exact simulations and analytical mapping to effective generalized quantum dimer models.

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arXiv2010

Valence bond entanglement entropy of frustrated spin chains

We extend the definition of the recently introduced valence bond entanglement entropy to arbitrary SU(2) wave functions of S=1/2 spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional J1-J2 spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We finally underline that the valence-bond entropy definition does depend on the choice of bipartition so that, for frustrated models, a "good" bipartition should be chosen, for instance according to the Marshall sign.

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arXiv2010

Generalized Hardcore Dimer Models approach to low-energy Heisenberg frustrated antiferromagnets: general properties and application to the kagome antiferromagnet

We propose a general non-perturbative scheme that quantitatively maps the low-energy sector of spin-1/2 frustrated Heisenberg antiferromagnets to effective Generalized Quantum Dimer Models. We develop the formal lattice independent frame and establish some important results on (i) the locality of the generated Hamiltonians (ii) how full resummations can be performed in this renormalization scheme. The method is then applied to the much debated kagome antiferromagnet for which a fully resummed effective Hamiltonian - shown to capture the essential properties and provide deep insights on the microscopic model [D. Poilblanc, M. Mambrini and D. Schwandt, arXiv:0912.0724] - is derived.

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arXiv2010

Effective Quantum Dimer Model for the Kagome Heisenberg Antiferromagnet: Nearby Quantum Critical Point and Hidden Degeneracy

The low-energy singlet dynamics of the Quantum Heisenberg Antiferromagnet on the Kagome lattice is described by a quantitative Quantum Dimer Model. Using advanced numerical tools, the latter is shown to exhibit Valence Bond Crystal order with a large 36-site unit cell and hidden degeneracy between even and odd parities. Evidences are given that this groundstate lies in the vicinity of a $\mathbb{Z}_2$ dimer liquid region separated by a Quantum Critical Point. Implications regarding numerical analysis and experiments are discussed.

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