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Electronic Green's function across the pseudogap to stripe transition in the $t$-$t'$-$J$ model
Authors:
Martin Ulaga,
Aritra Sinha,
Alexander Wietek
Abstract:
Superconducting domes in strongly correlated electronic systems are often accompanied by charge density waves and peculiar features in the electronic structure. The appearance of a pseudogap, in particular, and its relation to charge density waves remains insufficiently understood. Here, we investigate the electronic Green's function of the underdoped $t$-$t'$-$J$ model of the cuprate superconduct…
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Superconducting domes in strongly correlated electronic systems are often accompanied by charge density waves and peculiar features in the electronic structure. The appearance of a pseudogap, in particular, and its relation to charge density waves remains insufficiently understood. Here, we investigate the electronic Green's function of the underdoped $t$-$t'$-$J$ model of the cuprate superconductors using tensor network algorithms for finite temperature dynamics on cylinders of width $4$. We find the prominent momentum differentiation, the hallmark of the pseudogap, to be strongly dependent on $t'$, which develops into a momentum-dependent opening of a gap upon decreasing temperature, consistent with the formation of a Fermi arc at intermediate temperatures. The nodal gap around ${\bf k}=(π/2,π/2)$ closes and fills with increasing temperature as coherent stripe order melts into a regime of fluctuating charge clusters.
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Submitted 15 June, 2026;
originally announced June 2026.
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Incommensurate Spin-Density Waves in a Frustrated Maple-Leaf Lattice Ferromagnet
Authors:
Paul L. Ebert,
Yasir Iqbal,
Alexander Wietek
Abstract:
We study how ferromagnetism breaks down in the spin-$\tfrac12$ nearest-neighbor Heisenberg model on the maple-leaf lattice with ferromagnetic $J_t,J_d$ and antiferromagnetic $J_h$, motivated by the mixed ferro-antiferromagnetic interactions in Na$_2$Mn$_3$O$_7$. Exact diagonalization shows that the ferromagnetic boundary does not feature a zero-field spin-nematic phase on the clusters studied here…
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We study how ferromagnetism breaks down in the spin-$\tfrac12$ nearest-neighbor Heisenberg model on the maple-leaf lattice with ferromagnetic $J_t,J_d$ and antiferromagnetic $J_h$, motivated by the mixed ferro-antiferromagnetic interactions in Na$_2$Mn$_3$O$_7$. Exact diagonalization shows that the ferromagnetic boundary does not feature a zero-field spin-nematic phase on the clusters studied here, but an extended regime of incommensurate spin-density-wave correlations with continuously evolving ordering vector. The phase diagram also contains collinear Néel, canted $120^\circ$, and hexagonal-singlet regimes, separated by regions that remain difficult to classify from exact diagonalization alone. Variational tests of fully symmetric Gutzwiller-projected Abrikosov-fermion U(1) and $\mathbb{Z}_2$ states find no competitive spin-liquid description of the interior unresolved regions. By contrast, on the ruby-lattice boundary we identify a point between the collinear Néel and hexagonal-singlet phases where a projected $\mathbb{Z}_2$ Ansatz reproduces the finite-size energy and spin correlations with good accuracy.
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Submitted 12 May, 2026;
originally announced May 2026.
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Evolution of superconductivity from charge clusters to stripes in the $t$-$t'$-$J$ model
Authors:
Aritra Sinha,
Hannes Karlsson,
Martin Ulaga,
Alexander Wietek
Abstract:
Competition and coexistence of charge orders and superconductivity are hallmarks in many strongly correlated electron systems. Here, we unravel the precise role of charge fluctuations on the superconducting state in the $t$-$t'$-$J$ model of the high-temperature cuprate superconductors. Using finite-temperature tensor network simulations, we investigate thermal snapshots in the underdoped regime w…
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Competition and coexistence of charge orders and superconductivity are hallmarks in many strongly correlated electron systems. Here, we unravel the precise role of charge fluctuations on the superconducting state in the $t$-$t'$-$J$ model of the high-temperature cuprate superconductors. Using finite-temperature tensor network simulations, we investigate thermal snapshots in the underdoped regime where the ground state features a superconducting stripe phase. At intermediate temperatures, where stripes have melted and hole clustering is observed, we find that pairing correlations are tightly localized on the hole clusters. Upon entering the stripe regime at lower temperatures, pairing increasingly delocalizes across different hole clusters to ultimately become coherent across the full system in the ground state. This pair-charge locking gives rise to an intuitive picture of the parent state of the superconducting stripe phase: pairing is localized on hole clusters formed via hole attraction due to the onset of magnetic correlations at intermediate temperature. We discuss how this microscopic picture is consistent with a broad range of experimental observations in cuprate superconductors, including scanning tunneling microscopy (STM) evidence for local pairing above $T_c$ and nuclear magnetic resonance (NMR) signatures of charge clustering in the underdoped regime.
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Submitted 20 March, 2026;
originally announced March 2026.
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Cooper Condensation and Pair Wave Functions in Strongly Correlated Electrons
Authors:
Hannes Karlsson,
Johannes S. Hofmann,
Alexander Wietek
Abstract:
Identifying superconducting states of matter without prior assumptions is a central challenge in strongly correlated electron systems. We introduce a canonical framework for diagnosing the formation of Cooper pair condensates based on the Penrose-Onsager criterion, in which superconducting order is encoded in the spectral properties of the two-particle reduced density matrix (2RDM). Within this fo…
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Identifying superconducting states of matter without prior assumptions is a central challenge in strongly correlated electron systems. We introduce a canonical framework for diagnosing the formation of Cooper pair condensates based on the Penrose-Onsager criterion, in which superconducting order is encoded in the spectral properties of the two-particle reduced density matrix (2RDM). Within this formulation, the symmetry and structure of the condensate are obtained by projecting the 2RDM onto irreducible representations of the underlying symmetry group, enabling an unbiased identification of both conventional and exotic superconducting states. We demonstrate the power and versatility of the approach through applications to the two-dimensional Hubbard model, using both auxiliary-field quantum Monte Carlo (AFQMC) and the density matrix renormalization group (DMRG). For attractive interactions without a magnetic field, we reveal a clear finite-size scaling of the condensate fraction on square lattices of size up to $20\times 20$. The framework further provides direct access to the internal structure and extent of Cooper pairs, which we track across the BCS-BEC crossover. Moreover, it enables a clean diagnosis of the finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in a magnetic field. Finally, we apply the approach to a supersolid phase in the repulsive Hubbard model with an additional next-nearest neighbor hopping $t^\prime$, where a charge-density wave coexists with a superconductor. We confirm the fragmented nature of the condensate and uncover substantial pairing correlations in the triplet channel with $p$-wave spatial symmetry in addition to the dominant singlet $d$-wave pairing. Our results establish the 2RDM-based Penrose-Onsager framework as a broadly applicable and unbiased tool for characterizing superconducting order in correlated quantum matter.
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Submitted 26 January, 2026;
originally announced January 2026.
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Field-induced states and thermodynamics of the frustrated Heisenberg antiferromagnet on a square lattice
Authors:
Andreas Honecker,
M. E. Zhitomirsky,
Alexander Wietek,
Johannes Richter
Abstract:
We investigate the ground-state and finite-temperature properties of the $J_1$-$J_2$ Heisenberg antiferromagnet on the square lattice in the presence of an external magnetic field. We focus on the highly frustrated regime around $J_2 \approx J_1/2$. The $h$-$T$ phase diagram is investigated with particular emphasis on the finite-temperature transition into the "up-up-up-down" state that is stabili…
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We investigate the ground-state and finite-temperature properties of the $J_1$-$J_2$ Heisenberg antiferromagnet on the square lattice in the presence of an external magnetic field. We focus on the highly frustrated regime around $J_2 \approx J_1/2$. The $h$-$T$ phase diagram is investigated with particular emphasis on the finite-temperature transition into the "up-up-up-down" state that is stabilized by thermal and quantum fluctuations and manifests itself as a plateau at one half of the saturation magnetization in the quantum case. We also discuss the enhanced magnetocaloric effect associated to the ground-state degeneracy that arises at the saturation field for $J_2=J_1/2$. For reference, we first study the classical case by classical Monte Carlo simulations. Then we turn to the extreme quantum limit of spin-1/2 where we perform zero- and finite-temperature Lanczos calculations.
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Submitted 23 June, 2026; v1 submitted 20 January, 2026;
originally announced January 2026.
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Competing Paramagnetic Phases in the Maple-Leaf Heisenberg Antiferromagnet
Authors:
Paul L. Ebert,
Yasir Iqbal,
Alexander Wietek
Abstract:
We establish a remarkably rich ground state phase diagram in the maple-leaf lattice spin-$1/2$ Heisenberg antiferromagnet as a function of the three symmetry-inequivalent nearest-neighbor bonds using exact diagonalization and tower-of-states analysis on clusters up to $N=36$ sites. Besides a hexagonal plaquette state, a star-shaped valence bond solid state is discovered in close vicinity to the (c…
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We establish a remarkably rich ground state phase diagram in the maple-leaf lattice spin-$1/2$ Heisenberg antiferromagnet as a function of the three symmetry-inequivalent nearest-neighbor bonds using exact diagonalization and tower-of-states analysis on clusters up to $N=36$ sites. Besides a hexagonal plaquette state, a star-shaped valence bond solid state is discovered in close vicinity to the (canted) $120^\circ$ magnetic phase, strongly reminiscent of a de-confined critical point or Dirac spin liquid scenario on the triangular lattice antiferromagnets. Moreover, an exact dimer product-state is observed next to a collinear Néel-state, similar to the Shastry-Sutherland model. All identified phases compete in a parameter regime close to the isotropic point, providing a promising region for spin liquids to emerge. By analyzing Gutzwiller-projected wave-functions we identify a sliver of parameter regime where a gapped $\mathbb{Z}_{2}$ spin liquid Ansatz is in astonishing agreement with the exact $N=36$ ground state. This rich competition of paramagnetic phases demonstrates that the maple-leaf antiferromagnet is a promising platform for exotic states of matter and quantum critical phenomena.
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Submitted 21 January, 2026; v1 submitted 8 January, 2026;
originally announced January 2026.
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Thermodynamics of the Heisenberg antiferromagnet on the maple-leaf lattice
Authors:
Robin Schäfer,
Paul L. Ebert,
Noah Hassan,
Johannes Reuther,
David J. Luitz,
Alexander Wietek
Abstract:
We study the Heisenberg antiferromagnet on the maple-leaf lattice using several numerical approaches, focusing on the numerical linked-cluster expansion (NLCE), which exhibits an unconventional convergence extending to low and even zero temperatures. We evaluate thermodynamic properties as well as spin-spin correlations through the equal-time structure factor. Within NLCE the specific heat capacit…
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We study the Heisenberg antiferromagnet on the maple-leaf lattice using several numerical approaches, focusing on the numerical linked-cluster expansion (NLCE), which exhibits an unconventional convergence extending to low and even zero temperatures. We evaluate thermodynamic properties as well as spin-spin correlations through the equal-time structure factor. Within NLCE the specific heat capacity reveals a two-peak structure at $T_1 \approx 0.479\,J$ and $T_2 \approx 0.131\,J$, reminiscent of the corresponding result for the triangular lattice. At intermediate temperatures, the spin-spin structure factor develops features that reflect the absence of reflection symmetry in the lattice. The zero-temperature convergence of NLCE enables reliable estimates of the ground-state energy and points to a short-range correlated paramagnetic ground state composed of resonating hexagonal motifs. The NLCE results are benchmarked against Pseudo-Majorana Functional Renormalization Group, finite-temperature Lanczos, and classical Monte Carlo simulations.
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Submitted 5 April, 2026; v1 submitted 26 November, 2025;
originally announced November 2025.
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Krylov space dynamics of ergodic and dynamically frozen Floquet systems
Authors:
Luke Staszewski,
Asmi Haldar,
Pieter W. Claeys,
Alexander Wietek
Abstract:
In isolated quantum many-body systems periodically driven in time, the asymptotic dynamics at late times can exhibit distinct behavior such as thermalization or dynamical freezing. Understanding the properties of and the convergence towards infinite-time (nonequilibrium) steady states however remains a challenging endeavor. We propose a physically motivated Krylov space perspective on Floquet ther…
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In isolated quantum many-body systems periodically driven in time, the asymptotic dynamics at late times can exhibit distinct behavior such as thermalization or dynamical freezing. Understanding the properties of and the convergence towards infinite-time (nonequilibrium) steady states however remains a challenging endeavor. We propose a physically motivated Krylov space perspective on Floquet thermalization which offers a natural framework to study rates of convergence towards steady states and, simultaneously, an efficient numerical algorithm to evaluate infinite-time averages of observables within the diagonal ensemble. The effectiveness of our algorithm is demonstrated by applying it to the periodically driven mixed-field Ising model, reaching system sizes of up to 30 spins. Our method successfully resolves the transition between the ergodic and dynamically frozen phases and provides insight into the nature of the Floquet eigenstates across the phase diagram. Furthermore, we show that the long-time behavior is encoded within the localization properties of the Ritz vectors under the Floquet evolution, providing an accurate diagnostic of ergodicity.
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Submitted 22 October, 2025;
originally announced October 2025.
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Ab initio spin Hamiltonians and magnetism of Ce and Yb triangular-lattice compounds
Authors:
Leonid V. Pourovskii,
Rafael D. Soares,
Alexander Wietek
Abstract:
We calculate the crystal-field splitting, ground-state Kramers doublet and intersite exchange interactions within the ground-state doublet manifold using an ab initio Hubbard-I based approach for a representative set of Ce and Yb triangular-lattice compounds. These include the putative quantum spin liquids (QSL) RbCeO$_2$ and YbZn$_2$GaO$_5$ and the antiferromagnets KCeO$_2$ and KCeS$_2$. The calc…
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We calculate the crystal-field splitting, ground-state Kramers doublet and intersite exchange interactions within the ground-state doublet manifold using an ab initio Hubbard-I based approach for a representative set of Ce and Yb triangular-lattice compounds. These include the putative quantum spin liquids (QSL) RbCeO$_2$ and YbZn$_2$GaO$_5$ and the antiferromagnets KCeO$_2$ and KCeS$_2$. The calculated nearest-neighbor (NN) couplings are antiferromagnetic and exhibit noticeable anisotropy. The next-nearest-neighbor (NNN) couplings are ferromagnetic in the Ce systems and dominated by classical dipole-dipole interactions in the Yb case. Solving the resulting effective spin-1/2 models by exact diagonalization up to $N=36$ sites, we predict ordered magnetic ground states for all systems, including the two QSL candidates. We explore the phase space of an anisotropic NN + isotropic NNN triangular-lattice model finding that a significant antiferromagnetic NNN coupling is required to stabilize QSL phases, while the NN exchange anisotropy is detrimental to them. Our findings highlight a possibly important role of deviations from the perfect triangular model - like atomic disorder - in real triangular-lattice materials.
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Submitted 4 February, 2026; v1 submitted 3 September, 2025;
originally announced September 2025.
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Superfluid dome in the spatially modulated two-dimensional XY model
Authors:
Feng-Feng Song,
Aditya Chugh,
Hanggai Nuomin,
Naoki Kawashima,
Alexander Wietek
Abstract:
In strongly correlated electron systems, superconductivity and charge density waves often coexist in close proximity, suggesting a deeper relationship between these competing phases. Recent research indicates that these orders can intertwine, with the superconducting order parameter coupling to modulations in the electronic density. To elucidate this interplay, we study a two-dimensional XY model…
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In strongly correlated electron systems, superconductivity and charge density waves often coexist in close proximity, suggesting a deeper relationship between these competing phases. Recent research indicates that these orders can intertwine, with the superconducting order parameter coupling to modulations in the electronic density. To elucidate this interplay, we study a two-dimensional XY model with a periodic modulation of the coupling strength in one spatial direction. Using a combination of tensor network methods and Monte Carlo simulations, we reveal a non-monotonic, dome-like dependence of $T_c$ on the modulation wavelength, with the peak $T_c$ shifting to longer wavelengths as the modulation strength grows. The origin of this phenomenon is traced back to an effective pinning of vortices in the valleys of the modulation, confirmed by a comparison to modulated $q$-state clock models. These findings shed new light on the phase behavior of intertwined superconducting and charge-ordered states, offering a deeper understanding of their complex interactions.
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Submitted 19 June, 2025;
originally announced June 2025.
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XDiag: Exact Diagonalization for Quantum Many-Body Systems
Authors:
Alexander Wietek,
Luke Staszewski,
Martin Ulaga,
Paul L. Ebert,
Hannes Karlsson,
Siddhartha Sarkar,
Leyna Shackleton,
Aritra Sinha,
Rafael D. Soares
Abstract:
Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-sour…
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Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-source software package designed to combine advanced and efficient algorithms for ED with and without symmetry-adapted bases with user-friendly interfaces. Implemented in C++ for computational efficiency and wrapped in Julia for ease of use, XDiag provides a comprehensive toolkit for ED calculations. Key features of XDiag include the first publicly accessible implementation of sublattice coding algorithms for large-scale spin system diagonalizations, efficient Lin table algorithms for symmetry lookups, and random-hashing techniques for distributed memory parallelization. The library supports various Hilbert space types (e.g., spin-1/2, electron, and t-J models), facilitates symmetry-adapted block calculations, and automates symmetry considerations. The package is complemented by extensive documentation, a user guide, reproducible benchmarks demonstrating near-linear scaling on thousands of CPU cores, and over 20 examples covering ground-state calculations, spectral functions, time evolution, and thermal states. By integrating high-performance computing with accessible scripting capabilities, XDiag allows researchers to perform state-of-the-art ED simulations and explore quantum many-body phenomena with unprecedented flexibility and efficiency.
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Submitted 13 February, 2026; v1 submitted 5 May, 2025;
originally announced May 2025.
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Observation of emergent scaling of spin-charge correlations at the onset of the pseudogap
Authors:
Thomas Chalopin,
Petar Bojović,
Si Wang,
Titus Franz,
Aritra Sinha,
Zhenjiu Wang,
Dominik Bourgund,
Johannes Obermeyer,
Fabian Grusdt,
Annabelle Bohrdt,
Lode Pollet,
Alexander Wietek,
Antoine Georges,
Timon Hilker,
Immanuel Bloch
Abstract:
In strongly correlated materials, interacting electrons are entangled and form collective quantum states, resulting in rich low-temperature phase diagrams. Notable examples include cuprate superconductors, in which superconductivity emerges at low doping out of an unusual "pseudogap" metallic state above the critical temperature. The Fermi-Hubbard model, describing a wide range of phenomena associ…
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In strongly correlated materials, interacting electrons are entangled and form collective quantum states, resulting in rich low-temperature phase diagrams. Notable examples include cuprate superconductors, in which superconductivity emerges at low doping out of an unusual "pseudogap" metallic state above the critical temperature. The Fermi-Hubbard model, describing a wide range of phenomena associated with strong electron correlations, still offers major computational challenges despite its simple formulation. In this context, ultracold atoms quantum simulators have provided invaluable insights into the microscopic nature of correlated quantum states. Here, we use a quantum gas microscope Fermi-Hubbard simulator to explore a wide range of dopings and temperatures in a regime where a pseudogap is known to develop. By measuring multi-point correlation functions up to fifth order, we uncover a novel universal scaling behaviour in magnetic and higher-order spin-charge correlations characterised by a doping-dependent temperature scale. Accurate comparisons with determinant Quantum Monte Carlo and Minimally Entangled Typical Thermal States simulations confirm that this temperature scale is comparable to the pseudogap temperature T*. Our quantitative findings reveal a novel qualitative behaviour of magnetic properties and spin-charge correlations in an emergent pseudogap and pave the way towards the exploration of charge pairing and collective phenomena expected at lower temperatures.
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Submitted 27 January, 2026; v1 submitted 23 December, 2024;
originally announced December 2024.
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Forestalled Phase Separation as the Precursor to Stripe Order
Authors:
Aritra Sinha,
Alexander Wietek
Abstract:
Stripe order is a prominent feature in the phase diagram of the high-temperature cuprate superconductors. It has been confirmed as the lowest-energy state of the two-dimensional Fermi Hubbard model in certain parameter regimes. Upon increasing the temperature, stripes and the superconducting state give way to the enigmatic strange-metal and pseudogap regimes, whose precise nature remains a long-st…
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Stripe order is a prominent feature in the phase diagram of the high-temperature cuprate superconductors. It has been confirmed as the lowest-energy state of the two-dimensional Fermi Hubbard model in certain parameter regimes. Upon increasing the temperature, stripes and the superconducting state give way to the enigmatic strange-metal and pseudogap regimes, whose precise nature remains a long-standing puzzle. Using modern tensor network techniques, we discover a crucial aspect of these regimes. Infinite projected entangled pair states (iPEPS) simulations in the fully two-dimensional limit reveal a maximum in the charge susceptibility at temperatures above the stripe order. This maximum is located around filling $n=0.9$ and intensifies upon cooling. Using minimally entangled typical thermal states (METTS) on finite cylinders, we attribute the enhanced charge susceptibility to the formation of charge clusters, reminiscent of phase separation where the system is partitioned into hole-rich and hole-depleted regions. In contrast to genuine phase separation, the charge cluster sizes fluctuate without a divergent charge susceptibility. Hence, while this precursor state features clustering of charge carriers, true phase separation is ultimately forestalled at lower temperatures by the onset of stripe order.
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Submitted 12 September, 2025; v1 submitted 8 November, 2024;
originally announced November 2024.
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Quench dynamics of stripes and phase separation in the two-dimensional $t$-$J$ model
Authors:
Luke Staszewski,
Alexander Wietek
Abstract:
We investigate the fundamental dynamical process of an initial quench of the chemical potential of the two-dimensional $t$-$J$ model. Depending on the ground state phase, sharply different dynamical behavior of the charge distribution and entanglement properties are observed. In the stripe phase, the intertwining of the spin and charge density waves remains stable under time evolution. A ballistic…
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We investigate the fundamental dynamical process of an initial quench of the chemical potential of the two-dimensional $t$-$J$ model. Depending on the ground state phase, sharply different dynamical behavior of the charge distribution and entanglement properties are observed. In the stripe phase, the intertwining of the spin and charge density waves remains stable under time evolution. A ballistic spreading of the charge density is observed with a propagation speed that is only weakly dependent on the coupling ratio, $J/t$. Moreover, in the phase-separated regime for large $J/t$, we report a complete dynamical freezing of charge degrees of freedom within, where even under long time evolution the entanglement entropy remains bounded. Our results are obtained by combining large-scale exact diagonalizations and matrix product state techniques for time evolution.
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Submitted 21 October, 2024;
originally announced October 2024.
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Dynamical freezing in the thermodynamic limit: the strongly driven ensemble
Authors:
Asmi Haldar,
Anirban Das,
Sagnik Chaudhuri,
Luke Staszewski,
Alexander Wietek,
Frank Pollmann,
Roderich Moessner,
Arnab Das
Abstract:
The ergodicity postulate, a foundational pillar of Gibbsian statistical mechanics predicts that a periodically driven (Floquet) system in the absence of any conservation law heats to a featureless `infinite temperature' state. Here, we find--for a clean and interacting generic spin chain subject to a {\it strong} driving field--that this can be prevented by the emergence of {\it approximate but st…
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The ergodicity postulate, a foundational pillar of Gibbsian statistical mechanics predicts that a periodically driven (Floquet) system in the absence of any conservation law heats to a featureless `infinite temperature' state. Here, we find--for a clean and interacting generic spin chain subject to a {\it strong} driving field--that this can be prevented by the emergence of {\it approximate but stable} conservation-laws not present in the undriven system. We identify their origin: they do not necessarily owe their stability to familiar protections by symmetry, topology, disorder, or even high energy costs. We show numerically, {\it in the thermodynamic limit,} that when required by these emergent conservation-laws, the entanglement-entropy density of an infinite subsystem remains zero over our entire simulation time of several decades in natural units. We further provide a recipe for designing such conservation laws with high accuracy. Finally, we present an ensemble description, which we call the strongly driven ensemble incorporating these constraints. This provides a way to control many-body chaos through stable Floquet-engineering. Strong signatures of these conservation-laws should be experimentally accessible since they manifest in all length and time scales. Variants of the spin model we have used, have already been realized using Rydberg-dressed atoms.
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Submitted 14 October, 2024;
originally announced October 2024.
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Spectroscopy and complex-time correlations using minimally entangled typical thermal states
Authors:
Zhenjiu Wang,
Paul McClarty,
Dobromila Dankova,
Andreas Honecker,
Alexander Wietek
Abstract:
Tensor network states have enjoyed great success at capturing aspects of strong correlation physics. However, obtaining dynamical correlators at non-zero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correl…
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Tensor network states have enjoyed great success at capturing aspects of strong correlation physics. However, obtaining dynamical correlators at non-zero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correlators of physical operators in real time, we propose extensions where correlations are evaluated in the complex-time plane. The imaginary time component bounds the rate of entanglement growth and strongly alleviates the computational difficulty allowing the study of larger system sizes. To extract the physical correlator one must take the limit of purely real-time evolution. We present two routes to obtaining this information (i) via an analytic correlation function in complex time combined with a stochastic analytic continuation method to obtain the real-time limit and (ii) a hermitian correlation function that asymptotically captures the desired correlation function quantitatively without requiring effort of numerical analytic continuation. We show that these numerical techniques capture the finite-temperature dynamics of the Shastry-Sutherland model - a model of interacting spin one-half in two dimensions.
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Submitted 28 February, 2025; v1 submitted 28 May, 2024;
originally announced May 2024.
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Anomalous thermal broadening in the Shastry-Sutherland model and SrCu$_2($BO$_3)_2$
Authors:
Zhenjiu Wang,
Paul McClarty,
Dobromila Dankova,
Andreas Honecker,
Alexander Wietek
Abstract:
The quantum magnet SrCu$_2($BO$_3)_2$ and its remarkably accurate theoretical description, the spin-$1/2$ Shastry-Sutherland model, host a variety of intriguing phenomena such as a dimer ground state with a nearly flat band of triplon excitations, a series of magnetization plateaux, and a possible pressure-induced deconfined quantum critical point. One open puzzle originating from inelastic neutro…
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The quantum magnet SrCu$_2($BO$_3)_2$ and its remarkably accurate theoretical description, the spin-$1/2$ Shastry-Sutherland model, host a variety of intriguing phenomena such as a dimer ground state with a nearly flat band of triplon excitations, a series of magnetization plateaux, and a possible pressure-induced deconfined quantum critical point. One open puzzle originating from inelastic neutron scattering and Raman experiments is the anomalous broadening of the triplon modes at relatively low temperatures compared to the triplon gap $Δ$. We demonstrate that the experimentally observed broadening is captured by the Shastry-Sutherland model. To this end, we develop a numerical simulation method based on matrix-product states to simulate dynamical spectral functions at nonzero temperatures accurately. Perturbative calculations identify the origin of this phenomenon as a small energy scale compared to $Δ$ between single triplon and bound triplon states at the experimentally relevant model parameters.
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Submitted 28 May, 2024;
originally announced May 2024.
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Fragmented superconductivity in the Hubbard model as solitons in Ginzburg-Landau theory
Authors:
Niccolò Baldelli,
Hannes Karlsson,
Benedikt Kloss,
Matthew Fishman,
Alexander Wietek
Abstract:
The phenomena of superconductivity and charge density waves are observed in close vicinity in many strongly correlated materials. Increasing evidence from experiments and numerical simulations suggests both phenomena can also occur in an intertwined manner, where the superconducting order parameter is coupled to the electronic density. Employing density matrix renormalization group simulations, we…
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The phenomena of superconductivity and charge density waves are observed in close vicinity in many strongly correlated materials. Increasing evidence from experiments and numerical simulations suggests both phenomena can also occur in an intertwined manner, where the superconducting order parameter is coupled to the electronic density. Employing density matrix renormalization group simulations, we investigate the nature of such an intertwined state of matter stabilized in the phase diagram of the elementary $t$-$t^\prime$-$U$ Hubbard model in the strong coupling regime. Remarkably, the condensate of Cooper pairs is shown to be fragmented in the presence of a charge density wave where more than one pairing wave function is macroscopically occupied. Moreover, we provide conclusive evidence that the macroscopic wave functions of the superconducting fragments are well-described by soliton solutions of a Ginzburg-Landau equation in a periodic potential constituted by the charge density wave. In the presence of an orbital magnetic field, the order parameters are gauge invariant, and superconducting vortices are pinned between the stripes. This intertwined Ginzburg-Landau theory is proposed as an effective low-energy description of the stripe fragmented superconductor.
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Submitted 12 November, 2024; v1 submitted 21 July, 2023;
originally announced July 2023.
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Finite-temperature properties of the easy-axis Heisenberg model on frustrated lattices
Authors:
Martin Ulaga,
Jure Kokalj,
Alexander Wietek,
Andrej Zorko,
Peter Prelovšek
Abstract:
Motivated by recent experiments on a compound {displaying Ising-like short-range correlations on the triangular lattice, we study the anisotropic easy-axis spin-$1/2$ Heisenberg model on the triangular and kagome lattice} by performing numerical calculations of finite-temperature properties, in particular of static spin structure factor and of thermodynamic quantities, on systems with up to 36 sit…
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Motivated by recent experiments on a compound {displaying Ising-like short-range correlations on the triangular lattice, we study the anisotropic easy-axis spin-$1/2$ Heisenberg model on the triangular and kagome lattice} by performing numerical calculations of finite-temperature properties, in particular of static spin structure factor and of thermodynamic quantities, on systems with up to 36 sites. On the triangular lattice, the low-temperature spin structure factor {exhibits long-range} spin correlations in the whole range of anisotropies, whereas thermodynamic quantities reveal a crossover upon increasing the anisotropy, most pronounced in the vanishing generalized Wilson ratio in the easy-axis regime. In contrast, on the kagome lattice, the spin structure factor is short-range, and thermodynamic quantities evolve steadily between the easy-axis and the isotropic case, consistent with the interpretation in terms of {a} spin liquid.
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Submitted 5 October, 2023; v1 submitted 7 July, 2023;
originally announced July 2023.
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Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet
Authors:
Alexander Wietek,
Sylvain Capponi,
Andreas M. Läuchli
Abstract:
Quantum electrodynamics in $2+1$ dimensions (QED$_3$) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated $J_1$-$J_2$ Heisenberg model…
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Quantum electrodynamics in $2+1$ dimensions (QED$_3$) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated $J_1$-$J_2$ Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED$_3$, in the quantum spin liquid regime. This includes a large manifold of explicitly constructed monopole and bilinear excitations of QED$_3$, which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of an emergent valence bond solid (VBS), which suggests a scenario where only the critical point of a transition from the $120^\circ$ Néel order to a VBS is described by QED$_3$. Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to $N=48$ sites.
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Submitted 3 July, 2024; v1 submitted 2 March, 2023;
originally announced March 2023.
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Variational Benchmarks for Quantum Many-Body Problems
Authors:
Dian Wu,
Riccardo Rossi,
Filippo Vicentini,
Nikita Astrakhantsev,
Federico Becca,
Xiaodong Cao,
Juan Carrasquilla,
Francesco Ferrari,
Antoine Georges,
Mohamed Hibat-Allah,
Masatoshi Imada,
Andreas M. Läuchli,
Guglielmo Mazzola,
Antonio Mezzacapo,
Andrew Millis,
Javier Robledo Moreno,
Titus Neupert,
Yusuke Nomura,
Jannes Nys,
Olivier Parcollet,
Rico Pohle,
Imelda Romero,
Michael Schmid,
J. Maxwell Silvester,
Sandro Sorella
, et al. (8 additional authors not shown)
Abstract:
The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems,…
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The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems, identifying cases where state-of-the-art numerical approaches show limited accuracy, and future algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods toward a quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible.
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Submitted 22 October, 2024; v1 submitted 9 February, 2023;
originally announced February 2023.
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Topological States of Matter in Frustrated Quantum Magnetism
Authors:
Alexander Wietek
Abstract:
Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the proce…
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Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the procedure of diagonalizing the Hamiltonian matrix are proposed and suggest a randomized distributed memory parallelization strategy. Benchmarks of computations on various supercomputers with system size up to 50 spin-1/2 particles have been performed. Results concerning the emergence of a chiral spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented. The stability and extent of this phase are discussed. In an extended Heisenberg model on the triangular lattice, we establish another chiral spin liquid phase. We discuss the special case of the Heisenberg $J_1$-$J_2$ model and present a scenario where the critical point of phase transition from the 120-degree Néel to a putative $\mathbf{Z}_2$ spin liquid is described by a Dirac spin liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of freedom on the triangular lattice with an additional ring-exchange term is discussed. We present our contribution to the project and the final results that suggest a series of chiral spin liquid phases in an extended parameter range. Finally, we present preliminary data from a Quantum Monte Carlo study of an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and multi-column representations. We establish the phase boundary between the Néel ordered phase and the disordered phases. The disordered phase in the four-column representation is expected to be a two-dimensional analog of the Haldane phase for the spin-1 Heisenberg chain.
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Submitted 7 October, 2022;
originally announced October 2022.
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Unidirectional subsystem symmetry in a hole-doped honeycomb-lattice Ising magnet
Authors:
Sambuddha Sanyal,
Alexander Wietek,
John Sous
Abstract:
We study a model of a hole-doped collinear Ising antiferromagnet on the honeycomb lattice as a route toward the realization of subsystem symmetry. We find nearly exact conservation of dipole symmetry verified both numerically with exact diagonalization (ED) on finite clusters and analytically with perturbation theory. The emergent symmetry forbids the motion of single holes -- or fractons -- but a…
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We study a model of a hole-doped collinear Ising antiferromagnet on the honeycomb lattice as a route toward the realization of subsystem symmetry. We find nearly exact conservation of dipole symmetry verified both numerically with exact diagonalization (ED) on finite clusters and analytically with perturbation theory. The emergent symmetry forbids the motion of single holes -- or fractons -- but allows hole pairs -- or dipoles -- to move freely along a one-dimensional line, the antiferromagnetic direction, of the system; in the transverse direction, both fractons and dipoles are completely localized. This presents a realization of a `unidirectional' subsystem symmetry. By studying interactions between dipoles, we argue that the subsystem symmetry is likely to continue to persist up to finite (but probably small) hole concentrations.
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Submitted 30 September, 2022;
originally announced October 2022.
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Bose-Einstein condensation in honeycomb dimer magnets and $\rm Yb_2 \rm Si_2 \rm O_7$
Authors:
Chunhan Feng,
E. Miles Stoudenmire,
Alexander Wietek
Abstract:
An asymmetric Bose-Einstein condensation (BEC) dome was observed in a recent experiment on the quantum dimer magnet $\rm Yb_2Si_2O_7$, which is modeled by a "breathing" honeycomb lattice Heisenberg model with possible anisotropies. We report a remarkable agreement between key experimental features and predictions from numerical simulations of the magnetic model. Both critical fields, as well as cr…
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An asymmetric Bose-Einstein condensation (BEC) dome was observed in a recent experiment on the quantum dimer magnet $\rm Yb_2Si_2O_7$, which is modeled by a "breathing" honeycomb lattice Heisenberg model with possible anisotropies. We report a remarkable agreement between key experimental features and predictions from numerical simulations of the magnetic model. Both critical fields, as well as critical temperatures of the BEC dome, can be accurately captured, as well as the occurrence of two regimes inside the BEC phase. Furthermore, we investigate the role of anisotropies in the exchange coupling and the $g$-tensor. While we confirm a previous proposal that anisotropy can induce a zero temperature phase transition at magnetic fields smaller than the fully polarizing field strength, we find that this effect becomes negligible at temperatures above the anisotropy scale. Instead, the two regimes inside the BEC dome are found to be due to a non-linear magnetization behavior of the isotropic breathing honeycomb Heisenberg antiferromagnet. Our analysis is performed by combining the density matrix renormalization group (DMRG) method with the finite-temperature techniques of minimally entangled typical thermal states (METTS) and quantum Monte Carlo (QMC).
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Submitted 31 August, 2022;
originally announced September 2022.
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Tunable Stripe Order and Weak Superconductivity in the Moiré Hubbard Model
Authors:
Alexander Wietek,
Jie Wang,
Jiawei Zang,
Jennifer Cano,
Antoine Georges,
Andrew Millis
Abstract:
The moiré Hubbard model describes correlations in certain homobilayer twisted transition metal dichalcogenides. Using exact diagonalization and density matrix renormalization group methods, we find magnetic Mott insulating and metallic phases, which, upon doping exhibit intertwined charge and spin ordering and, in some regimes, pair binding of holes. The phases are highly tunable via an interlayer…
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The moiré Hubbard model describes correlations in certain homobilayer twisted transition metal dichalcogenides. Using exact diagonalization and density matrix renormalization group methods, we find magnetic Mott insulating and metallic phases, which, upon doping exhibit intertwined charge and spin ordering and, in some regimes, pair binding of holes. The phases are highly tunable via an interlayer potential difference. Remarkably, the hole binding energy is found to be highly tunable revealing an experimentally accessible regime where holes become attractive. In this attractive regime, we study the superconducting correlation function and point out the possibility of weak superconductivity.
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Submitted 14 November, 2022; v1 submitted 8 April, 2022;
originally announced April 2022.
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Order, Disorder and Monopole Confinement in the Spin-$1/2$ XXZ Model on a Pyrochlore Tube
Authors:
Chunhan Feng,
Alexander Wietek,
E. Miles Stoudenmire,
Rajiv R. P. Singh
Abstract:
We study the ground state and thermodynamic properties of the spin-half XXZ model, with an Ising interaction $J_z$ and a transverse exchange interaction $J_{x}$, on a pyrochlore tube obtained by joining together elementary cubes in a one-dimensional array. Periodic boundary conditions in the transverse directions ensure that the bulk of the system consists of corner-sharing tetrahedra, with the sa…
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We study the ground state and thermodynamic properties of the spin-half XXZ model, with an Ising interaction $J_z$ and a transverse exchange interaction $J_{x}$, on a pyrochlore tube obtained by joining together elementary cubes in a one-dimensional array. Periodic boundary conditions in the transverse directions ensure that the bulk of the system consists of corner-sharing tetrahedra, with the same local geometry as the pyrochlore lattice. We use exact diagonalization, the density matrix renormalization group (DMRG), and minimally entangled typical thermal states (METTS) methods to study the system. When $J_z$ is antiferromagnetic ($J_{z}>0$) and $J_x$ is ferromagnetic ($J_{x}<0$), we find a transition from a spin liquid to an XY ferromagnet, which has power-law correlations at $T=0$. For $J_{z}<0$ and $J_{x}>0$, spin-two excitations are found to have lower energy than spin-one at the transition away from the fully polarized state, showing evidence for incipient spin-nematic order. When both interactions are antiferromagnetic, we find a non-degenerate ground state with no broken symmetries and a robust energy gap. The low energy spectra evolve smoothly from predominantly Ising to predominantly XY interactions. In the spin-liquid regime of small $|J_{x}|$, we study the confinement of monopole-anti-monopole pairs and find that the confinement length scale is larger for $J_x<0$ than for $J_{x}>0$, although both length scales are very short. These results are consistent with a local spin-liquid phase for the Heisenberg antiferromagnet with no broken symmetries.
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Submitted 28 February, 2022;
originally announced March 2022.
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Fragmented Cooper pair condensation in striped superconductors
Authors:
Alexander Wietek
Abstract:
Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called "fragmented" if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in…
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Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called "fragmented" if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in striped superconductors. To this end, we investigate the nature of the superconducting ground state realized in the two-dimensional $t$-$t^\prime$-$J$ model. In the presence of charge density modulations, the condensate is shown to be fragmented and composed of partial condensates located on the stripes. The fragments of the condensates hybridize to form an extended macroscopic wave function across the system. The results are obtained from evaluating the singlet-pairing two-particle density matrix of the ground state on finite cylinders computed via the density matrix renormalization group (DMRG) method. Our results shed light on the intricate relation between stripe order and superconductivity in systems of strongly correlated electrons.
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Submitted 18 October, 2022; v1 submitted 11 February, 2022;
originally announced February 2022.
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Mott insulating states with competing orders in the triangular lattice Hubbard model
Authors:
Alexander Wietek,
Riccardo Rossi,
Fedor Šimkovic IV,
Marcel Klett,
Philipp Hansmann,
Michel Ferrero,
E. Miles Stoudenmire,
Thomas Schäfer,
Antoine Georges
Abstract:
The physics of the triangular lattice Hubbard model exhibits a rich phenomenology, ranging from a metal-insulator transition, intriguing thermodynamic behavior, and a putative spin liquid phase at intermediate coupling, ultimately becoming a magnetic insulator at strong coupling. In this multimethod study, we combine a finite-temperature tensor network method, minimally entangled thermal typical s…
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The physics of the triangular lattice Hubbard model exhibits a rich phenomenology, ranging from a metal-insulator transition, intriguing thermodynamic behavior, and a putative spin liquid phase at intermediate coupling, ultimately becoming a magnetic insulator at strong coupling. In this multimethod study, we combine a finite-temperature tensor network method, minimally entangled thermal typical states (METTS), with two Green-function-based methods, connected-determinant diagrammatic Monte Carlo and cellular dynamical mean-field theory, to establish several aspects of this model. We elucidate the evolution from the metallic to the insulating regime from the complementary perspectives brought by these different methods. We compute the full thermodynamics of the model on a width-four cylinder using METTS in the intermediate to strong coupling regime. We find that the insulating state hosts a large entropy at intermediate temperatures, which increases with the strength of the coupling. Correspondingly, and consistently with a thermodynamic Maxwell relation, the double occupancy has a minimum as a function of temperature which is the manifestation of the Pomeranchuk effect of increased localization upon heating. The intermediate coupling regime is found to exhibit both pronounced chiral as well as stripy antiferromagnetic spin correlations. We propose a scenario in which time-reversal symmetry-broken states compete with stripy-spin states at lowest temperatures.
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Submitted 19 October, 2021; v1 submitted 25 February, 2021;
originally announced February 2021.
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Quantum phase transition at non-zero doping in a random $t$-$J$ model
Authors:
Leyna Shackleton,
Alexander Wietek,
Antoine Georges,
Subir Sachdev
Abstract:
We present exact diagonalization results on finite clusters of a $t$-$J$ model of spin-1/2 electrons with random all-to-all hopping and exchange interactions. We argue that such random models capture qualitatively the strong local correlations needed to describe the cuprates and related compounds, while avoiding lattice space group symmetry breaking orders. The previously known spin glass ordere…
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We present exact diagonalization results on finite clusters of a $t$-$J$ model of spin-1/2 electrons with random all-to-all hopping and exchange interactions. We argue that such random models capture qualitatively the strong local correlations needed to describe the cuprates and related compounds, while avoiding lattice space group symmetry breaking orders. The previously known spin glass ordered phase in the insulator at doping $p=0$ extends to a metallic spin glass phase up to a transition $p=p_c \approx 1/3$. The dynamic spin susceptibility shows signatures of the spectrum of the Sachdev-Ye-Kitaev models near $p_c$. We also find signs of the phase transition in the entropy, entanglement entropy and compressibility, all of which exhibit a maximum near $p_c$. The electron energy distribution function in the metallic phase is consistent with a disordered extension of the Luttinger-volume Fermi surface for $p>p_c$, while this breaks down for $p<p_c$.
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Submitted 3 April, 2021; v1 submitted 11 December, 2020;
originally announced December 2020.
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Stripes, Antiferromagnetism, and the Pseudogap in the Doped Hubbard Model at Finite Temperature
Authors:
Alexander Wietek,
Yuan-Yao He,
Steven R. White,
Antoine Georges,
E. Miles Stoudenmire
Abstract:
The interplay between thermal and quantum fluctuations controls the competition between phases of matter in strongly correlated electron systems. We study finite-temperature properties of the strongly coupled two-dimensional doped Hubbard model using the minimally-entangled typical thermal states (METTS) method on width $4$ cylinders. We discover that a phase characterized by commensurate short-ra…
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The interplay between thermal and quantum fluctuations controls the competition between phases of matter in strongly correlated electron systems. We study finite-temperature properties of the strongly coupled two-dimensional doped Hubbard model using the minimally-entangled typical thermal states (METTS) method on width $4$ cylinders. We discover that a phase characterized by commensurate short-range antiferromagnetic correlations and no charge ordering occurs at temperatures above the half-filled stripe phase extending to zero temperature. The transition from the antiferromagnetic phase to the stripe phase takes place at temperature $T/t \approx 0.05$ and is accompanied by a step-like feature of the specific heat. We find the single-particle gap to be smallest close to the nodal point at $\mathbf{k}=(π/2, π/2)$ and detect a maximum in the magnetic susceptibility. These features bear a strong resemblance to the pseudogap phase of high-temperature cuprate superconductors. The simulations are verified using a variety of different unbiased numerical methods in the three limiting cases of zero temperature, small lattice sizes, and half-filling. Moreover, we compare to and confirm previous determinantal quantum Monte Carlo results on incommensurate spin-density waves at finite doping and temperature.
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Submitted 16 July, 2021; v1 submitted 22 September, 2020;
originally announced September 2020.
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Loop-gas description of the localized-magnon states on the kagome lattice with open boundary conditions
Authors:
Andreas Honecker,
Johannes Richter,
Jürgen Schnack,
Alexander Wietek
Abstract:
The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical…
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The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.
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Submitted 22 April, 2021; v1 submitted 24 August, 2020;
originally announced August 2020.
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SU$(3)_1$ Chiral Spin Liquid on the Square Lattice: a View from Symmetric PEPS
Authors:
Ji-Yao Chen,
Sylvain Capponi,
Alexander Wietek,
Matthieu Mambrini,
Norbert Schuch,
Didier Poilblanc
Abstract:
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 sp…
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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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Submitted 23 January, 2020; v1 submitted 31 December, 2019;
originally announced December 2019.
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Valence bond solid and possible deconfined quantum criticality in an extended kagome lattice Heisenberg antiferromagnet
Authors:
Alexander Wietek,
Andreas M. Läuchli
Abstract:
We present numerical evidence for the emergence of an extended valence bond solid (VBS) phase at $T=0$ in the kagome $S=1/2$ Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamond-like singlet covering pattern with a…
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We present numerical evidence for the emergence of an extended valence bond solid (VBS) phase at $T=0$ in the kagome $S=1/2$ Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamond-like singlet covering pattern with a $12$-site unit-cell. Our results suggest the possibility of a direct, possibly continuous, quantum phase transition from the neighboring $\mathbf{q}=0$ coplanar magnetically ordered phase into the VBS phase. Moreover, a second phase which breaks lattice symmetries, and is of likely spin-nematic type, is found close to the transition to the ferromagnetic phase. The results have been obtained using numerical Exact Diagonalization. We discuss implications of our results on the nature of nearest-neighbor Heisenberg antiferromagnet.
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Submitted 27 July, 2020; v1 submitted 7 August, 2019;
originally announced August 2019.
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Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase
Authors:
Alexander Wietek,
Philippe Corboz,
Stefan Wessel,
Bruce Normand,
Frédéric Mila,
Andreas Honecker
Abstract:
The thermodynamic properties of the Shastry-Sutherland model have posed one of the longest-lasting conundrums in frustrated quantum magnetism. Over a wide range on both sides of the quantum phase transition (QPT) from the dimer-product to the plaquette-based ground state, neither analytical nor any available numerical methods have come close to reproducing the physics of the excited states and the…
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The thermodynamic properties of the Shastry-Sutherland model have posed one of the longest-lasting conundrums in frustrated quantum magnetism. Over a wide range on both sides of the quantum phase transition (QPT) from the dimer-product to the plaquette-based ground state, neither analytical nor any available numerical methods have come close to reproducing the physics of the excited states and thermal response. We solve this problem in the dimer-product phase by introducing two qualitative advances in computational physics. One is the use of thermal pure quantum (TPQ) states to augment dramatically the size of clusters amenable to exact diagonalization. The second is the use of tensor-network methods, in the form of infinite projected entangled pair states (iPEPS), for the calculation of finite-temperature quantities. We demonstrate convergence as a function of system size in TPQ calculations and of bond dimension in our iPEPS results, with complete mutual agreement even extremely close to the QPT. Our methods reveal a remarkably sharp and low-lying feature in the magnetic specific heat around the QPT, whose origin appears to lie in a proliferation of excitations composed of two-triplon bound states. The surprisingly low energy scale and apparently extended spatial nature of these states explain the failure of less refined numerical approaches to capture their physics. Both of our methods will have broad and immediate application in addressing the thermodynamic response of a wide range of highly frustrated magnetic models and materials.
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Submitted 22 October, 2019; v1 submitted 28 June, 2019;
originally announced July 2019.
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NetKet: A Machine Learning Toolkit for Many-Body Quantum Systems
Authors:
Giuseppe Carleo,
Kenny Choo,
Damian Hofmann,
James E. T. Smith,
Tom Westerhout,
Fabien Alet,
Emily J. Davis,
Stavros Efthymiou,
Ivan Glasser,
Sheng-Hsuan Lin,
Marta Mauri,
Guglielmo Mazzola,
Christian B. Mendl,
Evert van Nieuwenburg,
Ossian O'Reilly,
Hugo Théveniaut,
Giacomo Torlai,
Alexander Wietek
Abstract:
We introduce NetKet, a comprehensive open source framework for the study of many-body quantum systems using machine learning techniques. The framework is built around a general and flexible implementation of neural-network quantum states, which are used as a variational ansatz for quantum wave functions. NetKet provides algorithms for several key tasks in quantum many-body physics and quantum tech…
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We introduce NetKet, a comprehensive open source framework for the study of many-body quantum systems using machine learning techniques. The framework is built around a general and flexible implementation of neural-network quantum states, which are used as a variational ansatz for quantum wave functions. NetKet provides algorithms for several key tasks in quantum many-body physics and quantum technology, namely quantum state tomography, supervised learning from wave-function data, and ground state searches for a wide range of customizable lattice models. Our aim is to provide a common platform for open research and to stimulate the collaborative development of computational methods at the interface of machine learning and many-body physics.
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Submitted 29 March, 2019;
originally announced April 2019.
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Sublattice Coding Algorithm and Distributed Memory Parallelization for Large-Scale Exact Diagonalizations of Quantum Many-Body Systems
Authors:
Alexander Wietek,
Andreas M. Läuchli
Abstract:
We present algorithmic improvements for fast and memory-efficient use of discrete spatial symmetries in Exact Diagonalization computations of quantum many-body systems. These techniques allow us to work flexibly in the reduced basis of symmetry-adapted wave functions. Moreover, a parallelization scheme for the Hamiltonian-vector multiplication in the Lanczos procedure for distributed memory machin…
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We present algorithmic improvements for fast and memory-efficient use of discrete spatial symmetries in Exact Diagonalization computations of quantum many-body systems. These techniques allow us to work flexibly in the reduced basis of symmetry-adapted wave functions. Moreover, a parallelization scheme for the Hamiltonian-vector multiplication in the Lanczos procedure for distributed memory machines avoiding load balancing problems is proposed. We demonstrate that using these methods low-energy properties of systems of up to 50 spin-1/2 particles can be successfully determined.
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Submitted 3 October, 2018; v1 submitted 13 April, 2018;
originally announced April 2018.
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Studying Continuous Symmetry Breaking using Energy Level Spectroscopy
Authors:
Alexander Wietek,
Michael Schuler,
Andreas M. Läuchli
Abstract:
Tower of States analysis is a powerful tool for investigating phase transitions in condensed matter systems. Spontaneous symmetry breaking implies a specific structure of the energy eigenvalues and their corresponding quantum numbers on finite systems. In these lecture notes we explain the group representation theory used to derive the spectral structure for several scenarios of symmetry breaking.…
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Tower of States analysis is a powerful tool for investigating phase transitions in condensed matter systems. Spontaneous symmetry breaking implies a specific structure of the energy eigenvalues and their corresponding quantum numbers on finite systems. In these lecture notes we explain the group representation theory used to derive the spectral structure for several scenarios of symmetry breaking. We give numerous examples to compute quantum numbers of the degenerate groundstates, including translational symmetry breaking or spin rotational symmetry breaking in Heisenberg antiferromagnets. These results are then compared to actual numerical data from Exact Diagonalization.
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Submitted 27 April, 2017;
originally announced April 2017.
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Chiral Spin Liquid and Quantum Criticality in Extended $S=1/2$ Heisenberg Models on the Triangular Lattice
Authors:
Alexander Wietek,
Andreas M. Läuchli
Abstract:
We investigate the $J_1$-$J_2$ Heisenberg model on the triangular lattice with an additional scalar chirality term and show that a chiral spin liquid is stabilized in a sizeable region of the phase diagram. This topological phase is situated in between a coplanar $120^\circ$ Néel ordered and a non-coplanar tetrahedrally ordered phase. Furthermore we discuss the nature of the spin-disordered interm…
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We investigate the $J_1$-$J_2$ Heisenberg model on the triangular lattice with an additional scalar chirality term and show that a chiral spin liquid is stabilized in a sizeable region of the phase diagram. This topological phase is situated in between a coplanar $120^\circ$ Néel ordered and a non-coplanar tetrahedrally ordered phase. Furthermore we discuss the nature of the spin-disordered intermediate phase in the $J_1$-$J_2$ model. We compare the groundstates from Exact Diagonalization with a Dirac spin liquid wavefunction and propose a scenario where this wavefunction describes the quantum critical point between the $120^\circ$ magnetically ordered phase and a putative $\mathbb{Z}_2$ spin liquid.
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Submitted 26 April, 2016;
originally announced April 2016.
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Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields
Authors:
Pierre Nataf,
Miklós Lajkó,
Alexander Wietek,
Karlo Penc,
Frédéric Mila,
Andreas M. Läuchli
Abstract:
We show that, in the presence of a $π/2$ artificial gauge field per plaquette, Mott insulating phases of ultra-cold fermions with $SU(N)$ symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by a multiplet of $N$ low-lying singlet excitations for periodic boundary conditions, and by chiral edge states described by the…
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We show that, in the presence of a $π/2$ artificial gauge field per plaquette, Mott insulating phases of ultra-cold fermions with $SU(N)$ symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by a multiplet of $N$ low-lying singlet excitations for periodic boundary conditions, and by chiral edge states described by the $SU(N)_1$ Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for $N$ between $3$ and $9$, and by a parton construction based on a set of $N$ Gutzwiller projected fermionic wave-functions with flux $π/N$ per triangular plaquette. Experimental implications are briefly discussed.
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Submitted 5 January, 2016;
originally announced January 2016.
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Nature of chiral spin liquids on the kagome lattice
Authors:
Alexander Wietek,
Antoine Sterdyniak,
Andreas M. Läuchli
Abstract:
We investigate the stability and the nature of the chiral spin liquids which were recently uncovered in extended Heisenberg models on the kagome lattice. Using a Gutzwiller projected wave function approach -- i.e. a parton construction -- we obtain large overlaps with ground states of these extended Heisenberg models. We further suggest that the appearance of the chiral spin liquid in the time-rev…
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We investigate the stability and the nature of the chiral spin liquids which were recently uncovered in extended Heisenberg models on the kagome lattice. Using a Gutzwiller projected wave function approach -- i.e. a parton construction -- we obtain large overlaps with ground states of these extended Heisenberg models. We further suggest that the appearance of the chiral spin liquid in the time-reversal invariant case is linked to a classical transition line between two magnetically ordered phases.
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Submitted 11 March, 2015;
originally announced March 2015.