Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1504.05332 · Physical Review B

SU(N) Heisenberg model with multicolumn representations

Tsuyoshi Okubo, Kenji Harada, Jie Lou, Naoki Kawashima

Published Oct 5, 2015• 10 citations
arXivHTMLPDF

Abstract

The $\mathrm{SU}(N)$ symmetric antiferromagnetic Heisenberg model with multicolumn representations on the two-dimensional square lattice is investigated by quantum Monte Carlo simulations. For the representation of a Young diagram with two columns, we confirm that a valence-bond solid (VBS) order appears as soon as the N\'eel order disappears at $N=10$, indicating no intermediate phase. In the case of the representation with three columns, there is no evidence for either the N\'eel or the VBS ordering for $N\ensuremath{\ge}15$. This is actually consistent with the large-$N$ theory, which predicts that the VBS state immediately follows the N\'eel state, because the expected spontaneous order is too weak to be detected.

DOI
10.1103/physrevb.92.134404

Related papers

Phase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice

Jonas Schwab, Francesco Parisen Toldin, Fakher F. Assaad

We investigate the ground state phase diagram of an SU($N$)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU($N$) described by a square Young tableau of $N/2$ rows and $2S$ columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any $S$ and even values of $N$. In the large-$N$ limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large $S$-limit, the semi-classical approximation points to Néel state. On a line set by $N=8S + 2$ in the $S$ versus $N$ phase diagram, we observe a variety of phases proximate to the Néel state. At $S = 1/2$ and $3/2$ we observe the aforementioned four fold degenerate valence bond solid state. At $S=1$ a two fold degenerate spin nematic state in which the C$_4$ lattice symmetry is broken down to C$_2$ emerges. Finally at $S=2$ we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18), $S=1/2$ boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by $\text{mod}(4,2S)$.

arXiv:2304.07329•Apr 14, 2023•0 citations

Quantum phase transitions in bilayer SU( N ) antiferromagnets

Ribhu K. Kaul

We present a detailed study of the destruction of SU($N$) magnetic order in square lattice bilayer antiferromagnets using unbiased quantum Monte Carlo numerical simulations and field theoretic techniques. We study phase transitions from an SU($N$) N\'eel state into two distinct quantum disordered ``valence-bond'' phases: a valence-bond liquid (VBL) with no broken symmetries and a lattice-symmetry-breaking valence-bond solid (VBS) state. For finite interlayer coupling, the cancellation of Berry phases between the layers has dramatic consequences on the two phase transitions: the N\'eel-VBS transition is first order for all $N\ensuremath{\ge}5$ accesible in our model, whereas the N\'eel-VBL transition is continuous for $N=2$ and first order for $N\ensuremath{\ge}4$; for $N=3$ the N\'eel-VBL transition show no signs of first-order behavior.

arXiv:1203.6677•Physical Review B•May 29, 2012•42 citations

Ground States of the SU(N) Heisenberg Model

Naoki Kawashima, Yuta Tanabe

The SU(N) Heisenberg model with various single-row representations is investigated by quantum Monte Carlo simulations. While the zero-temperature phase boundary agrees qualitatively with the theoretical predictions based on the 1/N expansion, some unexpected features are also observed. For N> or =5 with the fundamental representation, for example, it is suggested that the ground states possess exact or approximate U(1) degeneracy. In addition, for the representation of Young tableau with more than one column, the ground state shows no valence-bond-solid order even at N greater than the threshold value.

arXiv:cond-mat/0702599•Physical Review Letters•Jan 29, 2007•48 citations

SU ( N ) Kondo-Heisenberg chain: Phase diagram, Ising criticality, and the coexistence of heavy quasiparticles and valence bond solid order

Marcin Raczkowski, Fakher F. Assaad

We map out the ground-state phase diagram of a one-dimensional $\mathrm{SU}(N)$ Kondo-Heisenberg lattice model at half-filling and in the fully antisymmetric self-adjoint representation as a function of $\frac{1}{N}$ and Kondo coupling ${J}_{k}/t$. On the basis of auxiliary field quantum Monte Carlo (QMC) simulations with even $N$ up to 8, we show that the enlarged $\mathrm{SU}(N\ensuremath{\ge}4$) symmetry realizes a quantum phase transition separating a valence bond solid (VBS) phase occupying a weak-coupling part of the phase diagram and the Kondo insulator (KI) state dominating in the strong ${J}_{k}/t$ limit. Along the phase boundary, we always observe critical exponents that belong to a two-dimensional classical Ising universality class. We next trace the evolution of the composite fermion and spin spectral functions across the phase boundary and conclude that VBS order triggers a bond order wave state of conduction electrons, both coexisting with Kondo screening. Upon further reducing ${J}_{k}/t$, we observe that composite quasiparticles lose their spectral weight indicating that VBS order gradually liberates localized $f$ spins from forming Kondo singlets with conduction electron spins. We contrast the QMC results with a static large-$N$ approximation, and we show that in the limit of infinite degeneracy $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the order-disorder transition becomes of first order and is accompanied by a full decoupling of conduction electrons and localized $f$ spins. We complete our analysis by considering the limit of a vanishing Heisenberg coupling ${J}_{h}=0$, and we provide evidence that the enlarged $\mathrm{SU}(N)$ symmetry and low dimensionality of the Ruderman-Kittel-Kasuya-Yosida exchange interaction are not sufficient to induce VBS order in the conventional $\mathrm{SU}(N)$ Kondo chain, which hosts solely a KI phase. Finally, we quantify the role of increasing $N$ in reducing the separation of charge and spin timescales specific to the KI state, and we show that the latter is adiabatically connected to a trivial band insulator captured by the large-$N$ approximation.

arXiv:2403.03694•Physical review. B./Physical review. B•May 29, 2024•4 citations

Phase diagrams of S = 1 2 bilayer models of SU(2) symmetric antiferromagnets

Fan Zhang, Nisheeta Desai, Wenan Guo +1 more

We study the $T=0$ phase diagrams of models of bilayers of $S=1/2$ square lattices antiferromagnets with SU(2) Heisenberg symmetry that have 2, 4, and 6 spin exchanges. We study two families of bilayer models with distinct internal symmetries and, hence, different phase diagram topologies. A traditional bilayer model in which the interlayer interaction is Heisenberg so that the two layers can exchange spin (and energy) with each other, making it possible to achieve a simple dimerized valence bond liquid-like state. The resulting phase diagram is rich with Néel, valence bond solid and simple dimer phases, and both first-order and continuous transitions, which we demonstrate are consistent with the conventional Landau theory of order parameters. In the second family of models in which the layers can exchange only energy but no spin (reminiscent of the Ashkin-Teller coupling), the simple dimer state cannot occur. The phase diagrams reveal a number of phase transitions that are accessed for the first time. We find that the phase transition between Néel and VBS is first order in both the spin-spin and energy-energy coupled models, although they have strikingly distinct finite-size scaling behavior and that the transition from VBS to dimer in the spin-spin coupling model deviates from the expected scenario of an XY model with dangerously irrelevant four-fold anisotropy.

arXiv:2511.04101•Physical review. B./Physical review. B•Feb 17, 2026•0 citations

First-order Néel to columnar valence bond solid transition in a model square-lattice S=1 antiferromagnet

Julia Wildeboer, Nisheeta Desai, Jonathan D’Emidio +1 more

We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free $S=1$ square-lattice model. This is the same kind of transition that for $S=1/2$ has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice $S=1/2$ N\'eel-VBS transition have found consistency with the deconfined critical point scenario with no direct evidence for first-order behavior. In contrast to the $S=1/2$ case, in our quantum Monte Carlo simulations for the $S=1$ model, we present unambiguous evidence for a direct conventional first-order quantum phase transition. Classic signs of a first-order transition demonstrating coexistence including double-peaked histograms and switching behavior are observed. The sharp contrast from the $S=1/2$ case is remarkable; we hypothesize that this is a striking demonstration of the role of the size of the quantum spin in the phase diagram of two-dimensional lattice models.

arXiv:1808.04731•Physical review. B./Physical review. B•Jan 13, 2020•11 citations
The SU(N)\mathrm{SU}(N)SU(N) symmetric antiferromagnetic Heisenberg model with multicolumn representations on the two-dimensional square lattice is investigated by quantum Monte Carlo simulations. For the representation of a Young diagram with two columns, we confirm that a valence-bond solid (VBS) order appears as soon as the N'eel order disappears at N=10N=10N=10, indicating no intermediate phase. In the case of the representation with three columns, there is no evidence for either the N'eel or the VBS ordering for N\ensuremath≥15N\ensuremath{\ge}15N\ensuremath≥15. This is actually consistent with the large-NNN theory, which predicts that the VBS state immediately follows the N'eel state, because the expected spontaneous order is too weak to be detected.
Phase diagram of the SU(NNN) antiferromagnet of spin SSS on a square lattice
We investigate the ground state phase diagram of an SU(NNN)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(NNN) described by a square Young tableau of N/2N/2N/2 rows and 2S2S2S columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any SSS and even values of NNN. In the large-NNN limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large SSS-limit, the semi-classical approximation points to Néel state. On a line set by N=8S+2N=8S + 2N=8S+2 in the SSS versus NNN phase diagram, we observe a variety of phases proximate to the Néel state. At S=1/2S = 1/2S=1/2 and 3/23/23/2 we observe the aforementioned four fold degenerate valence bond solid state. At S=1S=1S=1 a two fold degenerate spin nematic state in which the C4_44​ lattice symmetry is broken down to C2_22​ emerges. Finally at S=2S=2S=2 we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18), S=1/2S=1/2S=1/2 boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by mod(4,2S)\text{mod}(4,2S)mod(4,2S).
We present a detailed study of the destruction of SU(NNN) magnetic order in square lattice bilayer antiferromagnets using unbiased quantum Monte Carlo numerical simulations and field theoretic techniques. We study phase transitions from an SU(NNN) N\'eel state into two distinct quantum disordered ``valence-bond'' phases: a valence-bond liquid (VBL) with no broken symmetries and a lattice-symmetry-breaking valence-bond solid (VBS) state. For finite interlayer coupling, the cancellation of Berry phases between the layers has dramatic consequences on the two phase transitions: the N\'eel-VBS transition is first order for all $N\ensuremath{\ge}5$ accesible in our model, whereas the N\'eel-VBL transition is continuous for N=2N=2N=2 and first order for $N\ensuremath{\ge}4$; for N=3N=3N=3 the N\'eel-VBL transition show no signs of first-order behavior.
The SU(N) Heisenberg model with various single-row representations is investigated by quantum Monte Carlo simulations. While the zero-temperature phase boundary agrees qualitatively with the theoretical predictions based on the 1/N expansion, some unexpected features are also observed. For N> or =5 with the fundamental representation, for example, it is suggested that the ground states possess exact or approximate U(1) degeneracy. In addition, for the representation of Young tableau with more than one column, the ground state shows no valence-bond-solid order even at N greater than the threshold value.
We map out the ground-state phase diagram of a one-dimensional SU(N)\mathrm{SU}(N)SU(N) Kondo-Heisenberg lattice model at half-filling and in the fully antisymmetric self-adjoint representation as a function of 1N\frac{1}{N}N1​ and Kondo coupling Jk/t{J}_{k}/tJk​/t. On the basis of auxiliary field quantum Monte Carlo (QMC) simulations with even NNN up to 8, we show that the enlarged $\mathrm{SU}(N\ensuremath{\ge}4$) symmetry realizes a quantum phase transition separating a valence bond solid (VBS) phase occupying a weak-coupling part of the phase diagram and the Kondo insulator (KI) state dominating in the strong Jk/t{J}_{k}/tJk​/t limit. Along the phase boundary, we always observe critical exponents that belong to a two-dimensional classical Ising universality class. We next trace the evolution of the composite fermion and spin spectral functions across the phase boundary and conclude that VBS order triggers a bond order wave state of conduction electrons, both coexisting with Kondo screening. Upon further reducing Jk/t{J}_{k}/tJk​/t, we observe that composite quasiparticles lose their spectral weight indicating that VBS order gradually liberates localized fff spins from forming Kondo singlets with conduction electron spins. We contrast the QMC results with a static large-NNN approximation, and we show that in the limit of infinite degeneracy $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the order-disorder transition becomes of first order and is accompanied by a full decoupling of conduction electrons and localized fff spins. We complete our analysis by considering the limit of a vanishing Heisenberg coupling Jh=0{J}_{h}=0Jh​=0, and we provide evidence that the enlarged SU(N)\mathrm{SU}(N)SU(N) symmetry and low dimensionality of the Ruderman-Kittel-Kasuya-Yosida exchange interaction are not sufficient to induce VBS order in the conventional SU(N)\mathrm{SU}(N)SU(N) Kondo chain, which hosts solely a KI phase. Finally, we quantify the role of increasing NNN in reducing the separation of charge and spin timescales specific to the KI state, and we show that the latter is adiabatically connected to a trivial band insulator captured by the large-NNN approximation.
We study the T=0T=0T=0 phase diagrams of models of bilayers of S=1/2S=1/2S=1/2 square lattices antiferromagnets with SU(2) Heisenberg symmetry that have 2, 4, and 6 spin exchanges. We study two families of bilayer models with distinct internal symmetries and, hence, different phase diagram topologies. A traditional bilayer model in which the interlayer interaction is Heisenberg so that the two layers can exchange spin (and energy) with each other, making it possible to achieve a simple dimerized valence bond liquid-like state. The resulting phase diagram is rich with Néel, valence bond solid and simple dimer phases, and both first-order and continuous transitions, which we demonstrate are consistent with the conventional Landau theory of order parameters. In the second family of models in which the layers can exchange only energy but no spin (reminiscent of the Ashkin-Teller coupling), the simple dimer state cannot occur. The phase diagrams reveal a number of phase transitions that are accessed for the first time. We find that the phase transition between Néel and VBS is first order in both the spin-spin and energy-energy coupled models, although they have strikingly distinct finite-size scaling behavior and that the transition from VBS to dimer in the spin-spin coupling model deviates from the expected scenario of an XY model with dangerously irrelevant four-fold anisotropy.
We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free S=1S=1S=1 square-lattice model. This is the same kind of transition that for S=1/2S=1/2S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice S=1/2S=1/2S=1/2 N\'eel-VBS transition have found consistency with the deconfined critical point scenario with no direct evidence for first-order behavior. In contrast to the S=1/2S=1/2S=1/2 case, in our quantum Monte Carlo simulations for the S=1S=1S=1 model, we present unambiguous evidence for a direct conventional first-order quantum phase transition. Classic signs of a first-order transition demonstrating coexistence including double-peaked histograms and switching behavior are observed. The sharp contrast from the S=1/2S=1/2S=1/2 case is remarkable; we hypothesize that this is a striking demonstration of the role of the size of the quantum spin in the phase diagram of two-dimensional lattice models.