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Physical review. B, Condensed matter

Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets

N. Read, Subir Sachdev A

Published Sep 1, 1990• 492 citations
Publisher

Abstract

We examine large-N limits of the nearest-neighbor SU(N) quantum antiferromagnets on bipartite lattices in d=1,2. In d=2 the model displays a transition from a N\'eel to a disordered phase. The properties of the disordered phase close to the phase boundary are crucially dependent upon the nature of ``hedgehog''-like instanton tunneling events. We calculate the Berry phases of the instantons and show that, at scales larger than the spin-correlation length, the system can be described by a Coulomb plasma of instantons with complex fugacities. The properties of the Coulomb plasma vary periodically with the ``spin'' ${\mathit{n}}_{\mathit{c}}$ of the states at each site, with periodicity given by the coordination number Z of the lattice [${\mathit{n}}_{\mathit{c}}$=2S for SU(2)]. For ${\mathit{n}}_{\mathit{c}}$\ensuremath{\ne}0 (mod Z) the disordered phase has a broken lattice symmetry with spin-Peierls order, while for ${\mathit{n}}_{\mathit{c}}$=0 (mod Z), the ground state is a valence-bond solid state with no broken symmetry. Related topological effects for the d=1 chain lead to spin-Peierls order for odd ${\mathit{n}}_{\mathit{c}}$. These results are for a class of models which have, at sites of the A sublattice, representations of SU(N) described by a Young tableau with a single row, and the conjugate on the B sublattice. Similar results are also obtained for representations with m rows, using U(m) gauge theory.

DOI
10.1103/physrevb.42.4568

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We examine large-N limits of the nearest-neighbor SU(N) quantum antiferromagnets on bipartite lattices in d=1,2. In d=2 the model displays a transition from a N'eel to a disordered phase. The properties of the disordered phase close to the phase boundary are crucially dependent upon the nature of hedgehog''-like instanton tunneling events. We calculate the Berry phases of the instantons and show that, at scales larger than the spin-correlation length, the system can be described by a Coulomb plasma of instantons with complex fugacities. The properties of the Coulomb plasma vary periodically with the spin'' nc{\mathit{n}}_{\mathit{c}}nc​ of the states at each site, with periodicity given by the coordination number Z of the lattice [nc{\mathit{n}}_{\mathit{c}}nc​=2S for SU(2)]. For nc{\mathit{n}}_{\mathit{c}}nc​\ensuremath{\ne}0 (mod Z) the disordered phase has a broken lattice symmetry with spin-Peierls order, while for nc{\mathit{n}}_{\mathit{c}}nc​=0 (mod Z), the ground state is a valence-bond solid state with no broken symmetry. Related topological effects for the d=1 chain lead to spin-Peierls order for odd nc{\mathit{n}}_{\mathit{c}}nc​. These results are for a class of models which have, at sites of the A sublattice, representations of SU(N) described by a Young tableau with a single row, and the conjugate on the B sublattice. Similar results are also obtained for representations with m rows, using U(m) gauge theory.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.
We examine a recently developed sine-Gordon model description of the non-N\'eel phase of quantum antiferromagnets on two-dimensional bipartite lattices. Using a spatial dimensionality d=1+\ensuremath{\epsilon} expansion we argue that the model always scales to its strong-coupling limit and displays spin Peierls or valence-bond-solid order. The structure of the theory in this strong-coupling limit bears a remarkable resemblance to a fermionic large-N limit of the nearest-neighbor SU(N) antiferromagnet.
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 generalize the SU(N=2) S=1/2 square-lattice quantum magnet with nearest-neighbor antiferromagnetic coupling (J(1)) and next-nearest-neighbor ferromagnetic coupling (J(2)) to arbitrary N. For all N>4, the ground state has valence-bond-solid order for J(2)=0 and Néel order for J(2)/J(1)≫1, allowing us access to the transition between these types of states for large N. Using quantum Monte Carlo simulations, we show that both order parameters vanish at a single quantum-critical point, whose universal exponents for large enough N (here up to N=12) approach the values obtained in a 1/N expansion of the noncompact CP(N-1) field theory. These results lend strong support to the deconfined quantum-criticality theory of the Néel-valence-bond-solid transition.
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.
We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) methods to study the phases and transitions displayed by a class of sign-free designer Hamiltonians for SU(N)\mathrm{SU}(N)SU(N) analogs of spin S=1S=1S=1 quantum antiferromagnets on the square lattice. The SU(N)\mathrm{SU}(N)SU(N) spins are generators of the single-row two-column representation (complex conjugate of single-row two-column representation) on AAA (BBB) sublattices of the square lattice, and the Hamiltonian is designed to explore the competition between the nearest-neighbor antiferromagnetic exchange couplings JJJ and four-spin interactions QQQ that favor a plaquette-ordered valence bond solid (p-VBS) ground state. We find that this state is indeed established at large Q/JQ/JQ/J for all N>3N>3N>3. For $3<N\ensuremath{\le}9$, the ground state exhibits a direct first-order quantum phase transition from a small-Q/JQ/JQ/J N\'eel ordered antiferromagnetic state to this large-Q/JQ/JQ/J p-VBS state. The ground state at Q/J=0Q/J=0Q/J=0 for $N\ensuremath{\ge}10$ has been previously reported to be a valence bond nematic state, dubbed the Haldane nematic in recent literature. For small nonzero Q/JQ/JQ/J and $N\ensuremath{\ge}10$, we additionally find an unusual intermediate state in which the four-site plaquette energy has a Bragg peak at wave vector $(\ensuremath{\pi},\ensuremath{\pi})$ with no accompanying Bragg peaks at wave vectors $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$. This $(\ensuremath{\pi},\ensuremath{\pi})$ state also appears to be metastable at Q/J=0Q/J=0Q/J=0, as evidenced by low-temperature histograms of the Haldane nematic and $(\ensuremath{\pi},\ensuremath{\pi})$ order parameters. Deep in the p-VBS phase of the ground state phase diagram, we find the temperature-driven melting of the p-VBS order is in the Ashkin-Teller universality class. In this regime, we identify an interesting signature of Ashkin-Teller criticality in four-site plaquette singlet correlations at wave vector $(\ensuremath{\pi},\ensuremath{\pi})$; this is in addition to the expected critical fluctuations of the conventional p-VBS order parameter at wave vectors $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$.