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arXiv:1304.0392 · Physical Review B

Berry phase investigation of spin- S ladders

Natalia Chepiga, Frédéric Michaud, Frédéric Mila

Published Nov 19, 2013• 19 citations
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Abstract

We investigate the properties of antiferromagnetic spin-$S$ ladders with the help of local Berry phases defined by imposing a twist on one or a few local bonds. In gapped systems with time-reversal symmetry, these Berry phases are quantized, hence able in principle to characterize different phases. In the case of a fully frustrated ladder where the total spin on a rung is a conserved quantity that changes abruptly upon increasing the rung coupling, we show that two Berry phases are relevant to detect such phase transitions: the rung Berry phase defined by imposing a twist on one rung coupling, and the twist Berry phase defined by twisting the boundary conditions along the legs. In the case of nonfrustrated ladders, we have followed the fate of both Berry phases when interpolating between standard ladders and dimerized spin chains, with the surprising conclusion that, at least far enough from dimerized chains, they define different domains in parameter space. A careful investigation of the spin gap and of edge states shows that a change of twist Berry phase is associated with a quantum phase transition at which the bulk gap closes, and at which, with appropriate boundary conditions, edge states appear or disappear, while a change of rung Berry phase is not necessarily associated with a quantum phase transition. The difference is particularly acute for regular ladders, in which the twist Berry phase does not change at all upon increasing the rung coupling from zero to infinity while the rung Berry phase changes $2S$ times. By analogy with the fully frustrated ladder, these changes are interpreted as crossovers between domains in which the rungs are in different states of total spin from 0 in the strong rung limit to $2S$ in the weak rung limit. This interpretation is further supported by the observation that these crossovers turn into real phase transitions as a function of rung coupling if one rung is strongly ferromagnetic, or equivalently if one rung is replaced by a spin $2S$ impurity.

DOI
10.1103/physrevb.88.184418

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We investigate the properties of antiferromagnetic spin-SSS ladders with the help of local Berry phases defined by imposing a twist on one or a few local bonds. In gapped systems with time-reversal symmetry, these Berry phases are quantized, hence able in principle to characterize different phases. In the case of a fully frustrated ladder where the total spin on a rung is a conserved quantity that changes abruptly upon increasing the rung coupling, we show that two Berry phases are relevant to detect such phase transitions: the rung Berry phase defined by imposing a twist on one rung coupling, and the twist Berry phase defined by twisting the boundary conditions along the legs. In the case of nonfrustrated ladders, we have followed the fate of both Berry phases when interpolating between standard ladders and dimerized spin chains, with the surprising conclusion that, at least far enough from dimerized chains, they define different domains in parameter space. A careful investigation of the spin gap and of edge states shows that a change of twist Berry phase is associated with a quantum phase transition at which the bulk gap closes, and at which, with appropriate boundary conditions, edge states appear or disappear, while a change of rung Berry phase is not necessarily associated with a quantum phase transition. The difference is particularly acute for regular ladders, in which the twist Berry phase does not change at all upon increasing the rung coupling from zero to infinity while the rung Berry phase changes 2S2S2S times. By analogy with the fully frustrated ladder, these changes are interpreted as crossovers between domains in which the rungs are in different states of total spin from 0 in the strong rung limit to 2S2S2S in the weak rung limit. This interpretation is further supported by the observation that these crossovers turn into real phase transitions as a function of rung coupling if one rung is strongly ferromagnetic, or equivalently if one rung is replaced by a spin 2S2S2S impurity.
We show that spin-S ladders undergo, for antiferromagnetic rung coupling, a series of 4S topological phase transitions when the ratio of the leg to rung coupling changes from strongly antiferromagnetic to strongly ferromagnetic. These phase transitions are characterized by a change of Berry phase on the rungs, and, with appropriate boundary conditions, by the appearance and disappearance of edge states without the bulk gap ever closing. The various phases can be distinguished by the number of valence bond singlets on the rungs, which decreases from 2S for strong rung coupling to 0 for dominant leg coupling. In particular, the isotropic spin-3/2 ladder with equal leg and rung coupling is explicitly shown to be in the valence bond solid phase with one singlet per bond.
Motivated by near-term experiments with ultracold alkaline-earth atoms confined to optical lattices, we establish numerically and analytically the phase diagram of two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladders. Two-leg ladders provide a rich and highly nontrivial extension of the single chain case on the way towards the relatively little explored two-dimensional situation. Focusing on the experimentally relevant limit of one fermion per site, antiferromagnetic exchange interactions, and $2\ensuremath{\le}N\ensuremath{\le}6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}$, strongly differ for even versus odd NNN. For even N=4N=4N=4 and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of N/2N/2N/2 rungs. For odd N=3N=3N=3 and 5, we find surprisingly rich phase diagrams exhibiting three distinct phases. For weak rung coupling, we obtain a VBC with a spatial period of NNN rungs, whereas for strong coupling we obtain a critical phase related to the case of a single chain. In addition, we encounter intermediate phases for odd NNN, albeit of a different nature for N=3N=3N=3 as compared to N=5N=5N=5. For N=3N=3N=3, we find a novel gapless intermediate phase with ${J}_{\ensuremath{\perp}}$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For N=5N=5N=5, there are strong indications for a narrow potentially gapped intermediate phase, whose nature is not entirely clear. Our results are based on (i) field theoretical techniques, (ii) qualitative symmetry considerations, and (iii) large-scale density matrix renormalization group (DMRG) simulations keeping beyond a million of states by fully exploiting and thus preserving the SU(N)\mathrm{SU}(N)SU(N) symmetry.
We study the fractionally quantized ZNZ_NZN​ Berry phase, γN=0,2πN,4πN,…,2(N−1)πNγ_N=0, {2π\over N}, {4π\over N},\ldots, {2(N-1)π\over N}γN​=0,N2π​,N4π​,…,N2(N−1)π​, to characterize local NNN-mer spin structures at magnetization plateaux in spin-1/2 Heisenberg multimer (NNN-mer) models, i.e., highly frustrated NNN-leg ladder models, which are generalizations of an orthogonal dimer chain and have exact ground states in the strong multimer coupling region. We demonstrate that all NNN types of Berry phases, which characterize magnetization-plateau phases, appear in a magnetic phase diagram when N=2N=2N=2 and 444. We show that magnetization plateau with magnetization ⟨m⟩\langle m\rangle⟨m⟩ and DDD-fold degenerated states has γN=π(⟨m⟩−1)Dγ_N=π(\langle m\rangle-1) DγN​=π(⟨m⟩−1)D, except for the Haldane phase with γN=0γ_N=0γN​=0. In addition, we find that a complementary ZNZ_NZN​ Berry phase becomes non-zero in the S=N/2S=N/2S=N/2 Haldane phase for N=2N=2N=2 and 4. Because the exact quantization of the ZNZ_NZN​ Berry phases is protected by the translational (or rotational) symmetry along the rung direction, the ZNZ_NZN​ Berry phase has the potential to be applied for a wide class of magnetization plateaux in coupled multimer systems.
A spin-1/2 frustrated two-leg ladder with four-spin exchange interaction is studied by quantized Berry phases. We found that the Berry phase successfully characterizes the Haldane phase in addition to the rung-singlet phase, and the dominant vector-chirality phase. The Hamiltonian of the Haldane phase is topologically identical to the S=1 antiferromagnetic Heisenberg chain. Decoupled models connected to the dominant vector-chirality phase revealed that the local object identified by the non-trivial (πππ) Berry phase is the direct product of two diagonal singlets.
Recently, by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators has been given (Hatsugai 2006 Preprint cond-mat/0603230). It requires that the ground state is gapped and is invariant under some anti-unitary operation. A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations. As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU (2) twist only at a specific link and define the Berry connection by the derivative. Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link. We numerically apply this scheme for several spin liquids and show its physical validity. For example, it implies that the Haldane phase of the S = 1 chains is characterized by uniform π quantized Berry phases.
A spin-12\frac{1}{2}21​ two-leg ladder with four-spin ring exchange is studied by quantized Berry phases, used as local-order parameters. Reflecting local objects, nontrivial $(\ensuremath{\pi})$ Berry phase is founded on a rung for the rung-singlet phase and on a plaquette for the vector-chiral phase. Since the quantized Berry phase is topologically invariant for gapped systems with the time-reversal symmetry, topologically identical models can be obtained by the adiabatic modification. The rung-singlet phase is adiabatically connected to a decoupled rung-singlet model and the vector-chiral phase is connected to a decoupled vector-chiral model. Decoupled models reveal that the local objects are a local singlet and a plaquette singlet, respectively.