Strong-coupling phases of the spin-orbit-coupled spin-1 Bose-Hubbard chain: Odd-integer Mott lobes and helical magnetic phases

Abstract

We study the odd integer filled Mott phases of a spin-1 Bose-Hubbard chain and determine their fate in the presence of a Raman induced spin-orbit coupling which has been achieved in ultracold atomic gases; this system is described by a quantum spin-1 chain with a spiral magnetic field. The spiral magnetic field initially induces helical order with either ferromagnetic or dimer order parameters, giving rise to a spiral paramagnet at large field. The spiral ferromagnet-to-paramagnet phase transition is in a novel universality class, with critical exponents associated with the divergence of the correlation length ν ≈ 2/3 and the order parameter susceptibility γ ≈ 1/2. We solve the effective spin model exactly using the density matrix renormalization group, and compare with both a large-S classical solution and a phenomenological Landau theory. We discuss how these exotic bosonic magnetic phases can be produced and probed in ultracold atomic experiments in optical lattices.Strongly-correlated quantum spin chains are an interacting many-body system that have been an instrumental platform to develop an understanding of topological properties [1], Berry phase effects [2], and quantum phase transitions [3]. One of the paradigmatic theoretical models in this context is the spin-1 bilinear-biquadratic Heisenberg chain [2-4](J is the unit of energy) which supports several conceptually important phases and phase transitions [5][6][7][8][9][10][11][12][13][14]. The physics associated with the Hamiltonian in Eq. (1) is not only theoretically tantalizing, but is also directly accessible to experiments. For example, the antiferromagnetic spin-1 chain (cos θ > 0) has been probed experimentally in insulating quantum magnets [15][16][17] that possess wellisolated, quasi-one-dimensional chains of S = 1 local moments. Interestingly, ultracold atomic gases are a natural platform for realizing the complementary ferromagnetic spin-1 chain (cos θ < 0), where the system is an effective description of the Mott insulating spin-1 Bose-Hubbard model [18][19][20][21][22][23]. Solid state experiments always contain at least some disorder that breaks the spin chain into segments [16], while the cold atomic gas setting is essentially pristine with no disorder in principle, so that the theoretical model defined by Eq. (1) (as well as its generalizations) can be studied directly. Until recently, it has been difficult to realize magnetic ordering in ultra-cold atomic gases [24,25], and a crucial element of physics is developing a theoretical understanding of the available strongly correlated phases. A virtue of the flexibility and the precise control in atomic and molecular experiments is that various physical effects can be engineered simulating desired features of solid-state systems despite the vastly different setting (i.e. atoms versus solids). Along these lines, recent experiments on cold atomic gases have demonstrated that, despite the atoms being electrically neutral, spin orbit coupling (SOC) can be engineered using two counter propagating...

Quantum spiral spin-tensor magnetism

“…ii ) We obtain and characterize the ground-state phase diagram of such a spin-1 chain numerically using the density-matrix renormalization group (DMRG) method. We find that both the spin-vector arrow (direction and length), the spin-tensor ellipsoid (direction and size) and their relative orientations are spiral along the chain, in contrast with previous studies for spiral spin-vector phases [34] or 2D nematic phases [41,42].…”

“…We restrict to spin-1 magnetic orders, which may be described by three crucial elements: i) spin-vector (represented by an arrow), ii ) spin-tensor (represented by an ellipsoid), and iii) the relative orientation between the arrow and ellipsoid. In previously studied spiral spinvector order [34], arrow length, ellipsoid size, and their relative orientations are uniformly fixed across the lattice chain, while the 2D nematic phase in a frustrated lattice [41,42] possesses vanishing spin-vector arrow and fixed ellipsoid size and directions.…”

“…Quantum magnetism with large spins, such as spin-1, has also received considerable attention in recent years, where the large spin could originate from, for instance, intrinsic orbital degrees of electrons or pseudo-spins defined by hyperfine states of cold atoms [24][25][26][27][28][29][30][31][32][33][34]. Mathematically, a full description of a large spin (≥ 1) involves not only rank-1 spin-vectors, but also high-rank spin-tensors, therefore it is expected that the resulting quantum magnetism may possess both spin-vector and tensor orders.…”

“…Mathematically, a full description of a large spin (≥ 1) involves not only rank-1 spin-vectors, but also high-rank spin-tensors, therefore it is expected that the resulting quantum magnetism may possess both spin-vector and tensor orders. So far, spin-vector magnetism of a strongly correlated spin-1 chain has been extensively studied, where the competition between spin interaction and Zeeman field (either uniform or spiral along the chain) leads to rich phase diagrams [34][35][36][37][38][39][40]. Certain nematic magnetic orders of spin-tensors (with vanishing spin-vector) have been investigated in 2D geometrically frustrated (e.g., triangle) lattices [41][42][43][44][45][46][47][48][49].…”

Emergent gauge field and the Lifshitz transition of spin-orbit coupled bosons in one dimension

“…The spin part of the Lagrangian () is now spatially uniform, and includes an isotropic ferromagnetic exchange, a Dzyaloshinskii-Moriya term, and an easy-axis anisotropy along , along with a uniform Zeeman field along . Similar 1D spin models have been studied previously in this context, though typically in the Mott-insulating limit on a lattice, where there is no back-action on the spin from charge fluctuations 23–28 . The charge part of the Lagrangian () now depends explicitly on spin from a dynamical vector potential αs 3 .…”