A strongly spin-orbital coupled systems could be in a magnetic ordered phase at zero field. However, a Zeeman field could drive it into different quantum or topological phases. In this work, starting from general symmetry principle, we construct vari
ous effective actions to study all these quantum phases and phase transitions which take different forms depending on the condensation momenta are commensurate or in-commensurate. We not only recover all these quantum phases and their excitations achieved by the microscopic calculations, but also discover several novel classes of quantum phase transitions with dynamic exponents $ z=1, z=2 $ and anisotropic ones $ (z_x=3/2, z_y=3) $ respectively. We determine the relations between the quantum spin and the order parameters of the effective actions which display rich spin-orbital structures. We find a new type of dangerously irrelevant operator we name type-II, in distinction from the known one we name type-I. We explore a new phenomena called order parameter fractionization where one complex order parameter split into two which is different than quantum spin fractionization into a spinon and a $ Z_2 $ flux. Finite temperature transitions are presented. The dynamic spin-spin correlation functions are evaluated. Thermal Hall conductivities are discussed. The cases with the $ U(1)_{soc} $ symmetry explicitly broken are briefly outlined. In view of recent experimental advances in generating 2d SOC for cold atoms in optical lattices, these new many-body phenomena can be explored in the near future cold atom experiments. Implications to various SOC materials such as MnSi, Fe$_{0.5}$Co$_{0.5}$Si, especially 4d Kitaev materials $alpha$-RuCl$_3$ in a Zeeman field are outlined.
We report that a possible Z2 quantum spin liquid (QSL) can be observed in a new class of frustrated system: spinor bosons subject to a pi flux in a square lattice. We construct a new class of Ginsburg-Landau (GL) type of effective action to classify
possible quantum or topological phases at any coupling strengths. It can be used to reproduce the frustrated SF with the 4 sublattice $ 90^{circ} $ coplanar spin structure plus its excitations in the weak coupling limit and the FM Mott plus its excitations in the strong coupling limit achieved in our previous work. It also establishes deep and intrinsic connections between the GL effective action and the order from quantum disorder (OFQD) phenomena in the weak coupling limit. Most importantly, it predicts two possible new phases at intermediate couplings: a FM SF phase or a frustrated magnetic Mott phase. We argue that the latter one is more likely and melts into a $ Z_2 $ quantum spin liquid (QSL) phase. If the heating issue can be under a reasonable control at intermediate couplings $ U/t sim 1 $, the topological order of the $ Z_2 $ QSL maybe uniquely probed by the current cold atom or photonic experimental techniques.
We develop a systematic and unified random matrix theory to classify Sachdev-Ye-Kitaev (SYK) and supersymmetric (SUSY) SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even $q$- and SUSY SYK with odd
$q$-body interaction, $N$ even or odd number of Majorana fermions are put on the same footing in the minimal Hilbert space, $Npmod 8$ and $qpmod 4$ double Bott periodicity are identified. Exact diagonalizations are performed to study both the bulk energy level statistics and hard edge behaviours. A new moment ratio of the smallest positive eigenvalue is introduced to determine hard edge index efficiently. Excellent agreements between the ED results and the symmetry classifications are demonstrated. Our complete and systematic methods can be transformed to map out more complicated periodic tables of SYK models with more degree of freedoms, tensor models and symmetry protected topological phases. Possible classification of charge neutral quantum black holes are hinted.
We report a new many body phenomena called Nearly order from quantum disorder phenomena (NOFQD). We demonstrate the NOFQD in the experimentally realized weakly interacting Quantum Anomalous Hall system of spinor bosons in an optical lattice. We esta
blish intrinsic connections between the phenomenological GL theory and the microscopic calculations on the effective potential. Connections with the bilayer quantum Hall system with a total filling factor $ u_T=1 $ are made. Some insightful analogy with $ NAdS_2/NCFT_1 $ ( where $ N $ also means nearly ) correspondence in the context of Sachdev-Ye-Kitaev models are hinted. Two types of OFQDs are classified, one response trivially, another non-trivially to a small deformation to the Hamiltonian leading to NOFQD. The NOFQD can be detected in the current cold atom bosonic quantum anomalous Hall experiments and may also appear in many other frustrated systems.
The random matrix theory (RMT) can be used to classify both topological phases of matter and quantum chaos. We develop a systematic and transformative RMT to classify the quantum chaos in the colored Sachdev-Ye-Kitaev (SYK) model first introduced by
Gross and Rosenhaus. Here we focus on the 2-colored case and 4-colored case with balanced number of Majorana fermion $N$. By identifying the maximal symmetries, the independent parity conservation sectors, the minimum (irreducible) Hilbert space, and especially the relevant anti-unitary and unitary operators, we show that the color degree of freedoms lead to novel quantum chaotic behaviours. When $N$ is odd, different symmetry operators need to be constructed to make the classifications complete. The 2-colored case only show 3-fold Wigner-Dyson way, and the 4-colored case show 10-fold generalized Wigner-Dyson way which may also have non-trivial edge exponents. We also study 2- and 4-colored hybrid SYK models which display many salient quantum chaotic features hidden in the corresponding pure SYK models. These features motivate us to develop a systematic RMT to study the energy level statistics of 2 or 4 un-correlated random matrix ensembles. The exact diagonalizations are performed to study both the bulk energy level statistics and the edge exponents and find excellent agreements with our exact maximal symmetry classifications. Our complete and systematic methods can be easily extended to study the generic imbalanced cases. They may be transferred to the classifications of colored tensor models, quantum chromodynamics with pairings across different colors, quantum black holes and interacting symmetry protected (or enriched) topological phases.
We investigate chaotic to integrable transition in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or $(q=2)^2$ SYK with an interaction $ sqrt{K} $ in type-II. T
hese models include hybrid Majorana fermion, complex fermion and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way by Random Matrix Theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $ N=infty $, $ 1/N $ expansions at a finite $ N $ and the CIT characterized by the RMT at a finite $ N $. The corresponding distinctions between the edge states and bulk states in Fock spaces are studied. Some future perspectives, especially the failure of the Zamoloddchikovs c-theorem in 1d CFT are outlined.
We study possible superfluid states of the Rashba spin-orbit coupled (SOC) spinor bosons with the spin anisotropic interaction $ lambda $ hopping in a square lattice. The frustrations from the non-abelian flux due to the SOC leads to novel spin-bond
correlated superfluids. By using a recently developed systematic order from quantum disorder analysis, we not only determine the true quantum ground state, but also evaluate the mass gap in the spin sector at $ lambda < 1 $, especially compute the the excitation spectrum of the Goldstone mode in the spin sector at $ lambda=1 $ which would be quadratic without the analysis. The analysis also leads to different critical exponents on the two sides of the 2nd order transition driven by a roton touchdown at $ lambda=1 $. The intimate analogy at $ lambda=1 $ with the charge neutral Goldstone mode in the pseudo-spin sector in the Bilayer quantum Hall systems at the total filling factor $ u_T=1 $ are stressed. The experimental implications and detections of these novel phenomena in cold atoms loaded on a optical lattice are presented.
We study possible many body phenomena in the Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice. There are various novel spin-bond correlated superfluids (SF) and quantum or topological phase transitions among these
SF phases. One transition is a first order one driven by roton droppings ( but with non-zero gaps $ Delta_R $ ) tuned by the Zeeman field $ h $. Another is a second order bosonic Lifshitz transition with the dynamic exponents $ z_x=z_y=2 $ and an accompanying $ [C_4 times C_4]_D $ symmetry breaking. It is driven by the softening of the superfluid Goldstone mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping strength. The two phase boundaries meet at a topological tri-critical (TT) point which separates the $ h=0 $ line into two SF phases with $ N=2 $ and $ N=4 $ condensation momenta respectively. At the $ h=0 $ line where the system has an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number of classically degenerate family of states on both sides. We perform a systematic order from quantum disorder analysis to find the quantum ground states, also calculate the roton gaps $ Delta_R $ generated by the order from disorder mechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases have the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry and break the $ [C_4 times C_4]_D $ symmetry, so they be distinguished only by the different topology of the BEC condensation momenta instead of by any differences in the symmetry breaking patterns. All these novel quantum or topological phenomena can be probed in the recent experimentally realized weakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {sl et.al}, Science 354, 83-88 (2016).
Inspired by recent experimental advances to generate Abelian flux for neutral cold atoms and photons moving in a lattice, we investigate the possible effects of the $ pi $ flux through a unit cell in the pseudo-spin 1/2 spinor boson Hubbard model in
a square lattice. We find that the $ pi $ flux induces a dramatic interplay between the charge and the spin which leads to a frustrated superfluid. We develop a new and systematic order from quantum disorder analysis to determine not only the true quantum ground state, but also the excitation spectrum. The superfluid ground state has a 4 sublattice $ 90^{circ} $ coplanar spin structure which supports 4 linear gapless modes with 3 different velocities. We speculate the transition from the weak coupling frustrated SF to the strong coupling Ferromagnetic Mott state to be in a new universality class of non-Ginsburg Landau type. These novel phenomena may be observed in these recent cold atom and photonic experiments.
It remains an open problem if there are universal scaling functions across a topological quantum phase transition (TPT) without an order parameter, but with extended Fermi surfaces (FS ). Here, we study a simple system of fermions hopping in a cubic
lattice subject to a Weyl type spin-orbit coupling (SOC). As one tunes the SOC parameter at the half filling, the system displays Weyl fermions and also various TPT due to the collision of particle-particle or hole-hole Weyl Fermi Surface (WFS). At the zero temperature, the TPT is found to be a third order one whose critical exponent is determined. We derive the scaling functions of the specific heat, compressibility and magnetic susceptibilities. In contrast to all the previous cases in quantum or topological transitions, although the leading terms are non-universal and cutoff dependent, the sub-leading terms satisfy universal scaling relations. The sub-leading scaling leads to the topological depletions (TD) which show non-analytic and non-Fermi liquid corrections in the quantum critical regime, can be easily distinguished from the analytic leading terms and detected experimentally. One can also form a topological Wilson ratio from the TD of two conserved quantities such as the specific heat and the compressibility. As a byproduct, we also find Type II Weyl fermions appearing as the TPT due to the collision of the extended particle-hole WFS. Experimental realizations and detections in cold atom systems and materials with SOC are discussed.
