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Motivated by recent progress on field-induced phase transitions inquasi-one-dimensional quantum antiferromagnets, we study the phase diagram of$S=1/2$ antiferromagnetic Heisenberg chains with Ising anisotropic interchaincouplings under a longitudinal magnetic field via large-scale quantum MonteCarlo simulations. The interchain interactions is shown to enhance longitudinalspin correlations to stabilize an incommensurate longitudinal spin density waveorder at low temperatures. With increasing field the ground state changes to acanted antiferromagnetic order until the magnetization fully saturates above aquantum critical point controlled by the $(3+2)$D XY universality. Increasingtemperature in the quantum critical regime the system experiences a fascinatingdimension crossover to a universal Tomonaga-Luttinger liquid. The calculatedNMR relaxation rate $1/T_1$ indicates this Luttinger liquid behavior survives abroad field and temperature regime. Our results determine the global phasediagram and quantitative features of quantum criticality of a general model forquasi-one-dimensional spin chain compounds, and thus lay down a concrete groundto the study on these materials.

Phase diagram and quantum criticality of Heisenberg spin chains with Ising-like interchain couplings -- Implication to YbAlO$_3$