Pair Density Wave in the Doped t−J Model with Ring Exchange on a Triangular Lattice

Abstract

In our previous work [Phys. Rev. Lett. 121, 046401 (2018)], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with fourspin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.A pair density wave (PDW) is a superconducting state in which Cooper pairs have finite momentum. The first example of PDW is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [1, 2] which can arise in superconductors in strong magnetic fields when the Fermi surface is split by Zeeman effect. Recently PDW has come into prominence in the context of underdoped Cuprate superconductors. The striped PDW was proposed as a mechanism for dynamically inter-layer decoupling observed in 1/8 hole doped La 2−x Ba x CuO 4 [3-6]. One of us has proposed fluctuating bi-directional PDW as the "mother state" that is responsible for many of the anomalous properties of the pseudo-gap regime [7]. Experimentally a direct observation of PDW has been made via local Cooper pair tunnelling in Bi 2 Sr 2 CaCu 2 O 8+x [8]. They found Cooper pair density modulation with period 4a 0 where a 0 is the length of unit cell, and the magnitude of the modulation is five percent of the uniform pairing background. This may be interpreted as a subsidiary PDW being generated by the period 4a 0 charge order together with the uniform d-wave pairing. In this sense the recent report [9] of short range period 8a 0 charge order in the vicinity of the vortex core is even more exciting, because it may be the signature of a hidden period 8a 0 PDW [10,11].Theoretically, there are very few microscopic models which are shown to have PDW ground states. Berg et al.[12] studied a Kondo-Heisenberg model with 1D electron gas coupled to a spin chain. They found a spin gapped phase with PDW correlations oscillating with period 2a 0 , which matches the period of the ordering tendency of the spin chain. An extended two-leg Hubbard-Heisenberg model is also found to have a spin gapped phase with a PDW [13]. In all these examples, the PDW is commensurate and can either be interpreted as a composite order between short range spin order with the same commensurate period and another short range triplet pairing order, or requires the specially tuned symmetry of π flux through the ladder plaquette. Dodaro et al. [14] searched for PDW in a more standard t-J model for doped cuprates, but they did not find any evidence of PDW ordering ev...

Gapless spin liquid and pair density wave of the Hubbard model on three-leg triangular cylinders

“…[13]. The signature of the PDW ordering was also observed in t-J model with four-spin ring exchange interaction on the four-leg triangular cylinder [14] and the t-J-like extension of the Kitaev model on the three-leg honeycomb cylinder. [17] However, there is no evidence of the PDW state found in the standard Hubbard model on systems wider than a two-leg ladder.…”

“…Although theoretically much is known about the properties of the PDW state, its realization in microscopic models remains still very few. [11][12][13][14][15][16] These include the one-dimensional (1D) Kondo-Heisenberg model [11], extended two-leg Hubbard-Heisenberg model [12] and an extended Hubbard model with a staggered spindependent magnetic flux per plaquette on a three-leg triangular lattice. [13].…”

Charge, bond, and pair density wave orders in a strongly correlated system

“…The importance of detecting the PDW state is underlined by the several exotic proposals of the celebrated pseudogap state that hinges on the quantum disordering [26] or fractionalization of the PDW state [27,28]. However, it is extremely difficult to stabilize such PDW sates as the ground state of a microscopic Hamiltonian [29][30][31][32][33].…”

Strong Coupling Limit of the Holstein-Hubbard Model

“…In one-dimensional systems -which can be reliably treated using DMRG -the closest one can get to such order is PDW quasi-long-range order with a divergent susceptibility. However, even this has only been seen to date for the case of commensurability-two PDW order in Heisenberg-Kondo chains [17,18]; other reported DMRG sightings [14,19,20] have found PDW correlations that fall off fast enough to yield only finite superconducting susceptibility.…”

“…However, it has been conjectured that a PDW could also arise from a strong-coupling mechanism and could be responsible for the dynamical layer decoupling phenomena observed in certain cuprate materials [4][5][6]. While PDW states have been shown to be energetically competitive in certain mean-field and variational calculations [7][8][9][10], and evidence has been sought in numerical studies of generalized t-J [4,[11][12][13][14] and Hubbard models [15,16], PDW long-range order has not been established in any controlled calculation in two or higher dimensions. In one-dimensional systems -which can be reliably treated using DMRG -the closest one can get to such order is PDW quasi-long-range order with a divergent susceptibility.…”