Pseudogap phase as fluctuating pair density wave
Zheng-Yuan Yue Zheng-Tao Xu Shuo Yang Zheng-Cheng Gu

APS/123-QED




These authors contribute equally.
Department of Physics, The Chinese University of Hong Kong, Sha Tin, New Territories, Hong Kong, China



These authors contribute equally.
State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China


shuoyang@tsinghua.edu.cn
State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
Frontier Science Center for Quantum Information, Beijing 100084, China
Hefei National Laboratory, Hefei 230088, China


zcgu@phy.cuhk.edu.hk
Department of Physics, The Chinese University of Hong Kong, Sha Tin, New Territories, Hong Kong, China






    The physical nature of pseudogap phase is one of the most important and intriguing problems towards understanding the key mechanism of high temperature superconductivity in cuprates.
    Theoretically, the square-lattice t-J model is widely believed to be the simplest toy model that captures the essential physics of cuprate superconductors. 
    We employ the Grassmann tensor product state approach to investigate uniform states in the underdoped (δ≲ 0.1) region. 
    In addition to the previously known uniform d-wave state, we discover a strongly fluctuating pair density wave (PDW) state with wave vector Q = (π, π). 
    This fluctuating PDW state weakly breaks the C_4 rotational symmetry of the square lattice and has a lower or comparable energy to the d-wave state (depending on doping and the t/J ratio), making it a promising candidate state for describing the pseudogap phase. 


Pseudogap phase as fluctuating pair density wave
    Zheng-Cheng Gu
    April 29, 2024
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*Introduction — 
High-temperature superconductivity in cuprate materials remains a challenging field in condensed matter physics, even decades after its discovery <cit.>. Cuprates are Mott insulators at half-filling, but they become superconducting upon doping <cit.>. 
The underdoped region of the cuprate phase diagram is particularly intriguing due to the emergence of the pseudogap below a certain temperature T^* <cit.>. It is now shown to be associated with the breaking of various symmetries, including the C_4 rotational symmetry of the CuO_2 planes <cit.>. 
Interestingly, the model compound HgBa_2CuO_4+δ (Hg1201, which has a tetragonal structure and only one CuO_2 plane per unit cell) <cit.> exhibits nematicity below T^* along the diagonal directions [110] or [11̅0] <cit.>. 
Such a rotational symmetry breaking is very different from the charge-density-wave (CDW) phase, which breaks translational symmetry. 
In the pseudogap phase, there is growing interest in the possible existence of a pair density wave (PDW) order <cit.>. PDW is characterized by the formation of Cooper pairs that carry a nonzero total momentum Q, resulting in a spatial modulation of the superconducting order parameter. Numerous experiments support the presence of PDW in cuprate superconductors <cit.>. However, theoretical understanding of the PDW state is based primarily on phenomenological models, and ongoing debates persist regarding its microscopic foundations and its relation to the pseudogap <cit.>.

In this work, we use the Grassmann tensor product state (TPS) method <cit.> (which can also be equivalently formulated in fermionic tensors <cit.>) to study the t-J model on a square lattice <cit.>, which is believed to capture the essential physics of the CuO_2 plane. 
Previous numerical studies using techniques such as Density Matrix Renormalization Group (DMRG) <cit.>, infinite Projected Entangled-Pair States (iPEPS) <cit.>, and Variational Monte Carlo (VMC), which utilizes a d-wave slave-boson projective wave function <cit.> have revealed various competing orders, including a d-wave superconducting (SC) order that coexists with antiferromagnetic (AFM) order, as well as partially filled stripe orders with varying periods. We report the discovery of a uniform and fluctuating PDW state with Q = (π, π) (coexisting with antiferromagnetic order in the doping range δ≲ 0.1). 
Important features of this state include: (a) the nearest-neighbor (NN) singlet pairing vanishes as the bond dimension D →∞, implying strong quantum fluctuation; (b) the hole density, staggered magnetization, and the NN singlet pairing magnitude are uniform; and (c) the C_4 rotational symmetry is weakly broken to mere reflections about the diagonal next-nearest-neighbor (NNN) bonds. These contrast sharply with previously found PDW states, which are essentially spatially modulated d-wave states (also called "antiphase" d-wave states) with a smaller but nonzero wave vector and a static singlet pairing pattern <cit.>. 
Our fluctuating PDW state has a 

lower or comparable energy to the uniform d-wave state at low doping (depending on doping and the t/J ratio).

These features suggest that the pseudogap phase may actually be a distinct quantum state, which strongly competes with the d-wave superconducting state at low doping.




  
*Model and methods —
The Hamiltonian of the well-known t-J model reads

    H = -t∑_⟨ ij⟩ , σ(
            c̃_iσ^†c̃_jσ + h.c.
            ) + J∑_⟨ ij⟩(
            𝐒_i ·𝐒_j
            - 1/4n̂_i n̂_j
        ),

Here ⟨ij|$⟩ sums over NN bonds.
    c̃_iσ 
    = c_iσ(
        1 - c^†_iσ̅
        c_iσ̅
    )
is the electron operator defined in the no-double occupancy subspace.
    n̂_i = ∑_σc_iσ^†c_iσ
is the electron number operator.
    𝐒_i = (1/2) ∑_α,β
    c^†_iα
    σ_αβ c_iβ
is the spin-1/2 operator, withσbeing Pauli matrices. 
The ground state|ψ⟩is taken as a two-dimensional translational invariant Grassmann TPS <cit.>, which is divided into two sub-latticesAandB, each generated by local Grassmann tensorsT_AorT_B. 
There are also Schmidt weightsΛon the four types of bonds (named1to4), which are diagonal dual Grassmann matrices.



    |ψ⟩ = 
        < g r a p h i c s >
    .

The arrow on each bond indicates the order of the two dual Grassmann numbers in the Grassmann metric used by the contraction <cit.>. 
All virtual indices have the same total dimensionDand the bosonic (even-parity) subspace dimensionD_e. 
The physical index is3dimensional, labeled as|↑⟩(spin-up),|↓⟩(spin-down) and|0⟩(empty site). 
The double occupancy state is excluded. 
We adopt the slave fermion point of view <cit.> and assign|↑⟩, |↓⟩to the bosonic subspace. 


Starting from an initialization of|ψ⟩, the ground state is obtained using the simple update (SU) method, which is an imaginary time evolution algorithm <cit.>.
Since nonuniform states (e.g. with stripe order or charge density wave order) break the translational symmetry, they are irrelevant to pseudogap physics. Furthermore, a previous study with the full update (FU) scheme <cit.> suggests that the energy of uniform states may eventually be comparable to that of nonuniform states asDincreases, especially with smaller doping. Therefore, we focus on uniform states, which are obtained by averaging the Schmidt weights after updating them all once. 
Dopingδis controlled by adding a chemical potential term-μ∑_i n̂_i(μ> 0) to the Hamiltonian, and a largerμleads to smaller doping. 
The evolution time step slowly decreases to ensure its convergence. 
The update stops when the change in the Schmidt weight is sufficiently small. 
We measure the resulting ground state with the variational uniform matrix product state (VUMPS) method <cit.>.
The measurement result converges when the dimension of the MPS virtual boundary bond isχ≳4D<cit.>.