Complementary pair-density-wave and d-wave-checkerboard

Physics

Physics Research Publications

Purdue University Year

Complementary pair-density-wave and

d-wave-checkerboard orderings in

high-temperature superconductors

K. Seo H. D. Chen

J. P. Hu

This paper is posted at Purdue e-Pubs.

http://docs.lib.purdue.edu/physics articles/906

Complementary pair-density-wave and d-wave-checkerboard orderings in high-temperaturesuperconductors

Kangjun Seo,1 Han-Dong Chen,2 and Jiangping Hu1

1Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

�Received 27 August 2007; revised manuscript received 29 June 2008; published 17 September 2008�

The competing orders in the particle-particle �P-P� channel and the particle-hole �P-H� channel have beenproposed separately to explain the pseudogap physics in cuprates. By solving the Bogoliubov-de Gennesequation self-consistently, we show that there is a general complementary connection between thed-wave-checkerboard �DWCB� order in the P-H channel and the pair-density-wave �PDW� order in the P-Pchannel. A small pair density localization generates DWCB and PDW orders simultaneously. The result sug-gests that suppressing superconductivity locally or globally through phase fluctuation should induce bothorders in underdoped cuprates. The presence of both DWCB and PDW orders with 4a�4a periodicity canexplain the checkerboard modulation observed in Fourier transform scanning tunneling spectroscopies fromscanning tunneling microscopy and the puzzling dichotomy between the nodal and antinodal regions as well asthe characteristic features such as nondispersive Fermi arc in the pseudogap state.

DOI: 10.1103/PhysRevB.78.094510 PACS number�s�: 74.25.Jb, 74.25.Dw, 74.72.�h

I. INTRODUCTION

An important characteristic of strongly correlated electronsystems is the existence of different instabilities that lead tomany competing orders. In high-temperature superconduct-ors, besides the superconducting phase, many competing or-ders, such as spin-density wave �SDW�,1,2 d-density wave�DDW�,3,4 pair density wave �PDW�,5,6 stripe,7 and so on,have been proposed to explain various experimental obser-vations. Those competing orders can be generally classifiedinto two categories: the orders in particle-particle �P-P� chan-nel and the orders in the particle-hole �P-H� channel. So far,most of theoretical works in cuprates have focused on theeffect of individual competing orders. However, the orders intwo channels are not completely independent of each other.In some cases, they must be correlated. In this paper, wedetailed study one of these examples, the extended s-wavePDW order in the P-P channel and the d-wave-checkerboard�DWCB� density order in the P-H channel.

The motivation of this study mainly comes from the re-cent experiments of scanning tunneling microscopy �STM�in cuprates. These experiments have revealed surprising yetimportant electronic structures in the high-temperature super-conductors. The Fourier transform scanning tunneling spec-troscopies �FT-STS� from STM have captured two differentgeneral features in both momentum and energy spaces.8–16

One feature is dispersive peaks in FT-STS,11,12 interpreted asinterference patterns caused by elastic scattering of quasipar-ticles from impurities.17 The other is nondispersive peaks, acheckerboard modulation observed in various different ma-terials and circumstances. The checkerboard structure wasfirst discovered locally in Bi2Sr2CaCu2O8+� �BSCCO� near avortex core.8,18 Then, it was found to be a characteristic ofthe large gap regions where the STM spectrum resemblesthat in the pseudogap phase.9,12,14 Later, in the pseudogapphase, a similar checkerboard pattern was also observed.13

Finally, the STM studies of Ca2−xNaxCuO2Cl2 revealed thepresence of a global long-range commensurate checkerboardorder independent of doping.10

There have been various theoretical proposals to explainthe nondispersive checkerboard modulations. Most of theseproposals are related to the competing orders. In thesetheories, the origin of the nondispersive modulations aretied to the existence of particular order parameters. Thetheories include pair density modulation,5,6,19–21 current-density modulation,22,23 spin modulation,2 stripe chargemodulation,7,24 impurity scattering,25 and so on.

Among the proposed mechanisms, the pair density wavehas been shown to capture important characteristics of thecheckerboard density modulation. The mechanism of PDWderives from high pairing energy scale in cuprates. It sug-gests that, unlike the superconductivity of normal BCS typesuperconductors that can be destroyed by breaking Cooperpairs, the superconductivity in cuprates can be more easilyweakened or destroyed by phase fluctuations than by pairbreaking. Based on this argument, pair density localization5

was first proposed to explain the local checkerboard modu-lation in the presence of impurity or vortex. Later, a globalpair density wave was proposed to explain the checkerboardphysics in the pseudogap state.6,13 It has also been shownthat the symmetry of the tunneling intensity can distinguishthe pair density modulation from the conventional densitymodulation.6 While the pair density modulation provides agood understanding of the experimental results, the theorydoes not cover two important characterizations of the tunnel-ing experiments, namely, the dichotomy between nodal andantinodal regions and the Fermic arc.

Recently, we have proposed a d-wave-checkerboard den-sity order in P-H channel.26 The DWCB can be viewed asa natural extension of the d-density wave order proposedto explain pseudogap physics3,4 and is only different fromthe latter in terms of order wave vectors. We have shown thatthe DWCB order must exist when the PDW order is presentin the global d-wave superconducting state. Moreover, wehave demonstrated that the DWCB captures many generalfeatures of the STM experimental results. It has been dem-onstrated that the DWCB order has little effect on the density

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of state at low energy in the superconducting phase, but hasa strong effect on the STM spectrum around the super-conducting gap at high energy. This result naturally explainsthe puzzling dichotomy between the nodal and antinodalregions observed in STM �Ref. 14� and angle-resolvedphotoemission spectroscopy �APRES�.27 The DWCB orderalso preserves in FT-STS spectrum at the wave vectorsQ= �� /2a ,0� , �0,� /2a�� the same symmetry as that ob-served in the experiments. Moreover, the DWCB preservesthe nodes in the single-particle spectrum and generates aFermi arc with little dispersion around the nodal points athigh temperature, which are consistent with the results fromARPES. The Fermi arc has been a signature of thepseudogap region and has been proposed to explain thecheckerboard pattern observed in the pseudogap state.28

Thus, the DWCB provides a physical origin of the Fermi arc.In this paper, by solving the Bogoliubov-de Gennes equa-

tion self-consistently, we further demonstrate the resultsdrawn in Ref. 26. We design a microscopic model with thespace modulated density-density interaction. We show thatfrom the self-consistent solutions of the Bogoliubov-deGennes equations, a weak spatially modulated density-density interactions can generate the DWCB and the PDWorders simultaneously in general. The combination of thesetwo orders captures many important features of the STMexperimental results. We organized the rest of this paper asfollows: In Sec. II we introduce the d-wave-checkerboarddensity order to study the electronic states in the disorderedd-wave superconducting states and the pseudogap phase at atemperature above Tc. In Sec. III we show how the comple-mentary connection between the density orderings in theparticle-particle and the particle-hole channels are closely re-lated in the order wave vector and the symmetries. In Sec. IVwe study the orders in cuprates by calculating a full self-consistent Bogoliubov-de Gennes �BdG� equation includingDWCB and PDW in the d-wave superconducting state. InSec. V we calculate the local density of states and the spec-tral weights in the presence of both DWCB and PDW andcompare the recent experimental data.

II. d-WAVE-CHECKERBOARD DENSITY ORDER INCUPRATES

In this section, we shall introduce the DWCB order anddiscuss its roles in different regions of the phase diagram inhigh-temperature superconductors. In Ref. 26, we have pro-posed a DWCB to explain the experimental results observedin the STM measurements in a global d-wave superconduct-ing state. It has been shown that the presence of DWCBand competition with d-wave superconducting �DSC� ordercan capture the physics in a disordered superconductingstate and in the pseudogap phase at a temperature above Tc.Here we present a more detailed discussion and analysis.First, we will show the average density of states �DOS�and the Fourier component at the order wave vector Q= �� /2,0� , �0,� /2�� calculated in a mixed state of DSC andDWCB orders show good agreements with experiments suchas the spatially modulated local density of state �LDOS� inthe disordered superconductors. Second, we will show that

the presence of the DWCB order with the d-wave supercon-ducting order at a higher temperature above Tc plays an im-portant role in understanding the physics of the pseudogapphase: an emergence of the energy-independent Fermi arcsabove Tc and the linear dependence of its length on tempera-ture.

The mean-field Hamiltonian for the system where theDWCB coexists with d-wave superconducting order is givenby

HMF = H0 + HDWCB, �1�

where

H0 = �k,�

�kck�† ck� + �

k�kck↑

† c−k↓† + H.c., �2�

HDWCB = �k,q,�

Wk,qck+q�† ck� + H.c. �3�

H0 is the mean-field Hamiltonian for the d-wave supercon-ducting state and thus �k=�0 /2�cos ky −cos ky�. The DWCBhas particular order wave vectors Q= �� /2,0� , �0,� /2��and can be written as Wk,q=Wk�q,Q with Wk=W0 /2�cos kx−cos ky�.

To obtain a clear picture of the DWCB, we rewriteHDWCB in the real-space configuration

HDWCB = �r

Im�W0��cos�x

2+ sin

�x

2�Jr

x − x ↔ y+ Re�W0��cos

�x

2− sin

�x

2+ 2 cos

�y

2�

�Brx − x ↔ y , �4�

where r= �x ,y�a, Brx�y�=��cr,�

† cr+ax�y�,�+H.c.� is the densityoperator defined in the nearest-neighbor bond between r andr+ax�y� with x�y� unit vectors along x�y� directions, and

Jrx�y�= i��cr,�

† cr+ax�y�,�−H.c.� is the current-density operator

defined in the same bond as Brx�y�. Figure 1 shows a static

pattern of the bond strength of the DWCB order Br�. It isclear that the DWCB order defined in Eq. �4� has 4a�4aperiodicity and dx2−y2 symmetry. Similar order parametershave been mentioned in Ref. 25.

A. DWCB in the disordered superconducting state

Based on the mean-field Hamiltonian, Eq. �1�, we havecalculated the averaged local density of states �� and theFourier component of local density states at Q= �� /2,0� , �0,� /2��, �Q��, in two band dispersions. Thecalculation results turned out to be rather general and insen-sitive to the bare band structures. The effective mean-fieldHamiltonian, Eq. �1�, can be rewritten by using Nambu for-malism

HMF = �k

k†H�k�k, �5�

where k= �ck↑ ,ck+Q↑ ,c−k↓† ,c−k−Q↓

† �† and

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H�k� =��k Wk �k 0

Wk� �k+Q 0 �k+Q

�k� 0 − �−k Wk+Q

�

0 �k+Q� Wk+Q − �−k+Q

. �6�

Then the retarded Green function is given by

G−1�k,� = � + i��I − H�k� , �7�

where I is the identity matrix with the same rank with H�k�.The averaged density of states and the Fourier component atQ can be calculated as the following, respectively:

�� = −1

��k

Im G11�k,� , �8�

�Q�� = −1

��k

Im G12�k,� . �9�

First, we performed calculations in the particle-hole sym-metric case, where the band dispersion is given by

�k = − t/2�cos kx + cos ky� − � . �10�

We chose t=125 meV and �=0. �0=40 meV, which is rel-evant for underdoped BSCCO. The imaginary part of the selfenergy �=5 meV was used for the numerical calculation.

Figure 2�a� shows the averaged DOS normalized by oneof the noninteracting Fermi liquids. In the absence of DWCBorder, i.e., W0=0, there are sharp coherence peaks at theenergy of superconducting gap as expected. As DWCB orderdevelops, the coherence peaks located at 40 meV are sup-pressed, while the spectrum at low energy remains un-changed. From W0=4 meV the prominent peak begins toappear within the superconducting gap. Note that even smallDWCB made a strong effect on the spectrum at high energyas W0 increases.

Figure 2�b� shows the Fourier components of LDOS atthe wave vectors Q= �� /2,0� , �0,� /2��. It is known6 that

�Q��=�Q�−� for a bond-centered P-H pairing, while�Q��=−�Q�−� for a site-centered P-H pairing such as aconventional charge-density wave �CDW�. As expected fromthe fact that the DWCB order is bond centered, �Q�� iseven with respect to and it shows good agreement withexperiment.9

We repeat our calculations with the more realistic banddispersion provided by Norman et al.29 and the result is dis-played at Fig. 3. The band energy dispersion is now modifiedas such

Cu

�3

�1

�1

�3

FIG. 1. The configuration of the bond density of the DWCBorder in the real space. It is manifestly shown that the pattern has4a�4a periodicity and dx2−y2 symmetry.

�150 �100 �50 0 50 100 1500

0.5

1

1.5

2

2.5

�150 �100 �50 0 50 100 150�0.1

�0.05

0

0.05

0.1

0.15

0.2

0.25

W0= 0 meV

W0= 4 meV

W0= 8 meV

W0= 2 meV

W0= 4 meV

W = 8 meV

ρQ

(ω)

ρQ

(ω)

ω(meV)

Q = (0, 0)

Q = (π/2, 0)(0, π/2)

(a)

(b)

FIG. 2. �a� Averaged local density of states �LDOS� are plottedfor various DWCB orders: W0=0,4, and 8 meV with t=−125 meV, t�=�=0. �b� The Fourier components of LDOS at Q= �� /2,0� , �0,� /2��.

�150 �100 �50 0 50 100 1500

0.5

1

1.5

2

2.5

3

W0= 0 meV

W0= 4 meV

W0= 8 meV

ω(meV)

ρQ

(ω)

Q = (0, 0)

FIG. 3. Averaged local density of states with the finite chemicalpotential included in the band dispersion provided by Norman et al.�Ref. 29�

COMPLEMENTARY PAIR-DENSITY-WAVE AND d-WAVE-… PHYSICAL REVIEW B 78, 094510 �2008�

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�k = t1/2�cos kx + cos ky� + t2 cos kx cos ky

+ t3/2�cos 2kx + cos 2ky�

+ t4/2�cos 2kx cos ky + cos kx cos 2ky�

+ t5 cos 2kx cos 2ky − � , �11�

where t1=−0.5951 eV, t2=0.1636 eV, t3=−0.0519 eV, t4=−01117 eV, and t5=0.0510 eV.29 The chemical potential� is now set to −0.1660 eV. Compared with the P-H sym-metric case, the effect of DWCB on the averaged DOS andthe Fourier component of LDOS at Q is insensitive to theenergy-band structure. Qualitatively the numerical results arestrikingly consistent with experimental results,16 and thelarge gap region can be interpreted in terms of the coexist-ence of weak �8–12 meV� DWCB and DSC orders.

B. DWCB in the pseudogap state

The results of LDOS have demonstrated the consistencybetween the coexistence of the DWCB and DSC orders andthe STM experimental results in the disordered supercon-ducting state. Now we will show that the presence of theDWCB order also captures important physics in thepseudogap phase at a temperature above Tc. While there stillhave been hot debates over the interpretation of the origin ofthe pseudogap phase, we will show how many experimentalobservations in this phase can be explained by interpretingthe pseudogap phase as the mixed state of the DWCB andDSC orders. In this paper, we will focus on two features ofthe Fermi arcs developed from the nodal point along theFermi surface: nondispersive energy independence and lineartemperature dependence.

The autocorrelation of ARPES data from BSCCO shownondispersive peaks in momentum space arising from thetips of the Fermi arcs in the pseudogap phase, while thesuperconducting state shows dispersion with binding

energy.28 For the temperature dependence of the Fermi arcsin the pseudogap phase, it is known that its length increaseslinearly with temperature in the range between Tc and T�,below which the material is believed to be in the pseudogapphase.30 If the pseudogap phase is strongly connected tophase fluctuations of d-wave superconductivity, the single-particle spectrum should reflect the DWCB order. Therefore,a robust Fermi arc feature should exist in the mixed DWCBand DSC phases at high temperature. We will show that thisis indeed the case.

In order to illustrate the emergence of the nondispersiveFermi arc in the pseudogap state with DWCB order coexist-ent with DSC, we have calculated the spectral functionA�k ,� as given by the imaginary part of the retarded Greenfunction

A�k,� = −1

�Im G11�k,� . �12�

First we have studied the energy dependence of the Fermiarcs in the pseudogap state. We have plotted the spectralfunction A�k ,� in the first Brillouin zone in Fig. 4 based onthe energy dispersion Eq. �10�. In Fig. 4�a�, the Fermi arcsare in red and one of the d-wave-checkerboard wave vectorsQ= �� /2,0� is shown as a white bar. The scattering wavevectors connecting the tips of each arc are nearly equal to theorder wave vectors of DWCB, �Q�=� /2, which is consistentwith the nondispersive Fermi arc in experimental observa-tions. The small dispersion is negligible when compared withthe elongation of the gapless part along the Fermi surfacefrom the nodal point in the DSC state without DWCB order.It may depend on the bandwidth of the calculation. Whenincluding more hopping terms in the energy band, the disper-sion shown in the simple band calculation will be reduced.

Now let us consider the temperature dependence ofA�k ,� at the Fermi level �=0 meV�. It can be calculated

50K 100K 150K 200K

QQQQ(a)

(b)

��0meV

Q QQQ

FIG. 4. �Color online� The spectral function given by Eq. �12� based on the energy dispersion Eq. �10�. �a� The energy dependence of theFermi arc along the Fermi surface. The white bars with the magnitude of �Q�=� /2 are displayed to show the Fermi arc is nondispersive. Thedeviations for the energies �=0,−10,−20,−30 meV� from �Q� are negligibly small by 8%, 2%, 0%, and 13% of �Q�, respectively. �b� Inthe presence of DSC��0=40 meV� and DWCB�W0=8 meV�, the spectral weights A�k ,=0 meV� are plotted with varying temperature�T=50, 100, 150, and 200�.


094510-4

in the pseudogap phase by taking the temperature depen-dence as an effect from the self-energy �. We have plottedthe spectral function A�k ,� as a function of the temperaturein the first Brillouin zone in Fig. 4�b�. At very low tempera-ture the Fermi surface is gapped except at the nodal point�� /2,� /2�. As the temperature rises, the nodal points growsignificantly along the Fermi surface with slight broadeningin the direction perpendicular to the Fermi surface. We willshow the detailed calculations and discuss more in Sec. V.

III. COMPLEMENTARY CONNECTION BETWEEN THEORDERS IN THE P-P AND P-H CHANNELS

In this section, we will give a general argument regardingthe complementary connection between the orders in theparticle-particle and the particle-hole channels in the disor-dered d-wave superconducting state at zero temperature aswell as in the pseudogap phase at high temperature above Tc.As shown above, a d-wave-checkerboard order in the P-Hchannel ck�

† ck+Q��= f�k� with Q= �� /2,0� , �0,� /2�� andf�k�=cos kx−cos ky can explain the experiments on cupratesin the disordered DSC state and in the pseudogap phase.Preserving the same symmetry in FT-STS spectrum as thatobserved in experiments, the DWCB generates Fermi arcswith little dispersion with the binding energy around nodalpoints at high temperature above Tc. Since the Fermi arc hasbeen a signature of the pseudogap, the DWCB provides adirect link between the competing order and the pseudogapphysics.

The DWCB, however, is not a completely independentmechanism. In fact, there are some intimate connections withother orders in the P-P channel. Due to the existence of acomplementary connection between orders in both channels,the DWCB is directly connected to the PDW order. The an-gular symmetry of one orderings is determined by the com-

bination of the angular symmetries of DSC and the comple-mentary order. In this section, we will study how thesedensity orderings in both channels are related in the modu-lation wave vector and the symmetry.

A. Connection between PDW and DDW

To show the connection, we will use the example of thed-density wave order. Since the density order in the P-Hchannel has been well studied, examining the complemen-tary connected ordering in the P-P channel will be a prelimi-nary step to generalize the complementary connection in thecase of DWCB.

The mean-field Hamiltonian in the DSC state coexistingwith DDW can be written as

H = �k�

�kck�† ck� + iWkck�

† ck+Q� + �k

�kck↑† c−k↓

† + H.c.

= �k

†�k�A�k��k� , �13�

where †�k�= �ck↑ ck+Q↑ c−k↓† c−k−Q↓

† �† and

A�k� =��k iWk �k 0

− iWk − �k 0 − �k

�k� 0 − �k iWk

0 − �k� − iWk �k

, �14�

where ck�† and ck� are the creation operator and destruction

operator of the single particle with spin �, respectively.�k is the energy dispersion of the single particle. �k=�0 /2�cos kx−cos ky� and Wk=W0 /2�cos kx−cos ky� are thed-wave superconducting order and the DDW, respectively.Then the Hamiltonian is diagonalized by the followingtransformation:

�ck↑

ck+Q↑

c−k↓†

c−k−Q↓†

=�− iWk

�2Ek�Ek − �k��k + Ek

�2Ek�Ek + �k�− iWk

�2Ek�Ek + �k��k − Ek

�2Ek�Ek − �k��k − Ek

�2Ek�Ek − �k�− iWk

�2Ek�Ek + �k��k + Ek

�2Ek�Ek + �k�− iWk

�2Ek�Ek − �k�

0�k

�2Ek�Ek + �k�0

�k

�2Ek�Ek − �k��k

�2Ek�Ek − �k�0

�k

�2Ek�Ek + �k�0

��1�k��2�k��3�k��4�k�

, �15�

where Ek=��k2 + ��k�2+Wk

2. The eigenvalues E��k� corre-sponding to the eigenvectors ��k� are given by

E1�k� = E2�k� = + Ek, �16�

E3�k� = E4�k� = − Ek. �17�

The ground state �� is defined by the following conditions:

�1�k�� = �2�k�� = 0, �18�


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�3†�k�� = �4

†�k�� = 0. �19�

Then nonzero expectation values of the bilinear operators ofcs are

ck↑† ck↑� = vk

2 ��k�2

��k�2 + Wk2 , �20�

ck↑† c−k↓

† � =�k

2Ek, �21�

ck↑† ck+Q↑� =

iWk

Ek, �22�

ck↑† c−k−Q↓

† � = vk2 iWk�k

��k�2 + Wk2 , �23�

where vk2 = 1

2 �1−�k

Ek�. Note that in a DSC state without DDW,

or Wk=0,

ck↑† ck↑� =

1

2�1 −�k

��k2 + ��k�2

� . �24�

Therefore, in the mixed state of the DSC ck↑† c−k↓

† ��kand DDW ck�

† ck+Q�� iWk, a PDW order with the samewave vector Q is expected to exist naturally, whose symme-try is an extended s wave manifestly given by

ck↑† c−k−Q↓

† � � iWk�k =iW0�0

4�cos kx − cos ky�2. �25�

This indicates that the mixed state of DDW and DSC leads tothe existence of PDW. This argument can also be applied tothe state of DSC coexisting with PDW leading to DDW.Therefore the mixed state of DSC and DDW is nothing but acomplementarily connected description of the mixed state ofDSC and PDW.

B. Connection between PDW and DWCB

Given the close resemblance between DDW and DWCB,it is natural to expect that similar results also hold for theDWCB order, since the difference between DDW andDWCB is the order wave vector Q. Due to the high energyscale of superconducting gap in high temperature supercon-ductors �HTS�, superconducting order �SC� can be destroyedor suppressed by phase fluctuations. Carrying out a similaranalysis as Gossamer superconductors,31 we can show howthe complementary connection between PDW and DWCBarises in the low energy effective Hamiltonian that describesa system with pair modulation induced by phase fluctuations.

We start from a BCS Hamiltonian on a two-dimensional�2D� square lattice

HBCS = �k,�

�kck,�† ck,� + �

k�kck,↑

† c−k,↓† + H.c. �26�

The superconducting vacuum is constructed by the Cooperpairs with opposite momentum and spin

��BCS� = UBCS�0� � e�k�kck,↑† c−k,↓

†�0� , �27�

where ��k�2=Ek−�k+�

Ek+�k−� and Ek=��k2 + ��k�2. A local pair fluctua-

tion induced by disorder such as a vortex, impurities, orother factors will lead to a new ground state with nonzerosupercurrent due to the uncertainty principle. In other words,a new ground state �� can be obtained from the BCSground state ��BCS� by applying a boost of the total momen-tum q of the Cooper pairs

�� = e��k,q� qck↑† c−k+q↓

† − q�c−k+q↓ck↑��BCS� , �28�

where � is a small parameter and q is the structural factorof pair fluctuations determined phenomenologically. For acommensurate checkerboard modulation, q will be peakedat the related total momenta q of the pair, that is q=Q= �� /2,0� , �0,� /2��: q��q,Q.

The effective Hamiltonian associated with �� as aground state can be obtained as, up to a first order �,

Heff = HBCS + Hkin + Hint + O��2� , �29�

where Hkin and Hint are additional terms generated byboosted kinetic and pair interaction terms, respectively, inEq. �26�. For a commensurate checkerboard modulation,

Hkin = − ��k,q�k q�ck,↑

† c−k+q,↓† + c−k+q,↑

† ck,↓† � + H.c.

= �k�k,Q�ck,↑

† c−k+q,↓† + c−k+q,↑

† ck,↓† � + H.c., �30�

where �k,Q=−��q�k q��k�q,Q and it represents an ex-tended s-wave density order in the P-P channel or PDW.Moreover,

Hint = � �k,q,�

q�kck,�† ck+q,� + H.c. = �

k,�Wkck,�

† ck+q,� + H.c.,

�31�

where Wk=��q q�k��k�q,Q and it represents a d-wavedensity order in the P-H channel DWCB. The above deriva-tion tells us that a d-wave-like ordering in the P-H channeleffectively leads to Cooper pairs with a finite center of massmomentum.

IV. SELF-CONSISTENT CALCULATION FOR PDW ANDDWCB

Due to the complementary connection between PDW andDWCB, it is suggestive to calculate a fully self-consistentBogoliubov-de Gennes equation including both orders simul-taneously. It has been argued that long-range interaction be-tween charge carriers is very important in cuprates. PDW canbe induced by the long-range interaction between hole pairs.In this section, we will show that a solution of a self-consistent calculation with modulating pair potential andnearest-neighbor interaction can lead to orderings in bothP-P and P-H channels.

We start from a full model Hamiltonian on a 2D lattice


094510-6

H = −1

2 �i,j,�

�tijci�† cj� + H.c.� + �

i,jVijninj − ��

i

ni,

�32�

where ni=ni↑+ni↓.In particular, we will consider nearest-neighbor hopping

tij = t and an attractive nearest-neighbor interaction Vij. Thechemical potential � is chosen such that it is P-H symmetriccase. The nearest-neighbor interactions between two oppo-site spins are considered to be attractive and modulating.They are given by a negative constant pair potential and along-range interaction modulated by Q= �� /2,0� , �0,� /2��,

Vij = − V0 + �Vij = − V0 + �Q

�Vij�cos Q · ri + cos Q · r j� ,

�33�

where

�Vij = �+ V1, j = i � x

− V1, j = i � y ,� �34�

with V0 and V1 positive constants. While the local behaviorof �Vij seems to be anisotropic at the center and ��2a , �2a�as seen in Fig. 5, the global behavior is isotropic in a sensethat there is no preferential direction for attraction enhancedby negative modulation or weakened by positive modulation.The anisotropic modulation is rotationally invariant with thecombination of translation by 4a.

Starting with this Hamiltonian, let us derive the BdGequations by introducing mean-field decoupling of thenearest-neighbor interaction terms. The quartic term ni�nj��has three different types of mean-field decouplings

ci�† ci�cj��

† cj�� ⇒ ci�† ci��cj��

† cj�� + cj��† cj��ci�

† ci�

+ ci�† cj��

† �cj��ci� + ci�† cj��

† cj��ci��

− ci�† cj��cj��

† ci� − ci�† cj��cj��

† ci�� .

�35�

The first two terms are nothing but the chemical-potentialterms. The decouplings in the second line are the usual BCSdecouplings. The last two terms are decouplings in the P-Hchannel, which are just the exchange terms. For interactionsbetween two particles with same spins, these exchange terms

effectively modified hopping terms. For interactions betweentwo particles with opposite spins, it corresponds to a bond-centered spin-density wave. From our full Hamiltonian, wecan argue that the spin rotational symmetry is not broken.Thus the SDW decouplings can be safely ignored. Also,since we are interested in d-wave pairing, the BCS decou-plings of triplet pairing will be dropped too. We then arriveat the following mean-field �MF� Hamiltonian:

HMF = −1

2 �i,j��

��t + Wij�ci�† cj� + H.c.� + �

i,j��ij

�1�ci↑† cj↓

†

+ H.c.� + �i,j�

��ij�2�ci↑

† cj↓† + H.c.� − �

i,j��ij�ni�,

�36�

where

�ij�1� = − V0�cj↓ci↑� + ci↓cj↑��/2, �37�

Wij = − �Vijcj�† ci�� , �38�

�ij�2� = �Vij�cj↓ci↑� + ci↓cj↑��/2, �39�

�ij� = ��ij − Vijcj−�† cj−�� , �40�

The pairing amplitude on a bond �i , j�, �ij�1�, stems from a

constant nearest-neighbor attraction −V0 and the densitywave pairings in the P-P and the P-H channels are caused bya long-range interaction �Vij. Since our major interest is tostudy the orderings in both channels at the same time, thechemical potential �ij� is taken to be a constant.

The Bogoliubov-de Gennes equation is given by

�H0 ��

� − H0��un�i�

vn�i� � = En�un�i�vn�i� � , �41�

where H0 and � are transfer matrices such that

H0x�i� = − �� j�t + Wji� + � ji�x�j� , �42�

�x�i� = � j�� ji

�1� + � ji�2��x�j� , �43�

where x�i� can be either un�i� or vn�i�.We numerically solve BdG eigenvalues En and eigenvec-

tors �un�i� ,vn�i�� on a lattice of N sites with a periodicboundary condition. We then calculate the pairing amplitudeand the density orderings in the P-P and the P-H channelswhich are, respectively, given by

�ij�1� = −

V0

2 �n

�un�j�vn��i� + un�i�vn

��j�� , �44�

�ij�2� =

�Vij

2 �n

�un�j�vn��i� + un�i�vn

��j�� , �45�

Wij = − �Vij�n

vn�j�vn��i� . �46�

Substituting these equations back into the BdG equation, Eq.�41�, and repeating the same process until self-consistency is

��

��

��

��

FIG. 5. A modulating nearest-neighbor interaction �Vij is plot-ted in a 2D square lattice.


094510-7

achieved for each of the local variables, we can obtain solu-tions of the BdG equations. In the calculation, we have usedBroyden’s method for efficient iteration.

From the self-consistent solution, we can then computethe interesting order parameters. The d-wave pairing ampli-tude �DSC� at a lattice site r is determined by four nearestneighbors �ij

�1�:

�DSC�r� = ��r,r+x�1� + �r,r−x

�1� − x ↔ y�/4. �47�

In the same way, an extended s-wave pairing density waveorder in the P-P channels �PDW� can be obtained by �ij

�2�:

�PDW�r� = ��r,r+x�2� + �r,r−x

�2� + x ↔ y�/4, �48�

and d-wave density ordering in the P-H channel �DWCB� is

WDWCB�r� = �Wr,r+x + Wr,r−x − x ↔ y�/4. �49�

We shall now present our results of full calculation. Theuniform part of the interaction is set to V0=−2.5t while themodulation part has amplitudes of V1 with the range of0.1t−1.0t. The solutions we have obtained are independentof the initial guesses for the local variables. In Fig. 6�a�, theorder parameters �DSC�r�, �PDW�r�, and WDWCB�r� in a 2Dreal space are plotted for V0=2.5t and V1=0.5t with t=125 meV. The P-H channel ordering Wr,r� shows d-wavesymmetry such that

Wr,r� = � W0�cos Q · r + cos Q · r�� , r� = r � x

− W0�cos Q · r + cos Q · r�� , r� = r � y ,�

�50�

where W0=0.06t=8.3 meV and the order wave vectors Q= �� /2,0� , �0, �� /2�� as shown in Fig. 6�b�.

The order in P-P channel has both d-wave �r,r��1� and ex-

tended s-wave �r,r��2� part. The extended s-wave part is a pure

modulation described by

�r,r��2� = ��1�cos Q · r + cos Q · r�� , r� = r � x

�1�cos Q · r + cos Q · r�� , r� = r � y ,� �51�

where �1=0.13t=16.3 meV and the same order wave vec-tors Q. We have also obtained the results of d-wave pairingamplitude or �r,r�x

�1� =�0 and �r,r�y�1� =−�0 with �0=0.28t. In

Fig. 7 the modulation amplitudes of �DSC�r�, �PDW�r�, andWDWCB�r� were plotted as a function of V1.

The self-consistent BdG calculation shows that there isindeed a complementary connection between the density or-der in the P-P channel �PDW� and in the P-H channel�DWCB�. Since PDW and DWCB are based on the samemodulating interaction �Vij, not only PDW and DWCBshould be present simultaneously with d-wave superconduc-tivity but also the symmetries and order wave vector Q ofboth orders should be closely related to each other. There-

kx

ky�pi

�pi/2

0

pi/2

pi

�pi

�pi/2

0

pi/2

pi

0

5000

10000

kx

ky

�pi�pi/2

0

pi/2

pi

�pi

�pi/2

0

pi/2

pi

0

500

1000

kx

ky�pi

�pi/2

0

pi/2

pi

�pi

�pi/2

0

pi/2

pi

0

200

400

600

Wdwcb(r)(a)

(b)

∆dsc(r) ∆pdw(r)

FIG. 6. �Color online� �a� The amplitude of the order parameters �DSC�r�, �PDW�r�, and WDWCB�r� defined by Eqs. �47�–�49�, respec-tively, plotted in 16�16 lattice sites. �DSC�r��0, �PDW=�1 cos Q ·r, and WDWCB�r�=W0 cos Q ·r with �0=34.2 meV, �1=16.3 meV,and W0=8.3 meV, respectively. t=125 meV is chosen to have the amplitudes in units of meV. �b� The Fourier transforms of the orderparameters are displayed in the first Brillouin zone. The prominent peaks are located at Q= �� /2,0� , �0, �� /2�� in both PDW andDWCB.


094510-8

fore, the mean-field solutions in the disordered d-wave su-perconductivity should include a d-wave density wave withQ in the P-H pairing and an extended s-wave density wavewith the same Q in the P-H pairing. Furthermore it is impor-tant to take both density orders into account on the samefooting when we try to understand the electronic states of thedisordered high Tc cuprates and the pseudogap state at hightemperature.

V. THE MIXED DWCB AND PDW STATE

Analytically, the general features in STM measurementscan be captured by the DWCB. Due to the anisotropy inher-ited from the d-wave factor of pairing, a weak DWCB orderhas a much stronger effect on the antinodal region than onthe nodal region. Thus it naturally explains the puzzling di-chotomy between the nodal and antinodal excitations inhigh-temperature superconductors. The local phase fluctua-tions of Cooper pairs lead to a local modulation of d-waveordering in the P-H channel �DWCB�, which strongly affectsthe antinodal single-particle excitations as well as an ex-tended s-wave order in the P-P channel �PDW�. Since bothPDW and DWCB are bond centered, �Q�� will be an evenfunction of .6 This symmetry distinguishes the PDW andDWCB orders from the typical CDW in the P-H channel.The existing experimental results are consistent with theeven case.

The purpose of this section is to determine the symmetriesof density orders in P-P and P-H channels and then study thepseudogap physics under the presence of the orderings in P-Pand P-H channels. Due to the complementary connection, ifthe PDW is an extended s wave, DWCB is a d-wave P-Hpairing. If the PDW is a d wave, the P-H pairing densityorder must be extended s wave. It is worth noting that thesymmetries of the orders are very important to explain thebehavior of the conductance spectra at low energy. We will

show that in DSC state the d-wave P-H pairing �DWCB� andthe extended s-wave P-P pairing �PDW� can explain the lowenergy spectra while the solution with other symmetries can-not sufficiently match the experimental results.

In Sec. II, we have shown that, even without PDW, thepresence of DWCB and DSC orders in a pseudogap state cancapture the features of the Fermi arc. To understand the tem-perature dependence of the length of the Fermi arc above Tcwe assumed the temperature dependence of the imaginarypart of the energy � in the Green function implicitly. Usingthe results of the self-consistent calculations, this assumptionof the temperature dependence of � is not necessary. Thepresence of the complementary connected d-wave orders inP-P and the extended s-wave order in P-H channels withDSC order naturally lead us to the temperature dependenceof the length of the Fermi arc within the Franz and Millismodel.32

A. Symmetries in P-P and P-H channels

In this subsection we study how the local density of statesis associated with the symmetries of the complementary con-nected orders in P-P and P-H channels in a disorderedd-wave superconducting state.

The local density of states in the DSC state in the pres-ence of P-P and P-H channel orders is calculated by

��r,� = �n

��un�r��2�� − En� + �vn�r��2�� + En�� .

�52�

The averaged DOS is given by

�� = �r

��r,� �53�

and the Fourier components at the wave vectors Q= �� /2,0� , �0,� /2�� by

�Q�� = �r

eiQ·r��r,� . �54�

We have calculated them in a simple band structure t=−125 meV and �=0.

For the d-wave P-H and extended s-wave P-P pairing, wehave used V0=2.5t and various V1 as in Eq. �55�, where orderparameters as solutions of the BdG calculation are given inmeV, and �h is the height of �� at =0 off of the super-conducting DOS displayed in a dashed line in Fig. 8.

V1 �DSC �PDW WDWCB �h

0.4t 33.0 12.7 3.3 0.0599

0.6t 35.7 19.9 9.9 0.1125

0.8t 39.6 27.5 13.0 0.1445

1.0t 44.2 35.6 16.1 0.1546

�55�

For the extended s-wave P-H and d-wave P-P channel or-ders, we have used V0=2.5t and V1 as in Eq. �56�, whereWSWCB denotes the extended s-wave P-H pairing for conve-nience.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50�0

�1

W0

Am

plit

udes

(meV

)

V1/t

FIG. 7. The amplitudes, �0, �1, and W0 of �DSC�r�, �PDW�r�,and WDWCB�r�, respectively, as a function of the modulation long-range interaction V1. t=125 meV is chosen to have the amplitudesin units of meV. For a d-wave superconducting �DSC� order �0

denoted by x along the curve for DSC, at each V1 represents themean value of �DSC�r� which is slightly modulated.


094510-9

V1 �DSC �PDW WSWCB �h

0.4t 31.5 11.5 6.6 0.1144

0.6t 32.7 17.3 10.0 0.2933

0.8t 34.2 22.9 13.3 0.4200

1.0t 36.2 28.4 16.6 0.3589

�56�

The BdG calculation shows that DWCB, which is d-waveP-H pairing order, is crucial to the physics in the disorderedsuperconductor. It captures the experimental observation ofconductance spectrum at both low and high energies. As ex-pected, the d-wave symmetry of the P-H pairing density or-der has little effect at low energy while it strongly affects thesuperconducting coherence peaks. Therefore we can con-clude that the competing order in the P-H channel must haved-wave symmetry. Due to the complementary connection inthe orders in P-H and P-P channels, it naturally leads to theexistence of the pair density order that has extended s-wavesymmetry.

In Fig. 8, to illustrate the above argument, we plotted theaveraged DOS in two cases: �a� d-wave P-H and extendeds-wave P-P channel orders and �b� s-wave P-H and extendedd-wave P-P channel orders in the d-wave superconductingstate ��DSC=33 meV�.

B. The DWCB and PDW in the pseudogap state

Now let us consider the spectral function A�k ,� in thepseudogap state to capture the features in the Fermi arc. First

we rewrite the mean-field Hamiltonian for PDW and DWCBin momentum space. Taking the Fourier transform of Eq.�36�, we obtain

HPDW = �r,�,�

�r,r+��2� cr,�

† cr+�,−�† + H.c.

= �p,q,�

��r,�

�r,r+��2� e−i�p+q�·re−iq·�cp,�

† cq,−�† , �57�

where � denotes the unit vectors for the nearest neighbors ofr. Since �r,r+�

�2� =�1 /2�Q�cos Q ·r+cos Q · �r+��, the termin the square bracket in Eq. �57� is rewritten as

�1 �r,Q,�

�cos Q · r + cos Q · �r + ��e−i�p+q�·re−iq·�

= �1�Q,�

�e−iq·� + eip·��p+q,Q = �1�Q

�cos px + cos�px + Qx�

+ x ↔ y��p+q,Q.

Thus HPDW in momentum space can be given by

HPDW = �k,Q,�

�k�2�ck,�

† c−k−Q,−�† + H.c., �58�

where �k�2�=�1�cos kx+cos�kx+Qx�+x↔y�.

The momentum space expressions for DWCB in HMF,Eq. �36�, can be obtained in the same way:

HDWCB = �k,Q,�

Wkck+Q,�† ck,� + H.c., �59�

where Wk=W0�cos kx+cos�kx+Qx�−x↔y�. Therefore themean-field Hamiltonian, Eq. �36�, can be recast in momen-tum space as

HMF = �k,�

��kck,�†ck,� + �k�1�ck,�

† c−k,−�† �

+ �k,Q,�

��k�2�ck,�

† c−k−Q,−�† + Wkck+Q,�

† ck,�� + H.c.,

�60�

where �k=−t /2�cos kx+cos ky�−� and �k�1�=�0 /2�cos kx

−cos ky�. In the same way as in Sec. II, it can be re-expressed in terms of the Nambu formalism as such

HMF = �k

k†H�k�k, �61�

where k= �ck↑ ,ck+Q↑ ,c−k↓† ,c−k−Q↓

† �† and

H�k� =��k Wk �k

�1� �k�2�

Wk� �k+Q �k+Q

�2� �k+Q�1�

�k�1�� k+Q

�2�� −k Wk+Q�

�k�2�� k+Q

�1�� Wk+Q �−k+Q

. �62�

The spectral function is given by the imaginary part of theGreen function

G11�k,� = �� + i� − H�k��11−1�� . �63�

The ground state �� is defined by the state without quasi-particles. In the pseudogap phase, it can be calculated withinthe model of Franz and Millis:32

FIG. 8. �a� The averaged DOS are plotted for �DSC=33 meV inthe case of d-wave P-H and extended s-wave P-P channel orders:�PDW=12.7 meV and WDWCB=3.3 meV. The arrow indicates �h,that is, DOS at =0 off of one for pure d-wave superconductingstate. �h=0.0599. �b� The averaged DOS in the mixed state of DSC��DSC=33 meV� with d-wave P-P and extended s-wave P-H chan-nel orders. Here we plotted one with �PDW=17.3 meV andWSWCB=10 meV. �h=0.2933.


094510-10

A�k,� = −1

�� d�P��Im G11�k, − �� , �64�

where � is the energy shift due to the presence of a uniformsupercurrent and P�� is the probability distribution of �calculated within a 2D XY model as such

P�� = �2�W�T�e−�2/2W2�T�. �65�

where W�T� is an order of pseudogap and an increasing func-tion of temperature above Tc.

Now we will study the energy dependence of the Fermiarc in the pseudogap state. One of the salient features of thepseudogap phase is that the Fermi arc is nondispersive,whereas it is dispersive in the superconducting phase. TheFermi arc along the Fermi surface is identified by the peak ofenergy distribution curve �EDC� at =0. Figure 9�b� dem-onstrates that the Fermi arc in green remains unchanged as is increased. In the superconducting state, however, the gap-less region develops as the energy decreases below the Fermilevel as seen in Fig. 9�a�.

While the Fermi arc is nondispersive, it is found that thelength of it is increasing linearly as a function oftemperature.30 Within the model of Franz and Millis, whenT�Tc, W�T� is an approximately linearly increasing functionwhen the order parameters remain fixed. Adopting the valuesof W�T� above Tc, we have calculated A�k ,� as a functionof temperature.

In Fig. 10�a�, EDC are plotted from the nodal to antinodalpoint along the Fermi surface. Clearly it shows a gaplessnodal point and gapped antinodal point as indicated bydarker curves. But, contrary to a superconducting state wherethe Fermi surface is gapped except at the nodal point, there isa region, the so-called Fermi arc, where peaks still survive at=0 and thus the superconducting gap is closed. The curveon top is the EDC at the nodal point �� /2,� /2� perpendicu-lar to the Fermi surface. It is manifest that the size of thearrows is increasing as the temperature rises. Figure 10�b�shows the Fermi arc in the first Brillouin zone. As in the Fig.9 the green color represents the gapless region of the Fermisurface, that is, the Fermi arc.

VI. CONCLUSION AND DISCUSSION

We have presented a detailed analysis of competing or-ders in cuprates which explains the checkerboard pattern ob-served in STM spectra. We have shown that in general thereis a complementary connection between the orders in the P-Pand the P-H channels in both the superconducting state andthe pseudogap state. The symmetries of the orders in bothchannels are closely related to each other. In the d-wavesuperconducting state, the presence of the d-wave �extendeds-wave� P-H density order implies the coexistence of theextended s-wave �d-wave� P-P pairing order. The self-consistent calculations in the disordered superconductingstate result in solutions with uniform d-wave superconduct-ing order, a P-H channel checkerboard density order with theorder wave vector Q, and a P-P channel pairing density wavewith the same Q. The symmetries of the channel orders arecrucial to the effects on the density of states in STM experi-ments. We have found that d-wave-checkerboard density or-der in the P-P and the extended s-wave pair density wave arein good agreement with the experimental data in the super-conducting phase.

��

0meV

-10meV

-20meV

-30meV

0meV

-10meV

-20meV

-30meV

FIG. 9. �Color online� The energy dependence of the Fermi arc.�a� shows the development of the gapless region as a function ofenergy in the superconducting state without DWCB. As in-creases �from top to bottom�, the scattering wave vectors connect-ing one tip of the green curve to another tip decrease. However, in�b� the gapless region remains unchanged as increases, that is, theFermi arc is nondispersive.

��

�

�

T=Tc T=1.2Tc T=1.5Tc T=1.8Tc T=2Tc

FIG. 10. �Color online� The temperature dependence of theFermi arc above Tc. �a� EDCs along the Fermi surface from thenodal point �top� to antinodal point �bottom� are plotted. The arrowsindicate the gapless part in the Fermi surface and show the length ofthe Fermi arc linearly depends on the temperature in the pseudogapstate. �b� The Fermi arc as a function of the temperature T at aboveTc.


094510-11

The coexistence of orders in both P-P and P-H channelsare important to understand the pseudogap phase which canbe considered as a state which maintains pairing amplitudewithout phase coherence. The fluctuation of phase wouldnaturally lead to the DWCB and PDW orders. We haveshown that the presence of DWCB and PDW results in anondispersive Fermi arc and our calculations show the lineardependence of the length of the Fermi arc at temperaturesabove Tc. Moreover, the effect of the presence of PDW andDWCB on single-particle spectra is much larger in the anti-nodal direction than in the nodal direction. These results alsoexplain the dichotomy between the nodal and antinodalsingle-particle spectra in the cuprates.

The connections of the P-P and P-H channel orders areimportant in formulating the effective low energy theory in

cuprates. From our results in this paper, the low energy ef-fective theory should include both orders. So far, most theo-ries have treated them independently. It is also clear that thepresence of both orders can result in new physics in transportand thermal properties. A detailed study of these effects willbe reported elsewhere.

ACKNOWLEDGMENTS

J.P.H. and K.S. are supported by National Scientific Foun-dation under Grant No. Phy-0603759. H.D.C. is supportedby the U.S. Department of Energy, Division of MaterialsSciences under Grant No. DEFG02-91ER45439, through theFrederick Seitz Materials Research Laboratory at the Univer-sity of Illinois at Urbana-Champaign.

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Complementary pair-density-wave and d-wave-checkerboard