Lifshitz transition in two-dimensional spin-density wave models

Jie LinMaterials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

�Received 27 July 2010; revised manuscript received 20 September 2010; published 9 November 2010�

We argue that both pocket-disappearing and neck-disrupting types of Lifshitz transitions can be realized intwo-dimensional spin-density wave models for underdoped cuprates, and study both types of transitions withimpurity scattering treated in the self-consistent Born approximation. We first solve for the electron self-energyfrom the self-consistent equation, and then study the low-temperature electrical conductivity and thermopower.Close to the Lifshitz transition, the thermopower is strongly enhanced. For the pocket-disappearing type, it hasa sharp peak while for the neck-disrupting type, it changes sign at the transition, with its absolute value peakedon both sides of the transition. We discuss possible applications to underdoped cuprates.

DOI: 10.1103/PhysRevB.82.195110 PACS number�s�: 72.15.Jf, 75.30.Fv, 74.72.Kf

I. INTRODUCTION

One of the cornerstones of metallic physics is the conceptof the Fermi surface,1,2 the surface in momentum space sepa-rating filled states from empty ones at zero temperature. Ap-plying external perturbations, such as hydrostatic pressure,anisotropic strain, or a change in the chemical composition,the Fermi surface can change its shape or more drastically itstopology. This change in the Fermi surface topology is usu-ally referred to as a Lifshitz transition,3 and has a profoundimpact on material properties.4–7 Recently, the Lifshitz tran-sition has invoked revived interest in the context ofcuprates,8 iron-arsenic superconductors,9 cold-atomsystems,10 bilayer graphene,11 and heavy-fermion metals andorganic conductors.12,13

It is known that a density wave instability can change theFermi surface topology. In the density wave-ordered state,the Fermi surface evolves with the strength of the coherentdensity wave potential. Well-known examples are the under-doped cuprates. In the case of the electron-doped cuprateswith commensurate �� ,�� spin-density wave order, it hasbeen shown that the hole pockets around �� /2,� /2� disap-pear as one tunes the magnitude of the density-wave poten-tial or the electron doping,14 as shown in Fig. 1. �We take thein-plane lattice constant a=1.� This is an example of the

pocket-disappearing Lifshitz transition.3 Another example ofthis type is found in the site-centered antiphase stripe model,as shown in Refs 15 and 16, where both the electron pocketand the hole pockets can disappear independently by tuningthe strength of the spin and charge potentials. This pocket-disappearing Lifshitz transition has important consequencesfor the Hall effect.14–16 Recently, it was realized that theneck-disrupting Lifshitz transition is also possible in a stripemodel for the underdoped cuprates,17 as shown in Fig. 2,which provides a simple explanation for the disappearance ofquantum oscillations and the associated diverging cyclotronmass in YBa2Cu3O6+x.

18

In the single-particle picture, the energy dispersion ofband electrons �p has extrema in the first Brillouin zone �lo-cal maxima, local minima, and/or saddle points� at energiesdenoted generically as �c. As the chemical potential � isvaried through �c, the Fermi surface changes its topology anda Lifshitz transition takes place; for a local maximum orminimum, a pocket appears or disappears, corresponding tothe pocket-disappearing-type transition, and for a saddlepoint, the Fermi surface changes its connection, correspond-ing to a neck-disrupting-type transition.

The cuprate band structure has saddle points located atM1,2 in Fig. 1.19 Upon hole doping, when � reaches theenergy of the saddle points, a neck-disrupting Lifshitz tran-sition occurs, characterized by a logarithmically divergentdensity of states. There have been extensive studies on the(a) (b)

Γ Γ

X

M2

M1

FIG. 1. �Color online� The Fermi surfaces for two different val-ues of the spin potential in the commensurate spin-density wavemodel for electron-doped cuprates. This shows an example of thepocket-disappearing Lifshitz transition. Compared to �a�, there is anextra small Fermi pocket represented by the thick solid line in �b�.The momentum region where the new pocket forms is bounded bythe dashed circle, and is defined as the L region in the main text. Inall Fermi surface plots in this paper, �= �0,0�, X= �� ,��, M1

= �0,��, and M2= �� ,0�, where the planar lattice constant a=1.

(a) (b)

Γ Γ

X

M2

M1

FIG. 2. �Color online� The Fermi surfaces for two different val-ues of the stripe potential in the stripe model with ordering wavevector �5� /6,�� for hole-doped cuprates. This shows an exampleof the neck-disrupting Lifshitz transition. From �a� to �b�, the Fermisurface represented by the thick solid line changes its topology. Themomentum region where this happens is bounded by the dashedbox, and is defined as the L region in the main text.

PHYSICAL REVIEW B 82, 195110 �2010�

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van Hove singularity in the cuprate band dispersion, and itseffects on material properties, as reviewed in Ref. 20. For theband structure obtained in Ref. 19, this Lifshitz transitionoccurs at a relatively large hole doping. On the other hand, inthe spin-density wave model, the neck-disrupting Lifshitztransition can happen in the underdoped regime.17

The electron-electron �e-e� interaction may modify thesingle-particle picture. In particular, the e-e interaction incuprates is believed to be strong, and may hold the key tounderstanding the mechanism of high-temperature supercon-ductivity and other properties.21 Here, we take the point ofview in which the spin-density wave order is driven by theCoulomb interaction. Since the Lifshitz transitions in Figs. 1and 2 occur in states with well-established order �corre-sponding to relatively large density wave potentials�, we as-sume a Fermi-liquid description for the electrons on theCuO2 planes.22 In the framework of the Fermi liquid theory,the effect of e-e interaction on the Lifshitz transition hasbeen studied.7,23 In this paper, we concentrate on the effect ofimpurity scattering.

To lowest order in the electron-impurity interaction, theimpurity scattering rate is proportional to the single-particledensity of states. In three dimensions, the density of statesacquires a ����−�c� nonanalytic contribution in the vicinityof Lifshitz transitions, which has important effects on thethermodynamic and kinetic properties of the system.4,24 Thecorrection to the impurity scattering rate vanishes, albeit in anonanalytical way, as the transition is approached. The situ-ation is different in two spatial dimensions, where for thepocket-disappearing-type transition, the density of states hasa finite jump, and for the neck-disrupting type, the density ofstates has a logarithmic singularity �ln��−�c�.25 The loga-rithmically diverging density of states presents conceptualchallenges; it was pointed out that a system with divergingdensity of states at the Fermi energy is unstable to densitywave ordering.26 Furthermore, when the impurity scatteringis large, perturbation theory cannot be applied.

In this paper, we study Lifshitz transitions in two-dimensional systems, treating the impurity scattering in theself-consistent Born approximation, and calculate the trans-port coefficients in the low-temperature limit. We keep bothreal and imaginary parts of the electron self-energy in solv-ing the self-consistent equation. For the neck-disrupting-typetransition, the divergence in the density of states is cut off bythe impurity potential, and this approach leads to a concep-tually consistent picture. Restricting to the low-temperaturelimit allows us to study the impurity effects and neglect thescattering from phonons and other collective excitations. Thecorrections to the self-consistent Born approximation consistof diagrams with crossed impurity lines, and as discussed inRefs 4 and 7, they are negligible for the study of the electronself-energy. Their contribution to two-particle processes,such as the weak localization effect,27 will not be considered.An external magnetic field suppresses the weak localizationeffect. A recent discussion on the localization effect fromexperimental point of view can be found in Ref. 8.

The rest of the paper is organized as follows. In Sec II, wesolve the electron self-energy in the self-consistent Born ap-proximation. In Sec III, we study the low-temperature elec-trical conductivity and thermopower close to the Lifshitz

transition. Sec IV is a summary where we discuss the mainresults and their applications to underdoped cuprates. Sometechnical details are contained in Appendices A–C. Through-out this paper, we take �=kB=1.

II. SELF-CONSISTENT BORN APPROXIMATIONFOR IMPURITY SCATTERING

We use a mean-field approximation for the spin-densitywave state in underdoped cuprates, in which electrons moveon a square lattice subject to additional periodic potentialsdue to density wave ordering. This results in a multibandsituation with a band dispersion depending on density wavepotentials.14,22,28 Close to the transition, this dependence canbe neglected, and we assume a rigid band model. The chemi-cal potential is the tuning parameter for the transition.

For impurity scattering, we assume pointlike scatterers,such that the electron-impurity interaction matrix element isa constant in momentum space, denoted as u. As a result, theelectron self-energy in the self-consistent Born approxima-tion is independent of momentum,29

��i�n� = u2� d2p

�2��2

1

i�n − �p + � − ��i�n�, �1�

where �p is the electron band dispersion and � is the chemi-cal potential. We study the situation where �p has only oneextremal point at momentum pL with energy �c close to �,and assume that when several extrema are close to �, theireffects are additive. We separate the momentum integral intoa region close to pL, labeled by “L” �the regions bounded bythe dashed lines in Figs. 1 and 2�, and a region far away frompL, labeled by “R, ”and assume that there are extensiveFermi surface pieces in the R region. For the momentumintegral in the R region, the usual arguments apply,30 and weobtain

u2�R

d2p

�2��2

1

i�n − �p + � − ��i�n�� − i0 sign �n, �2�

where we have assumed, a posteriori, that ���i�n�+ i�n�Dwith D on the order of the bandwidth which is the largestenergy scale in our calculation, and 0=�u2�̄0 with �̄0 thedensity of electronic states in the R region. 0 can be esti-mated from the scattering rate far from the Lifshitz transi-tion. We assume weak impurity scattering, 0D. After ana-lytical continuation i�n→�+ i0+, the retarded self-energy�R��� can be written as �R���=−i0+��R���. Denoting R���and −1��� as real and imaginary parts of ��R���, we have

�R��� = R��� − it��� ,

where t���=0+1���. From Eqs. �1� and �2�, R��� andt��� are determined from the integral equations

R��� = u2�L

d2p

�2��2

� + � − R��� − �p

�� + � − R��� − �p2 + t2 , �3�

JIE LIN PHYSICAL REVIEW B 82, 195110 �2010�

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t��� = 0 + u2�L

d2p

�2��2

t����� + � − R��� − �p2 + t

2 . �4�

In the low-temperature regime �Tt�, one can use the ex-pansion R����R0+R0�� and t����t0+t0� �. Further calcu-lation requires a knowledge of �p in the L region which isdifferent for the two types of Lifshitz transitions and is dis-cussed in the following subsections.

A. Pocket-disappearing Lifshitz transition

For the pocket-disappearing Lifshitz transition, the disper-sion relation in the L region can be expanded as, taking pL asthe origin and px,y along the principle axes,

�p = �c � px2

2mx+

py2

2my� , �5�

where we choose �c=0 in the calculation. The upper signcorresponds to the emergence of an electron pocket for � 0 and the lower sign corresponds to that of a hole pocketfor ��0. In this paper, we assume an electron pocket; theother case can be studied in the same way. The L region isdefined by �kx

2+ky2��, where kx= px /�2mx and ky

= py /�2my.Assuming �2�t, Eqs. �3� and �4� become

R��� = − U ln�2

��� + � − R���2 + t���2, �6�

t��� = 0 + U��

2+ tan−1 � + � − R���

t��� �� , �7�

where U=u2�mxmy /2� and is of the same order as 0. In thefollowing discussion, we find that it is convenient to measureenergies in units of U.

From Eq. �6�, we obtain

� = �� − R0� − U ln�2

��� − R0�2 + t02

, �8�

showing that � is a monotonically increasing function of �−R0. As can be seen from Eqs. �6� and �7�, t0, t0� , and R0 alldepend on �−R0. As a result, we absorb R0 into �. From Eq.�7�, we obtain

t0 = 0 + U��

2+ tan−1 �

t0� �9�

which is shown in Fig. 3. t0� and 1−R0� can be solved fromEqs. �6� and �7�, and are shown in Fig. 4.

Since �tan−1 x��� /2, t����U. For ��U, t0�0+�U, �R0���U /�1, and �t0� ��U2 /�21. In this limit, theelectron pocket in the L region is well established, with anenergy scale large compared to impurity broadening, and thiscorresponds to the usual situation for impurity scatteringwhere the imaginary part of �R��� is proportional to thesingle-particle density of states, and the real part is negli-gible. For −��U, t0�0, �R0���U / ���1, and �t0� ��U2 /�21. In this limit, the states in the L region are far

below the chemical potential and are inactive. For ����U,t0 is solved from Eq. �9�, �R0���O�1�, and �t0� ��O�1�. Atthe transition point ��=0�, t0=0+ �

2 U, R0�=1− �1+U2 /t0

2 �−1, and t0� = Ut0

�1+U2 /t02 �−1. Close to the Lifshitz

transition, the real part of the self-energy is not negligible.In Fig. 3, t0 �solid line� is compared to the result from

perturbation theory t0pt =0+U����� �dashed line�, where

��x�=1 for x 0 and 0 otherwise, which is obtained by sub-stituting ��i�n�= i0+ sign �n into the integrand in Eq. �1�.There is a jump in t0

pt at the transition, originating from thatin the density of states when a new Fermi pocket appears intwo-dimensional systems. From Fig. 3, we note that for large���, t0�t0

pt and the self-consistency is not essential whilefor small ���, the jump in t0

pt is replaced by a smooth cross-over in t0. Since the width of the crossover region and thejump in t0 are both of order U, we expect t0� �O�1� closeto the Lifshitz transition �����U�.

We note that in all cases t�U and 1−R0��1, and thus���i�n�+ i�n�D. This justifies the calculation of the integralin Eq. �2�.

B. Neck-disrupting Lifshitz transition

For the neck-disrupting Lifshitz transition, the dispersionrelation �p in the L region can be expanded as, choosing pLas the origin and px,y along the principle axes,

�p � �c +px

2

2mx−

py2

2my, �10�

where �c=0. For convenience, the L region is defined as−��kx , ky �� where kx= px /�2mx and ky = py /�2my. UsingAppendix A, Eqs. �3� and �4� become

-40 -20 0 20 40µ/U

1

2

3

4

γ t0(U

)

FIG. 3. The impurity scattering rate t0 close to the pocket-disappearing Lifshitz transition. Solid line: t0 solved from the self-consistent Born approximation, Eq. �9�. Dashed line: t0

pt =0

+U����� in perturbation theory. Here, 0=U.

-40 -20 0 20 40µ/U

0.8

1

1.2

1.4

1-R

’ 0

-40 -20 0 20 40µ/U

0

0.2

0.4

γ’ t0

(a) (b)

FIG. 4. �a�: 1−R0� and �b�: t0� close to the pocket-disappearingLifshitz transition. Here 0=U.

LIFSHITZ TRANSITION IN TWO-DIMENSIONAL SPIN-… PHYSICAL REVIEW B 82, 195110 �2010�

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R��� = U tan−1� + � − R���t���

, �11�

t��� = 0 + U ln�2

�t���2 + �� + � − R���2. �12�

From Eq. �11�,

� = � − R0 + tan−1� − R0

t0, �13�

which is again a monotonically increasing function. As dis-cussed above, R0 is absorbed into �. From Eq. �12�,

t0 = 0 + U ln�2

��2 + t02

, �14�

which is shown in Fig. 5. We now discuss the role played by�, introduced for the expansion of �p in the vicinity of theband extrema in Eqs. �5� and �10�. One expects that �2

�D. In the calculation, we assume that U�2D. Close tothe pocket-disappearing transition, � can be chosen in anobvious way, as suggested by Fig. 1. Furthermore, � onlyappears in Eq. �8�, which is absorbed into �. The neck-disrupting transition is more subtle; in Fig. 2, the introduc-tion of � seems to be arbitrary, and furthermore � appearsexplicitly in Eq. �12�. When �2D such that Eq. �10� is alsoapplicable in part of the R region, 0 also depends on �, andthis dependence cancels the � dependence in 1. Thus, t isindependent of the choice of �.

In Eq. �14�, for ����t0, t0�0+U ln �2

��� . In this limit,R0�� U2

�2 ln �2

��� 1, and �t0� ��U / ���1. On the other hand,

when ����t0�U ln�2

U , �R0���1 / ln�2

U , and �t0� ��1 / ln�2

U .For �=0, t0� =0. In Fig. 5, t0 �solid line� is compared tot0

pt =0+U ln �2

��� from perturbation theory �dashed line�.25

The logarithmic divergence in t0pt is replaced by a finite peak

with height �U ln�2

U . 1−R0�, and t0� are shown in Fig. 6,from which we see that R0� and t0� are small ��0.1� in theentire range plotted. However, because ln x increases ratherslowly for x�1, they are not negligible at ����U ln�2

U evenfor �2=104U.

Since t0�U ln�2

U and 1−R0��1, ���i�n�+ i�n�D is sat-isfied for a weak impurity potential U�2. This justifies theevaluation of the momentum integral in Eq. �2�.

III. LOW-TEMPERATURE ELECTRICAL CONDUCTIVITYAND THERMOPOWER CLOSE TO THE LIFSHITZ

TRANSITION

In order to study the thermopower, defined in Eq. �B7�,close to the Lifshitz transition, we first calculate the electricalconductivity �xx and the thermoelectric transport coefficient�xx. We separate the momentum integrals in Eqs. �B5� and�B6� into R and L regions, and introduce

�xxR,L

�xxR,L� =

e

2��

−�

� dx

cosh2�x/2� e

x�

��R,L

d2p

�2��2 �vpx�2�Im GR�p,xT��2, �15�

where GR�p ,��= 1�1−R0���−�p+�+it��� . Then, �xx=�xx

R +�xxL and

�xx=�xxR +�xx

L .In the R region, we define the function FR��� as

FR��� = �R

d2p

�2��2���p − � − ���vpx�2, �16�

in terms of which, �xxR and �xx

R can be written as

�xxR

�xxR � =

e

2��

−�

� dx

cosh2�x/2� e

x�� d�FR���

�� t�xT��� − �1 − R0��xT2 + t�xT�2�2

. �17�

The integral is dominated by x�1, and for Tt, in order toobtain a nonzero �xx

R , we expand the integrand to order xT.One contribution comes from the expansion FR����FR0+FR0� �, where FR0� �FR0 /�2 and is negligible. By neglectingcontributions of order 1 /�2, we are calculating the singularcontribution to �xx due to the Lifshitz transition. Thus,

�xxR

�xxR � =

e

4FR0�

−�

� dx

cosh2�x/2� e

x� 1

t�xT�. �18�

In Appendix C, we argue that, close to the Lifshitz tran-sition, �xx

L and �xxL are both negligible. As a result, �xx and

�xx are given by Eq. �18�. Performing the remaining integra-tion,

-100 -50 0 50 100µ/U

6

7

8

9

γ t0(U

)

FIG. 5. The impurity relaxation rate t0 as a function of � closeto the neck-disrupting Lifshitz transition. Solid line: t0 solved fromthe self-consistent Born approximation, Eq. �14�. Dashed line: t0

pt

=0+U ln �2

��� from perturbation theory. Here, 0=U and �2=104U.

-100 -50 0 50 100µ/U

0.9

0.95

1

1-R

’ 0

-100 -50 0 50 100µ/U

-0.05

0

0.05

γ’ t0

(a) (b)

FIG. 6. �a�: 1−R0� and �b�: t0� close to the neck-disrupting Lif-shitz transition. Here, 0=U and �=104U.

JIE LIN PHYSICAL REVIEW B 82, 195110 �2010�

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�xx � e2FR0/t0, �19�

and

�xx � −�2

3eTFR0

t0�

t02 . �20�

Equations �19� and �20� can be used for both types of Lif-shitz transitions; both �xx and �xx are determined by states inthe R region, and the effect of the transition is manifestedthrough t���.

The resistivity �xx=1 /�xx�t0. Thus, up to scale factors,�xx is given by the solid curve in Fig. 3 for the pocket-disappearing transition and the solid curve in Fig. 5 for theneck-disrupting transition.

We now give an estimate of �xxN and �xx

N for a normalmetallic state, which are obtained from Eq. �17� with FR���replaced by F��� which is defined by Eq. �16� with a mo-mentum integration over the entire Brillouin zone. We notethat F�FR. Since dF��� /d��F /D and d /d�� /D with�U, �xx

N �e2F /U and ��xxN �� �2

3 eT FU

1D . The thermopower

of a normal metallic state is then �QN�= ��xxN � /�xx

N � �2

3e T /D,which is usually small.31

Close to Lifshitz transitions, the singular contribution tothe thermopower Q is

Qsing =�xx

�xx� −

�2

3eT

t0�

t0= −

�2

3eT d ln t���

d��

�=0. �21�

Figure 7�a� shows −Qsing /T close to the pocket-disappearingtransition, which is characterized by a sharp peak for ����U, arising from the sharp peak in t0� �Fig. 4�b�. As dis-cussed in Sec. II, close to the transition, t0� �O �1�. As aconsequence, �Qsing /QN��D /U�1, suggesting a strong en-hancement over the normal state value for ����U. Figure7�b� shows −Qsing /T close to the neck-disrupting Lifshitztransition, from which we see that Qsing changes sign at thetransition, and �Qsing� has two peaks at �� �U ln�2

U . For����t0, �Qsing /QN�� D/U

�ln�2

U�2

�1, and for ����t0,

�Qsing /QN�� D/���

ln�2

���

�1. Thus, close to the neck-disrupting

transition, the thermopower is strongly enhanced comparedto the normal state value.

We now give a qualitative discussion on relating � to thedoping x which is usually easier to control experimentally.Assuming a rigid-band model �the density wave potential isfixed�, close to xL �the critical doping�,

x − xL � A�/t1, �22�

where t1 is the leading hopping integral for the cuprate banddispersion.14,19 In Eq. �22�, we have neglected complicationsdue to Lifshitz transitions; in the pocket-disappearing type,A+ for x xL is different from A− for x�xL with A+ /A−�O�1�, and in the neck-disrupting type, A has a logarithmicdivergence as �→0, which is cut off by the impurity scat-tering. For a qualitative estimation, we assume Eq. �22�, andthus obtain

�/U = �x − xL�/x0, �23�

where x0�AU / t1�0 / t1, where A�1 since the total bandwidth is t1.

IV. SUMMARY AND DISCUSSION

In this paper, we have discussed the effects of impurityscattering on Lifshitz transitions in two-dimensional systems,with applications to underdoped cuprates. We solved theelectron self-energy from the self-consistent Born approxi-mation, and studied the low-temperature electrical conduc-tivity and thermopower. We find that the conductivity and thethermopower are determined by states in the R region, awayfrom the momentum where the Fermi surface changes at thetransition. The effects of the transition are manifested by theimpurity scattering relaxation rate and its derivative. Close tothe transition, the electrical resistivity is proportional to therelaxation rate t0, and the thermopower is proportional tothe derivative of the logarithm of the relaxation rate and isstrongly enhanced compared to that of a normal metal,�Q /QN�� t1 /0 where QN and 0 are, respectively, the ther-mopower and the relaxation rate away from the transition,and t1 is the leading hopping integral of the cuprate bandstructure.19

Recent thermopower measurements on both electron-doped cuprates32 and hole-doped cuprates�La1.6−xNd0.4SrxCuO4 �Ref. 33� and YBa2Cu3O6.67 in a mag-netic field34 have provided evidence for spin-density waveground states and quantum phase transitions into such states.In this paper, we see that by tuning the system through Lif-shitz transitions, via, e.g., doping, the thermopower showscharacteristic behavior. This provides another perspective onthe spin-density wave state in underdoped cuprates. Forelectron-doped cuprates, a commensurate �� ,�� spin-densitywave state has been proposed as the ground state when thesuperconductivity is suppressed.35 In this scenario, the dop-ing level at which the hole pocket disappears and a pocket-disappearing Lifshitz transition occurs depends on how fastthe spin potential grows with underdoping. Hall effect calcu-lations suggest that this happens at a doping xL�0.12.14 Inthis paper, we argue that the signatures in the thermopower�Fig. 7�a� can be used to determine the value of xL. For thehole-doped materials YBa2Cu3O6+x, the Fermi surface recon-struction due to the stripe order is more complicated.15,28

-20 -10 0 10 20µ/U

0

0.1

0.2

0.3

-Qsi

ng/T

(arb.units

)

-60 -40 -20 0 20 40 60µ/U

-0.5

0

0.5(a) (b)

FIG. 7. The singular part of the low-temperature thermopower,−Qsing /T, for the pocket-disappearing Lifshitz transition �a� and forthe neck-disrupting Lifshitz transition �b�. 0=U and �2=104U.

LIFSHITZ TRANSITION IN TWO-DIMENSIONAL SPIN-… PHYSICAL REVIEW B 82, 195110 �2010�

195110-5

Recent Hall effect measurement suggests a pocket-disappearing transition,8 and the disappearance of quantumoscillations has been interpreted as due to a neck-disruptingtransition.17 Our results suggest that a measurement of thelow-temperature thermopower may provide more informa-tion about this material.

One advantage of our proposal is that Lifshitz transitionstake place deep in the ordered states, and thus quantum fluc-tuations are expected to be not important. The potential dis-advantage is that, as shown in Fig. 7, the signatures in thethermopower are prominent only in the close vicinity of thetransition. Equation �23� suggests that the doping x is variedat the scale of x0�0 / t1, which may present challenges inexperimental studies.

We note that our approach also applies to other density-wave orders proposed for the underdoped cuprates, e.g., thed-density wave order.36,37 It can also be extended to studyother two-dimensional systems close to Lifshitz transitions.

ACKNOWLEDGMENTS

The author thanks Alex Levchenko for discussions, and A.J. Millis and M. R. Norman for advice and reading of themanuscript. This work was supported by the U.S. DOE, Of-fice of Science, under Contract No. DE-AC02-06CH11357and by the Center for Emergent Superconductivity, an En-ergy Frontier Research Center funded by the U.S. DOE, Of-fice of Science, under Award No. DE-AC0298CH1088.

APPENDIX A: THE INTEGRALS IN THENECK-DISRUPTING LIFSHITZ TRANSITION

In this appendix, we present a general method to calculateintegrals of the form

I�F = �−�

� �−�

�

dkxdkyf�kx2 − ky

2�F�kx,ky� , �A1�

which appears in the discussion of the neck-disrupting Lif-shitz transition. As usual, we make the transformation�dkxdky→�d�dkt

1v , where �=kx

2−ky2 which can be viewed as

the energy variable, v=2�kx2+ky

2, and dkt2=dkx

2+dky2 is the

line element along the constant energy contour.For � 0, the constant energy contour is kx= ���+ky

2

with −��ky �� while for ��0, the constant energy con-tour is ky = ��kx

2+ ��� with −��kx��, as plotted in Fig. 8.Thus,

I�F =1

2�

� 0d��

−�

�

dkyf���

���� + ky2�F��ky

2 + ���,ky�

+ F�− �ky2 + ���,ky�

+1

2�

��0d��

−�

�

dkxf���

���� + kx2�F�kx,�kx

2 + ����

+ F�kx,− �kx2 + ���� . �A2�

As an example, we consider the case F=1 and f���=���−��, where I�F is proportional to the single-particle densityof states,

I�F =� d��−�

�

dk1

���� + k2��� − �� � 2 ln

�

����, �A3�

for ������.

APPENDIX B: TRANSPORT COEFFICIENTS IN LINEARRESPONSE THEORY

In this appendix, we define the conductivity and the ther-moelectric coefficients, and discuss their calculations in lin-ear response theory. In the linear response regime, the elec-tric current J flowing in response to an applied externalelectric field E and/or temperature gradient �T is given by

Ja = �abEb − �ab�bT , �B1�

where a ,b=x ,y and repeated indices represent summation.Using the Kubo formula,38

�ab = − lim�→0

Im�Qeeab���

�, �ab = −

1

Tlim�→0

Im�Qeqab���

�,

�B2�

where Qee,eqab ��� is obtained from Qee,eq

ab �i��� by analyticalcontinuation i��→�+ i0+, with

Qeeab�i��� = 2e2T�

i�n

� d2p

�2��2vpavp

bG�p,i�n�G�p,i�n + i��� ,

�B3�

and

Qeqab�i��� = 2eT�

i�n

� d2p

�2��2 �i�n + i��/2�

�vpavp

bG�p,i�n�G�p,i�n + i��� . �B4�

Note that in the present approximation, the self-energy��i�n� has no momentum dependence. As a result, we haveused bare electric and heat current vertices in Eqs. �B3� and�B4�. The frequency summation and the analytical continua-tion are standard.29 The results are

�ab =e2

2��

−�

� dx

cosh2�x/2�� d2p

�2��2vpavp

b�Im GR�p,xT��2,

�B5�

and

kx

kx

ky

ky

ξ>0 ξ<0

FIG. 8. The constant energy contour for kx2−ky

2=�.

JIE LIN PHYSICAL REVIEW B 82, 195110 �2010�

195110-6

�ab =e

2��

−�

� xdx

cosh2�x/2�� d2p

�2��2vpavp

b�Im GR�p,xT��2.

�B6�

Equations �B5� and �B6� are derived for a single-band sys-tem. It applies also to the multiband situation �e.g., the spin-density wave ordered state� when the impurity-scattering-induced interband transition can be neglected.

In the quasiparticle approximation,29 �Im GR�p ,���2

�����−�p+�� /2, Eqs. �B5� and �B6� lead to the familiarresults for �ab and �ab obtained from the Boltzmann equa-tion in the relaxation time approximation where the relax-ation time ��1 /2. This gives a justification for the electricand heat current operators we have used.

The thermopower Q is defined as the coefficient of theelectric field E generated by an applied temperature gradientin the absence of the electric current. From Eq. �B1�,

Q =Ex

�xT=

�xx

�xx. �B7�

APPENDIX C: THE CALCULATION OF �xxL AND �xx

L

In this appendix, we give a detailed discussion on thesmallness of �xx

L and �xxL . The calculation is different for the

two types of Lifshitz transitions, and is discussed in the fol-lowing two subsections.

1. Pocket-disappearing Lifshitz transition

For the pocket-disappearing type with �p in the L regiongiven by Eq. �5�, after rescaling momenta px,y =�2mx,ykx,y,�xx

L , and �xxL in Eq. �15� become

�xxL

�xxL � =

e

2��

−�

� dx

cosh2�x/2� e

x� 1

�2�my

mx�

�k���

�d2kkx

2t2

��k2 − �1 − R0��xT − �2 + t2�2 .

We first consider �xxL . In the limit Tt0,

�xxL �

e2

2�2�my

mx�1 + �

2+ tan−1 �

t0� �

t0�

with corrections �T2. For both −��t0 and ����t0, �xxL

�e2�my /mx. In these two cases, �xxL �xx

R since we expectthat FR0 /t0 is large for a weak impurity scattering potential.

For ��t0, �xxL � e2

2��my

mx

�t0

. In this limit, �xxL gives a small

correction to �xxR since FR0�D�� close to the transition.

We thus argue that �xxL is always negligible compared to �xx

R .For �xx

L , we find

�xxL �

�T

6�my

mx�

2+ tan−1 �

t0+

t0�

t02 + �2�

�1 − R0�

t0−

�

t02 t0� � .

For ����t0, �xxL �1 /t0 and �xx

R �FR0 /t02 , and thus ��xx

L � ��xx

R �. For −��t0, ��xxL � ��xx

R � since the states in the Lregion are far below the Fermi energy and are not contribut-ing to the transport properties. For ��t0, �xx

L � �

t02 t0, and

��xxL � ��xx

R � since ���FR0. Thus, �xxL is negligible compared

to �xxR .

2. Neck-disrupting Lifshitz transition

For the neck-disrupting type with �p in the L region givenby Eq. �10�, we calculate the momentum integral using themethod outlined in Appendix A,

�xxL

�xxL � =

e

2��

−�

� dx

cosh2�x/2� e

x� 2

�2�my

mx

�� d�t

2��2 + � ln �

���� ���� − �1 − R0��xT − �2 + t

2�2 .

For �xxL ,

�xxL �

2e2

�2 �my

mx

�2

t0+

e2

�2�my

mx

�

t0ln

�2

��2 + t02

,

with corrections of order T2. Compared to �xxR , the first term

is smaller by a factor ��2 /D1. Since t0 , ����2, thesecond term is smaller than the first one. Thus, �xx

L can beneglected.

For �xxL , to leading order in T, we obtain

�xxL �

1

3eT�my

mx� 1

t0ln

�2

�t02 + �2

−�

t02 + �2 + Ut0

���1 − R0�� ,

where we have neglected the term ��2t0� /t02 and the term

��

t02 t0� ln �2

�t02 +�2 , both of which give contributions smaller

than �xxR . For ����t0, �xx

L � 1U �xx

R where �xxR �

FR0

U21

�ln�2

U�3

.

For ����t0, �xxL � 1

U �xxR where �xx

R �FR0

U���1

�ln�2

��� �2. However,

since t0� =0 for �=0, �xxL is potentially important for ���

t0. In this limit, �xxL � 1

U . We note that in the normal me-tallic state, �xx

N � FD

1U , and since F /D�O�1� for a generic

dispersion �p, �xxL ��xx

N . We thus argue that �xxL can be ne-

glected for the calculation of the singular contribution to �xx.

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