LT 21 Proceedings of the 21st International Conference on Low Temperature Physics Prague, August 8-14, 1996

Part $5 - LT Properties of Solids 2: Correlated electrons

D y n a m i c s o f s i n g l e - p a r t i c l e a n d c o l l e c t i v e e x c i t a t i o n s f r o m Q u a n t u m M o n t e C a r l o

E.G.Klepfish, C.E. Creffieid, P.E. Kornilovitch, E.R. Pike and Sarben Sarkar

Departmcnt of Physics, King's College London, London WC2R 2LS, UK

Singular value decomposition is applied to the analytic continuation into real time of the Matsubara Green's functions obtained in Monte Carlo sinmlations. The spectral density of single-particle excitations, and dy- nanfical spin susceptibility are calculated for a two-dimensional Ilubbard model. The Quantum Monte Carlo results reveal a two-band single-particle spectrum with an extended saddle point. The dynamical spin suscep- tibility while contradicting the prediction of long-range antiferromagnetic order based on the random-phase approximation, is in qualitative agreement with self-consistent renormalization results for the same lattice size and temperature as the simulations.

1. I N T R O D U C T I O N

Quantum Monte Carlo (QMC) sinmlations have been widely applied to explore models of strongly correlated electrons and their relevance to the mech- anism of high-temperature superconductivity. The most phenonaenologically relevant in this research is an attempt to derive dynamical properties of these systems with subsequent comparison to the experi- mental data. QMC sinmlations supply direct infor- mation only about the ixnaginary-time dependence of the correlation functions. The investigation of the dynamics requires analytic continuation of the imaginary-tilne (Matsubara) Green's functions into real time. This is an ill-posed problem, hence the contradictory results reported recently in the liter- ature [1],[2] for the single-particle Green's functions and the largely unexplored area of the frequency de- pendence of the two-particle Green's functions [3],[4].

We present results based on application of the singular value decomposition (SVD) teclmique for the spectral density of single-particle excitations and spin-spin Green's function for a two-dimensional Itubbard model at half filling. Our results in the intermediate-coupling regime indicate the existence of a two-band structure in the single-l)article spec- trum of excitations at finite temperature, indicated by a two-peak structure in the spectral weight fimc- tion and the density of states. These results cannot bc explained within the spin-density wave random- l)hase approximation (SDW-RPA) since the imagi- nary part of the longitudinal spin susceptibility does not show a gap at zero frequency corresponding to

the gap between the two single-particle bands in this approximation. By applying an improved version of Moriya's self-consistent reuormalization theory [5] we find the static spin susceptibility is in qualita- tive agreement with the zero frequency limit of the QMC results. This calculation accounts for a renor- realization of the two-particle irreducible diagrams and satisfies the self-consistency relation between the free energy as a functional in dynamical susceptibil- ity and the static susceptibility. It also enables us to derive analytically the susceptibility of the simulated finite system.

2. RESULTS

Analytic continuation of Matsubara Green's functions GAB(r) into real time amounts to a so- lution of the following ill-posed inverse problem:

/ .~o e - - W r

GaB(r) = dw i + e -"~# f(w); (0 < r < fl) o o

(1) f(w) being the imaginary part of the retarded Green's function. For A and B being the electron creation and annihilation operators the sign in the denominator is +, and for spin operators - . SVD on the integral operator K acting on f(w) in Eq. (1) yields tim set of singular values "" {Oti}i= 1 with corre- sponding singular functions ui(w) and singular vec- tors vi(r) satisfying

/Cuk = akv~; h:*vk = akUk. (2)

Czechoslovak Joumal of Physics, Vol. 46 (1996), Suppl. S5 2655

The solution of Eq.(1) is given by:

n , 1

= ~ - < C, Vk > ttk( ' ) (3) f(w) O'~: k = l

where <, > denotes the inner product in the data space [2].

Fig.1 shows the spectrum of single particle exci- tations over the Brillouin zone identified as tim loca- tion of tim peaks of the spectral weight function at corresponding values of the lattice momentum. The two-band structure remains persistent with increas- ing lattice size up to 12'. We note a clear sign of a fiat band in the vicinity of lattice momentum/7 = (Tr, 0) (similar behaviour was seen at the point (0, r)).

._% tlJ

, , i

. V ' o.o " �9 I .~ ." ....... .

| " -

- 4 . 0 F 1~1 M X F

Figure 1: Spectrum of single-particle excitations. Lower band - ,; upper band - , . Dotted line - r = -2t(cos k= + cos kv); solid line - Mean-field solution.

1 5 . 0 , , ,

. . . . . . . . 8x8

- 5 . 0 ' ' ' ' ' ' -4J -2.0 0.0 2.0 4.0

Figure 2: hnaginary part of X ~ T(w) at f = (,-r, 7r) .

10.0 3

5.0 F-

0.0

20.0

15.0

10.0

5.0

0.0

q I

z 2

u ~ , o §

~ o §

_ _ ~ e _ ~ a e ~ a @ - " , i~ @ _~ _~ _ F X M F

E l 12

@ o Q

Figure 3: Static spin susceptibility vs. mentum (lattice 10 2) .

lattice mo-

]n Fig.2 we present the imaginary part of tim time ordered spin-spin Green's function. The SDW-RPA

predicts a gap at w < 2A where A is tile half distance bctween the single-particle bands. This gap is clearly absent.

Tile comparison with the improved self-consistent renormalisation calculation is presented in Fig.3. Qualitative agreement between simulation results and the analytical calculation is transparent.

3. C O N C L U S I O N S Our method of analytic continuation reveals im-

portant features of tile spectrum of single-particle and collective excitations in the Itubbard model. We observe a flat band at. the corners of the magnetic Brilouin zone in accordance with experimental data on angle-resolved photoemission [6]. Contrary to the conclusions of Ref. [1] the gap in this spectrum is persistent at lattices with linear dimension bigger than the antiferromagnetic correlation length and at finite temperature. This result cannot therefore be attributed to finite-size effects leading to an effective zero-temperature situation in which the rotational symmetry is broken due to the SDW-RPA scenario. The inadequacy of the SDW-RPA treatment is il- lustrated by the absence of the gap in the longitu- dinal spin-spin correlation function. The qualitative agreement of the QMC evaluated static spin suscepti- bility with the self-consistent renormalisation theory points to the essence of including the paramagnon interaction in the description of the magnetic prop- erfies, with subsequent relevance to the full picture of the dynamics of the excitations in the Hubbard model.

R E F E R E N C E S [1] S.R. White, Plays. Rev. B44, (1991) 4670; M.

Veki6 and S.R. White, Plays. Rev. B47, (1993) 1160;

[2] C.E. Creffield, E.G. Klepfish, E.R. Pike and Sar- ben Sarkar, Phys. Rev. Lett. 75 (1995) 517;

[3] M. Vekid et aL, Phys. Rev. Lett. 74 (1995) 2367; H. Endres et al. , unpublished;

[4] C.E. Creffield, P.E. Kornilovitch, E.G. Klepfish, E.R. Pike and Sarben Sarkar, unpublished;

[5] T6ru Moriya, Spin Fluctuations in Initerant Electron. Magnetism Spr inger -Ver lag (1985); I1. ltasegawa and T6ru Moriya, Jour. Phys. Soc. Japan 36, (1974) 1542;

[6] D.S. Dessau et ai., Plays. Rev. Lett. 71 (1993) 2781; D.M. King et aL, Plays. Rev. Lett. 73, (1994) 3298.

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