The discovery by Tranquada et al. of an ordered phase of charged domain walls
in the high-Tc cuprates leads us to consider the possible existence of a
quantum domain-wall liquid. We propose minimal models for the quantization, by
meandering fluctuations, of isolated charged domain walls. These correspond to
lattice string models. The simplest model of this kind, a directed lattice
string, can be mapped onto a quantum spin chain or on a classical
two-dimensional solid-on-solid surface model. The model exhibits a rich phase
diagram, containing several rough phases with low-lying excitations as well as
ordered phases which are gapped.Comment: 4 two-column pages, including the 3 Postscript figure

B. Keimer

D. Poilblanc

D. S. Fisher

G. An

G. Shirane

H. E. Viertiö

H. Eskes

H. J. M. van Bemmel

H. J. Schulz

J. A. Vergés

J. M. Tranquada

J. Zaanen

M. den Nijs

M. Inui

M. Kato

P. Prelovšek

R. Grimberg

S. M. Hayden

S. N. Coopersmith

S.-W. Cheong

T. E. Mason

T. Giamarchi

U. Löw

V. Barzykin

V. J. Emery

W. van Saarloos

English

arXiv

Crossref

Quantizing charged magnetic domain walls: Strings on a lattice

Eskes, H.

Grimberg, R.

Saarloos, W. van

Zaanen, J.

Leiden University Scholary Publications

Quantizing charged magnetic domain walls: Strings on a latticeH. Eskes, R. Grimberg, W. van Saarloos, and J. ZaanenInstitute Lorentz, Leiden University, P.O. Box 9506, NL-2300 RA Leiden, The Netherlands~Received 19 March 1996; revised manuscript received 17 April 1996!Recent experiments have shown the existence of an ordered stripe phase in the cuprates, suggesting that thenearby metallic phase is best described as a domain-wall fluid. As a minimal model to describe the melting ofcharged domain walls we introduce quantum lattice strings. This model is related to a generalized quantumspin chain. The string exhibits a rich phase diagram, containing several rough phases with low-lying excita-tions as well as gapped ordered phases. In this scenario the occurrence of the stripe phase is related to atransition from the quantum-rough~metal! to the classical flat phase.@S0163-1829~96!51526-2#The study of high-Tc superconductors has caused a crisisof the conventional paradigm of metal physics, the Fermi-liquid theory which insists that the current in metals is car-ried by quasiparticles with all properties of electrons exceptthat their effective interactions vanish at the Fermi surface. Itseems now widely agreed that this theory fails fundamentallyin the context of the high-Tc cuprates. Indeed, a strong casecan be made that any theory starting with a particlelikevacuum is bound to fail.1 From a theoretical viewpoint thissuggests that the Fermi-liquid fixed point can be unstable andin recent years there have been many investigations aimed atthe possible breakdown of Fermi-liquid theory or at identi-fying new phases of strongly correlated fermions.The present study of quantum domain walls is motivatedby recent compelling experimental evidence2 suggesting thatnature has indeed chosen to realize a collective phase whosebasic ingredient is an intuitively simple and easily identifi-able many-particle bound state: a charged domain wall. Mi-croscopically these domain walls in two dimensions consistof holes bound in a linear stringlike fashion, separating anti-ferromagnetically ordered regions. Across a wall the antifer-romagnetic order parameter points in opposite directions~seeFig. 1!. Tranquadaet al. find a freezing of these domainwalls in a so-called striped phase at a doping concentrationof x51/8, in the middle of the superconducting regime inLa1.48Nd0.4Sr0.12CuO4.2These domain walls were found some time ago to be thegeneric semiclassical~spin S→`) mean-field solutions ofmodels for doped Mott-Hubbard insulators with short-range~on-site! Coulomb interactions only.3 As is well known, themotion of a hole in an antiferromagnetic background is ham-pered by the spin order. In domain walls the holes still haveroom to fluctuate, so that their kinetic energy is low, while atthe same time the number of broken~spin! bonds per hole issmall. Domain walls also survive in more sophisticatedtreatments,4 and their formation is favored by Coulomb5 andelectron-phonon interactions.6 Domain wall striped phaseswere found before in two-dimensional, doped La2NiO4 ,7where one is most likely approaching the classical limit moreclosely6 than in the cuprates.As argued in the paper by Tranquadaet al.,2 the occur-rence of the striped phase in the cuprate is suggestive ofdominating domain wall correlations in the fluid~metallic,superconducting! phases as well. After all, the freezing isexplained2 in terms of a simple commensuration effect,which is inactive in the absence of charged domain wallcorrelations. Moreover, the spin fluctuations in the metallicstate8 closely match the static spin incommensurations in theordered state.9These observations force one to consider the possibilitythat the metallic state is a fluid of charged domain walls: aphase with strong dynamical hole correlations which arereminiscent of a disordered domain wall network. We arguedbefore10 that if the ~collective! dynamics of such a domainwall fluid is assumed to be driven by thermal fluctuations,one obtains a natural explanation for the anomalous mag-netic dynamics in the metallic state. This analysis indicatesthat at not too low temperatures there should be a crossoverto a low temperature regime where the fluid becomes domi-nated by quantum fluctuations—obviously, because theground state is a superconductor and not a striped phase.Virtually all questions concerning the precise nature of thisconjecturedquantum domain wall fluidare, at present, wideopen.In this paper, we focus on the quantum meandering fluc-tuations of a single wall on a lattice. It turns out that a singlewall displays a surprisingly rich behavior and it appears asthe natural building block to describe both the metallic andordered phases. In developing a model we are guided byseveral observations.~i! From Hartree-Fock calculations,3 itis known that the energy associated with adding or removinga hole from a domain wall is rather high. Hence we considerthe number of holes in a wall to be fixed.~ii ! The well-known fact that long wavelength shape fluctuations of fluidinterfaces~i.e., capillary waves! and membranes have a lowenergy, suggests that these are important low energy modesof a domain wall as well. Moreover, mean field calculationsindicate that the ‘‘stiffness’’ associated with these modes issmall: kinks have a low creation energy, while their tunnel-FIG. 1. Schematic illustration of a domain wall of holes sepa-rating two antiferromagnetically ordered domains. Note that in re-ality the hole density may vary more gradually across the wall.PHYSICAL REVIEW B 1 JULY 1996-IIVOLUME 54, NUMBER 2540163-1829/96/54~2!/724~4!/$10.00 R724 © 1996 The American Physical Societying rate to neighboring sites is appreciable.11 ~iii ! From thetheory of interfaces, it is well known that meandering fluc-tuations are not properly incorporated in a mean field theory,and in order to understand the commensurability effect of alattice ~roughening transition!, one has to start with a propermicroscopic lattice model.~iv! Quantum mechanical domainwall fluctuations reduce the kinetic energy of a hole. Forwalls in an antiferromagnetic background~Fig. 1!, the lowenergy fluctuations without spin frustration are those wherethe holes preserve their stringlike order~as beads on astring!; in other words, where the wall does not break up intotwo disconnected pieces.~v! The latter fluctuations dominatethe dynamics of spins near a wall. It is therefore reasonableto neglect in a first stage the interaction between spin wavesin the antiferromagnetic domains and the wall degrees offreedom.The model we propose for the strong hole-binding limit ofquantum domain walls is that ofquantum lattice strings.These are connected strings of ‘‘holes’’ on a square lattice,successive holesl and l11 having either a distance 1 orA2. Let (h lx ,h ly) be the position of holel . We write theclassical interaction as a sum of nearest and next-nearestneighbor discretized string tensions,Hcl5(lFKd~ uh l11x 2h lxu21!d~ uh l11y 2h lyu21!1 (i , j502Li jd~ uh l12x 2h lxu2 i !d~ uh l12y 2h lyu2 j !G1M(l ,md~h lx2hmx !d~h ly2hmy !. ~1!The relevant local configurations are shown in Fig. 2~a!. Thelast term is an excluded volume type interaction; the physi-cally relevant limit isM→`, so that holes cannot occupythe same site (L00 is now irrelevant!. We neglect extremecurvature,L105`, and chooseL2050. Therefore the modelis parameterized byK,L11,L21, and L22. The strings arequantized by introducing conjugate momentap la ,@h la ,pmb #5 id l ,mda,b . A term einp lxwill cause holel to hopa distancen in the x direction. Therefore the hopping con-tribution in its simplest, nearest-neighbor form12 isHqu52t(l ,aPstra ~ l !cos~p la!. ~2!HerePstra ( l ) is a projector restricting the motion of holel inthe a direction to string configurations@Fig. 2~a!#.13 Bymeans of the Suzuki-Trotter mapping the quantum problemHcl1Hqu translates to the problem of two coupled~111!-dimensional restricted solid-on-solid~RSOS! models. Thetwo RSOS height flavors correspond toh l ,kx andh l ,ky , k be-ing the imaginary-time Trotter index. The ‘‘steps’’h l11,ka2h l ,ka andh l ,k11a 2h l ,ka are restricted to 0,61.An important simplification occurs by restricting the al-lowed configurations to those with lx5 l . In this directedstring problem the bonds between neighboring holes alwaysstep to the right@see Fig. 2~b!#—in interface language: thestrings have no ‘‘overhangs.’’ By using the Suzuki-Trotter ortransfer matrix formulation, one can write the ground stateproblem of a directed lattice string in terms of the statisticalmechanics of a classical solid-on-solid surface model.14 Inthis mapping, one may think of the quantum lattice string astracing out a two-dimensional world sheet in space-time; thisformulation has the advantage that it allows a rather naturaland direct translation of various phases of the two-dimensional surface problem into those of quantum strings—e.g., rough surfaces correspond to lattice strings whose quan-tum fluctuations are so strong that they meander, even in theground state.The equivalence between RSOS problems and spin alge-bras was extensively discussed in the seminal work by denNijs and Rommelse.14 As the links between neighboringholes can only point in three directions@Fig. 2~a!#, theHamiltonian can equivalently be formulated in terms of aspin-1 model, withSlz51,0,21 corresponding with the linkbetween sitel and l11 pointing up, horizontal, and down,respectively. In terms of the string-tension parameters,H5(lF S L222 22L21D ~SlzSl21z !21 L222 SlzSl21z1~K12L21!~Slz!21t2~Sl1Sl212 1Sl2Sl211 !G . ~3!In the special caseL2254L21 the Hamiltonian reduces to thewell-known spin-1 chain with single-site anisotropyD5(K12L21)/t and Ising anisotropyJ5L22/2t. The phasediagram as given by den Nijs and Rommelse14 is shown inthe inset in Fig. 3. However, forL22.4L21 we find severalextra phases to occur. Using a combination of quantumMonte Carlo, exact diagonalization, and knowledge of thecritical behavior at the various transitions14,15we arrive at thezero-temperature phase diagram forL22/222L2155 shownin Fig. 3.There are several ordered phases in Fig. 3. They are listedin Table I. For large negativeL22 diagonal walls are foundFIG. 2. ~a! Energies~of the central hole! and tunneling ampli-tudes of the various local-bond configurations. The tunneling pro-cess is between the dashed configuration and the one drawn in full.~b! Typical configuration of a rough directed string on a squarelattice.54 R725QUANTIZING CHARGED MAGNETIC DOMAIN WALLS: . . .~phase I! and for largeK the walls are horizontal and flat~phase II!. PositiveL22 ~phase III! favors~not very realistic!zigzag walls~or antiferromagnetism in spin language!. Apartfrom these, two new phases occur. Phase VIII corresponds toa 22.5° slanted phase of alternating diagonal and horizontalbonds~alternatingSz51 andSz50). Phase X is similar butnow the diagonals themselves have an alternating up step,down step order. The excitation spectra of these phases aregapped. Alternating order of diagonal and horizontal bondsis favored byL22/222L21.0.The remaining phases are quantum disordered. They canbe characterized by the presence and/or absence of order ofthe diagonal and horizontal bonds. Starting from the flatphase~phase II! and loweringK (L22,0) the quantum me-andering fluctuations become dominant and the lattice stringundergoes a roughening transition.16 Like rough two-dimensional interfaces, the vertical displacementsh ly of thehole have logarithmically diverging spatial correlations^(h ly2hmy )2&; lnul2mu. These rough strings have low lyingexcitations ~capillary waves in interface language, spinwaves in spin language!. In phase IV both the diagonal andhorizontal bonds are disordered. Phase VI occurs at largenegativeK. Here only virtual pairs of horizontal bonds canoccur. The model can be mapped on the anisotropic Heisen-berg spin-1/2XXZ model and phase VI corresponds to ananisotropy parameter21,D,1 of this model. Phase V isthe disordered flat or Haldane phase. In this region up stepsof the strings are followed by down steps, but the location ofthese up and down steps has no long range order because thehorizontal bonds are not ordered. As a result, the latticestrings are macroscopically flat. In terms of the height vari-ables the order parameter is local and is given by^exp(iphly)(hl11y 2hly)&.14There are two new rough phases, again occurring whenL22/222L21 is sufficiently large and positive. Region VIIis a rough slanted phase: up steps are diluted by horizon-tal bonds with positional disorder. The average angle ofthe string is smaller than 45° and the deviations from theaverage are again logarithmic. Phase IX is a rough phasewhere the even bonds are horizontal and the odd bonds arediagonal with up-down disorder. On average, the string ishorizontal.The results summarized in the phase diagram give a clearanswer to whether and how quantum-domain walls can‘‘melt’’ 11: they can roughen either via a conventionalKosterlitz-Thouless transition~from phase II to IV!, a first-order KDP transition~phase I and IV!,14 or a Pokrovsky-Talapov transition~phase I to VII!. Because of the largehopping probability in the cuprates the parameter region ofinterest isJ andD small, J,0. The relevant point is thatdomain walls come in two varieties: classically ordered andquantum rough. Such rough walls in our view are the build-ing blocks of the conjectured domain-wall fluid and theirlow-lying collective excitations may be responsible foranomalous features of the metallic state. They give a naturalexplanation of the crossover scale proportional to the tem-perature as observed in the imaginary part of thesusceptibility.10 The structural deformation of the LTT phasewill suppress the diagonal segments, leading to an effectiveincrease of theK parameter. Therefore we envision that theformation of the stripe ordering in the La compounds2 irelated to a transition from phase IV to phase II. At thetransition meandering fluctuations become gapped. However,one should note that even in the most rough phase~phase IV!the meandering fluctuations are only logarithmic at low tem-peratures, and therefore the domain walls will be very sus-TABLE I. A schematic representation of the different phases.FIG. 3. Phase diagram of the directed lattice-string problemwith L22/222L2155, as a function ofJ5L22/2 andD5K12L21.The inset shows the phase diagram forL2254L21.R726 54ESKES, GRIMBERG, van SAARLOOS, AND ZAANENceptible to wall-wall interactions and long-range Coulombforces, which will promote ordering.17 Reconnectioneffects,13 however, will favor a fluid phase and compete withthe above ordering mechanisms. Other crucial questions thatneed to be addressed is whether the domain wall scenario iscompatible with the existence of fermionlike excitations, asseen, e.g., in photoemission experiments and in studies of thet-J model, and how exchange with free holes can be incor-porated and affects the behavior.H.E. is supported by the Stichting voor FundamenteelOnderzoek der Materie~FOM!, which is financially sup-ported by the Nederlandse Organisatie voor Wetenschap-pelijk Onderzoek~NWO!, and J.Z. acknowledges support bythe Dutch Royal Academy of Sciences~KNAW !.1V. J. Emery and S. Kivelson, Phys. Rev. Lett.74, 3253~1995!.2J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S.Uchida, Nature375, 561 ~1995!.3J. Zaanen and O. Gunnarson, Phys. Rev. B40, 7391 ~1989!; D.Poilblanc and T. M. Rice,ibid. 39, 9749 ~1989!; H. J. Schulz,Phys. Rev. Lett.64, 1445 ~1990!; M. Kato et al., J. Phys. Soc.Jpn.59, 1047~1990!; M. Inui and P. B. Littlewood, Phys. Rev.B 44, 4415~1991!; J. A. Vergés, F. Guinea, and E. Louis,ibid.46, 3562~1992!.4P. Prelovsˇek and X. Zotos, Phys. Rev. B47, 5984 ~1993!; T.Giamarchi and C. Lhuillier,ibid. 43, 12 943~1991!; G. An andJ. M. J. van Leeuwen,ibid. 44, 9410~1991!; H. J. M. van Bem-mel et al., Phys. Rev. Lett.72, 2442~1994!.5U. Löw, V. J. Emery, K. Fabricius, and S. A. Kivelson, Phys.Rev. Lett.72, 1918~1994!.6J. Zaanen and P. B. Littlewood, Phys. Rev. B50, 7222~1994!.7J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo,Phys. Rev. Lett.73, 1003~1994!.8S.-W. Cheonget al., Phys. Rev. Lett.67, 1791 ~1991!; G.Shirane, R. J. Birgeneau, Y. Endoh, and M. A. Kastner, PhysicaB 197, 158 ~1994!; T. E. Mason, G. Aeppli, and H. A. Mook,Phys. Rev. Lett.68, 1414~1992!; S. M. Haydenet al., ibid. 66,821 ~1991!; B. Keimeret al., Phys. Rev. Lett.67, 1930~1991!.9V. Barzykin, D. Pines, and D. Thelen, Phys. Rev. B50, 16 052~1994!.10J. Zaanen, M. L. Horbach, and W. van Saarloos, Phys. Rev. B53,8671 ~1996!.11H. E. Viertiö and T. M. Rice, J. Phys. Condens. Matter6, 7091~1994!.12More generally, it is natural to take the hopping amplitudes of thelast two configurations in Fig. 2~a! different.13An interesting and possibly important extension of~1! is to in-clude terms describing self-avoidance and/or reconnection pro-cesses, the latter leading to string loops splitting off.14M. den Nijs and K. Rommelse, Phys. Rev. B40, 4709~1989!.15H. Eskes, R. Grimberg, W. van Saarloos, and J. Zaanen~unpub-lished!.16Note that, while it was argued by D. S. Fisher and J. D. Weeks,Phys. Rev. Lett.50, 1077~1983!, that quantum fluctuations can-not drive a two-dimensional interface rough atT50, our resultsshow that this does not extend to one-dimensional interfaces,i.e., strings.17S. N. Coopersmithet al., Phys. Rev. B25, 349 ~1982!.54 R727QUANTIZING CHARGED MAGNETIC DOMAIN WALLS: . . .

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Quantizing Charged Magnetic Domain Walls: Strings on a Lattice

Quantizing Charged Magnetic Domain Walls: Strings on a Lattice

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Leiden University Scholary Publications

Leiden University Scholary Publications

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