Quantum Matter Without Quasiparticles

Quantum matter without quasiparticles

M.I.T. March 13, 2014 ! Subir Sachdev

Talk online: sachdev.physics.harvard.edu HARVARD William Witczak-Krempa Perimeter Erik Sorensen McMaster Andrew Lucas Harvard

Raghu Mahajan Stanford Sean Hartnoll Matthias Punk Koenraad Schalm Stanford Innsbruck Leiden Foundations of quantum many body theory: 1. Ground states connected adiabatically to Metal Metal independent electron states carrying a current 2. Quasiparticle structure of excited states

Metals

SuperconductorMetal InsulatorMetal carrying E a current

k Superconductor Insulator E

k Foundations of quantum many body theory: 1. Ground states connected adiabatically to Metal Metal independent electron states carrying a current 2. Boltzmann-Landau theory of quasiparticles

Metals

SuperconductorMetal InsulatorMetal carrying E a current

k Superconductor Insulator E

k Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Boltzmann-Landau theory of quasiparticles

Famous examples: The fractional quantum Hall effect of electrons in two dimensions (e.g. in graphene) in the presence of a strong magnetic ﬁeld. The ground state is described by Laughlin’s wavefunction, and the excitations are quasiparticles which carry fractional charge. Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Boltzmann-Landau theory of quasiparticles

Famous examples: Electrons in one dimensional wires form the Luttinger liquid. The quanta of density oscillations (“phonons”) are a quasiparticle basis of the low- energy Hilbert space. Similar comments apply to magnetic insulators in one dimension. Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle2. No quasiparticlesstructure of excited states Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle2. No quasiparticlesstructure of excited states

Only 2 examples:

1. Conformal ﬁeld theories in spatial dimension d >1 ! 2. Quantum critical metals in dimension d=2 Iron pnictides: a new class of high temperature superconductors

Ishida, Nakai, and Hosono arXiv:0906.2045v1 Resistivity ⇢ + AT ↵ ⇠ 0 T BaFe2(As1-xPx)2 0 TSDW

α"

2.0

AF 1.0 +nematicSDW Superconductivity 0

α" 2.0 Neel (AF) and “nematic” order

AF 1.0 +nematicSDW Superconductivity 0

α"

2.0

AF 1.0 +nematicSDW Superconductivity 0 Superconductor S. Kasahara, T. Shibauchi,Bose K. Hashimoto, condensate K. Ikada, S. Tonegawa, R.of Okazaki, pairs H. Shishido, of electrons H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010) Resistivity ⇢ + AT ↵ ⇠ 0 T BaFe2(As1-xPx)2 0 TSDW

α"

2.0

AF 1.0 +nematicSDW Superconductivity 0 Ordinary metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,(Fermi liquid) Physical Review B 81, 184519 (2010) Resistivity ⇢ + AT ↵ ⇠ 0 T BaFe2(As1-xPx)2 0 TSDW

no quasiparticles, α" Landau-Boltzmann theory 2.0 does not apply

AF 1.0 +nematicSDW Superconductivity 0

Metal Tc

no quasiparticles, α" Landau-Boltzmann theory 2.0 does not apply

AF 1.0 +nematicSDW Superconductivity 0

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010) Outline

1. The simplest model without quasiparticles Superﬂuid-insulator transition of ultracold bosonic atoms in an optical lattice (Conformal ﬁeld theories in 2+1 dimensions) !

2. Strange metals in the high Tc superconductors Non-quasiparticle transport at the Ising-nematic quantum critical point Outline

1. The simplest model without quasiparticles Superﬂuid-insulator transition of ultracold bosonic atoms in an optical lattice (Conformal ﬁeld theories in 2+1 dimensions) !

2. Strange metals in the high Tc superconductors Non-quasiparticle transport at the Ising-nematic quantum critical point Superﬂuid-insulator transition

Ultracold 87Rb atoms - bosons

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). T

Quantum critical

TKT =0 =0 h i6 h i Superfluid Insulator 0 g cc g a complex ﬁeld representing the Bose-Einstein! condensate of the superﬂuid T

Quantum critical

TKT =0 =0 h i6 h i Superfluid Insulator 0 g cc g = d2rdt @ 2 c2 2 V ( ) S | t | |rr | Z 2 V ( )=( )⇥ 2 + u 2 ⇤ c | | | | T

Quantum critical

TKT =0 =0 h i6 h i Superfluid Insulator 0 g cc g LETTER RESEARCH

a The results for selected lattice depths V are shown in Fig. 2b. We 1 V j/j 1 0 c observe a gapped response with an asymmetric overall shape that will be analysed in the following paragraphs. Notably, the maximum observed temperature after modulation is well below the ‘melting’ 25 temperature for a Mott insulator in the atomic limit , Tmelt < 0.2U/kB (kB,Boltzmann’sconstant),demonstratingthatourexperimentsprobe the quantum gas in the degenerate regime. To obtain numerical values for the onset of spectral response, we fitted each spectrum with an error function centred at a frequency n0 (Fig. 2b, black lines). With j Nambu– approaching jc, the shift of the gap to lower frequencies is already Goldstone Higgs mode visible in the raw data (Fig. 2b) and becomes even more apparent for mode the fitted gap n0 as a function of j/jc (Fig. 2a, filled circles). The n0 values are in quantitative agreement with a prediction for the Higgs gap n at Im(Ψ ) SF commensurate filling (solid line): Re(Ψ ) LETTER RESEARCHLETTER RESEARCH 1=2 hn =U~ 3p2{4 1zj=j j=j {1 1=2 SF ðÞc ðÞc 2 j/jc * 1 a The results for selected lattice depths V0 are shown in Fig. 2b. We a 1 V Thej/j results Here 1 h fordenotes selected Planck’s latticeÂÃÀÁ constant. depthsffiffiffi ThisV0 are value shown is based in Fig. on an 2b. analysis We of 1 V j/j 1 c 7,16 c observe avariations gappedobserve response around a a gappedwith mean-field an response asymmetric state with(throughout overall an asymmetric shape the that manuscript, overall will shape that will be analysed in the following paragraphs. Notably, the maximumLETTER RESEARCH 2 2 2 2 be analysedwe have in rescaled the followingjc in the paragraphs. theoretical calculations Notably, the to match maximum the value 26 = d rdt @t c r V ( ) jc<0:06 obtainedobserved from temperature quantum Monte after modulation Carlo simulations is well). below the ‘melting’ S | | |r | observed temperature after modulation is well below the ‘melting’25 Z The sharpnesstemperature of the for spectral a Mott onset insulator can be in quantified25 the atomic by the limit width, Tmelt of < 0.2U/kB ⇥ 2 2 2 ⇤ temperature for a Mott insulator in the atomic limit , Tmelt < 0.2U/kB V ( )=( ) + u the fitted( errorkB,Boltzmann’sconstant),demonstratingthatourexperimentsprobe function, which is shown as vertical dashed lines in a c (k ,Boltzmann’sconstant),demonstratingthatourexperimentsprobeThe results for selected lattice depths V0 are shown in Fig. 2b. We 1 | | | | V B j/j Fig. 1 2a. Approachingthe quantum the gas critical in the degenerate point, the regime. spectral To onset obtain becomes numerical values T the quantumc gas in the degenerate regime. To obtain numerical values sharper, andforobserve the the width onset of normalized a spectral gapped response, to response the centre we fitted frequency with each an spectrumn0 asymmetricremains with an error overall shape that will 3 j/jc ,for 1 the onset of spectral response, we fitted each spectrum with an error constant (Supplementaryfunctionbe analysed centred Fig. at 3). in a The frequency the constancy followingn0 of(Fig. this ratio 2b, paragraphs. black indicates lines). Notably, With j the maximum Particles and holes correspond function centred at a frequency n (Fig. 2b, black lines). With j Nambu– that the widthapproaching of the spectraljc, the0 shift onset of scales the gap with to the lower distance frequencies to the is already to the 2 normal modes in the observed temperature after modulation is well below the ‘melting’ Nambu– QuantumGoldstone approachingHiggs modecriticaljc, point thevisible shift in the in of the same the raw way gap data as to the (Fig. lower gap 2b) frequency. frequencies and becomes is even already more apparent for oscillation of about = 0. mode 25 Goldstone Higgs mode visible in theWe raw observethe datatemperature fitted similar (Fig. gap 2b) gappedn0 andas a becomes responsesfor function a Mott of even inj/ thejc insulator more(Fig. Mott 2a, apparent insulating filled in circles). the for regime atomic The n0 values limit , Tmelt < 0.2U/kB mode critical the fitted(Supplementary gap n0 asare a function in quantitative Information of j/jc (Fig. agreement and 2a, Fig. filled 5a), with circles). with a prediction the The gapn0 closing forvalues the con- Higgs gap nSF at Im(Ψ ) (kB,Boltzmann’sconstant),demonstratingthatourexperimentsprobe Re( ) are in quantitativetinuouslycommensurate when agreement approaching with filling a prediction the (solid critical line): for point the (Fig. Higgs 2a, gap opennSF circles).at TKT Ψ Im(Ψ ) b commensurateWe interpret fillingthe this (solid as quantum a line): result of combined gas in the particle degenerate and hole excitations regime. To obtain numerical values Re(Ψ Lattice ) loading Modulation Hold time Ramp to Temperature p 1=2 1=2 atomic measurement with a frequencyfor the givenh onsetnSF by=U the~ of Mott spectral3 excitation2{4 response,1z gapj=jc that closes wej=jc fitted{ at1 the each spectrum with an error 20=0 2 =0 j/j * 1 1=2 ðÞ1=2 ðÞ limit ctransitionhnSF=UHere point~ h163denotes.p The2{ fitted4 Planck’s1z gapsj=jc are constant. consistentj=jc This{1 with value the is Mott based gap on an analysis of h i6 h i function centredðÞÂÃÀÁffiffiffi atðÞ a frequency n0 (Fig. 2b, black lines). With j 2 Insulator j/jc * 1 7,16 Superfluid Here h denotes Planck’svariations constant. around This a mean-field value is based state1=2 on an(throughout analysis1=2 of the manuscript, ) ÂÃÀÁffiffiffi p r hn =U~ 1z 12 2{17 j=j 1{j=j Nambu– approachingMI 7,16jc, the shiftc of thec gap to lower frequencies is already 0 E 15 variations aroundwe a mean-field have rescaled statejc in the(throughout theoreticalðÞ the calculations manuscript, to match the value Goldstone T = 200 ms j 0:06 obtained from quantum Monte Carlo simulations16 26). tot Higgswe have modewhere rescalednMIjcis

density transforms) into a function with a minimum at Y 5 0 (panels 2 and 3). p 20 r limit the low-frequency16 edgehnMI of= theU~ experimental1z 12 2 data{17 canj=j bec interpreted1{j=jc as Simultaneously,E 15 the curvature in the radial direction decreases, leadingtransition to a point . The fitted gaps are consistent with the MottðÞ gap the fitted error function, which is shown as16 vertical dashed lines in Ttot = 200 ms extractingwhere the frequencyn is the Mott of this gap mode, as predicted which by explains mean-field the goodtheory (Fig. 2a, characteristic reduction of the excitation frequency for the Higgs mode. In the MI ÂÃÀÁ1=2 ffiffiffi 1=2

) quantitative agreementFig. 2a.p with Approaching the prediction for the the critical homogeneous point, com- the spectral onset becomes r disordered (Mott insulating) phase, two gapped modes exist, respectively hnMI=Udashed~ 1z line).12 2{17 j=jc 1{j=jc E 15 10 mensurate filling in Fig. 2a. Modes at differentðÞ frequencies from the correspondingT to particle= 200 ms and hole excitations in our case (red and blue arrow in Thesharper, observed and softening the width of the normalized onset of16 spectral to the response centre in the frequency n0 remains tot where nMI is the MottÂÃ gap asÀÁ predictedffiffiffi by mean-field theory (Fig. 2a, 3panel 3). b, The Higgs mode can be excited with a periodic modulation of the j/jclowest-lying , 1 superfluid amplitude-like regime mode has led broaden to an the identification spectrum only of theabove experimental

Lattice depth ( A = 0.03V coupling j, which amounts to a ‘shaking’ of the classical energy density dashed line).the0 onset of spectralconstant response. (Supplementary Fig. 3). The constancy of this ratio indicates 10 5 V0 signal with a response from collective excitations of Higgs type. To potential. In the experimental sequence, this is realized by a modulation ofThe the observedAn eigenmodegain softeningthat further analysis, the of insight the width however, onset into of of the does the spectral full not spectral in-trap yield response response,any onset information in the we scales calculated with the the distance to the optical lattice potential (see main text for details). t 5 1/nmod; Er, latticesuperfluid recoil about regime the finite has led spectral to an width identification of the modes, of the which experimental stems from the16,22

Lattice depth ( A = 0.03V eigenspectrum of the system in a Gutzwiller approach (Methods energy. Tmod = 200 critical point in the same way as the gap frequency. 5 V0 0 signal withinteraction a responseand between Supplementary from amplitude collective and Information). excitations phase excitations. of The Higgs result We type. will is consider a To series of discrete 0 100 200 300 400gain furtherthe question insight of into theWe the spectral observe full in-trap width similar by response, analysing gapped we the calculated low-, responses intermediate- the in the Mott insulating regime eigenfrequencies (Fig. 3a), and the corresponding eigenmodes show system with a recently developed schemeTime ( basedms) on single-atom- and high-frequency parts of the response separately.16,22 We begin by T = 20 eigenspectrum ofin-trap the(Supplementary system superfluid in a Gutzwiller density Information distributions, approach which and(Methods are Fig. reminiscent 5a), with of the the gap closing con- resolved detection24. It is themod high sensitivity of this method that examining the low-frequency part of the response, which is expected 0 Figure 1 | Illustration of the Higgs mode and experimentaland sequence. Supplementaryvibrational Information). modes The of a drum result (Fig. is a 3b). series The of frequency discrete of the lowest- 0 allowed100 us to reduce200 the modulation300 amplitude400 by almost an order to be governedtinuously by a process when coupling approaching a virtually excited the critical amplitude point (Fig. 2a, open circles). a, Classical energy density V as a function of the order parametereigenfrequenciesY. Within the lying (Fig. 3a), amplitude-like and the corresponding eigenmode n0,G eigenmodesclosely follows show the long-wave- b of magnitude comparedTime with (ms) earlier experiments20,21 and to stay well mode to a pair of phase modes with opposite momenta. As a result, Lattice loadingordered (superfluid) Modulation phase, Nambu–Goldstone Hold time Ramp and Higgs to modesTemperaturein-trap arise superfluid from length densityWe prediction interpret distributions, for this homogeneous which as a are result reminiscent commensurate of combined of the filling particlenSF over and a hole excitations within thephase linear and amplitude response regime modulations (Supplementary (blue and red Information). arrows in panel 1). Asthe the response of a strongly interacting, two-dimensional superfluid is Figure 1 | Illustration of the Higgs mode and experimental sequence.atomic measurementvibrational modeswide ofwith a range drum a of frequency (Fig. couplings 3b). Thej/jc givenuntil frequency the by response the of the Mott rounds lowest- excitation off in the vicinity gap that closes at the 5 20 coupling j J/U (see main text) approaches the critical limit value jc, the energy of the critical point due to16 the finite size of the system (Fig. 3c). Fitting a, Classical energy density V as a function of the order parameter Y. Within the lying amplitude-like eigenmode26n JULY0,G closely 2012 |follows VOL 487 the | long-wave- NATURE | 455 density transforms into a function with a minimum at Y 5 0 (panels 2 and 3). thetransition low-frequency point edge of. the The experimental fitted gaps data are can beconsistent interpreted with as the Mott gap ordered (superfluid) phase,Simultaneously, Nambu–Goldstone the curvature and in Higgs the radial modes©2012 directionMacmillan arise from decreases, Publisherslength leading Limited. prediction to a All rights for reserved homogeneous commensurate filling nSF over a phase and amplitude modulations (blue and red arrows in panel 1). As the extracting the frequency of this mode, which explains the good characteristic reduction of the excitation frequency for the Higgswide mode. range In of the couplings j/jc until the response rounds off in the vicinity 1=2 1=2

) p couplingr j 5 J/U (seedisordered main text) (Mott approaches insulating) the critical phase, value two gappedjc, the energy modes exist,of respectively the critical pointquantitative due to the agreement finiteh sizenMI of with=U the~ the system prediction1z (Fig.12 3c). for2 the Fitting{ homogeneous17 j=jc com-1{j=jc E 15 ðÞ density transforms intocorresponding a function with to particle a minimum and hole at Y excitations5 0 (panels in our 2 and case 3). (red andthe blue low-frequency arrow in mensurate edge of the filling experimental in Fig. 2a. data Modes can be at different interpreted frequencies as from the Simultaneously, the curvature in theT radial = 200 direction ms decreases, leading to a 16 panel 3). b, Thetot Higgs mode can be excited with a periodic modulationextracting of the the frequencylowest-lyingwhere ofn amplitude-likeMI thisis mode, the Mott which modeÂÃ gap explains broaden asÀÁ predicted the theffiffiffi spectrumgood by mean-field only above theory (Fig. 2a, characteristic reductioncoupling of the excitationj, which amounts frequency to a for ‘shaking’ the Higgs of the mode. classical In the energy density the onset of spectral response. quantitative agreementdashed with the line). prediction for the homogeneous com- disordered (Mott insulating)potential. phase, In the two experimental gapped modes sequence, exist, this respectively is realized by a modulation of the An eigenmode analysis, however, does not yield any information 10 mensurate filling in Fig. 2a. Modes at different frequencies from the corresponding to particleoptical and lattice hole excitations potential (see in our main case text (red for and details). blue arrowt 5 1/ innmod; Er, lattice recoil aboutThe the finite observed spectral width softening of the modes, of the which onset stems of from spectral the response in the panel 3). b, The Higgsenergy. mode can be excited with a periodic modulation of the lowest-lying amplitude-likeinteractionsuperfluid modebetween regime broaden amplitude the has and spectrum led phase to excitations. only an above identification We will consider of the experimental

coupling Lattice depth ( j, which amounts to a ‘shaking’ of the classical energy density A =the 0.03 onsetV0 of spectralthe response. question of the spectral width by analysing the low-, intermediate- potential. In the experimental sequence, this is realized by a modulation of the signal with a response from collective excitations of Higgs type. To 5 systemV0 with a recently developed scheme based on single-atom-An eigenmodeand analysis, high-frequency however, does parts not of the yield response any information separately. We begin by optical lattice potential (see main text for details). t 5 1/n ; E , lattice recoil resolved detection24. It is the highmod sensitivityr of thisabout method the that finite spectralexamininggain width further the of low-frequency the insight modes, whichintopart of the stems the response, full from in-trap the which response, is expected we calculated the energy. allowed us to reduce the modulation amplitude by almostinteraction an order betweento amplitude beeigenspectrum governed and by phase a process of excitations. the coupling system We a will virtually in consider a Gutzwiller excited amplitude approach16,22 (Methods of magnitude compared with earlierTmod = experiments 20 20,21 andthe to question stay well of themode spectral to a width pair of by phase analysing modes the with low-, opposite intermediate- momenta. As a result, system0 with a recentlywithin developed the linear response scheme regime based (Supplementary on single-atom- Information).and high-frequencytheand partsresponse Supplementary of the of a response strongly interacting, separately. Information). two-dimensional We begin The by result superfluid is is a series of discrete 0 100 200 300 400 resolved detection24. It is the high sensitivity of this method that examining the low-frequencyeigenfrequencies part of the response, (Fig. 3a), which and is expected the corresponding eigenmodes show allowed us to reduce the modulation amplitudeTime (ms by) almost an order to be governed by ain-trap process superfluidcoupling a26 virtually JULYdensity 2012 excited distributions, | VOL amplitude 487 | NATURE which | are 455 reminiscent of the of magnitude compared with earlier experiments20,21 and to©2012 stay wellMacmillanmode Publishers to a pair Limited. of phase All rights modes reserved with opposite momenta. As a result, Figurewithin the 1 | linearIllustration response of regime the Higgs (Supplementary mode and Information). experimental sequence.the response of a stronglyvibrational interacting, modes two-dimensional of a drum superfluid (Fig. 3b). is The frequency of the lowest- a, Classical energy density V as a function of the order parameter Y. Within the lying amplitude-like eigenmode n0,G closely follows the long-wave- ordered (superfluid) phase, Nambu–Goldstone and Higgs modes arise from 26 JULY 2012 | VOL 487 | NATURE | 455 ©2012 Macmillan Publishers Limited. All rights reservedlength prediction for homogeneous commensurate filling nSF over a phase and amplitude modulations (blue and red arrows in panel 1). As the wide range of couplings j/jc until the response rounds off in the vicinity coupling j 5 J/U (see main text) approaches the critical value jc, the energy of the critical point due to the finite size of the system (Fig. 3c). Fitting density transforms into a function with a minimum at Y 5 0 (panels 2 and 3). the low-frequency edge of the experimental data can be interpreted as Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the extracting the frequency of this mode, which explains the good disordered (Mott insulating) phase, two gapped modes exist, respectively quantitative agreement with the prediction for the homogeneous com- corresponding to particle and hole excitations in our case (red and blue arrow in mensurate filling in Fig. 2a. Modes at different frequencies from the panel 3). b, The Higgs mode can be excited with a periodic modulation of the lowest-lying amplitude-like mode broaden the spectrum only above coupling j, which amounts to a ‘shaking’ of the classical energy density the onset of spectral response. potential. In the experimental sequence, this is realized by a modulation of the An eigenmode analysis, however, does not yield any information optical lattice potential (see main text for details). t 5 1/nmod; Er, lattice recoil about the finite spectral width of the modes, which stems from the energy. interaction between amplitude and phase excitations. We will consider the question of the spectral width by analysing the low-, intermediate- system with a recently developed scheme based on single-atom- and high-frequency parts of the response separately. We begin by resolved detection24. It is the high sensitivity of this method that examining the low-frequency part of the response, which is expected allowed us to reduce the modulation amplitude by almost an order to be governed by a process coupling a virtually excited amplitude of magnitude compared with earlier experiments20,21 and to stay well mode to a pair of phase modes with opposite momenta. As a result, within the linear response regime (Supplementary Information). the response of a strongly interacting, two-dimensional superfluid is

Ground state = b† 0 | i i | i i Y Excitations of the insulator: Excitations of the insulator: Excitations of the insulator: Excitations of the insulator: Excitations of the insulator: Excitations of the insulator: LETTER RESEARCH

26 JULY 2012 | VOL 487 | NATURE | 455 ©2012 Macmillan Publishers Limited. All rights reserved = d2rdt @ 2 c2 2 V ( ) S | t | |rr | Z LETTER RESEARCH ⇥ 2 2 2 ⇤ V ( )=( c) + u a The results for selected lattice depths V are shown in Fig. 2b. We 1 V | | j/j 1| | 0 T c observe a gapped response with an asymmetric overall shape that will be analysed in the following paragraphs. Notably, the maximum observed temperature after modulation is well below the ‘melting’ 25 temperature for a Mott insulator in the atomic limit , Tmelt < 0.2U/kB (kB,Boltzmann’sconstant),demonstratingthatourexperimentsprobe the quantum gas in the degenerate regime. To obtain numerical values Quantumfor the onset of spectral response, we fitted each spectrum with an error function centred at a frequency n0 (Fig. 2b, black lines). With j Nambu– approaching jc, the shift of the gap to lower frequencies is already Goldstone Higgs modecriticalvisible in the raw data (Fig. 2b) and becomes even more apparent for mode the fitted gap n0 as a function of j/jc (Fig. 2a, filled circles). The n0 values are in quantitative agreement with a prediction for the Higgs gap n at TKT Im(Ψ ) SF Re(Ψ ) commensurate filling (solid line): 1=2 1=2 hnSF=U~ 3p2{4 1zj=jc j=jc{1 j/j * 1 ðÞðÞ 2 =0 c =0 Here h denotes Planck’sÂÃÀÁ constant.ffiffiffi This value is based on an analysis of h i6 variations aroundh i a mean-field state7,16 (throughout the manuscript, we have rescaledInsulatorjc in the theoretical calculations to match the value Superfluid 26 jc<0:06 obtained from quantum Monte Carlo simulations ). The sharpness of the spectral onset can be quantified by the width of 0 the fitted error function, which is shown as vertical dashed lines in g Fig. 2a. Approaching the critical point,g the spectral onset becomes sharper, and the width normalized to the centre frequency n0 remains j/j , 1c 3 c cconstant (Supplementary Fig. 3). The constancy of this ratio indicates that the width of the spectral onset scales with the distance to the critical point in the same way as the gap frequency. We observe similar gapped responses in the Mott insulating regime (Supplementary Information and Fig. 5a), with the gap closing con- tinuously when approaching the critical point (Fig. 2a, open circles). b Lattice loading Modulation Hold time Ramp to Temperature We interpret this as a result of combined particle and hole excitations atomic measurement with a frequency given by the Mott excitation gap that closes at the 20 limit transition point16. The fitted gaps are consistent with the Mott gap

1=2 1=2

) p r hnMI=U~ 1z 12 2{17 j=jc 1{j=jc

E 15 ðÞ T = 200 ms 16 tot where nMI is the MottÂÃ gap asÀÁ predictedffiffiffi by mean-field theory (Fig. 2a, dashed line). 10 The observed softening of the onset of spectral response in the superfluid regime has led to an identification of the experimental Lattice depth ( A = 0.03V0 5 V0 signal with a response from collective excitations of Higgs type. To gain further insight into the full in-trap response, we calculated the eigenspectrum of the system in a Gutzwiller approach16,22 (Methods Tmod = 20 0 and Supplementary Information). The result is a series of discrete 0 100 200 300 400 eigenfrequencies (Fig. 3a), and the corresponding eigenmodes show Time (ms) in-trap superfluid density distributions, which are reminiscent of the Figure 1 | Illustration of the Higgs mode and experimental sequence. vibrational modes of a drum (Fig. 3b). The frequency of the lowest- a, Classical energy density V as a function of the order parameter Y. Within the lying amplitude-like eigenmode n0,G closely follows the long-wave- ordered (superfluid) phase, Nambu–Goldstone and Higgs modes arise from length prediction for homogeneous commensurate filling nSF over a phase and amplitude modulations (blue and red arrows in panel 1). As the wide range of couplings j/jc until the response rounds off in the vicinity coupling j 5 J/U (see main text) approaches the critical value jc, the energy of the critical point due to the finite size of the system (Fig. 3c). Fitting density transforms into a function with a minimum at Y 5 0 (panels 2 and 3). the low-frequency edge of the experimental data can be interpreted as Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the extracting the frequency of this mode, which explains the good disordered (Mott insulating) phase, two gapped modes exist, respectively quantitative agreement with the prediction for the homogeneous com- corresponding to particle and hole excitations in our case (red and blue arrow in mensurate filling in Fig. 2a. Modes at different frequencies from the panel 3). b, The Higgs mode can be excited with a periodic modulation of the lowest-lying amplitude-like mode broaden the spectrum only above coupling j, which amounts to a ‘shaking’ of the classical energy density the onset of spectral response. potential. In the experimental sequence, this is realized by a modulation of the An eigenmode analysis, however, does not yield any information optical lattice potential (see main text for details). t 5 1/nmod; Er, lattice recoil about the finite spectral width of the modes, which stems from the energy. interaction between amplitude and phase excitations. We will consider the question of the spectral width by analysing the low-, intermediate- system with a recently developed scheme based on single-atom- and high-frequency parts of the response separately. We begin by resolved detection24. It is the high sensitivity of this method that examining the low-frequency part of the response, which is expected allowed us to reduce the modulation amplitude by almost an order to be governed by a process coupling a virtually excited amplitude of magnitude compared with earlier experiments20,21 and to stay well mode to a pair of phase modes with opposite momenta. As a result, within the linear response regime (Supplementary Information). the response of a strongly interacting, two-dimensional superfluid is

1=2 1=2

) p across the superfluid-insulator transition of r hnMI=U~ 1z 12 2{17 j=jc 1{j=jc E 15 ðÞ T = 200 ms 16 tot where nMI is the MottÂÃ gap asÀÁ predictedffiffiffi by mean-field theory (Fig. 2a, ultracold atoms in a 2-dimensional optical dashed line). 10 The observed softening of the onset of spectral response in the lattice: superfluid regime has led to an identification of the experimental

RESEARCH LETTER Lattice depth ( A = 0.03V 0 5 V0 signal with a response from collective excitations of Higgs type. To gain further insight into the full in-trap response, we calculated the Response to modulation of lattice depth scales eigenspectrum of the system in a Gutzwiller approach16,22 (Methods Tmod = 20 as expected from the LHP pole 0 and Supplementary Information). The result is a series of discrete j 0 100 200 300 400 eigenfrequencies (Fig. 3a), and the corresponding eigenmodes show Time (ms) in-trap superfluid density distributions, which are reminiscent of the 0 0.03 0.06 0.09 0.12Figure 0.15 1 | Illustration of the Higgs mode and experimental sequence. vibrational modes of a drum (Fig. 3b). The frequency of the lowest- a 1.2 a, Classical energyb density0.17V as a function of the order parameter Y. Within the lying amplitude-like eigenmode n0,G closely follows the long-wave- ordered (superfluid) phase, Nambu–Goldstone1 and Higgs modesV arise0 = from 8Er length prediction for homogeneous commensurate filling nSF over a Mott phase and amplitude modulations (blue and red arrows in panel 1). As the wide range of couplings j/jc until the response rounds off in the vicinity Superfuid 1 0.15 j/jc = 2.2 coupling j 5 J/U (see main text) approaches0 the critical value jc, the energy of the critical point due to the finite size of the system (Fig. 3c). Fitting 1 Insulator Y 5 density transforms into0.13 a function with a minimum at 0 (panels 2 and 3). the low-frequency edge of the experimental data can be interpreted as Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the extracting the frequency of this mode, which explains the good disordered (Mott insulating)0.11 phase, two gapped modes exist, respectively quantitative agreement with the prediction for the homogeneous com- 0.8 corresponding to particle and hole excitations in our case (red and blue arrow in mensurate filling in Fig. 2a. Modes at different frequencies from the panel 3). b, The Higgs0.18 mode can be excited with a periodic modulation of the lowest-lying amplitude-like mode broaden the spectrum only above 2 V0 = 9Er 2 coupling j, which amounts0.16 to a ‘shaking’ of the classical energy density the onset of spectral response. potential. In the experimentalU sequence,0 this is realized by a modulationj/j = of 1.6 the / U c An eigenmode analysis, however, does not yield any information /

0.6 optical lattice potentialT (see main text for details). t 5 1/nmod; Er, lattice recoil 0 0.14 about the finite spectral width of the modes, which stems from the B energy.

k interaction between amplitude and phase excitations. We will consider h 0.12 the question of the spectral width by analysing the low-, intermediate- 0.4 3 system with a recently developed scheme based on single-atom- and high-frequency parts of the response separately. We begin by 24 resolved detection0.18. It is3 the high sensitivity of this methodV = that10E examining the low-frequency part of the response, which is expected allowed us to reduce the modulation amplitude by almost0 an order r to be governed by a process coupling a virtually excited amplitude 20,21 j/j = 1.2 of magnitude compared0.16 with earlier experiments and to stayc well mode to a pair of phase modes with opposite momenta. As a result, 0.2 within the linear response regime0 (Supplementary Information). the response of a strongly interacting, two-dimensional superfluid is 0.14 26 JULY 2012 | VOL 487 | NATURE | 455 0 0.12 ©2012 Macmillan Publishers Limited. All rights reserved 0 0.5 1 1.5 2 2.5 0 400 800 mod (Hz) j/jc

Figure 2 | Softening of the HiggsManuel mode. Endres, a, The Takeshi fitted Fukuhara, gap values Davidhn0/U Pekker, Marcregion. Cheneau,b, Temperature Peter Schaub, response Christian to lattice Gross, modulation (circles and connecting (circles) show a characteristic softeningEugene closeDemler, to the Stefan critical Kuhr, point and in Immanuel quantitative Bloch, blueNature line) 487 and, 454 fit with(2012). an error function (solid black line) for the three different agreement with analytic predictions for the Higgs and the Mott gap (solid line points labelled in a. As the coupling j approaches the critical value jc, the change and dashed line, respectively; see text). Horizontal and vertical error bars in the gap values to lower frequencies is clearly visible (from panel 1 to panel 3). denote the experimental uncertainty of the lattice depths and the fit error for the Vertical dashed lines mark the frequency U/h corresponding to the on-site centre frequency of the error function, respectively (Methods). Vertical dashed interaction. Each data point results from an average of the temperatures over lines denote the widths of the fitted error function and characterize the ,50 experimental runs. Error bars, s.e.m. sharpness of the spectral onset. The blue shading highlights the superfluid expected to diverge at low frequencies, if the probe in use couples response is expected to scale as n3 at low frequencies3,6,9,17. longitudinally to the order parameter2,4,5,9 (for example to the real part Combining this result with the scaling dimensions of the response of Y, if the equilibrium value of Y was chosen along the real axis), as is function for a rotationally symmetric perturbation coupling to Y 2, the case for neutron scattering. If, instead, the coupling is rotationally we expect the low-frequency response to be proportionalj j to 2 22 3 invariant (for example through coupling to Y ), as expected for (1 2 j/jc) n (ref. 9 and Methods). The experimentally observed sig- lattice modulation, such a divergence couldj bej avoided and the nal is consistent with this scaling at the ‘base’ of the absorption feature (Fig. 4). This indicates that the low-frequency part is dominated by only a few in-trap eigenmodes, which approximately show the generic c a 1 scaling of the homogeneous system for a response function describing V = 9.5E 1.2 1 0 r coupling to Y 2. U 0.8 j/jc = 1.4 /

j j

h 0.4 In the intermediate-frequency regime, it remains a challenge to 2 0,G construct a first-principles analytical treatment of the in-trap system 0 1 1.5 2 2.5 0.5 j/j including all relevant decay and coupling processes. Lacking such a d c 0,G theory, we constructed a heuristic model combining the discrete spec- E/U 0.04 4 Δ trum from the Gutzwiller approach (Fig. 3a) with the line shape for a , Line strength (a.u.) Line strength 3 0.02 homogeneous system based on an O(N) field theory in two dimen- T/U 3,6 Δ 0 0 B sions, calculated in the large-N limit (Methods). An implicit assump-

0 0.5 0.6 0.7 0.8 0.9 1 1.1 k 0 0.5 1 1.5 2 tion of this approach is a continuum of phase modes, which is hmod/U hmod/U b 1 2 3 4 0.7 0.5

uid density 0.3

f 0.03 0.1 0.03 0.02 Super T/U

Δ 0.01 B

Figure 3 | Theory of in-trap response. a, A diagonalization of the trapped k 0 system in a Gutzwiller approximation shows a discrete spectrum of amplitude- T/U 0.02 Δ like eigenmodes. Shown on the vertical axis is the strength of the response to a B k

2 0 0.2 0.4 0.6 0.8 1 ) modulation of j. Eigenmodes of phase type are not shown (Methods) and n0,G c /U denotes the gap as calculated in the Gutzwiller approximation. a.u., arbitrary j/j mod 0.01 units. b, In-trap superfluid density distribution for the four amplitude modes (1 – with the lowest frequencies, as labelled in a. In contrast to the superfluid density, the total density of the system stays almost constant (not shown). c, Discrete amplitude mode spectrum for various couplings j/jc. Each red circle 0 corresponds to a single eigenmode, with the intensity of the colour being proportional to the line strength. The gap frequency of the lowest-lying mode 0 0.2 0.4 0.6 0.8 1 follows the prediction for commensurate filling (solid line; same as in Fig. 2a) mod/U until a rounding off takes place close to the critical point due to the finite size of the system. d, Comparison of the experimental response at V0 5 9.5Er (blue Figure 4 | Scaling of the low-frequency response. The low-frequency circles and connecting blue line; error bars, s.e.m.) with a 2 3 2 cluster mean- response in the superfluid regime shows a scaling compatible with the 22 3 field simulation (grey line and shaded area) and a heuristic model (dashed line; prediction (1 2 j/jc) n (Methods). Shown is the temperature response 2 for details see text and Methods). The simulation was done for V0 5 9.5Er (grey rescaled with (1 2 j/jc) for V0 5 10Er (grey), 9.5Er (black), 9Er (green), 8.5Er line) and for V0 5 (1 6 0.02) 3 9.5Er (shaded grey area), to account for the (blue) and 8Er (red) as a function of the modulation frequency. The black line is experimental uncertainty in the lattice depth, and predicts the energy a fit of the form anb with a fitted exponent b 5 2.9(5). The inset shows the same absorption per particle DE. data points without rescaling, for comparison. Error bars, s.e.m.

456 | NATURE | VOL 487 | 26 JULY 2012 ©2012 Macmillan Publishers Limited. All rights reserved = d2rdt @ 2 c2 2 V ( ) S | t | |rr | Z 2 V ( )=( )⇥ 2 + u 2 ⇤ c | | | | T Quantum state with “long-range” quantum entanglement and no quasipartices.Quantum critical

TKT =0 =0 h i6 h i Superfluid Insulator 0 g cc g T

Quantum critical

TKT

Superfluid Insulator 0 g cc g T

Quantum critical “Boltzmann” Boltzmann theory of Nambu- TKT theory of Goldstone-Higgs particles/holes excitations and vortices Superfluid Insulator 0 g cc g CFT3 at T>0 T

Quantum critical

TKT

Superfluid Insulator 0 g cc g CFT3 at T>0 T

NoQuantum quasiparticles: Boltzmanncritical theory does not apply TKT

Superfluid Insulator 0 g cc g Traditional CMT

Identify quasiparticles and their dispersions ! Compute scattering matrix elements of quasiparticles (or of collective modes) ! These parameters are input into a quantum Boltzmann equation ! Deduce dissipative and dynamic properties at non- zero temperatures Traditional CMT

T(!) ⇠

/T Quasiparticle view of quantum criticality (Boltzmann equation): Electrical transport for a (weakly) interacting CFT3

2 (1/(u⇤) ) Re[(!)] O 2 ((u⇤) ), O where u⇤ is the ﬁxed point interaction

1 ~! K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). kBT Quasiparticle view of quantum criticality (Boltzmann equation): Electrical transport for a (weakly) interacting CFT3

e2 ~! (!,T)= ⌃ ;⌃ a universal function h kBT ! ✓ ◆ Universal conductivity e2/h 2 (1/(u⇤) ) ⇠ Re[(!)] Universal time scale ~/kBT O ⇠ 2 ((u⇤) ), O where u⇤ is the ﬁxed point interaction

1 ~! K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). kBT Traditional CMT

Identify quasiparticles Start with strongly interacting and their dispersions CFT without particle- or wave- ! like excitations Compute scattering ! matrix elements of Compute scaling dimensions quasiparticles (or of and OPE co-efficients of collective modes) operators of the CFT ! ! These parameters are Relate OPE co-efficients to input into a quantum couplings of an effective Boltzmann equation gravitational theory on AdS ! ! Deduce dissipative and Non-zero T dynamics of CFT dynamic properties at non- maps to dynamics of a “horizon” zero temperatures in gravitational theory Basic characteristics of CFTs Ordinary quantum field theories are characterized by their particle spectrum, and the S-matrices describing interactions between the particles. The analog of these concepts for CFTs are the primary operators Oa(x) and their operator product expan- sions (OPEs). Each primary operator is associ- ated with a scaling dimension a,definedbythe (T = 0) expectation value (for the simplest case of scalar operators):

ab Oa(x)Ob(0) = h i x 2a | | Basic characteristics of CFTs The OPE describes what happens when two operators come together at a single spacetime point (considering scalar operators only)

fabc lim Oa(x0)Ob(x)Oc(0) = a+b+c x0 x h i x ! | |

The values of a,fabc determine (in principle) all observable properties{ } of the CFT, as constrained by a complex set of conformal Ward identities.

For the Wilson-Fisher CFT3, systematic methods exist to compute (in principle) all the a,fabc , and we will assume this data is known.Thisknowl-{ } edge will be taken as an input to the computation of the ﬁnite T dynamics Traditional CMT Dynamics without quasiparticles

AdS 4 R2,1 The symmetry group Minkowski of isometries of AdS4 CFT3 maps to the group of conformal sym- metries of the CFT3

zr AdS/CFT correspondence at zero temperature

AdS 4 R2,1 The symmetry group Minkowski of isometries of AdS4 CFT3 maps to the group of conformal sym- metries of the CFT3

A classical gravitational theory on AdS4 encodes the CFT3 data of a,fabc , and allows computation of CFT3 cor- relators{ consistent} with all conformal Ward identities Traditional CMT Dynamics without quasiparticles

r There is a family of rh solutions of Einstein’s equations which are AdS4 as r 0, but which! have horizons at A “horizon”, similar to the r = rh. surface of a black hole ! Gauge-gravity duality at non-zero temperatures

r A CFT3 at a temperature rh T 1/rh equal to⇠ the Hawking temperature of the horizon.

A “horizon”, similar to the surface of a black hole ! Gauge-gravity duality at non-zero temperatures

r A CFT3 at a temperature rh T 1/rh equal to⇠ the Hawking temperature of the horizon.

A “horizon”, similar to the surface of a black hole ! Dissipation and friction in the CFT3 = waves falling past the horizon Traditional CMT Dynamics without quasiparticles

Current Current J⌫ Jµ OPE co-ecient . Obeys bound 1/12. | |

Stress-energy tensor T⇢

Conductivity at T>0determinedby “scattering” of current by thermal stress-energy tensor.

R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Physical Review B 87, 085138 (2013). Quasiparticle view of quantum criticality (Boltzmann equation): Electrical transport for a (weakly) interacting CFT3

1 ~! K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). kBT AdS4 theory of quantum criticality