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spin- heisenberg chains adopt a non - magnetic ground state , which qualitatively can be seen as a linear superposition of states representing all the possible ways of forming singlets in the system.@xcite with quantum fluctuations suppressing any long - range order , the translational symmetry is preserved . at large length scales ( low energies ) _ any _ amount of disorder in the exchange parameters is predicted to dominate over the quantum or thermal fluctuations@xcite and the resulting system is known as `` random heisenberg chain '' ( rhc ) . in practice , the disorder leads to an inhomogeneous ground state by associating to every random configuration of exchange paths a unique way of forming singlets , regardless of the distance between the involved spins and of their interactions . this new type of ground state , specific to random heisenberg chains , is called _ random - singlet state _ ( rss).@xcite    it is widely accepted@xcite that the interest in rhcs originally arose from a novel approach to deal with random exchanges in the isotropic spin- heisenberg model , introduced in 1979 by dasgupta and ma.@xcite their new physical insight into the effects of disorder was later developed into a formal theory by fisher@xcite and , subsequently , applied to a large variety of problems involving magnets with quenched disorder.@xcite however , it is only the recent combined availability of materials representing physical realizations of disordered quasi-1d quantum magnets@xcite and of novel stochastic numerical methods@xcite that made possible the first quantitative studies concerning the impact of disorder on materials whose magnetic properties are well described by an antiferromagnetic heisenberg - chain hamiltonian [ see eq .  ] . the aim of the present work was to identify a model material featuring the properties of an rhc and , through experimental and numerical methods , to demonstrate that the chosen rhc hamiltonian describes its physical behavior . here we present a set of field- and temperature - dependent magnetization and @xmath3si nuclear magnetic resonance ( nmr ) data of the electronic insulator bacu@xmath0sigeo@xmath1.@xcite its relevant structural unit is shown in fig . [ fig : struct_basge ] and its physical properties can be discussed in terms of the one - dimensional hamiltonian : @xmath4 \nonumber \\ & -   g \mu_{\mathrm{b } } \sum_{i}h s^z_i +   \mu_{\mathrm{b } } h_{\perp } \sum_{i } ( -1)^i s^x_i . \label{rhc1}\end{aligned}\ ] ] here @xmath5 represent the random exchange couplings along the spin - chain sites @xmath6 , @xmath7 is the bohr magneton , @xmath8 the uniform applied magnetic field , @xmath9 a locally induced transverse staggered field,@xcite and the spin operators refer to a spin @xmath10 . ( color online ) sketch of the relevant structural unit in bacu@xmath0sigeo@xmath1 highlighting the different @xmath11 and @xmath12 exchange couplings . spin chains run along the crystallographic @xmath13-axis.,scaledwidth=42.0% ]    in order to achieve a description of the large length - scale / low - energy physics contained in eq .   with @xmath14 , dasgupta and ma developed a _ real - space _ renormalization procedure . they started from the local bonds rather than from the large spatial blocks , as is usually done for homogeneous systems.@xcite starting from the distribution of unrenormalized exchange couplings @xmath15 , where @xmath16 is the energy cutoff , an effective low - energy theory is constructed by tracking , via flow equations , the behavior of @xmath15 as the cutoff @xmath17 is progressively reduced . in the rss theory , the flow of the distribution of unrenormalized bond energies @xmath18 depends on the considered energy scale and temperature.@xcite as a result , the static magnetic properties are dominated by a temperature - dependent population of effectively paramagnetic moments @xmath19 . their characteristic energy- ( @xmath20 ) and temperature - dependent average spatial separation @xmath21 diverges as @xmath22 , leading to a quasi long - range ordered ground state.@xcite in the expression for @xmath23 , @xmath24 is the largest exchange coupling in the chain prior to the renormalization.@xcite due to weak residual interchain interactions , a magnetic order at a nel temperature @xmath25 k is established also in the random heisenberg chains , albeit with a lower ordering temperature in comparison with the disorder - free case.@xcite in the present work we focus our attention on the local and bulk static magnetic properties of bacu@xmath0sigeo@xmath1  for @xmath26 and @xmath27 . the experimental work on bacu@xmath0sigeo@xmath1  began with studies on tunable super - exchange interactions in spin - chain systems by yamada _ et al_. [ ] . across the series from bacu@xmath0si@xmath0o@xmath1 to bacu@xmath0ge@xmath0o@xmath1  the bacu@xmath0(si@xmath28ge@xmath29)@xmath0o@xmath1 compounds ( @xmath30 ) crystallize with a _ pnma _ space group . for @xmath31 , corresponding to maximum disorder in the exchange couplings , the values of the lattice parameters are @xmath32  , @xmath33  , and @xmath34 ,@xcite in between the parameters of the two end members of the series . the spin chains run along the crystallographic @xmath13 axis , as shown in fig . [ fig : struct_basge ] .    early magnetization measurements of polycrystalline samples revealed the often observed broad maximum of @xmath35 , also known as the bonner - fisher peak.@xcite this maximum shifts linearly with @xmath36 towards higher temperatures as the ge concentration is enhanced . from this feature it was initially deduced that the high - temperature properties of bacu@xmath0sigeo@xmath1   effectively reflect those of a standard spin- heisenberg chain with @xmath37 mev , where @xmath38 mev and @xmath39 mev represent the intrachain exchange interactions for no and for complete ge substitution , respectively . the enhancement of the afm exchange , resulting from a higher ge content , was explained in terms of a change of the cu - o - cu bonding angle , from @xmath40 = 124@xmath41 in bacu@xmath0si@xmath0o@xmath1  to 135@xmath41 in the ge parent compound.@xcite    earlier bulk measurements already revealed two distinct features:@xcite ( _ i _ ) a low - temperature divergence of the spin susceptibility in bacu@xmath0sigeo@xmath1 , interpreted as a simple curie - type behavior due to uncompensated magnetic moments,@xcite and ( _ ii _ ) a low - temperature magnetic order , expected even in the case of random heisenberg chains ( rhcs),@xcite with an onset at @xmath42 k in bacu@xmath0sigeo@xmath1.@xcite    in zero magnetic field , bacu@xmath0si@xmath0o@xmath1  is known to order antiferromagnetically at @xmath43 k.@xcite the more than tenfold reduction of @xmath44 in bacu@xmath0sigeo@xmath1 , may be related to the fact that the introduction of ge modifies not only the intrachain , but also the interchain coupling which , along the @xmath45-axis , changes from fm- to afm - type.@xcite the latter implies that for bacu@xmath0sigeo@xmath1  the interchain coupling alternates in sign and , therefore , it averages out to a mean - field value.@xcite    the interest in bacu@xmath0sigeo@xmath1  grew significantly as soon as results of magnetic - susceptibility measurements on single crystals appeared in the literature.@xcite they suggested a logarithmic dependence of the low - temperature spin susceptibility,@xcite consistent with predictions of the rhc model.@xcite subsequent inelastic neutron - scattering ( ins ) studies focused on the measurement of the energy dependence of the correlation length @xmath46 , which at zero - temperature is inversely proportional to the free - spin concentration @xmath47 , with @xmath48 the energy of the probing neutrons . in ins , @xmath49 is extracted from the inverse width of the energy - integrated scattering intensity @xmath50 which , in case of a disordered phase , is typically of lorentzian shape.@xcite following some earlier misinterpretation of the data due to problems with background subtraction,@xcite the ins results were found to be quantitatively consistent with a luttinger - liquid behavior reflecting a disorder - free spin chain with a single effective coupling @xmath51 mev.@xcite it was concluded that the expected rhc - related physics must manifest itself at energies much lower than those accessible by neutron experiments . in a previous nmr study@xcite we already pointed out how the spin dynamics of bacu@xmath0sigeo@xmath1 , probed at the characteristic larmor energy @xmath52 @xmath53ev ( with @xmath54 the nmr resonance frequency ) , turns out to be notably different with respect to that of the regular heisenberg chain bacu@xmath0si@xmath0o@xmath1 . on the other hand , considering existing data on bacu@xmath0si@xmath0o@xmath1  and bacu@xmath0ge@xmath0o@xmath1 , a low - temperature divergence of the magnetic susceptibility also in bacu@xmath0sigeo@xmath1  is not really surprising.@xcite in addition , since ins failed to detect any sort of disorder - induced spin dynamics in this compound , the existence of a random - singlet state ( rss  the ground state of the isotropic rhc hamiltonian ) was considered questionable . in the present work we show how the effect of disorder on the _ static _ magnetic properties can be quantitatively modeled and compared with experiments , thereby supporting the rhc scenario . ( color online ) field and temperature dependences of the magnetic susceptibility @xmath55 per cu ion in bacu@xmath0sigeo@xmath1  ( and in bacu@xmath0si@xmath0o@xmath1 , at 1 t ) with @xmath56 in normalized units , assuming @xmath57.@xcite the fit of @xmath58 with an rhc model was adopted from ref .  , while the @xmath35 curves for chains with a single isotropic exchange value were calculated as in ref .  . qmc simulations of the @xmath55 susceptibility at @xmath59 and 14 t for a simple @xmath11@xmath12 rhc model is also shown ( see text for details).,scaledwidth=45.0% ]    as a first step , we extended the measurements of the magnetic susceptibility @xmath55 to wider temperature and field ranges than those covered in previous studies . single crystals of bacu@xmath0sigeo@xmath1  were grown by the floating - zone technique . one of them was cut and the smaller 6.5-mg part was used in magnetometry measurements ; the remaining 20-mg piece was used for the nmr investigations . the magnetization of the chosen sample was measured by applying a magnetic field @xmath56 . magnetic susceptibility data @xmath60 are displayed as bold solid lines in fig . [ fig : rhc_chi_meas ] . for @xmath61 t and at temperatures @xmath62 k , the measurements were performed using a standard dc squid magnetometer , whereas to cover temperatures down to 0.5 k we employed a squid device with a @xmath63he insert . for all the other fields and for temperatures @xmath62 k we used a vibrating - sample magnetometer ( vsm ) . since in the latter case data are collected `` on the fly '' , i.e. , while the temperature is being swept at 0.25 k / min , this causes a small ( known ) temperature difference between sample and thermometer position . in addition , a certain dispersion in the magnetization values was reduced by applying a moving - average filter with a 200-mk width to the raw experimental data . magnetic hysteresis , defined as the difference between the zero - field - cooled and field - cooled magnetization data , was observed for @xmath64 mk . the hysteretic behavior is probably related to ferromagnetic domains with a rather small net magnetic moment present in the ordered low - temperature phase . since only the zero - field - cooled scans are relevant for our discussion , the field - cooled data are not shown .      as shown in fig . [ fig : rhc_chi_meas ] , for temperatures above 30 k the @xmath55 curves are field - independent and , for @xmath65 k , they approach the theoretical prediction for an _ isotropic _ heisenberg chain @xcite with an exchange parameter @xmath37 mev ( see the green dashed line in fig . [ fig : rhc_chi_meas ] ) . for comparison , the red dashed line indicates the calculated @xmath35 for an isotropic chain with @xmath66 mev , corresponding to the @xmath67 case . we first discuss two major differences between the magnetic susceptibilities of bacu@xmath0si@xmath0o@xmath1   ( @xmath67 ) and bacu@xmath0sigeo@xmath1  ( @xmath31 ) : ( _ i _ ) our previous successful attempt to interpret the low - temperature increase / divergence of @xmath35 in bacu@xmath0si@xmath0o@xmath1  and bacu@xmath0ge@xmath0o@xmath1  as simply being caused by a local transverse staggered field ( ltsf,@xcite see sec . [ ssec : nmr_analysis ] ) is inadequate for treating the more complex case of bacu@xmath0sigeo@xmath1 . first of all , the magnetic susceptibility @xmath68 is strongly _ reduced _ even by moderate fields . as a result , the magnetization deviates from the linear response with respect to @xmath8 ( see fig .  [ fig : m_h_qmc ] and ref .  ) , expected either for an isotropic chain in the ll regime , or for the sine - gordon ( sg ) model in the high - temperature limit , the latter describing the physics when an ltsf is present.@xcite    the sg model , even at intermediate temperatures , is inadequate to describe the physics of bacu@xmath0sigeo@xmath1 . indeed , in this case @xmath35 is known to exhibit a symmetric peak centered at a temperature @xmath69 , roughly corresponding to half of the field - induced gap @xmath70.@xcite to exemplify this fact , we consider the hamiltonian in eq .   with @xmath71 and @xmath27 . in the reduced units @xmath72 , with @xmath73 , this model features a gap @xmath70 in the spin - excitation spectrum which , for @xmath74 , scales as:@xcite    ( color online ) first derivative of the sample magnetization vs.  temperature , @xmath75 , in bacu@xmath0sigeo@xmath1  for @xmath76 . the graphic compares several cross sections of the color map shown in fig . [ fig : phase_bound ] . the assigned errors reflect the line - slope uncertainty for fits within the binning range ( see text for details ) . the shown qmc simulation results were obtained _ without _ using any free parameters.,scaledwidth=48.0% ]    @xmath77    equation   implies that for @xmath78 ( the value estimated for bacu@xmath0si@xmath0o@xmath1@xcite if @xmath56 ) , the ltsf yields @xmath79 mev at @xmath80 t , corresponding to @xmath81 k. on the other hand , as can be seen in fig . [ fig : rhc_chi_meas ] , the susceptibility recorded at 14 t tends to saturate at low temperatures , rather than display a peak at @xmath69 . ( _ ii _ ) the inadequacy of the sg model in interpreting the @xmath55 data of bacu@xmath0sigeo@xmath1   is particularly evident from the @xmath82-dependence of @xmath83 , the inflection point which marks the start of saturation ( corresponding to the maximum of @xmath84 in a fixed field @xmath8  see fig . [ fig : phase_bound_cuts ] ) . to overcome noise problems in the calculation of @xmath84 from raw magnetization data , the experimental points were linearly interpolated using a 200-mk binning and the resulting line slope was taken as the derivative with respect to temperature . the results of this procedure are shown as colored circles in fig . [ fig : phase_bound_cuts ] for four selected fields . a @xmath82 color map of the @xmath84 values is shown in fig . [ fig : phase_bound]a . here the inflection points ( assigned with an error corresponding to the peak width at 95% of the peak height in fig .  [ fig : phase_bound_cuts ] ) are shown as open circles . the resulting @xmath85 values follow a linear field dependence , such that @xmath86 . besides requiring a nonuniversal dimensionless prefactor,@xcite the field dependence of @xmath83 does not follow the @xmath87 power law that the small @xmath88 value would imply for @xmath89.@xcite the calculated dependences of @xmath90 for @xmath91 with @xmath92 or 0.052 are both also shown in fig . [ fig : phase_bound]a . the first @xmath13 value represents the ltsf proportionality constant in bacu@xmath0si@xmath0o@xmath1.@xcite because of the similar local environments , it seems reasonable to use the same value also for bacu@xmath0sigeo@xmath1 . the reason for choosing @xmath93 instead , will be clarified in sec . [ sec : nmr1 ] . ( color online ) ( a ) first derivative of the sample magnetization vs.  temperature , @xmath75 , in bacu@xmath0sigeo@xmath1  for @xmath76 ( data scaled using @xmath94).@xcite the open circles represent cross - over temperatures . for details on data treatment and for a description of the plotted lines see text . ( b ) qmc simulation of the @xmath11@xmath12 rhc model for a system of 6000 spins , using 20 realizations of disorder for each @xmath82-point . the color map was generated using the same data treatment as in panel ( a).,scaledwidth=48.0% ]    in view of these two inadequacies , we modified our previous approach to the experimental data analysis@xcite and instead chose to compare the experimental results with the predictions of the rhc model hamiltonian in eq . .@xcite a first comparison is carried out by fixing @xmath95 and by assuming the @xmath96 values to be uniformly distributed between the two limits , @xmath97 and @xmath98 .    in case of nonzero temperatures and magnetic fields , the renormalization flow of the decimation procedure is interrupted either at the thermal energy @xmath99 , or at the magnetic energy @xmath100.@xcite in the first case , the resulting susceptibility is:@xcite @xmath101 in the second case , the system is driven away from the zero - field fixed point into a regime with field - aligned undecimated spins , a saturated magnetization and zero entropy.@xcite in spin-@xmath102 afm - rhcs the decimation procedure involving the total set of spins leaves the renormalized magnetization magnitude at @xmath103 , unlike , for example , in an rhc with fm - afm coupling.@xcite for this reason , the onset of saturation of the low - temperature magnetization occurs at @xmath104.@xcite a fit to the susceptibility data at the smallest applied field , using the theoretical random - singlet expression for @xmath105 , is shown in fig . [ fig : rhc_chi_meas ] as a black dashed line . a rather good match is found above 4 k with a cutoff @xmath106 mev .    to achieve a more quantitative comparison , we carried out a series of quantum monte carlo ( qmc ) simulations based on eq .  , where @xmath96 couplings alternate randomly from @xmath107  mev to @xmath108  mev ( we call this a @xmath11@xmath12 model ) , mimicking the situation expected for bacu@xmath0sigeo@xmath1 . we chose 6000 spin sites with randomly distributed but equally probable @xmath11 and @xmath12 couplings . the simulations for @xmath55 were averaged over at least 25 random realizations of disorder , as obtained from a directed - loop algorithm @xcite within the alps 2.0 package.@xcite technical details related to the qmc simulations are discussed in app . [ app : a ] . disorder in the exchange couplings was imposed by assuming an equal number ( @xmath109 , with @xmath110 as the chain length ) of @xmath111 and @xmath112 values , randomly permuted to construct the rhc chain that was to be simulated . selected results of these simulations are shown as red and yellow circles in fig . [ fig : rhc_chi_meas ] , as interconnected diamonds in fig . [ fig : phase_bound_cuts ] , and as a color map in fig . [ fig : phase_bound]b . in view of the simplicity of the model , the qualitative and quantitative agreement of our parameter - free qmc simulations with the data is remarkable . incidentally , the best fit of the rs prediction to the low-@xmath113 qmc results in an applied field @xmath61 t requires @xmath114 . this corresponds to the largest energy scale in the system before the decimation starts , thus reinforcing the pertinence of the theory used to capture the magnetic properties of the system . as may be seen in figs . [ fig : phase_bound_cuts ] and [ fig : phase_bound]b , by applying the moving - average procedure described above also to the qmc results , numerical calculations predict the same free - spin - like linear field - dependence for @xmath83 as observed experimentally . this is an important result , because the existence of paramagnetic entities at @xmath115 k in a strongly - correlated system , with a characteristic energy scale corresponding to a temperature exceeding 200 k , is highly counterintuitive . although the measurements of @xmath55 and the comparison with qmc calculations already provide a strong indication for the formation of an rs phase in bacu@xmath0sigeo@xmath1  at low temperatures , it is tempting to directly verify the presence of a random - singlet state on a local scale . with decreasing temperature an increasing number of spins form pairs by adopting a nonmagnetic singlet ground state . the divergent low - temperature tail of @xmath55 is due to the residual unpaired spins . the tendency to form a random - singlet ground state should be directly manifest in nmr data , reflecting a growing number of @xmath2si sites which experience a zero net transferred local field . height - normalized @xmath2si nmr lines , as recorded in a 7-t external field applied along the crystallographic @xmath116 axis , are plotted in fig . [ fig : nmr_lines ] . the spectra were obtained by superposing several acquisitions in the frequency - sweep mode , as described in ref .  . the effects of disorder are evident already by comparing the high - temperature line shapes of bacu@xmath0sigeo@xmath1  and bacu@xmath0si@xmath0o@xmath1  ( see inset of fig .  [ fig : nmr_lines ] ) . we note that the @xmath2si resonance of bacu@xmath0sigeo@xmath1 , apart from being broader than that of its pristine counterpart , is also shifted to higher frequencies . we recall that nmr data of bacu@xmath0si@xmath0o@xmath1  ( taken at @xmath117 t , with @xmath56 ) exhibit a positive orbital shift of @xmath118 mhz , with a _ negative _ hyperfine - coupling to the longitudinal magnetization.@xcite    ( color online ) temperature dependence of the @xmath2si nmr lines of the disordered heisenberg chain compound bacu@xmath0sigeo@xmath1  in a magnetic field of 7.05 t applied along the @xmath116-axis . the zero frequency marks the resonance of @xmath2si nuclear spins in a standard reference sample such as si(ch@xmath119)@xmath120.@xcite the white dashed line marks the position of the resonance at room temperature , while the yellow points represent the position of the maxima , displaying a shift towards higher frequencies at low temperatures . the inset compares the cases of maximum and no disorder at high temperature . the green dashed line marks the orbital shift as obtained from ref . ( see text for details).,scaledwidth=45.0% ]    as shown in fig . [ fig : rhc_chi_meas ] , the high - temperature magnetization of bacu@xmath0sigeo@xmath1  is equivalent to that of an isotropic chain with @xmath121 mev and is distinctly smaller than that of bacu@xmath0si@xmath0o@xmath1 , well modelled by choosing @xmath122 mev . consequently , the negative hyperfine shift in case of a sample with disorder is smaller and , hence , the resonance is located closer to @xmath123 ( see inset of fig . [ fig : nmr_lines ] ) . this result is ultimately an independent confirmation of the validity of the model which allowed us to extract the parameter @xmath123 in ref .  . we emphasize that , due to a random variation of the hyperfine interactions , a broadening of the resonance in the @xmath31 case may well be of structural origin , rather than due to magnetic disorder . realistically , both structural and magnetic disorder have to be taken into account and one of the aims of the analysis outlined below is to disentangle them . another striking feature of the nmr signals is the temperature - dependent shift of the line maxima . in fig . [ fig : nmr_lines ] this shift is evidenced by comparing the sequence of yellow points , marking the maxima , with the vertical dashed line , representing the peak position of the 300-k line . it indicates a growing nonzero uniform magnetization as the temperature is lowered , in obvious contradiction to the rs - phase hypothesis , where the formation of singlets would imply the absence of a local magnetization . as expected , unlike to what is observed in bacu@xmath0si@xmath0o@xmath1,@xcite the nmr data show no indications of a phase transition around @xmath115 k for the disordered compound .          in order to interpret the recorded nmr line shapes and positions , we compare the experimental data with the results of qmc simulations of the @xmath11@xmath12 model , addressed in sec . [ sec : magnqmc ] . in fig . [ fig : lines_nmr_simulation]a we show the temperature - dependent , height - normalized histograms , which reflect the occurrences of longitudinal local magnetization values @xmath124 in @xmath125 @xmath7 intervals for @xmath117 t ( @xmath126 is the spin quantization axis along which the external field was applied ) . the histograms were obtained by using a single configuration of disorder in a system of @xmath127 spins . we note that at @xmath128 k the @xmath124 values are distributed around a nonzero , albeit small , uniform magnetization . in this regime the details of disorder are not essential for describing the physics . at low temperatures , however , a broadened magnetization peak appears , centered at zero local moment , as expected in the case of a continuous formation of singlets . at the same time we observe weak but extremely broad tails ( see top histogram in fig .  [ fig : lines_nmr_simulation]a ) , indicating the presence of incompletely compensated spins , which are progressively polarized as the temperature is reduced .    as shown in fig . [ fig : lines_nmr_simulation]b , the local - moment distribution obtained from these simulations ( which do not include an ltsf ) does not match the experimental nmr data . the formation of random singlets implies that at a significant number of sites the moments of the cu atoms are quenched at low temperatures , corresponding to a zero shift of the simulated nmr line , in disagreement with observations . it turns out that it is essential to consider the combined effect of random exchange interactions and an ltsf , namely to set @xmath129 in the hamiltonian . to include an ltsf we use the same approximation made in the case of no disorder,@xcite i.e. , we assume the total local magnetization @xmath130 to be the sum of a non - uniform longitudinal and a transverse contribution , in the form : @xmath131 where for the magnitude of the local transverse magnetization we write @xmath132 , and analogously for the longitudinal component , with @xmath133.@xcite    ( color online ) field dependence of the magnetization of bacu@xmath0sigeo@xmath1  at 2 and 10 k ( dots ) and comparison with qmc simulations of the @xmath134 rhc model without ( @xmath95 ) and with ( @xmath135 ) ltsf . the @xmath13 coefficient refers to the staggered field value @xmath136 , with @xmath8 the uniform field . also shown is the field dependence of the ground state magnetization ( @xmath137 k ) of an isotropic heisenberg spin- chain , as from ref .  . the magnetization is expressed in normalized units ( @xmath138 is the total number of copper sites and @xmath7 is the bohr magneton ) , assuming @xmath94.,scaledwidth=48.0% ]    because of this particular approximation , two qmc simulations , one for @xmath139 and another for @xmath140 , can be carried out separately on a diagonal basis . + with @xmath27 , an ltsf acts on paired spins by mixing the nonmagnetic singlet states @xmath141 with the magnetic triplet states @xmath142 and @xmath143 , thereby inducing a nonzero magnetization also in the ground state of the spin pair.@xcite if both the uniform field and the ltsf are considered , in a first - order perturbation approximation , the new ground state can be written as:@xcite @xmath144 the uniform field appearing in the denominator can be neglected if @xmath145 , which is the case for the strongest coupled singlets in the rs phase . therefore , at temperatures @xmath146 the longitudinal local magnetization varies linearly with the applied field . here min(@xmath16 ) refers to the minimum exchange coupling value of the already formed singlets .    the distribution of local - magnetization values in a @xmath11@xmath12 rhc model which includes an ltsf is shown in fig . [ fig : lines_nmr_simulation]c . clearly , the presence of an ltsf substantially modifies the magnetization profiles shown in fig . [ fig : lines_nmr_simulation]a . by assuming that the local symmetry of the oxygen atoms surrounding a cu@xmath147 ion is largely unaffected by the si - to - ge substitution , the same value of @xmath92 obtained for bacu@xmath0si@xmath0o@xmath1  ought to be valid also in our case . as seen in fig . [ fig : lines_nmr_simulation]c , at @xmath128 k and @xmath117 t , the chosen ltsf is too weak to induce a nonzero magnetization . however , as the temperature is progressively lowered , singlets with a _ distribution _ of effective exchange couplings are formed and a staggered field @xmath148 acting on them creates a distributed local magnetization of the form @xmath149 . therefore , the histograms in fig .  [ fig : lines_nmr_simulation]c split and broaden with decreasing temperature . the symmetrical splitting reflects the alternating sign of the ltsf from site to site , with a mean absolute value of the local moment which progressively shifts away from @xmath150 .    in order to compare the simulated local magnetization profiles with the nmr line shapes , structural details have to be considered . eight configurations of disorder in systems with @xmath127 sites were simulated . in order to reproduce 6000 unit cells , the local moments were then arranged into four different chains , with two inequivalent copper atoms each . at this first stage , the dipolar coupling was neglected , but we considered the hyperfine coupling to both the longitudinal and the transverse magnetization . following the notation of ref . , for @xmath56 , the local hyperfine field @xmath151 at a general @xmath2si nuclear site can be written as : @xmath152 where the index @xmath6 runs over the four nearest - neighbor copper sites . with this notation , the relative nmr resonance frequency @xmath153 can be written as @xmath154 , with @xmath155 as the @xmath2si nuclear gyromagnetic ratio . from our previous study@xcite of bacu@xmath0si@xmath0o@xmath1  we can fix the parameters @xmath156 t/@xmath7 , @xmath157 t/@xmath7 , whereas @xmath118 mhz at 7 t. each of the above hyperfine parameters , which couple the silicon nuclear magnetism to the longitudinal and transverse electronic magnetization of copper , represents the sum of four individual copper - to - silicon couplings . given the broken translational symmetry due to disorder , the individual cu - si couplings are essential for a quantitative comparison between data and theory . unfortunately , these can not be evaluated by experiment , leaving six unknown free parameters in the model .    for a comparison of the simulated nmr lines with data recorded at 5 k , we used the following parameters ( in t/@xmath7 units ) : @xmath158 , @xmath159 for the coupling to the transverse magnetization , and @xmath160 , @xmath161 , @xmath162 for the coupling to the longitudinal one . the sign of the parameters @xmath163 changes according to the considered si nucleus . by making use of them and with no other free parameters , we succeeded in simulating the complete temperature dependence of the nmr lines , which account for the simultaneous presence of both an ltsf and a uniform magnetic field . the result is shown in fig .  [ fig : lines_nmr_simulation]d , where the agreement between data and the theoretical model is again remarkable .    to be more specific , for a comparison with the experiment , we calculated the line shapes by convoluting the distribution of local fields at the @xmath2si sites with a 2-khz wide gaussian , which is equivalent to the width of the line profile of pure bacu@xmath0si@xmath0o@xmath1 , shown in the inset of fig .  [ fig : nmr_lines ] . the choice of the optimal binning range , which strongly affects the shape of the tails of the @xmath124 histograms in fig . [ fig : lines_nmr_simulation ] , was done by employing the algorithms proposed in refs .  . note that the lines displayed in fig . [ fig : lines_nmr_simulation]d show that the averaging process over four cu sites removes the splitting in the original histograms in fig . [ fig : lines_nmr_simulation]c . in addition , the simulated lines shift with temperature , as indeed observed in the experiment . the minimally - shifted line shape at @xmath128 k reflects magnetic disorder . by comparing figs . [ fig : lines_nmr_simulation]b and [ fig : lines_nmr_simulation]d , it may be seen that the inclusion of an ltsf in the simulations significantly improves the agreement between theory and experiment . therefore , from both a macroscopic and a local point of view , nmr data confirm that the static magnetic properties of bacu@xmath0sigeo@xmath1  are those of an rhc , with the addition of a residual ltsf . to cross check the consistency of the proposed model , it seems natural to consider the ltsf - induced effects also in the magnetometry data . since the influence of the ltsf grows with the applied field @xmath8 , we consider the calculated @xmath164 curves ( shown by dotted lines in fig .  [ fig : m_h_qmc ] ) at two representative temperatures , @xmath165 k and 10 k. the calculation of @xmath164 at @xmath165 k , using the average exchange coupling @xmath166 mev , reflects the _ linear _ behavior expected for an isotropic heisenberg chain in its ll regime@xcite and differs substantially from the experimental data . in fact , this oversimplified model completely neglects the field - induced alignment of the still uncompensated spins that , in a random heisenberg chain , survive even at the lowest temperatures . on the other hand , qmc simulations using the @xmath11@xmath12 model discussed in sec .  [ sec : magnqmc ] and in ref . show significant improvements over the simplified `` average-@xmath16 '' model ( solid blue and red lines in fig . [ fig : m_h_qmc ] for @xmath165 k and 10 k , respectively ) . the next refinement was to include both a uniform and a staggered magnetization in the @xmath11@xmath12 model and combine them@xcite to obtain : @xmath167,\ ] ] with @xmath168 the rhc magnetization in a uniform field . as can be seen in fig . [ fig : m_h_qmc ] , the qmc calculation for @xmath169 with @xmath78 unexpectedly fails to reproduce the data . a good agreement with the magnetization scans at both temperatures was , however , obtained using @xmath170 . the discrepancy in the prefactor is most probably due to the fact that the current model neglects the effects of structural disorder in bacu@xmath0sigeo@xmath1 . a more refined model taking into account such subtleties , by including a site - dependent @xmath13 value , was not implemented in the present analysis . nonetheless , the essence of rhc physics , which dominates the magnetic properties of the probed system , is born out by the combined investigation of bulk magnetometry , nmr measurements and qmc simulations . taken together they strongly suggest that bacu@xmath0sigeo@xmath1   is an excellent realization of a random heisenberg chain model , as originally claimed in ref . , thus providing the ideal testing ground for analytic theories modeling a random - singlet ground state . in conclusion , we considered the bacu@xmath0sigeo@xmath1  compound , where spin-@xmath102 cu@xmath147 ions interact via random exchange couplings , as a representative of rhc systems with bond disorder . we demonstrate that this spin - chain material has a ground state which is very well described on the basis of the rhc hamiltonian in eq .  . by using temperature- and magnetic field - dependent microscopic @xmath3si nmr measurements , combined with macroscopic magnetic - susceptibility data and results of detailed numerical quantum monte carlo simulations , we find a coherent description of the physical properties of the system , compatible with that of an rs phase . in particular , the considerable broadening of the nmr lines and the divergent magnetic response at low temperatures , combined with a nonlinear increase of the magnetization as a function of field at low @xmath113 , all reflect the presence of unpaired spins with arbitrary small couplings even at temperatures close to zero . specific refinements in the simulations of @xmath11-@xmath12 model have shown that the addition of a local transverse staggered field  ltsf ( similar to that present in the parent compound bacu@xmath0si@xmath0o@xmath1)@xcite  is essential also for reproducing the observed nmr line shifts . since the magnetization of a singlet in an ltsf is proportional to the local field , the detection of the field distribution by future @xmath171cu nmr measurements may provide a direct way to explore the @xmath16 distribution in an rhc . in addition , besides the existing stretched - exponential nmr relaxation data,@xcite the spectral properties of random heisenberg chains would be more directly accessible if neutron scattering experiments in a lower energy range would be possible . the authors thank a.  feiguin ( northwestern university , boston ) , k.  pra ( epfl , lausanne ) , and o.  zaharko ( psi , villigen ) for useful discussions . the bacu@xmath0sigeo@xmath1  samples used in this work were prepared in the early 2000 s in the group of prof . k.  uchinokura at the university of tokyo . this work was financially supported in part by the schweizerische nationalfonds zur frderung der wissenschaftlichen forschung ( snf ) and the nccr research pool manep of snf . in order to correctly interpret the numerical results , it is necessary to mention the theoretical expectations for the error bars in qmc simulations ( i.e. , specific to this method ) , in relation to the self - averaging induced errors ( i.e. , specific to disorder).@xcite for a spin chain of length @xmath110 , one can define the ratio @xmath172 , where @xmath173 is any macroscopic variable subject to self - averaging and @xmath174 denotes the same quantity averaged over the realizations of disorder.@xcite by using @xmath175  monte carlo updates " ( the number of random updates of the system performed by the algorithm before a new configuration is generated ) and @xmath176 independent monte carlo measurements in a system of 6000 spins , the relative error for @xmath177 in low - temperature qmc simulations was found to be 10 times lower than @xmath178 , the error due to the average on all the realizations of disorder . therefore , in our case @xmath178 is the dominant simulation error . whenever the correlation length is @xmath179 , self - averaging can be justified based on the central - limit theorem.@xcite in case of critical systems , where the correlation length tends to diverge ( @xmath180 ) , the only way to obtain meaningful results in the presence of disorder consists in increasing the system size @xmath110 . by simulating systems with different spin - chain lengths @xmath110 , we find that @xmath181 , with @xmath182 the standard deviation of the local spin susceptibility . this implies the correct evaluation of the thermodynamic limit even in the problematic @xmath183 case . incidentally , since @xmath182 is smaller at higher temperatures and magnetic fields , this justifies the smaller error bars in these regions . a. zheludev and t. roscilde , arxiv:1305.1194 [ cond-mat.str-el ] ( 2013 ) . s. ward , p. bouillot , h. ryll , k. kiefer , k. w. krmer , ch . regg , c. kollath and t. giamarchi , j. phys . : condens . matter * 25 * , 014004 ( 2013 ) . g. simutis , s. gvasaliya , m. mnsson , a. l. chernyshev , a. mohan , s. singh , c. hess , a. t. savici , a. i. kolesnikov , a. piovano , t. perring , i. zaliznyak , b. bchner , a. zheludev , accepted in phys . * 111 * ( 2013 ) . b.  bauer et al ( alps collaboration ) , j.  stat . p05001 ( 2011 ) .
the disordered quasi-1d magnet bacu@xmath0sigeo@xmath1 is considered as one of the best physical realizations of the random heisenberg chain model , which features an irregular distribution of the exchange parameters and whose ground state is predicted to be the scarcely investigated _ random - singlet _ state ( rss ) . based on extensive @xmath2si nmr and magnetization studies of bacu@xmath0sigeo@xmath1 , combined with numerical quantum monte carlo simulations , we obtain remarkable quantitative agreement with theoretical predictions of the random heisenberg chain model and strong indications for the formation of a random - singlet state at low temperatures in this compound . as a local probe , nmr is a well - adapted technique for studying the magnetism of disordered systems . in this case 
 it also reveals an additional local transverse staggered field ( ltsf ) , which affects the low - temperature properties of the rss . 
 the proposed model hamiltonian satisfactorily accounts for the temperature dependence of the nmr line shapes .