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Density fluctuations in the 1D Bose gas

Panfil, M.K.

Publication date

2013

Link to publication

Citation for published version (APA):

Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.

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Chapter 5

Finite temperature density-density

correlations

In this chapter we study density-density correlations at finite temperatures. We also switch gears and instead of an analytic approach we use exact numerics. The approach is based on the ABACUS algorithm 2.3.1 and is an extension of the zero temperature results [38]. This chapter is based solely on arXiv:soon .

The studies presented here have two main aims: experimental and more theoretical. As described earlier in Chapter2in the Bragg spectroscopy setting (see section2.4) experi-mentalists have a direct access to momentum and energy resolved correlation functions. In a typical procedure we would fix the momentum and vary the energy which results in a certain lineshape ofS(k, ω) as a function of ω. We would like to predict this lineshape theoretically. As we have seen in the previous chapter (section4.2) this might be a dif-ficult task. The effective universal theory is able to predict this lineshape only around the threshold singularities1. We could hope that resolving to methods directly using the Lieb-Liniger Hamiltonian would make the situation better, unfortunately this is not the case. The momentum resolved correlation functions are notoriously difficult to calculate. The only known results at finite temperatures involve simply either perturbative expan-sion aroundc =∞ [33] or Bogolyubov approximation for small interaction strength [16]. Thus one of the aims of this study is to provide quantitative predictions for the lineshape S(k, ω) at finite temperatures and at arbitrary value of interaction strength c > 0. Recalling the discussion about the Mermin-Wagner-Hohenberg theorem 1.2, the zero temperature correlation functions follow an algebraic decay in real space, the 1D Bose

1

Please notice that the effective theory itself does not give any prediction on the size of the interval around the singularity. In the worst case scenario it could happen that the predictions are valid only in a tiny neighborhood of the singularity.

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gas exhibits a quasi long-range order. Turning on finite temperature destroys completely any, even quasi, order and ultimately leads to an exponential decay of correlations. Therefore it is interesting to see how the signatures of the zero temperature correlation are washed out by the thermal fluctuations and how different are correlations in both cases. Using the ABACUS and the TBA we investigate the most interesting situations where neither interactions nor temperature are extreme. This is characterized by a strong competition between the quantum correlations and the thermal fluctuations, results of which are presented below.

This chapter is organized in the following way. We will start with a quick reminder of the ABACUS method and we put it in the context of finite temperature correlations. We also describe how to turn these ideas into a computationally feasible approach. Then we present the results, first in momentum space as these are naturally produced by the ABACUS and later in real space. We conclude by addressing points raised in this introduction.

5.1

ABACUS and the Saddle-point Method

Let us start with the exposition of the method. Recall the Lehmann representation of the finite temperature correlation function (2.34)

ST(k, ω) = 2π LZ X Ω,λ∈HN e−(EΩ−hNΩ)/T|hΩ|ˆρ k|λi|2δ (ω− (Eλ− EΩ)) , (5.1)

whereZ =Pλexp [−β (Hλ− µNλ)] is the partition function.

In principle we have all the ingredients necessary for numerical evaluation of (5.1) at any finite number of particles N . The Hilbert space is spanned by different choices of N quantum numbers. From there and the Bethe equations (2.19) the rapidities follow, which in turn determine the energies (2.17a) and the form factor of the density operator (3.67). Thus both the partition function as well as the double Hilbert space sum are computable.

At least in theory.

In practice the double Hilbert space sum is impossible to handle for any reasonable number of particles. This simply follows from the dimensionality of the Hilbert space. However we can easily turn (5.1) into a less computationally expensive problem. This can be achieved by effectively reducing by one the Hilbert space sums appearing in both the numerator and denominator of (5.1). Actually for the partition function we have already achieved that. Recall the computation of the partition function in the thermodynamic

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Finite temperature correlations 115

limit (3.2.4 in Chapter 3). We have shown there that in the thermodynamic limit the partition function can be written as an functional integral over densities of rapidities ρ(λ) and holes ρh(λ). We have also shown that since the Gibbs free energy (3.119b) is

an extensive quantity the partition function can be approximated by the saddle point method and yields (3.119a)

Z = exp(−F [ρ, ρh])× (1 + O(1/L)) (5.2)

where the Gibbs free energy (3.119b) equals

F [ρ, ρh] = L Z ∞ −∞  ρ(λ) λ2− (λ)− T (ρ(λ) + ρh(λ)) log (1 + exp (−(λ)/T ))  dλ, (5.3) and should be evaluated at ρ(λ) and ρh(λ) maximizing its value. Thus we see that the

partition function can be evaluated faithfully without invoking directly the Hilbert space sum. The summation is performed for us by the saddle-point approximation.

Fortunately the same idea can be used in the numerator of the correlation function (5.1). The form factor of the density operator does not depend on the microscopic details and summation over the microscopic states|λi can be written again as the functional integral

X Ω∈H e−(EΩ−µNΩ)/T|hλ|ˆρ k|Ωi|2δ(ω− Eλ+ EΩ) Th. Lim. −−−−−→ Z Dρ e−F [ρ]/T|hλ|ˆρk|Ωρi|2δ (ω− Eλ+ Eρ) , (5.4)

where |Ωρi is one of the microscopic states that in the thermodynamic limit follow the distribution of rapidities given by ρ(λ). For the correlation function we can write

ST(k, ω) = 1 Z Z Dρ e−βF [ρ]Sρ(k, ω), (5.5) where Sρ(k, ω) = 2π L X λ |hλ|ˆρ(0)|Ωρi|2δ (ω− Eλ+ Eρ) . (5.6)

Again since the Gibbs free energy is an extensive quantity for large system sizes we can use the saddle-point approximation in the evaluation of the functional integrals. On the other handSρ(k, ω) is finite in the thermodynamic limit (it is just a correlation function

computed on a specific state and therefore it is an intensive quantity) and does not in-fluence the saddle-point state. Thus the saddle-point state follows only from minimizing the Gibbs free energy and therefore describes simply the state of thermal equilibrium.

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Denoting the equilibrium density of rapidities byρT(λ) for the finite temperature

corre-lation function we write

ST(k, ω) = 2π L X µ |hµ|ˆρk|λρTi| 2δ(ω − Eµ+ ET) (1 +O(1/L)) , (5.7)

where the partition function canceled against the same factor coming from the numerator. In the limit of T → 0 the saddle-point state falls back to the ground state and Eq. (5.7) is consistent with the zero temperature limit of Eq. (5.1). The finite-size corrections are of order of1/L and come from the saddle-point approximation and the evaluation of Eq. (5.7) in the finite system. We will come back to them later while examining the results. Now let us discuss how Eq. (5.7) can be evaluated.

Looking at Eq. (5.7) it is easy to understand the general algorithm that was used. First we solved the saddle-point equations to obtain thermodynamic distribution of rapidities (following the discussion after eq. (3.119)) which for any fixed N can be approximated by a discrete state Ti given by a properly chosen set of quantum numbers. We then scanned through the Hilbert space of all relevant excitations (also with negative energy) and performed the summation. The precision of the calculation we can easily estimate using the f-sum rule (2.36)

Z ∞ −∞

2πωS(k, ω) = nk

2. (5.8)

To assure that the computed correlation was indeed thermal we employed the detailed balance relation (S(k, ω) = e−βωS(k,−ω), (2.35)) which combined with the f-sum rule

yields Z ∞ 0 dω 2π ωS(k, ω)  1− e−βω= nk2. (5.9) Finally repeating calculations for different system sizes explicitly shows the convergence towards the thermodynamic limit.

5.2

Momentum Space

The fullk and ω dependent density-density correlation function for various temperatures and interaction strengths is plotted on Fig. 5.1. The f-sum rule saturation is presented in Tab. 5.1. The ω dependence of the correlation is shown in Figs. 5.2 and 5.3 where fixed momentum cuts (at k = kF and k = 2kF ) through the correlation function are

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Finite temperature correlations 117 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −2 0 2 4 6 8 10 ω [k 2 F] c = 1, T = 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −2 0 2 4 6 8 10 c = 1, T = 1/2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −2 0 2 4 6 8 10 ω [k 2 F] c = 4, T = 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −2 0 2 4 6 8 10 c = 4, T = 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k[kF] −2 0 2 4 6 8 10 ω [k 2 F] c = 16, T = 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k[kF] −2 0 2 4 6 8 10 c = 16, T = 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Figure 5.1: The full dynamic correlator plotted for the intermediate values of interac-tionc = 1, 4, 16 and for zero [38] and relatively high temperature. As the temperature increases the correlation becomes smeared but stays approximately within the same region in the k− ω plane. The exception being the small correlation region at low momentum and negative energy visible forc = 16.

As pointed out in Chapter2the phase transition in the 1D Bose gas occurs atc = 0 and T = 0. However we can still distinguish different physical regimes with smooth transition between them [31,66,67]. For example when interaction is strong and dominates over the temperature we are in a regime of finite temperature fermionization. On the other hand when the temperature dominates over the interactions we approach the ideal gas. Here we would like to set our attention on the intermediate regime: neither the temperature nor the interactions dominate. For weak interactions this regime is described by a quasi-condensate and can be analyzed from the point of view of Bogolyubov theory with T √c 12. We extend this scaling to arbitrary values of interaction parameterc.

Results presented here are forT /√c = 1/4, 1/2 and as is clear from the fixed momentum

2

According to our choice of units and performing all calculations at n = 1 the temperature of degeneracy Td= ~2n2/(2m) = 1 and γ = c/n = c.

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k = kF k = 2kF c = 1 T = 1/4 0.989, 0.971 0.980 T = 1/2 0.999, 0.984 0.998 c = 4 T = 1/2 0.984, 0.959 0.954 T = 1 0.996, 0.962 0.992 c = 16 T = 1 0.978, 0.972 0.919 T = 2 0.997, 0.981 0.992

Table 5.1: The levels of saturation of the f-sum rule (first entry) and (if different) of the f-sum rule combined with the detailed balance relation (second entry) for the intermediate interaction strengths and two values of momenta.

plots (see Figs. 5.2and5.3) cover a large fraction of phase space with the zero tempera-ture curves setting one extreme. The other extreme is at high temperatempera-tures (T /√c 1), when the gas enters a decoherent regime. The correlation function is then dominated by thermal fluctuations and can be treated by a high temperature expansion (for exam-ple see [31]). As this regime is already well understood we focus here on intermediate temperatures where thermal fluctuations are already large but not yet dominant.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 S (k = kF , ω )

Dynamic Structure Factor

c = 1, T = 0 c = 4, T = 0 c = 16, T = 0 c = 64, T = 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 S (k = kF , ω ) c = 1, T = 1/4c = 4, T = 1/2 c = 16, T =1 c = 64, T = 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ω[k2 F] 0.0 0.2 0.4 0.6 0.8 1.0 S (k = kF , ω ) c = 1, T = 1/2c = 4, T = 1 c = 16, T = 2 c = 64, T = 4

Figure 5.2: Fixed momentum cuts through the correlation function for c = 1, 4, 16, 64 and at increasing values of temperatures from the top to the bottom. Finite temperature drastically modifies the lineshape of the correlation. The upper threshold singularities are washed out and the correlation becomes almost symmetric around its maximum. (inset) Approach towards the thermodynamic limit (waiting for the final data).

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Finite temperature correlations 119 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 S (k = 2kF , ω )

Dynamic Structure Factor

c = 1, T = 0 c = 4, T = 0 c = 16, T = 0 c = 64, T = 0 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 S (k = 2kF , ω ) c = 1, T = 1/4 c = 4, T = 1/2 c = 16, T =1 c = 64, T = 2 0 2 4 6 8 10 ω[k2 F] 0.0 0.1 0.2 0.3 0.4 0.5 S (k = 2kF , ω ) c = 1, T = 1/2 c = 4, T = 1 c = 16, T = 2 c = 64, T = 4

Figure 5.3: The same as Fig. 5.2 but now for k = 2kF. We observe again a

sym-metrization of the correlation which fork = 2kF extends to negative values ofω. The

lower threshold of excitations becomes washed out as well what signals diminishing role of umklapp excitations. Results agree with perturbative expansion in 1/γ (dots) obtained by Brand and Cherny [33]

Integration overω of the dynamical correlator yields the static correlation function (2.42) (see Fig. 5.4) ST(k) = Z ω −ω dω 2πST(k, ω). (5.10) Recall from Chapter 2 that the static correlator in the low momentum limit and at zero temperature can be computed exactly (2.45). This follows from the linearity of the spectrum at small k and from the f-sum rule (2.36). Thanks to the detailed balance relation (2.35) we can perform similar calculations also at finite temperature. To this end we write the static correlator as

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whereBk6= 0 only at finite temperatures as at zero temperature the gas is in the lowest

lying state. Using the f-sum rule (2.36) and detailed balance (2.35) yields [68]

S(0) =    |k| vs +O(k 2) T = 0, 2T v2 s +O(k 2) T > 0, (5.12)

where for a comparison we included also the zero temperature results. We see that at finite temperatures the static correlator is non-zero at k = 0 contrary to the zero temperature case. This can be easily understood by realizing that static correlator at k = 0 really measures number of equivalent states. At finite temperatures there are many equivalent states since the entropy is non-zero. However as entropy vanishes with decreasing temperature so does also S(k = 0).

The static correlator plotted in Fig. 5.4agrees with this low momentum prediction.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 S (k )

Static Structure Factor

c=1/4, kBT = 0 c=1/4, kBT=1/8 c=1/4, kBT=1/4 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 S (k ) c=16, kBT=0 c=16, kBT=1 c=16, kBT=2 0.0 0.5 1.0 1.5 2.0 2.5 k[kF] 0.0 0.2 0.4 0.6 0.8 1.0 S (k ) c=256, kBT=0 c=256, kBT=4 c=256, kBT=8

Figure 5.4: Static structure factor for 3 representative values of the interaction strength (c = 1/4, c = 16, c = 256). In the weakly interacting regime results agree with the prediction of Bogolyubov theory (red dots) [16]. The k → 0 limit of S(k) agrees with the hydrodynamic predictions Eq. 5.12(black dashed lines)

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Finite temperature correlations 121

5.3

Real Space

We move now to real space. The Fourier transform of the static correlator yields the pair correlation function (see Fig. 5.5)

S(x) = 1 L

X

k

e−ikxS(k), (5.13)

describing the probability of two particles to be a distance x apart.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 S (x ) Real-space Correlation c=1, kBT=0 c=1, kBT=1/4 c=1, kBT=1/2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 S (x ) c=16, kBT=0 c=16, kBT=1 c=16, kBT=2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x[n−1] 0.0 0.2 0.4 0.6 0.8 1.0 S (x ) c=256, kBT=0 c=256, kBT=4 c=256, kBT=8

Figure 5.5: The density-density correlation function in real space (see Eq. (5.13)). The points are the Luttinger liquid predictions supplied with the zero-temperature prefactorA1(see eq. (5.14) and discussion below it).

The Luttinger liquid theory at finite temperatures predicts exponential decay of the correlation function. The time-dependence of the chord function d(x± vst) (4.20) can

be traded for the temperature ±t → i/T and leading asymptotic behavior follows from (4.18) and (4.19) [58] S(x) = n2 1− K 2π2  π/LT n sinh(πx/LT) 2 + A1cos(2kFx)  π/LT n sinh(πx/LT) 2K! + (. . . ) , (5.14)

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where . . . represent terms that decay faster with a distance. The prefactor A1 is now

also a function of temperature andLT = vs/T is a thermal length. Recall from previous

chapter that at zero temperature the prefactorA1 can be explicitly computed from the

scaling limit of a single, specific matrix element of the density operator (4.27). At finite temperature the situation is more complicated because there is no such relation. At low temperatures, however, the prefactor is expected to be temperature independent [36]. For the values of temperature considered here (which go beyond the low-temperature limit) we find that the zero temperature prefactor gives predictions consistent with our results (see Fig. 5.5).

5.4

Conclusions

We presented results for the finite temperature correlation function of the 1D Bose gas obtained through exact numerical methods. The results cover the regime of physical parameters that is important for both theoretical and experimental reasons and which is difficult to access through other, analytical or numerical methods. We showed that for intermediate temperatures the correlation function carries zero temperature charac-teristics such as signs of threshold singularities and exponential decay closely resembling the power-law decay. Still the exact lineshape of the correlation is significantly modified signalizing the importance of thermal fluctuations. The exact and quantitative nature of the results will allow in the future for parameter-free fit with experimental predictions.

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